time travel articles

Is Time Travel Possible?

We all travel in time! We travel one year in time between birthdays, for example. And we are all traveling in time at approximately the same speed: 1 second per second.

We typically experience time at one second per second. Credit: NASA/JPL-Caltech

NASA's space telescopes also give us a way to look back in time. Telescopes help us see stars and galaxies that are very far away . It takes a long time for the light from faraway galaxies to reach us. So, when we look into the sky with a telescope, we are seeing what those stars and galaxies looked like a very long time ago.

However, when we think of the phrase "time travel," we are usually thinking of traveling faster than 1 second per second. That kind of time travel sounds like something you'd only see in movies or science fiction books. Could it be real? Science says yes!

This image from the Hubble Space Telescope shows galaxies that are very far away as they existed a very long time ago. Credit: NASA, ESA and R. Thompson (Univ. Arizona)

How do we know that time travel is possible?

More than 100 years ago, a famous scientist named Albert Einstein came up with an idea about how time works. He called it relativity. This theory says that time and space are linked together. Einstein also said our universe has a speed limit: nothing can travel faster than the speed of light (186,000 miles per second).

Einstein's theory of relativity says that space and time are linked together. Credit: NASA/JPL-Caltech

What does this mean for time travel? Well, according to this theory, the faster you travel, the slower you experience time. Scientists have done some experiments to show that this is true.

For example, there was an experiment that used two clocks set to the exact same time. One clock stayed on Earth, while the other flew in an airplane (going in the same direction Earth rotates).

After the airplane flew around the world, scientists compared the two clocks. The clock on the fast-moving airplane was slightly behind the clock on the ground. So, the clock on the airplane was traveling slightly slower in time than 1 second per second.

Credit: NASA/JPL-Caltech

Can we use time travel in everyday life?

We can't use a time machine to travel hundreds of years into the past or future. That kind of time travel only happens in books and movies. But the math of time travel does affect the things we use every day.

For example, we use GPS satellites to help us figure out how to get to new places. (Check out our video about how GPS satellites work .) NASA scientists also use a high-accuracy version of GPS to keep track of where satellites are in space. But did you know that GPS relies on time-travel calculations to help you get around town?

GPS satellites orbit around Earth very quickly at about 8,700 miles (14,000 kilometers) per hour. This slows down GPS satellite clocks by a small fraction of a second (similar to the airplane example above).

GPS satellites orbit around Earth at about 8,700 miles (14,000 kilometers) per hour. Credit: GPS.gov

However, the satellites are also orbiting Earth about 12,550 miles (20,200 km) above the surface. This actually speeds up GPS satellite clocks by a slighter larger fraction of a second.

Here's how: Einstein's theory also says that gravity curves space and time, causing the passage of time to slow down. High up where the satellites orbit, Earth's gravity is much weaker. This causes the clocks on GPS satellites to run faster than clocks on the ground.

The combined result is that the clocks on GPS satellites experience time at a rate slightly faster than 1 second per second. Luckily, scientists can use math to correct these differences in time.

If scientists didn't correct the GPS clocks, there would be big problems. GPS satellites wouldn't be able to correctly calculate their position or yours. The errors would add up to a few miles each day, which is a big deal. GPS maps might think your home is nowhere near where it actually is!

In Summary:

Yes, time travel is indeed a real thing. But it's not quite what you've probably seen in the movies. Under certain conditions, it is possible to experience time passing at a different rate than 1 second per second. And there are important reasons why we need to understand this real-world form of time travel.

If you liked this, you may like:

April 26, 2023

Is Time Travel Possible?

The laws of physics allow time travel. So why haven’t people become chronological hoppers?

By Sarah Scoles

yuanyuan yan/Getty Images

In the movies, time travelers typically step inside a machine and—poof—disappear. They then reappear instantaneously among cowboys, knights or dinosaurs. What these films show is basically time teleportation .

Scientists don’t think this conception is likely in the real world, but they also don’t relegate time travel to the crackpot realm. In fact, the laws of physics might allow chronological hopping, but the devil is in the details.

Time traveling to the near future is easy: you’re doing it right now at a rate of one second per second, and physicists say that rate can change. According to Einstein’s special theory of relativity, time’s flow depends on how fast you’re moving. The quicker you travel, the slower seconds pass. And according to Einstein’s general theory of relativity , gravity also affects clocks: the more forceful the gravity nearby, the slower time goes.

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing . By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

“Near massive bodies—near the surface of neutron stars or even at the surface of the Earth, although it’s a tiny effect—time runs slower than it does far away,” says Dave Goldberg, a cosmologist at Drexel University.

If a person were to hang out near the edge of a black hole , where gravity is prodigious, Goldberg says, only a few hours might pass for them while 1,000 years went by for someone on Earth. If the person who was near the black hole returned to this planet, they would have effectively traveled to the future. “That is a real effect,” he says. “That is completely uncontroversial.”

Going backward in time gets thorny, though (thornier than getting ripped to shreds inside a black hole). Scientists have come up with a few ways it might be possible, and they have been aware of time travel paradoxes in general relativity for decades. Fabio Costa, a physicist at the Nordic Institute for Theoretical Physics, notes that an early solution with time travel began with a scenario written in the 1920s. That idea involved massive long cylinder that spun fast in the manner of straw rolled between your palms and that twisted spacetime along with it. The understanding that this object could act as a time machine allowing one to travel to the past only happened in the 1970s, a few decades after scientists had discovered a phenomenon called “closed timelike curves.”

“A closed timelike curve describes the trajectory of a hypothetical observer that, while always traveling forward in time from their own perspective, at some point finds themselves at the same place and time where they started, creating a loop,” Costa says. “This is possible in a region of spacetime that, warped by gravity, loops into itself.”

“Einstein read [about closed timelike curves] and was very disturbed by this idea,” he adds. The phenomenon nevertheless spurred later research.

Science began to take time travel seriously in the 1980s. In 1990, for instance, Russian physicist Igor Novikov and American physicist Kip Thorne collaborated on a research paper about closed time-like curves. “They started to study not only how one could try to build a time machine but also how it would work,” Costa says.

Just as importantly, though, they investigated the problems with time travel. What if, for instance, you tossed a billiard ball into a time machine, and it traveled to the past and then collided with its past self in a way that meant its present self could never enter the time machine? “That looks like a paradox,” Costa says.

Since the 1990s, he says, there’s been on-and-off interest in the topic yet no big breakthrough. The field isn’t very active today, in part because every proposed model of a time machine has problems. “It has some attractive features, possibly some potential, but then when one starts to sort of unravel the details, there ends up being some kind of a roadblock,” says Gaurav Khanna of the University of Rhode Island.

For instance, most time travel models require negative mass —and hence negative energy because, as Albert Einstein revealed when he discovered E = mc 2 , mass and energy are one and the same. In theory, at least, just as an electric charge can be positive or negative, so can mass—though no one’s ever found an example of negative mass. Why does time travel depend on such exotic matter? In many cases, it is needed to hold open a wormhole—a tunnel in spacetime predicted by general relativity that connects one point in the cosmos to another.

Without negative mass, gravity would cause this tunnel to collapse. “You can think of it as counteracting the positive mass or energy that wants to traverse the wormhole,” Goldberg says.

Khanna and Goldberg concur that it’s unlikely matter with negative mass even exists, although Khanna notes that some quantum phenomena show promise, for instance, for negative energy on very small scales. But that would be “nowhere close to the scale that would be needed” for a realistic time machine, he says.

These challenges explain why Khanna initially discouraged Caroline Mallary, then his graduate student at the University of Massachusetts Dartmouth, from doing a time travel project. Mallary and Khanna went forward anyway and came up with a theoretical time machine that didn’t require negative mass. In its simplistic form, Mallary’s idea involves two parallel cars, each made of regular matter. If you leave one parked and zoom the other with extreme acceleration, a closed timelike curve will form between them.

Easy, right? But while Mallary’s model gets rid of the need for negative matter, it adds another hurdle: it requires infinite density inside the cars for them to affect spacetime in a way that would be useful for time travel. Infinite density can be found inside a black hole, where gravity is so intense that it squishes matter into a mind-bogglingly small space called a singularity. In the model, each of the cars needs to contain such a singularity. “One of the reasons that there's not a lot of active research on this sort of thing is because of these constraints,” Mallary says.

Other researchers have created models of time travel that involve a wormhole, or a tunnel in spacetime from one point in the cosmos to another. “It's sort of a shortcut through the universe,” Goldberg says. Imagine accelerating one end of the wormhole to near the speed of light and then sending it back to where it came from. “Those two sides are no longer synced,” he says. “One is in the past; one is in the future.” Walk between them, and you’re time traveling.

You could accomplish something similar by moving one end of the wormhole near a big gravitational field—such as a black hole—while keeping the other end near a smaller gravitational force. In that way, time would slow down on the big gravity side, essentially allowing a particle or some other chunk of mass to reside in the past relative to the other side of the wormhole.

Making a wormhole requires pesky negative mass and energy, however. A wormhole created from normal mass would collapse because of gravity. “Most designs tend to have some similar sorts of issues,” Goldberg says. They’re theoretically possible, but there’s currently no feasible way to make them, kind of like a good-tasting pizza with no calories.

And maybe the problem is not just that we don’t know how to make time travel machines but also that it’s not possible to do so except on microscopic scales—a belief held by the late physicist Stephen Hawking. He proposed the chronology protection conjecture: The universe doesn’t allow time travel because it doesn’t allow alterations to the past. “It seems there is a chronology protection agency, which prevents the appearance of closed timelike curves and so makes the universe safe for historians,” Hawking wrote in a 1992 paper in Physical Review D .

Part of his reasoning involved the paradoxes time travel would create such as the aforementioned situation with a billiard ball and its more famous counterpart, the grandfather paradox : If you go back in time and kill your grandfather before he has children, you can’t be born, and therefore you can’t time travel, and therefore you couldn’t have killed your grandfather. And yet there you are.

Those complications are what interests Massachusetts Institute of Technology philosopher Agustin Rayo, however, because the paradoxes don’t just call causality and chronology into question. They also make free will seem suspect. If physics says you can go back in time, then why can’t you kill your grandfather? “What stops you?” he says. Are you not free?

Rayo suspects that time travel is consistent with free will, though. “What’s past is past,” he says. “So if, in fact, my grandfather survived long enough to have children, traveling back in time isn’t going to change that. Why will I fail if I try? I don’t know because I don’t have enough information about the past. What I do know is that I’ll fail somehow.”

If you went to kill your grandfather, in other words, you’d perhaps slip on a banana en route or miss the bus. “It's not like you would find some special force compelling you not to do it,” Costa says. “You would fail to do it for perfectly mundane reasons.”

In 2020 Costa worked with Germain Tobar, then his undergraduate student at the University of Queensland in Australia, on the math that would underlie a similar idea: that time travel is possible without paradoxes and with freedom of choice.

Goldberg agrees with them in a way. “I definitely fall into the category of [thinking that] if there is time travel, it will be constructed in such a way that it produces one self-consistent view of history,” he says. “Because that seems to be the way that all the rest of our physical laws are constructed.”

No one knows what the future of time travel to the past will hold. And so far, no time travelers have come to tell us about it.

Time travel: Is it possible?

Science says time travel is possible, but probably not in the way you're thinking.

Albert Einstein's theory

• General relativity and GPS
• Wormhole travel
• Alternate theories

Science fiction

Is time travel possible? Short answer: Yes, and you're doing it right now — hurtling into the future at the impressive rate of one second per second. 

You're pretty much always moving through time at the same speed, whether you're watching paint dry or wishing you had more hours to visit with a friend from out of town. 

But this isn't the kind of time travel that's captivated countless science fiction writers, or spurred a genre so extensive that Wikipedia lists over 400 titles in the category "Movies about Time Travel." In franchises like " Doctor Who ," " Star Trek ," and "Back to the Future" characters climb into some wild vehicle to blast into the past or spin into the future. Once the characters have traveled through time, they grapple with what happens if you change the past or present based on information from the future (which is where time travel stories intersect with the idea of parallel universes or alternate timelines). 

Related: The best sci-fi time machines ever

Although many people are fascinated by the idea of changing the past or seeing the future before it's due, no person has ever demonstrated the kind of back-and-forth time travel seen in science fiction or proposed a method of sending a person through significant periods of time that wouldn't destroy them on the way. And, as physicist Stephen Hawking pointed out in his book " Black Holes and Baby Universes" (Bantam, 1994), "The best evidence we have that time travel is not possible, and never will be, is that we have not been invaded by hordes of tourists from the future."

Science does support some amount of time-bending, though. For example, physicist Albert Einstein 's theory of special relativity proposes that time is an illusion that moves relative to an observer. An observer traveling near the speed of light will experience time, with all its aftereffects (boredom, aging, etc.) much more slowly than an observer at rest. That's why astronaut Scott Kelly aged ever so slightly less over the course of a year in orbit than his twin brother who stayed here on Earth. 

Related: Controversially, physicist argues that time is real

There are other scientific theories about time travel, including some weird physics that arise around wormholes , black holes and string theory . For the most part, though, time travel remains the domain of an ever-growing array of science fiction books, movies, television shows, comics, video games and more. 

Einstein developed his theory of special relativity in 1905. Along with his later expansion, the theory of general relativity , it has become one of the foundational tenets of modern physics. Special relativity describes the relationship between space and time for objects moving at constant speeds in a straight line. 

The short version of the theory is deceptively simple. First, all things are measured in relation to something else — that is to say, there is no "absolute" frame of reference. Second, the speed of light is constant. It stays the same no matter what, and no matter where it's measured from. And third, nothing can go faster than the speed of light.

From those simple tenets unfolds actual, real-life time travel. An observer traveling at high velocity will experience time at a slower rate than an observer who isn't speeding through space. 

While we don't accelerate humans to near-light-speed, we do send them swinging around the planet at 17,500 mph (28,160 km/h) aboard the International Space Station . Astronaut Scott Kelly was born after his twin brother, and fellow astronaut, Mark Kelly . Scott Kelly spent 520 days in orbit, while Mark logged 54 days in space. The difference in the speed at which they experienced time over the course of their lifetimes has actually widened the age gap between the two men.

"So, where[as] I used to be just 6 minutes older, now I am 6 minutes and 5 milliseconds older," Mark Kelly said in a panel discussion on July 12, 2020, Space.com previously reported . "Now I've got that over his head."

General relativity and GPS time travel

The difference that low earth orbit makes in an astronaut's life span may be negligible — better suited for jokes among siblings than actual life extension or visiting the distant future — but the dilation in time between people on Earth and GPS satellites flying through space does make a difference. 

Read more: Can we stop time?

The Global Positioning System , or GPS, helps us know exactly where we are by communicating with a network of a few dozen satellites positioned in a high Earth orbit. The satellites circle the planet from 12,500 miles (20,100 kilometers) away, moving at 8,700 mph (14,000 km/h). 

According to special relativity, the faster an object moves relative to another object, the slower that first object experiences time. For GPS satellites with atomic clocks, this effect cuts 7 microseconds, or 7 millionths of a second, off each day, according to the American Physical Society publication Physics Central .  

Read more: Could Star Trek's faster-than-light warp drive actually work?

Then, according to general relativity, clocks closer to the center of a large gravitational mass like Earth tick more slowly than those farther away. So, because the GPS satellites are much farther from the center of Earth compared to clocks on the surface, Physics Central added, that adds another 45 microseconds onto the GPS satellite clocks each day. Combined with the negative 7 microseconds from the special relativity calculation, the net result is an added 38 microseconds. 

This means that in order to maintain the accuracy needed to pinpoint your car or phone — or, since the system is run by the U.S. Department of Defense, a military drone — engineers must account for an extra 38 microseconds in each satellite's day. The atomic clocks onboard don’t tick over to the next day until they have run 38 microseconds longer than comparable clocks on Earth.

Given those numbers, it would take more than seven years for the atomic clock in a GPS satellite to un-sync itself from an Earth clock by more than a blink of an eye. (We did the math: If you estimate a blink to last at least 100,000 microseconds, as the Harvard Database of Useful Biological Numbers does, it would take thousands of days for those 38 microsecond shifts to add up.) 

This kind of time travel may seem as negligible as the Kelly brothers' age gap, but given the hyper-accuracy of modern GPS technology, it actually does matter. If it can communicate with the satellites whizzing overhead, your phone can nail down your location in space and time with incredible accuracy. 

Can wormholes take us back in time?

General relativity might also provide scenarios that could allow travelers to go back in time, according to NASA . But the physical reality of those time-travel methods is no piece of cake. 

Wormholes are theoretical "tunnels" through the fabric of space-time that could connect different moments or locations in reality to others. Also known as Einstein-Rosen bridges or white holes, as opposed to black holes, speculation about wormholes abounds. But despite taking up a lot of space (or space-time) in science fiction, no wormholes of any kind have been identified in real life. 

Related: Best time travel movies

"The whole thing is very hypothetical at this point," Stephen Hsu, a professor of theoretical physics at the University of Oregon, told Space.com sister site Live Science . "No one thinks we're going to find a wormhole anytime soon."

Primordial wormholes are predicted to be just 10^-34 inches (10^-33 centimeters) at the tunnel's "mouth". Previously, they were expected to be too unstable for anything to be able to travel through them. However, a study claims that this is not the case, Live Science reported . 

The theory, which suggests that wormholes could work as viable space-time shortcuts, was described by physicist Pascal Koiran. As part of the study, Koiran used the Eddington-Finkelstein metric, as opposed to the Schwarzschild metric which has been used in the majority of previous analyses.

In the past, the path of a particle could not be traced through a hypothetical wormhole. However, using the Eddington-Finkelstein metric, the physicist was able to achieve just that.

Koiran's paper was described in October 2021, in the preprint database arXiv , before being published in the Journal of Modern Physics D.

Alternate time travel theories

While Einstein's theories appear to make time travel difficult, some researchers have proposed other solutions that could allow jumps back and forth in time. These alternate theories share one major flaw: As far as scientists can tell, there's no way a person could survive the kind of gravitational pulling and pushing that each solution requires.

Infinite cylinder theory

Astronomer Frank Tipler proposed a mechanism (sometimes known as a Tipler Cylinder ) where one could take matter that is 10 times the sun's mass, then roll it into a very long, but very dense cylinder. The Anderson Institute , a time travel research organization, described the cylinder as "a black hole that has passed through a spaghetti factory."

After spinning this black hole spaghetti a few billion revolutions per minute, a spaceship nearby — following a very precise spiral around the cylinder — could travel backward in time on a "closed, time-like curve," according to the Anderson Institute. 

The major problem is that in order for the Tipler Cylinder to become reality, the cylinder would need to be infinitely long or be made of some unknown kind of matter. At least for the foreseeable future, endless interstellar pasta is beyond our reach.

Time donuts

Theoretical physicist Amos Ori at the Technion-Israel Institute of Technology in Haifa, Israel, proposed a model for a time machine made out of curved space-time — a donut-shaped vacuum surrounded by a sphere of normal matter.

"The machine is space-time itself," Ori told Live Science . "If we were to create an area with a warp like this in space that would enable time lines to close on themselves, it might enable future generations to return to visit our time."

Amos Ori is a theoretical physicist at the Technion-Israel Institute of Technology in Haifa, Israel. His research interests and publications span the fields of general relativity, black holes, gravitational waves and closed time lines.

There are a few caveats to Ori's time machine. First, visitors to the past wouldn't be able to travel to times earlier than the invention and construction of the time donut. Second, and more importantly, the invention and construction of this machine would depend on our ability to manipulate gravitational fields at will — a feat that may be theoretically possible but is certainly beyond our immediate reach.

Time travel has long occupied a significant place in fiction. Since as early as the "Mahabharata," an ancient Sanskrit epic poem compiled around 400 B.C., humans have dreamed of warping time, Lisa Yaszek, a professor of science fiction studies at the Georgia Institute of Technology in Atlanta, told Live Science .  

Every work of time-travel fiction creates its own version of space-time, glossing over one or more scientific hurdles and paradoxes to achieve its plot requirements. 

Some make a nod to research and physics, like " Interstellar ," a 2014 film directed by Christopher Nolan. In the movie, a character played by Matthew McConaughey spends a few hours on a planet orbiting a supermassive black hole, but because of time dilation, observers on Earth experience those hours as a matter of decades. 

Others take a more whimsical approach, like the "Doctor Who" television series. The series features the Doctor, an extraterrestrial "Time Lord" who travels in a spaceship resembling a blue British police box. "People assume," the Doctor explained in the show, "that time is a strict progression from cause to effect, but actually from a non-linear, non-subjective viewpoint, it's more like a big ball of wibbly-wobbly, timey-wimey stuff." 

Long-standing franchises like the "Star Trek" movies and television series, as well as comic universes like DC and Marvel Comics, revisit the idea of time travel over and over. 

Related: Marvel movies in order: chronological & release order

Here is an incomplete (and deeply subjective) list of some influential or notable works of time travel fiction:

Books about time travel:

• Rip Van Winkle (Cornelius S. Van Winkle, 1819) by Washington Irving
• A Christmas Carol (Chapman & Hall, 1843) by Charles Dickens
• The Time Machine (William Heinemann, 1895) by H. G. Wells
• A Connecticut Yankee in King Arthur's Court (Charles L. Webster and Co., 1889) by Mark Twain
• The Restaurant at the End of the Universe (Pan Books, 1980) by Douglas Adams
• A Tale of Time City (Methuen, 1987) by Diana Wynn Jones
• The Outlander series (Delacorte Press, 1991-present) by Diana Gabaldon
• Harry Potter and the Prisoner of Azkaban (Bloomsbury/Scholastic, 1999) by J. K. Rowling
• Thief of Time (Doubleday, 2001) by Terry Pratchett
• The Time Traveler's Wife (MacAdam/Cage, 2003) by Audrey Niffenegger
• All You Need is Kill (Shueisha, 2004) by Hiroshi Sakurazaka

Movies about time travel:

• Planet of the Apes (1968)
• Superman (1978)
• Time Bandits (1981)
• The Terminator (1984)
• Back to the Future series (1985, 1989, 1990)
• Star Trek IV: The Voyage Home (1986)
• Bill & Ted's Excellent Adventure (1989)
• Groundhog Day (1993)
• Galaxy Quest (1999)
• The Butterfly Effect (2004)
• 13 Going on 30 (2004)
• The Lake House (2006)
• Meet the Robinsons (2007)
• Hot Tub Time Machine (2010)
• Midnight in Paris (2011)
• Looper (2012)
• X-Men: Days of Future Past (2014)
• Edge of Tomorrow (2014)
• Interstellar (2014)
• Doctor Strange (2016)
• A Wrinkle in Time (2018)
• The Last Sharknado: It's About Time (2018)
• Avengers: Endgame (2019)
• Tenet (2020)
• Palm Springs (2020)
• Zach Snyder's Justice League (2021)
• The Tomorrow War (2021)

Television about time travel:

• Doctor Who (1963-present)
• The Twilight Zone (1959-1964) (multiple episodes)
• Star Trek (multiple series, multiple episodes)
• Samurai Jack (2001-2004)
• Lost (2004-2010)
• Phil of the Future (2004-2006)
• Steins;Gate (2011)
• Outlander (2014-2023)
• Loki (2021-present)

Games about time travel:

• Chrono Trigger (1995)
• TimeSplitters (2000-2005)
• Kingdom Hearts (2002-2019)
• Prince of Persia: Sands of Time (2003)
• God of War II (2007)
• Ratchet and Clank Future: A Crack In Time (2009)
• Sly Cooper: Thieves in Time (2013)
• Dishonored 2 (2016)
• Titanfall 2 (2016)
• Outer Wilds (2019)

Additional resources

Explore physicist Peter Millington's thoughts about Stephen Hawking's time travel theories at The Conversation . Check out a kid-friendly explanation of real-world time travel from NASA's Space Place . For an overview of time travel in fiction and the collective consciousness, read " Time Travel: A History " (Pantheon, 2016) by James Gleik. 

Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

Get the Space.com Newsletter

Breaking space news, the latest updates on rocket launches, skywatching events and more!

Vicky Stein is a science writer based in California. She has a bachelor's degree in ecology and evolutionary biology from Dartmouth College and a graduate certificate in science writing from the University of California, Santa Cruz (2018). Afterwards, she worked as a news assistant for PBS NewsHour, and now works as a freelancer covering anything from asteroids to zebras. Follow her most recent work (and most recent pictures of nudibranchs) on Twitter. 

Ed Stone, who led NASA's iconic Voyager project for 50 years, dies at 88

NASA funds tech development for life-hunting Habitable Worlds Observatory

This long-studied star is actually a stellar duo: 'We were absolutely stunned'

Most Popular

• 2 At the heart of this distant galaxy lies not 1, but 2 jet-blasting black holes
• 3 Solar eclipse 2024: Live updates
• 4 Who is Dr. Kovich in 'Star Trek: Discovery'? The mystery explained
• 5 10 top tips for planning your 2027 solar eclipse trip

• Skip to main content
• Keyboard shortcuts for audio player

• LISTEN & FOLLOW
• Apple Podcasts
• Google Podcasts
• Amazon Music
• Amazon Alexa

Your support helps make our show possible and unlocks access to our sponsor-free feed.

Paradox-Free Time Travel Is Theoretically Possible, Researchers Say

Matthew S. Schwartz

A dog dressed as Marty McFly from Back to the Future attends the Tompkins Square Halloween Dog Parade in 2015. New research says time travel might be possible without the problems McFly encountered. Timothy A. Clary/AFP via Getty Images hide caption

A dog dressed as Marty McFly from Back to the Future attends the Tompkins Square Halloween Dog Parade in 2015. New research says time travel might be possible without the problems McFly encountered.

"The past is obdurate," Stephen King wrote in his book about a man who goes back in time to prevent the Kennedy assassination. "It doesn't want to be changed."

Turns out, King might have been on to something.

Countless science fiction tales have explored the paradox of what would happen if you went back in time and did something in the past that endangered the future. Perhaps one of the most famous pop culture examples is in Back to the Future , when Marty McFly goes back in time and accidentally stops his parents from meeting, putting his own existence in jeopardy.

But maybe McFly wasn't in much danger after all. According a new paper from researchers at the University of Queensland, even if time travel were possible, the paradox couldn't actually exist.

Researchers ran the numbers and determined that even if you made a change in the past, the timeline would essentially self-correct, ensuring that whatever happened to send you back in time would still happen.

"Say you traveled in time in an attempt to stop COVID-19's patient zero from being exposed to the virus," University of Queensland scientist Fabio Costa told the university's news service .

"However, if you stopped that individual from becoming infected, that would eliminate the motivation for you to go back and stop the pandemic in the first place," said Costa, who co-authored the paper with honors undergraduate student Germain Tobar.

"This is a paradox — an inconsistency that often leads people to think that time travel cannot occur in our universe."

A variation is known as the "grandfather paradox" — in which a time traveler kills their own grandfather, in the process preventing the time traveler's birth.

The logical paradox has given researchers a headache, in part because according to Einstein's theory of general relativity, "closed timelike curves" are possible, theoretically allowing an observer to travel back in time and interact with their past self — potentially endangering their own existence.

But these researchers say that such a paradox wouldn't necessarily exist, because events would adjust themselves.

Take the coronavirus patient zero example. "You might try and stop patient zero from becoming infected, but in doing so, you would catch the virus and become patient zero, or someone else would," Tobar told the university's news service.

In other words, a time traveler could make changes, but the original outcome would still find a way to happen — maybe not the same way it happened in the first timeline but close enough so that the time traveler would still exist and would still be motivated to go back in time.

"No matter what you did, the salient events would just recalibrate around you," Tobar said.

The paper, "Reversible dynamics with closed time-like curves and freedom of choice," was published last week in the peer-reviewed journal Classical and Quantum Gravity . The findings seem consistent with another time travel study published this summer in the peer-reviewed journal Physical Review Letters. That study found that changes made in the past won't drastically alter the future.

Bestselling science fiction author Blake Crouch, who has written extensively about time travel, said the new study seems to support what certain time travel tropes have posited all along.

"The universe is deterministic and attempts to alter Past Event X are destined to be the forces which bring Past Event X into being," Crouch told NPR via email. "So the future can affect the past. Or maybe time is just an illusion. But I guess it's cool that the math checks out."

• time travel
• grandfather paradox

Is time travel possible? Why one scientist says we 'cannot ignore the possibility.'

A common theme in science-fiction media , time travel is captivating. It’s defined by the late philosopher David Lewis in his essay “The Paradoxes of Time Travel” as “[involving] a discrepancy between time and space time. Any traveler departs and then arrives at his destination; the time elapsed from departure to arrival … is the duration of the journey.”

Time travel is usually understood by most as going back to a bygone era or jumping forward to a point far in the future . But how much of the idea is based in reality? Is it possible to travel through time? 

Is time travel possible?

According to NASA, time travel is possible , just not in the way you might expect. Albert Einstein’s theory of relativity says time and motion are relative to each other, and nothing can go faster than the speed of light , which is 186,000 miles per second. Time travel happens through what’s called “time dilation.”

Time dilation , according to Live Science, is how one’s perception of time is different to another's, depending on their motion or where they are. Hence, time being relative. 

Learn more: Best travel insurance

Dr. Ana Alonso-Serrano, a postdoctoral researcher at the Max Planck Institute for Gravitational Physics in Germany, explained the possibility of time travel and how researchers test theories. 

Space and time are not absolute values, Alonso-Serrano said. And what makes this all more complex is that you are able to carve space-time .

“In the moment that you carve the space-time, you can play with that curvature to make the time come in a circle and make a time machine,” Alonso-Serrano told USA TODAY. 

She explained how, theoretically, time travel is possible. The mathematics behind creating curvature of space-time are solid, but trying to re-create the strict physical conditions needed to prove these theories can be challenging. 

“The tricky point of that is if you can find a physical, realistic, way to do it,” she said. 

Alonso-Serrano said wormholes and warp drives are tools that are used to create this curvature. The matter needed to achieve curving space-time via a wormhole is exotic matter , which hasn’t been done successfully. Researchers don’t even know if this type of matter exists, she said.

“It's something that we work on because it's theoretically possible, and because it's a very nice way to test our theory, to look for possible paradoxes,” Alonso-Serrano added.

“I could not say that nothing is possible, but I cannot ignore the possibility,” she said. 

She also mentioned the anecdote of  Stephen Hawking’s Champagne party for time travelers . Hawking had a GPS-specific location for the party. He didn’t send out invites until the party had already happened, so only people who could travel to the past would be able to attend. No one showed up, and Hawking referred to this event as "experimental evidence" that time travel wasn't possible.

What did Albert Einstein invent?: Discoveries that changed the world

Just Curious for more? We've got you covered

USA TODAY is exploring the questions you and others ask every day. From "How to watch the Marvel movies in order" to "Why is Pluto not a planet?" to "What to do if your dog eats weed?" – we're striving to find answers to the most common questions you ask every day. Head to our Just Curious section to see what else we can answer for you. 

Is time travel even possible? An astrophysicist explains the science behind the science fiction

Published: Nov 13, 2023

By: Magazine Editor

Written by Adi Foord , assistant professor of physics , UMBC

Curious Kids is a series for children of all ages. If you have a question you’d like an expert to answer, send it to [email protected] .

Will it ever be possible for time travel to occur? – Alana C., age 12, Queens, New York

Have you ever dreamed of traveling through time, like characters do in science fiction movies? For centuries, the concept of time travel has captivated people’s imaginations. Time travel is the concept of moving between different points in time, just like you move between different places. In movies, you might have seen characters using special machines, magical devices or even hopping into a futuristic car to travel backward or forward in time.

But is this just a fun idea for movies, or could it really happen?

The question of whether time is reversible remains one of the biggest unresolved questions in science. If the universe follows the laws of thermodynamics , it may not be possible. The second law of thermodynamics states that things in the universe can either remain the same or become more disordered over time.

It’s a bit like saying you can’t unscramble eggs once they’ve been cooked. According to this law, the universe can never go back exactly to how it was before. Time can only go forward, like a one-way street.

Time is relative

However, physicist Albert Einstein’s theory of special relativity suggests that time passes at different rates for different people. Someone speeding along on a spaceship moving close to the speed of light – 671 million miles per hour! – will experience time slower than a person on Earth.

People have yet to build spaceships that can move at speeds anywhere near as fast as light, but astronauts who visit the International Space Station orbit around the Earth at speeds close to 17,500 mph. Astronaut Scott Kelly has spent 520 days at the International Space Station, and as a result has aged a little more slowly than his twin brother – and fellow astronaut – Mark Kelly. Scott used to be 6 minutes younger than his twin brother. Now, because Scott was traveling so much faster than Mark and for so many days, he is 6 minutes and 5 milliseconds younger .

Some scientists are exploring other ideas that could theoretically allow time travel. One concept involves wormholes , or hypothetical tunnels in space that could create shortcuts for journeys across the universe. If someone could build a wormhole and then figure out a way to move one end at close to the speed of light – like the hypothetical spaceship mentioned above – the moving end would age more slowly than the stationary end. Someone who entered the moving end and exited the wormhole through the stationary end would come out in their past.

However, wormholes remain theoretical: Scientists have yet to spot one. It also looks like it would be incredibly challenging to send humans through a wormhole space tunnel.

Paradoxes and failed dinner parties

There are also paradoxes associated with time travel. The famous “ grandfather paradox ” is a hypothetical problem that could arise if someone traveled back in time and accidentally prevented their grandparents from meeting. This would create a paradox where you were never born, which raises the question: How could you have traveled back in time in the first place? It’s a mind-boggling puzzle that adds to the mystery of time travel.

Famously, physicist Stephen Hawking tested the possibility of time travel by throwing a dinner party where invitations noting the date, time and coordinates were not sent out until after it had happened. His hope was that his invitation would be read by someone living in the future, who had capabilities to travel back in time. But no one showed up.

As he pointed out : “The best evidence we have that time travel is not possible, and never will be, is that we have not been invaded by hordes of tourists from the future.”

Telescopes are time machines

Interestingly, astrophysicists armed with powerful telescopes possess a unique form of time travel. As they peer into the vast expanse of the cosmos, they gaze into the past universe. Light from all galaxies and stars takes time to travel, and these beams of light carry information from the distant past. When astrophysicists observe a star or a galaxy through a telescope, they are not seeing it as it is in the present, but as it existed when the light began its journey to Earth millions to billions of years ago. https://www.youtube.com/embed/QeRtcJi3V38?wmode=transparent&start=0 Telescopes are a kind of time machine – they let you peer into the past.

NASA’s newest space telescope, the James Webb Space Telescope , is peering at galaxies that were formed at the very beginning of the Big Bang, about 13.7 billion years ago.

While we aren’t likely to have time machines like the ones in movies anytime soon, scientists are actively researching and exploring new ideas. But for now, we’ll have to enjoy the idea of time travel in our favorite books, movies and dreams.

This article is republished from The Conversation under a Creative Commons license. Read the original article and see more than 250 UMBC articles available in The Conversation.

Tags: CNMS , Physics , The Conversation

Related Posts

• Science & Tech

“Hidden” sex differences in neurological reward pathways suggest opportunity for improved psychiatric therapeutics

Ph.D. candidate Emily Faber improving climate modeling with NOAA fellowship

STEM BUILD at UMBC leads to lasting institutional change, benefiting STEM students and beyond

Search UMBC Search

• Accreditation
• Consumer Information
• Equal Opportunity
• Privacy PDF Download
• Web Accessibility

Search UMBC.edu

National Geographic content straight to your inbox—sign up for our popular newsletters here

Time travel for travelers? It’s tricky.

Scientific theories suggest it’s possible to travel through time. But the reality isn’t so clear.

Time travel has fascinated scientists and writers for at least 125 years. The concept feels especially intriguing now, when physical travel is limited. Here, a photo illustration of Tokyo’s Robot Restaurant captures the idea of speeding through time.

I’m stuck at home, you’re stuck at home, we’re all stuck at home. Jetting off to some fun-filled destination like we used to might not be in the cards for a little while yet. But what about travelling through time? And not just the boring way, where we wait for the future to arrive one second at a time. What if you could zip through time at will, travelling forward to the future or backward to the past as easily as pushing buttons on the dashboard of a souped-up DeLorean, just like in the movie Back to the Future ?

Time travel has been a fantasy for at least 125 years. H.G. Wells penned his groundbreaking novel, The Time Machine , in 1895, and it’s something that physicists and philosophers have been writing serious papers about for almost a century.

What really kick-started scientific investigations into time travel was the notion, dating to the closing years of the 19th century, that time could be envisioned as a dimension, just like space. We can move easily enough through space, so why not time?

At the end of the 19th century, scientists thought of time as a dimension like space, where travelers can go anywhere they want. This photo illustration of Tokyu Plaza in Tokyo’s Omotesando Harajuku evokes the feeling of visiting endless destinations.

“In space, you can go wherever you want, so maybe in time you can similarly go anywhere you want,” says Nikk Effingham, a philosopher at the University of Birmingham in the United Kingdom . “From there, it’s a short step to time machines.”

( Why are people obsessed with time travel? Best-selling author James Gleick has some ideas .)

Dueling theories

Wells was a novelist, not a physicist, but physics would soon catch up. In 1905, Albert Einstein published the first part of his relativity theory, known as special relativity . In it, space and time are malleable; measurements of both space and time depend on the relative speed of the person doing the measuring.

A few years later, the German mathematician Hermann Minkowski showed that, in Einstein’s theory, space and time could be thought of as two aspects of a single four-dimensional entity known as space-time . Then, in 1915, Einstein came up with the second part of his theory, known as general relativity . General relativity renders gravity in a new light: Instead of thinking of it as a force, general relativity describes gravity as a bending or warping of space-time.

But special relativity is enough to get us started in terms of moving through time. The theory “establishes that time is much more similar to space than we had previously thought,” says Clifford Johnson, a physicist at the University of Southern California. “So maybe everything we can do with space, we can do with time.”

Well, almost everything. Special relativity doesn’t give us a way of going back in time, but it does give us a way of going forward— and at a rate that you can actually control. In fact, thanks to special relativity, you can end up with two twins having different ages, the famous “twin paradox.”

Suppose you head off to the Alpha Centauri star system in your spaceship at a really high speed (something close to the speed of light), while your twin remains on Earth. When you come back home, you’ll find you’re now much younger than your twin. It’s counterintuitive, to say the least, but the physics, after more than a century, is rock solid.

“It is absolutely provable in special relativity that the astronaut who makes the journey, if they travel at very nearly the speed of light, will be much younger than their twin when they come back,” says Janna Levin, a physicist at Barnard College in New York . Interestingly, time appears to pass just as it always does for both twins; it’s only when they’re reunited that the difference reveals itself.

Maybe you were both in your 20s when the voyage began. When you come back, you look just a few years older than when you left, while your twin is perhaps now a grandparent. “My experience of the passage of time is utterly normal for me. My clocks tick at the normal rate, I age normally, movies run at the right pace,” says Levin. “I’m no further into my future than normal. But I’ve travelled into my twin’s future.”

( To study aging, scientist are looking to outer space .)

With general relativity, things really start to get interesting. In this theory, a massive object warps or distorts space and time. Perhaps you’ve seen diagrams or videos comparing this to the way a ball distorts a rubber sheet . One result is that, just as travelling at a high speed affects the rate at which time passes, simply being near a really heavy object—like a black hole —will affect one’s experience of time. (This trick was central to the plot of the 2014 film, Interstellar , in which Matthew McConaughey’s character spends time in the vicinity of a massive black hole. When he returns home, he finds that his young daughter is now elderly.)

To get around the “grandfather paradox,” some scientists theorize there could be multiple timelines. In these images of Nakagin Capsule Tower in Tokyo, Japan, time seems to pass at different rates.

But black holes are just the beginning. Physicists have also speculated about the implications of a much more exotic structure known as a wormhole . Wormholes, if they exist, could connect one location in space-time with another. An astronaut who enters a wormhole in the Andromeda Galaxy in the year 3000 might find herself emerging from the other end in our own galaxy, in the year 2000. But there’s a catch: While we have overwhelming evidence that black holes exist in nature—astronomers even photographed one last year—wormholes are far more speculative.

“You can imagine building a bridge from one region of space-time to another region of space-time,” explains Levin, “but it would require kinds of mass and energy that we don’t really know exist in reality, things like negative energy.” She says it’s “mathematically conceivable” that structures such as wormholes could exist, but they may not be part of physical reality.

There’s also the troubling question of what happens to our notions of cause and effect if backward time travel were possible. The most famous of these conundrums is the so-called “ grandfather paradox .” Suppose you travel back in time to when your grandfather was a young man. You kill him (perhaps by accident), which means your parent won’t be born, which means you won’t be born. Therefore, you won’t be able to travel through time and kill your grandfather.

Multiple timelines?

Over the years, physicists and philosophers have pondered various resolutions to the grandfather paradox. One possibility is that the paradox simply proves that no such journeys are possible; the laws of physics, somehow, must prevent backward time travel. This was the view of the late physicist Stephen Hawking , who called this rule the “ chronology protection conjecture .” (Mind you, he never specified the actual physics behind such a rule.)

But there are also other, more intriguing, solutions. Maybe backward time travel is possible, and yet time travelers can’t change the past, no matter how hard they try. Effingham, whose book Time Travel: Probability and Impossibility was published earlier this year, puts it this way: “You might shoot the wrong person, or you might change your mind. Or, you might shoot the person you think is your grandfather, but it turns out your grandmother had an affair with the milkman, and that’s who your grandfather was all along; you just didn’t know it.”

Which also means the much-discussed fantasy of killing Hitler before the outbreak of World War II is a non-starter. “It’s impossible because it didn’t happen,” says Fabio Costa, a theoretical physicist at the University of Queensland in Australia . “It’s not even a question. We know how history developed. There is no re-do.”

In fact, suggests Effingham, if you can’t change the past, then a time traveler probably can’t do anything . Your mere existence at a time in which you never existed would be a contradiction. “The universe doesn’t care whether the thing you’ve changed is that you’ve killed Hitler, or that you moved an atom from position A to position B,” Effingham says.

But all is not lost. The scenarios Effingham and Costa are imagining involve a single universe with a single “timeline.” But some physicists speculate that our universe is just one among many . If that’s the case, then perhaps time travelers who visit the past can do as they please, which would shed new light on the grandfather paradox.

( The Big Bang could have led to the creation of multiple universes, scientists say .)

“Maybe, for whatever reason, you decide to go back and commit this crime [of killing your grandfather], and so the world ‘branches off’ into two different realities,” says Levin. As a result, “even though you seem to be altering your past, you’re not really altering it; you’re creating a new history.” (This idea of multiple timelines lies at the heart of the Back to the Future movie trilogy. In contrast, in the movie 12 Monkeys , Bruce Willis’s character makes multiple journeys through time, but everything plays out along a single timeline.)

More work to be done

What everyone seems to agree on is that no one is building a time-travelling DeLorean or engineering a custom-built wormhole anytime soon. Instead, physicists are focusing on completing the work that Einstein began a century ago.

After more than 100 years, no one has figured out how to reconcile general relativity with the other great pillar of 20th century physics: quantum mechanics . Some physicists believe that a long-sought unified theory known as quantum gravity will yield new insight into the nature of time. At the very least, says Levin, it seems likely “that we need to go beyond just general relativity to understand time.”

Meanwhile, it’s no surprise that, like H.G. Wells, we continue to daydream about having the freedom to move through time just as we move through space. “Time is embedded in everything we do,” says Johnson. “It looms large in how we perceive the world. So being able to mess with time—I’m not surprised we’re obsessed with that, and fantasize about it.”

Related Topics

You may also like.

Why it's high time for slow travel in Gstaad, Switzerland

Scared to scuba? Here are 5 reasons it’s finally time to learn.

Introducing nat geo kids book bundle.

Step inside the factory where the NFL’s footballs are made

Volcanoes blow smoke rings. Here's how they do it.

The true history of Einstein's role in developing the atomic bomb

The monarch butterfly’s spots may be its superpower

What is a sonic boom—and is it dangerous?

• Perpetual Planet
• Environment

History & Culture

• History & Culture
• Race in America
• Mind, Body, Wonder
• Paid Content
• Terms of Use
• Privacy Policy
• Your US State Privacy Rights
• Children's Online Privacy Policy
• Interest-Based Ads
• About Nielsen Measurement
• Do Not Sell or Share My Personal Information
• Nat Geo Home
• Attend a Live Event
• Book a Trip
• Inspire Your Kids
• Shop Nat Geo
• Visit the D.C. Museum
• Learn About Our Impact
• Support Our Mission
• Advertise With Us
• Customer Service
• Renew Subscription
• Manage Your Subscription
• Work at Nat Geo
• Sign Up for Our Newsletters
• Contribute to Protect the Planet

Copyright © 1996-2015 National Geographic Society Copyright © 2015-2024 National Geographic Partners, LLC. All rights reserved

To revisit this article, visit My Profile, then View saved stories .

• Backchannel
• Newsletters
• WIRED Insider
• WIRED Consulting

Oliver Franklin-Wallis

A brief history of time travel

Subscribe to WIRED

Of all time travel's paradoxes, here's the strangest of them all: hop on a TARDIS back to 1894 and the concept didn't even exist. "Time travel is a new idea," explains New York-based author James Gleick, 62. "It's a very modern myth." Gleick's entertaining Time Travel: A History , out in hardback in February, quantum leaps from HG Wells's The Time Machine - the original - via Proust and alt-history right up to your Twitter timeline. Until we get the DeLorean working for real, fellow travellers, consider it the next best thing.

The Mahabharata

Time travel appears in Hindu text The Mahabharata, and in stories such as Washington Irving's Rip Van Winkle (1819) - but it usually only involved a one-way trip. "People fell asleep, and woke 
up in the future," says Gleick.

HG Wells's The Time Machine

"The idea of time travel with volition, in either direction, didn't arrive until Wells," says Gleick. It explains that time is a dimension - something not widely accepted until Einstein's theories in 1905.

Henri Bergson's Time And Free Will

Bergson's thesis is published soon after Wells's novel. "Bergson is a friend of Marcel Proust," says Gleick. Soon Proust et al are jumping on the idea of time travel to explore free will - and influencing new sci-fi in return.

Time Capsules

The idea of preserving a time stamp only arose in the 1930s in Scientific American. "It's the most pedestrian form of time travel: sending something into the future at a rate of one minute per minute."

Robert A Heinlein's By His Bootstraps

Heinlein's short story, published in Astounding Science Fiction, introduces the idea of a character appearing in multiple timelines, meeting themselves amid complex - and funny - paradoxes.

William Gibson's The Peripheral

Gleick cites Gibson's unique twist on the genre: "We can't send people, but what if you could send information back to the past?" It's a chilling new take. "It shows how our cultural conception of time is changing."

This article was originally published by WIRED UK

By Kim Zetter

By Chris Baraniuk

By Mark Harris

By Vittoria Elliott

Dennis Mersereau

Paul Sutter, Ars Technica

Emily Mullin

Rhett Allain

Steve Nadis

Jonathan O'Callaghan

• Table of Contents
• Random Entry
• Chronological
• Editorial Information
• About the SEP
• Editorial Board
• How to Cite the SEP
• Special Characters
• Advanced Tools
• Support the SEP
• PDFs for SEP Friends
• Make a Donation
• SEPIA for Libraries
• Entry Contents

Bibliography

Academic tools.

• Friends PDF Preview
• Author and Citation Info
• Back to Top

Time Travel and Modern Physics

Time travel has been a staple of science fiction. With the advent of general relativity it has been entertained by serious physicists. But, especially in the philosophy literature, there have been arguments that time travel is inherently paradoxical. The most famous paradox is the grandfather paradox: you travel back in time and kill your grandfather, thereby preventing your own existence. To avoid inconsistency some circumstance will have to occur which makes you fail in this attempt to kill your grandfather. Doesn’t this require some implausible constraint on otherwise unrelated circumstances? We examine such worries in the context of modern physics.

1. Paradoxes Lost?

2. topology and constraints, 3. the general possibility of time travel in general relativity, 4. two toy models, 5. slightly more realistic models of time travel, 6. the possibility of time travel redux, 7. even if there are constraints, so what, 8. computational models, 9. quantum mechanics to the rescue, 10. conclusions, other internet resources, related entries.

• Supplement: Remarks and Limitations on the Toy Models

Modern physics strips away many aspects of the manifest image of time. Time as it appears in the equations of classical mechanics has no need for a distinguished present moment, for example. Relativity theory leads to even sharper contrasts. It replaces absolute simultaneity, according to which it is possible to unambiguously determine the time order of distant events, with relative simultaneity: extending an “instant of time” throughout space is not unique, but depends on the state of motion of an observer. More dramatically, in general relativity the mathematical properties of time (or better, of spacetime)—its topology and geometry—depend upon how matter is arranged rather than being fixed once and for all. So physics can be, and indeed has to be, formulated without treating time as a universal, fixed background structure. Since general relativity represents gravity through spacetime geometry, the allowed geometries must be as varied as the ways in which matter can be arranged. Alongside geometrical models used to describe the solar system, black holes, and much else, the scope of variation extends to include some exotic structures unlike anything astrophysicists have observed. In particular, there are spacetime geometries with curves that loop back on themselves: closed timelike curves (CTCs), which describe the possible trajectory of an observer who returns exactly back to their earlier state—without any funny business, such as going faster than the speed of light. These geometries satisfy the relevant physical laws, the equations of general relativity, and in that sense time travel is physically possible.

Yet circular time generates paradoxes, familiar from science fiction stories featuring time travel: [ 1 ]

• Consistency: Kurt plans to murder his own grandfather Adolph, by traveling along a CTC to an appropriate moment in the past. He is an able marksman, and waits until he has a clear shot at grandpa. Normally he would not miss. Yet if he succeeds, there is no way that he will then exist to plan and carry out the mission. Kurt pulls the trigger: what can happen?
• Underdetermination: Suppose that Kurt first travels back in order to give his earlier self a copy of How to Build a Time Machine. This is the same book that allows him to build a time machine, which he then carries with him on his journey to the past. Who wrote the book?
• Easy Knowledge: A fan of classical music enhances their computer with a circuit that exploits a CTC. This machine efficiently solves problems at a higher level of computational complexity than conventional computers, leading (among other things) to finding the smallest circuits that can generate Bach’s oeuvre—and to compose new pieces in the same style. Such easy knowledge is at odds with our understanding of our epistemic predicament. (This third paradox has not drawn as much attention.)

The first two paradoxes were once routinely taken to show that solutions with CTCs should be rejected—with charges varying from violating logic, to being “physically unreasonable”, to undermining the notion of free will. Closer analysis of the paradoxes has largely reversed this consensus. Physicists have discovered many solutions with CTCs and have explored their properties in pursuing foundational questions, such as whether physics is compatible with the idea of objective temporal passage (starting with Gödel 1949). Philosophers have also used time travel scenarios to probe questions about, among other things, causation, modality, free will, and identity (see, e.g., Earman 1972 and Lewis’s seminal 1976 paper).

We begin below with Consistency , turning to the other paradoxes in later sections. A standard, stone-walling response is to insist that the past cannot be changed, as a matter of logic, even by a time traveler (e.g., Gödel 1949, Clarke 1977, Horwich 1987). Adolph cannot both die and survive, as a matter of logic, so any scheme to alter the past must fail. In many of the best time travel fictions, the actions of a time traveler are constrained in novel and unexpected ways. Attempts to change the past fail, and they fail, often tragically, in just such a way that they set the stage for the time traveler’s self-defeating journey. The first question is whether there is an analog of the consistent story when it comes to physics in the presence of CTCs. As we will see, there is a remarkable general argument establishing the existence of consistent solutions. Yet a second question persists: why can’t time-traveling Kurt kill his own grandfather? Doesn’t the necessity of failures to change the past put unusual and unexpected constraints on time travelers, or objects that move along CTCs? The same argument shows that there are in fact no constraints imposed by the existence of CTCs, in some cases. After discussing this line of argument, we will turn to the palatability and further implications of such constraints if they are required, and then turn to the implications of quantum mechanics.

Wheeler and Feynman (1949) were the first to claim that the fact that nature is continuous could be used to argue that causal influences from later events to earlier events, as are made possible by time travel, will not lead to paradox without the need for any constraints. Maudlin (1990) showed how to make their argument precise and more general, and argued that nonetheless it was not completely general.

Imagine the following set-up. We start off having a camera with a black and white film ready to take a picture of whatever comes out of the time machine. An object, in fact a developed film, comes out of the time machine. We photograph it, and develop the film. The developed film is subsequently put in the time machine, and set to come out of the time machine at the time the picture is taken. This surely will create a paradox: the developed film will have the opposite distribution of black, white, and shades of gray, from the object that comes out of the time machine. For developed black and white films (i.e., negatives) have the opposite shades of gray from the objects they are pictures of. But since the object that comes out of the time machine is the developed film itself it we surely have a paradox.

However, it does not take much thought to realize that there is no paradox here. What will happen is that a uniformly gray picture will emerge, which produces a developed film that has exactly the same uniform shade of gray. No matter what the sensitivity of the film is, as long as the dependence of the brightness of the developed film depends in a continuous manner on the brightness of the object being photographed, there will be a shade of gray that, when photographed, will produce exactly the same shade of gray on the developed film. This is the essence of Wheeler and Feynman’s idea. Let us first be a bit more precise and then a bit more general.

For simplicity let us suppose that the film is always a uniform shade of gray (i.e., at any time the shade of gray does not vary by location on the film). The possible shades of gray of the film can then be represented by the (real) numbers from 0, representing pure black, to 1, representing pure white.

Let us now distinguish various stages in the chronological order of the life of the film. In stage \(S_1\) the film is young; it has just been placed in the camera and is ready to be exposed. It is then exposed to the object that comes out of the time machine. (That object in fact is a later stage of the film itself). By the time we come to stage \(S_2\) of the life of the film, it has been developed and is about to enter the time machine. Stage \(S_3\) occurs just after it exits the time machine and just before it is photographed. Stage \(S_4\) occurs after it has been photographed and before it starts fading away. Let us assume that the film starts out in stage \(S_1\) in some uniform shade of gray, and that the only significant change in the shade of gray of the film occurs between stages \(S_1\) and \(S_2\). During that period it acquires a shade of gray that depends on the shade of gray of the object that was photographed. In other words, the shade of gray that the film acquires at stage \(S_2\) depends on the shade of gray it has at stage \(S_3\). The influence of the shade of gray of the film at stage \(S_3\), on the shade of gray of the film at stage \(S_2\), can be represented as a mapping, or function, from the real numbers between 0 and 1 (inclusive), to the real numbers between 0 and 1 (inclusive). Let us suppose that the process of photography is such that if one imagines varying the shade of gray of an object in a smooth, continuous manner then the shade of gray of the developed picture of that object will also vary in a smooth, continuous manner. This implies that the function in question will be a continuous function. Now any continuous function from the real numbers between 0 and 1 (inclusive) to the real numbers between 0 and 1 (inclusive) must map at least one number to itself. One can quickly convince oneself of this by graphing such functions. For one will quickly see that any continuous function \(f\) from \([0,1]\) to \([0,1]\) must intersect the line \(x=y\) somewhere, and thus there must be at least one point \(x\) such that \(f(x)=x\). Such points are called fixed points of the function. Now let us think about what such a fixed point represents. It represents a shade of gray such that, when photographed, it will produce a developed film with exactly that same shade of gray. The existence of such a fixed point implies a solution to the apparent paradox.

Let us now be more general and allow color photography. One can represent each possible color of an object (of uniform color) by the proportions of blue, green and red that make up that color. (This is why television screens can produce all possible colors.) Thus one can represent all possible colors of an object by three points on three orthogonal lines \(x, y\) and \(z\), that is to say, by a point in a three-dimensional cube. This cube is also known as the “Cartesian product” of the three line segments. Now, one can also show that any continuous map from such a cube to itself must have at least one fixed point. So color photography can not be used to create time travel paradoxes either!

Even more generally, consider some system \(P\) which, as in the above example, has the following life. It starts in some state \(S_1\), it interacts with an object that comes out of a time machine (which happens to be its older self), it travels back in time, it interacts with some object (which happens to be its younger self), and finally it grows old and dies. Let us assume that the set of possible states of \(P\) can be represented by a Cartesian product of \(n\) closed intervals of the reals, i.e., let us assume that the topology of the state-space of \(P\) is isomorphic to a finite Cartesian product of closed intervals of the reals. Let us further assume that the development of \(P\) in time, and the dependence of that development on the state of objects that it interacts with, is continuous. Then, by a well-known fixed point theorem in topology (see, e.g., Hocking & Young 1961: 273), no matter what the nature of the interaction is, and no matter what the initial state of the object is, there will be at least one state \(S_3\) of the older system (as it emerges from the time travel machine) that will influence the initial state \(S_1\) of the younger system (when it encounters the older system) so that, as the younger system becomes older, it develops exactly into state \(S_3\). Thus without imposing any constraints on the initial state \(S_1\) of the system \(P\), we have shown that there will always be perfectly ordinary, non-paradoxical, solutions, in which everything that happens, happens according to the usual laws of development. Of course, there is looped causation, hence presumably also looped explanation, but what do you expect if there is looped time?

Unfortunately, for the fan of time travel, a little reflection suggests that there are systems for which the needed fixed point theorem does not hold. Imagine, for instance, that we have a dial that can only rotate in a plane. We are going to put the dial in the time machine. Indeed we have decided that if we see the later stage of the dial come out of the time machine set at angle \(x\), then we will set the dial to \(x+90\), and throw it into the time machine. Now it seems we have a paradox, since the mapping that consists of a rotation of all points in a circular state-space by 90 degrees does not have a fixed point. And why wouldn’t some state-spaces have the topology of a circle?

However, we have so far not used another continuity assumption which is also a reasonable assumption. So far we have only made the following demand: the state the dial is in at stage \(S_2\) must be a continuous function of the state of the dial at stage \(S_3\). But, the state of the dial at stage \(S_2\) is arrived at by taking the state of the dial at stage \(S_1\), and rotating it over some angle. It is not merely the case that the effect of the interaction, namely the state of the dial at stage \(S_2\), should be a continuous function of the cause, namely the state of the dial at stage \(S_3\). It is additionally the case that path taken to get there, the way the dial is rotated between stages \(S_1\) and \(S_2\) must be a continuous function of the state at stage \(S_3\). And, rather surprisingly, it turns out that this can not be done. Let us illustrate what the problem is before going to a more general demonstration that there must be a fixed point solution in the dial case.

Forget time travel for the moment. Suppose that you and I each have a watch with a single dial neither of which is running. My watch is set at 12. You are going to announce what your watch is set at. My task is going to be to adjust my watch to yours no matter what announcement you make. And my actions should have a continuous (single valued) dependence on the time that you announce. Surprisingly, this is not possible! For instance, suppose that if you announce “12”, then I achieve that setting on my watch by doing nothing. Now imagine slowly and continuously increasing the announced times, starting at 12. By continuity, I must achieve each of those settings by rotating my dial to the right. If at some point I switch and achieve the announced goal by a rotation of my dial to the left, I will have introduced a discontinuity in my actions, a discontinuity in the actions that I take as a function of the announced angle. So I will be forced, by continuity, to achieve every announcement by rotating the dial to the right. But, this rotation to the right will have to be abruptly discontinued as the announcements grow larger and I eventually approach 12 again, since I achieved 12 by not rotating the dial at all. So, there will be a discontinuity at 12 at the latest. In general, continuity of my actions as a function of announced times can not be maintained throughout if I am to be able to replicate all possible settings. Another way to see the problem is that one can similarly reason that, as one starts with 12, and imagines continuously making the announced times earlier, one will be forced, by continuity, to achieve the announced times by rotating the dial to the left. But the conclusions drawn from the assumption of continuous increases and the assumption of continuous decreases are inconsistent. So we have an inconsistency following from the assumption of continuity and the assumption that I always manage to set my watch to your watch. So, a dial developing according to a continuous dynamics from a given initial state, can not be set up so as to react to a second dial, with which it interacts, in such a way that it is guaranteed to always end up set at the same angle as the second dial. Similarly, it can not be set up so that it is guaranteed to always end up set at 90 degrees to the setting of the second dial. All of this has nothing to do with time travel. However, the impossibility of such set ups is what prevents us from enacting the rotation by 90 degrees that would create paradox in the time travel setting.

Let us now give the positive result that with such dials there will always be fixed point solutions, as long as the dynamics is continuous. Let us call the state of the dial before it interacts with its older self the initial state of the dial. And let us call the state of the dial after it emerges from the time machine the final state of the dial. There is also an intermediate state of the dial, after it interacts with its older self and before it is put into the time machine. We can represent the initial or intermediate states of the dial, before it goes into the time machine, as an angle \(x\) in the horizontal plane and the final state of the dial, after it comes out of the time machine, as an angle \(y\) in the vertical plane. All possible \(\langle x,y\rangle\) pairs can thus be visualized as a torus with each \(x\) value picking out a vertical circular cross-section and each \(y\) picking out a point on that cross-section. See figure 1 .

Figure 1 [An extended description of figure 1 is in the supplement.]

Suppose that the dial starts at angle \(i\) which picks out vertical circle \(I\) on the torus. The initial angle \(i\) that the dial is at before it encounters its older self, and the set of all possible final angles that the dial can have when it emerges from the time machine is represented by the circle \(I\) on the torus (see figure 1 ). Given any possible angle of the emerging dial, the dial initially at angle \(i\) will develop to some other angle. One can picture this development by rotating each point on \(I\) in the horizontal direction by the relevant amount. Since the rotation has to depend continuously on the angle of the emerging dial, circle \(I\) during this development will deform into some loop \(L\) on the torus. Loop \(L\) thus represents all possible intermediate angles \(x\) that the dial is at when it is thrown into the time machine, given that it started at angle \(i\) and then encountered a dial (its older self) which was at angle \(y\) when it emerged from the time machine. We therefore have consistency if \(x=y\) for some \(x\) and \(y\) on loop \(L\). Now, let loop \(C\) be the loop which consists of all the points on the torus for which \(x=y\). Ring \(I\) intersects \(C\) at point \(\langle i,i\rangle\). Obviously any continuous deformation of \(I\) must still intersect \(C\) somewhere. So \(L\) must intersect \(C\) somewhere, say at \(\langle j,j\rangle\). But that means that no matter how the development of the dial starting at \(I\) depends on the angle of the emerging dial, there will be some angle for the emerging dial such that the dial will develop exactly into that angle (by the time it enters the time machine) under the influence of that emerging dial. This is so no matter what angle one starts with, and no matter how the development depends on the angle of the emerging dial. Thus even for a circular state-space there are no constraints needed other than continuity.

Unfortunately there are state-spaces that escape even this argument. Consider for instance a pointer that can be set to all values between 0 and 1, where 0 and 1 are not possible values. That is, suppose that we have a state-space that is isomorphic to an open set of real numbers. Now suppose that we have a machine that sets the pointer to half the value that the pointer is set at when it emerges from the time machine.

Figure 2 [An extended description of figure 2 is in the supplement.]

Suppose the pointer starts at value \(I\). As before we can represent the combination of this initial position and all possible final positions by the line \(I\). Under the influence of the pointer coming out of the time machine the pointer value will develop to a value that equals half the value of the final value that it encountered. We can represent this development as the continuous deformation of line \(I\) into line \(L\), which is indicated by the arrows in figure 2 . This development is fully continuous. Points \(\langle x,y\rangle\) on line \(I\) represent the initial position \(x=I\) of the (young) pointer, and the position \(y\) of the older pointer as it emerges from the time machine. Points \(\langle x,y\rangle\) on line \(L\) represent the position \(x\) that the younger pointer should develop into, given that it encountered the older pointer emerging from the time machine set at position \(y\). Since the pointer is designed to develop to half the value of the pointer that it encounters, the line \(L\) corresponds to \(x=1/2 y\). We have consistency if there is some point such that it develops into that point, if it encounters that point. Thus, we have consistency if there is some point \(\langle x,y\rangle\) on line \(L\) such that \(x=y\). However, there is no such point: lines \(L\) and \(C\) do not intersect. Thus there is no consistent solution, despite the fact that the dynamics is fully continuous.

Of course if 0 were a possible value, \(L\) and \(C\) would intersect at 0. This is surprising and strange: adding one point to the set of possible values of a quantity here makes the difference between paradox and peace. One might be tempted to just add the extra point to the state-space in order to avoid problems. After all, one might say, surely no measurements could ever tell us whether the set of possible values includes that exact point or not. Unfortunately there can be good theoretical reasons for supposing that some quantity has a state-space that is open: the set of all possible speeds of massive objects in special relativity surely is an open set, since it includes all speeds up to, but not including, the speed of light. Quantities that have possible values that are not bounded also lead to counter examples to the presented fixed point argument. And it is not obvious to us why one should exclude such possibilities. So the argument that no constraints are needed is not fully general.

An interesting question of course is: exactly for which state-spaces must there be such fixed points? The arguments above depend on a well-known fixed point theorem (due to Schauder) that guarantees the existence of a fixed point for compact, convex state spaces. We do not know what subsequent extensions of this result imply regarding fixed points for a wider variety of systems, or whether there are other general results along these lines. (See Kutach 2003 for more on this issue.)

A further interesting question is whether this line of argument is sufficient to resolve Consistency (see also Dowe 2007). When they apply, these results establish the existence of a solution, such as the shade of uniform gray in the first example. But physicists routinely demand more than merely the existence of a solution, namely that solutions to the equations are stable—such that “small” changes of the initial state lead to “small” changes of the resulting trajectory. (Clarifying the two senses of “small” in this statement requires further work, specifying the relevant topology.) Stability in this sense underwrites the possibility of applying equations to real systems given our inability to fix initial states with indefinite precision. (See Fletcher 2020 for further discussion.) The fixed point theorems guarantee that for an initial state \(S_1\) there is a solution, but this solution may not be “close” to the solution for a nearby initial state, \(S'\). We are not aware of any proofs that the solutions guaranteed to exist by the fixed point theorems are also stable in this sense.

Time travel has recently been discussed quite extensively in the context of general relativity. General relativity places few constraints on the global structure of space and time. This flexibility leads to a possibility first described in print by Hermann Weyl:

Every world-point is the origin of the double-cone of the active future and the passive past [i.e., the two lobes of the light cone]. Whereas in the special theory of relativity these two portions are separated by an intervening region, it is certainly possible in the present case [i.e., general relativity] for the cone of the active future to overlap with that of the passive past; so that, in principle, it is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world-line (in particular, that of my body), although it has a timelike direction at every point, to return to the neighborhood of a point which it has already once passed through. (Weyl 1918/1920 [1952: 274])

A time-like curve is simply a space-time trajectory such that the speed of light is never equaled or exceeded along this trajectory. Time-like curves represent possible trajectories of ordinary objects. In general relativity a curve that is everywhere timelike locally can nonetheless loop back on itself, forming a CTC. Weyl makes the point vividly in terms of the light cones: along such a curve, the future lobe of the light cone (the “active future”) intersects the past lobe of the light cone (the “passive past”). Traveling along such a curve one would never exceed the speed of light, and yet after a certain amount of (proper) time one would return to a point in space-time that one previously visited. Or, by staying close to such a CTC, one could come arbitrarily close to a point in space-time that one previously visited. General relativity, in a straightforward sense, allows time travel: there appear to be many space-times compatible with the fundamental equations of general relativity in which there are CTC’s. Space-time, for instance, could have a Minkowski metric everywhere, and yet have CTC’s everywhere by having the temporal dimension (topologically) rolled up as a circle. Or, one can have wormhole connections between different parts of space-time which allow one to enter “mouth \(A\)” of such a wormhole connection, travel through the wormhole, exit the wormhole at “mouth \(B\)” and re-enter “mouth \(A\)” again. CTCs can even arise when the spacetime is topologically \(\mathbb{R}^4\), due to the “tilting” of light cones produced by rotating matter (as in Gödel 1949’s spacetime).

General relativity thus appears to provide ample opportunity for time travel. Note that just because there are CTC’s in a space-time, this does not mean that one can get from any point in the space-time to any other point by following some future directed timelike curve—there may be insurmountable practical obstacles. In Gödel’s spacetime, it is the case that there are CTCs passing through every point in the spacetime. Yet these CTCs are not geodesics, so traversing them requires acceleration. Calculations of the minimal fuel required to travel along the appropriate curve should discourage any would-be time travelers (Malament 1984, 1985; Manchak 2011). But more generally CTCs may be confined to smaller regions; some parts of space-time can have CTC’s while other parts do not. Let us call the part of a space-time that has CTC’s the “time travel region” of that space-time, while calling the rest of that space-time the “normal region”. More precisely, the “time travel region” consists of all the space-time points \(p\) such that there exists a (non-zero length) timelike curve that starts at \(p\) and returns to \(p\). Now let us turn to examining space-times with CTC’s a bit more closely for potential problems.

In order to get a feeling for the sorts of implications that closed timelike curves can have, it may be useful to consider two simple models. In space-times with closed timelike curves the traditional initial value problem cannot be framed in the usual way. For it presupposes the existence of Cauchy surfaces, and if there are CTCs then no Cauchy surface exists. (A Cauchy surface is a spacelike surface such that every inextendable timelike curve crosses it exactly once. One normally specifies initial conditions by giving the conditions on such a surface.) Nonetheless, if the topological complexities of the manifold are appropriately localized, we can come quite close. Let us call an edgeless spacelike surface \(S\) a quasi-Cauchy surface if it divides the rest of the manifold into two parts such that

• every point in the manifold can be connected by a timelike curve to \(S\), and
• any timelike curve which connects a point in one region to a point in the other region intersects \(S\) exactly once.

It is obvious that a quasi-Cauchy surface must entirely inhabit the normal region of the space-time; if any point \(p\) of \(S\) is in the time travel region, then any timelike curve which intersects \(p\) can be extended to a timelike curve which intersects \(S\) near \(p\) again. In extreme cases of time travel, a model may have no normal region at all (e.g., Minkowski space-time rolled up like a cylinder in a time-like direction), in which case our usual notions of temporal precedence will not apply. But temporal anomalies like wormholes (and time machines) can be sufficiently localized to permit the existence of quasi-Cauchy surfaces.

Given a timelike orientation, a quasi-Cauchy surface unproblematically divides the manifold into its past (i.e., all points that can be reached by past-directed timelike curves from \(S)\) and its future (ditto mutatis mutandis ). If the whole past of \(S\) is in the normal region of the manifold, then \(S\) is a partial Cauchy surface : every inextendable timelike curve which exists to the past of \(S\) intersects \(S\) exactly once, but (if there is time travel in the future) not every inextendable timelike curve which exists to the future of \(S\) intersects \(S\). Now we can ask a particularly clear question: consider a manifold which contains a time travel region, but also has a partial Cauchy surface \(S\), such that all of the temporal funny business is to the future of \(S\). If all you could see were \(S\) and its past, you would not know that the space-time had any time travel at all. The question is: are there any constraints on the sort of data which can be put on \(S\) and continued to a global solution of the dynamics which are different from the constraints (if any) on the data which can be put on a Cauchy surface in a simply connected manifold and continued to a global solution? If there is time travel to our future, might we we able to tell this now, because of some implied oddity in the arrangement of present things?

It is not at all surprising that there might be constraints on the data which can be put on a locally space-like surface which passes through the time travel region: after all, we never think we can freely specify what happens on a space-like surface and on another such surface to its future, but in this case the surface at issue lies to its own future. But if there were particular constraints for data on a partial Cauchy surface then we would apparently need to have to rule out some sorts of otherwise acceptable states on \(S\) if there is to be time travel to the future of \(S\). We then might be able to establish that there will be no time travel in the future by simple inspection of the present state of the universe. As we will see, there is reason to suspect that such constraints on the partial Cauchy surface are non-generic. But we are getting ahead of ourselves: first let’s consider the effect of time travel on a very simple dynamics.

The simplest possible example is the Newtonian theory of perfectly elastic collisions among equally massive particles in one spatial dimension. The space-time is two-dimensional, so we can represent it initially as the Euclidean plane, and the dynamics is completely specified by two conditions. When particles are traveling freely, their world lines are straight lines in the space-time, and when two particles collide, they exchange momenta, so the collision looks like an “\(X\)” in space-time, with each particle changing its momentum at the impact. [ 2 ] The dynamics is purely local, in that one can check that a set of world-lines constitutes a model of the dynamics by checking that the dynamics is obeyed in every arbitrarily small region. It is also trivial to generate solutions from arbitrary initial data if there are no CTCs: given the initial positions and momenta of a set of particles, one simply draws a straight line from each particle in the appropriate direction and continues it indefinitely. Once all the lines are drawn, the worldline of each particle can be traced from collision to collision. The boundary value problem for this dynamics is obviously well-posed: any set of data at an instant yields a unique global solution, constructed by the method sketched above.

What happens if we change the topology of the space-time by hand to produce CTCs? The simplest way to do this is depicted in figure 3 : we cut and paste the space-time so it is no longer simply connected by identifying the line \(L-\) with the line \(L+\). Particles “going in” to \(L+\) from below “emerge” from \(L-\) , and particles “going in” to \(L-\) from below “emerge” from \(L+\).

Figure 3: Inserting CTCs by Cut and Paste. [An extended description of figure 3 is in the supplement.]

How is the boundary-value problem changed by this alteration in the space-time? Before the cut and paste, we can put arbitrary data on the simultaneity slice \(S\) and continue it to a unique solution. After the change in topology, \(S\) is no longer a Cauchy surface, since a CTC will never intersect it, but it is a partial Cauchy surface. So we can ask two questions. First, can arbitrary data on \(S\) always be continued to a global solution? Second, is that solution unique? If the answer to the first question is \(no\), then we have a backward-temporal constraint: the existence of the region with CTCs places constraints on what can happen on \(S\) even though that region lies completely to the future of \(S\). If the answer to the second question is \(no\), then we have an odd sort of indeterminism, analogous to the unwritten book: the complete physical state on \(S\) does not determine the physical state in the future, even though the local dynamics is perfectly deterministic and even though there is no other past edge to the space-time region in \(S\)’s future (i.e., there is nowhere else for boundary values to come from which could influence the state of the region).

In this case the answer to the first question is yes and to the second is no : there are no constraints on the data which can be put on \(S\), but those data are always consistent with an infinitude of different global solutions. The easy way to see that there always is a solution is to construct the minimal solution in the following way. Start drawing straight lines from \(S\) as required by the initial data. If a line hits \(L-\) from the bottom, just continue it coming out of the top of \(L+\) in the appropriate place, and if a line hits \(L+\) from the bottom, continue it emerging from \(L-\) at the appropriate place. Figure 4 represents the minimal solution for a single particle which enters the time-travel region from the left:

Figure 4: The Minimal Solution. [An extended description of figure 4 is in the supplement.]

The particle “travels back in time” three times. It is obvious that this minimal solution is a global solution, since the particle always travels inertially.

But the same initial state on \(S\) is also consistent with other global solutions. The new requirement imposed by the topology is just that the data going into \(L+\) from the bottom match the data coming out of \(L-\) from the top, and the data going into \(L-\) from the bottom match the data coming out of \(L+\) from the top. So we can add any number of vertical lines connecting \(L-\) and \(L+\) to a solution and still have a solution. For example, adding a few such lines to the minimal solution yields:

Figure 5: A Non-Minimal Solution. [An extended description of figure 5 is in the supplement.]

The particle now collides with itself twice: first before it reaches \(L+\) for the first time, and again shortly before it exits the CTC region. From the particle’s point of view, it is traveling to the right at a constant speed until it hits an older version of itself and comes to rest. It remains at rest until it is hit from the right by a younger version of itself, and then continues moving off, and the same process repeats later. It is clear that this is a global model of the dynamics, and that any number of distinct models could be generating by varying the number and placement of vertical lines.

Knowing the data on \(S\), then, gives us only incomplete information about how things will go for the particle. We know that the particle will enter the CTC region, and will reach \(L+\), we know that it will be the only particle in the universe, we know exactly where and with what speed it will exit the CTC region. But we cannot determine how many collisions the particle will undergo (if any), nor how long (in proper time) it will stay in the CTC region. If the particle were a clock, we could not predict what time it would indicate when exiting the region. Furthermore, the dynamics gives us no handle on what to think of the various possibilities: there are no probabilities assigned to the various distinct possible outcomes.

Changing the topology has changed the mathematics of the situation in two ways, which tend to pull in opposite directions. On the one hand, \(S\) is no longer a Cauchy surface, so it is perhaps not surprising that data on \(S\) do not suffice to fix a unique global solution. But on the other hand, there is an added constraint: data “coming out” of \(L-\) must exactly match data “going in” to \(L+\), even though what comes out of \(L-\) helps to determine what goes into \(L+\). This added consistency constraint tends to cut down on solutions, although in this case the additional constraint is more than outweighed by the freedom to consider various sorts of data on \({L+}/{L-}\).

The fact that the extra freedom outweighs the extra constraint also points up one unexpected way that the supposed paradoxes of time travel may be overcome. Let’s try to set up a paradoxical situation using the little closed time loop above. If we send a single particle into the loop from the left and do nothing else, we know exactly where it will exit the right side of the time travel region. Now suppose we station someone at the other side of the region with the following charge: if the particle should come out on the right side, the person is to do something to prevent the particle from going in on the left in the first place. In fact, this is quite easy to do: if we send a particle in from the right, it seems that it can exit on the left and deflect the incoming left-hand particle.

Carrying on our reflection in this way, we further realize that if the particle comes out on the right, we might as well send it back in order to deflect itself from entering in the first place. So all we really need to do is the following: set up a perfectly reflecting particle mirror on the right-hand side of the time travel region, and launch the particle from the left so that— if nothing interferes with it —it will just barely hit \(L+\). Our paradox is now apparently complete. If, on the one hand, nothing interferes with the particle it will enter the time-travel region on the left, exit on the right, be reflected from the mirror, re-enter from the right, and come out on the left to prevent itself from ever entering. So if it enters, it gets deflected and never enters. On the other hand, if it never enters then nothing goes in on the left, so nothing comes out on the right, so nothing is reflected back, and there is nothing to deflect it from entering. So if it doesn’t enter, then there is nothing to deflect it and it enters. If it enters, then it is deflected and doesn’t enter; if it doesn’t enter then there is nothing to deflect it and it enters: paradox complete.

But at least one solution to the supposed paradox is easy to construct: just follow the recipe for constructing the minimal solution, continuing the initial trajectory of the particle (reflecting it the mirror in the obvious way) and then read of the number and trajectories of the particles from the resulting diagram. We get the result of figure 6 :

Figure 6: Resolving the “Paradox”. [An extended description of figure 6 is in the supplement.]

As we can see, the particle approaching from the left never reaches \(L+\): it is deflected first by a particle which emerges from \(L-\). But it is not deflected by itself , as the paradox suggests, it is deflected by another particle. Indeed, there are now four particles in the diagram: the original particle and three particles which are confined to closed time-like curves. It is not the leftmost particle which is reflected by the mirror, nor even the particle which deflects the leftmost particle; it is another particle altogether.

The paradox gets it traction from an incorrect presupposition. If there is only one particle in the world at \(S\) then there is only one particle which could participate in an interaction in the time travel region: the single particle would have to interact with its earlier (or later) self. But there is no telling what might come out of \(L-\): the only requirement is that whatever comes out must match what goes in at \(L+\). So if you go to the trouble of constructing a working time machine, you should be prepared for a different kind of disappointment when you attempt to go back and kill yourself: you may be prevented from entering the machine in the first place by some completely unpredictable entity which emerges from it. And once again a peculiar sort of indeterminism appears: if there are many self-consistent things which could prevent you from entering, there is no telling which is even likely to materialize. This is just like the case of the unwritten book: the book is never written, so nothing determines what fills its pages.

So when the freedom to put data on \(L-\) outweighs the constraint that the same data go into \(L+\), instead of paradox we get an embarrassment of riches: many solution consistent with the data on \(S\), or many possible books. To see a case where the constraint “outweighs” the freedom, we need to construct a very particular, and frankly artificial, dynamics and topology. Consider the space of all linear dynamics for a scalar field on a lattice. (The lattice can be though of as a simple discrete space-time.) We will depict the space-time lattice as a directed graph. There is to be a scalar field defined at every node of the graph, whose value at a given node depends linearly on the values of the field at nodes which have arrows which lead to it. Each edge of the graph can be assigned a weighting factor which determines how much the field at the input node contributes to the field at the output node. If we name the nodes by the letters a , b , c , etc., and the edges by their endpoints in the obvious way, then we can label the weighting factors by the edges they are associated with in an equally obvious way.

Suppose that the graph of the space-time lattice is acyclic , as in figure 7 . (A graph is Acyclic if one can not travel in the direction of the arrows and go in a loop.)

Figure 7: An Acyclic Lattice. [An extended description of figure 7 is in the supplement.]

It is easy to regard a set of nodes as the analog of a Cauchy surface, e.g., the set \(\{a, b, c\}\), and it is obvious if arbitrary data are put on those nodes the data will generate a unique solution in the future. [ 3 ] If the value of the field at node \(a\) is 3 and at node \(b\) is 7, then its value at node \(d\) will be \(3W_{ad}\) and its value at node \(e\) will be \(3W_{ae} + 7W_{be}\). By varying the weighting factors we can adjust the dynamics, but in an acyclic graph the future evolution of the field will always be unique.

Let us now again artificially alter the topology of the lattice to admit CTCs, so that the graph now is cyclic. One of the simplest such graphs is depicted in figure 8 : there are now paths which lead from \(z\) back to itself, e.g., \(z\) to \(y\) to \(z\).

Figure 8: Time Travel on a Lattice. [An extended description of figure 8 is in the supplement.]

Can we now put arbitrary data on \(v\) and \(w\), and continue that data to a global solution? Will the solution be unique?

In the generic case, there will be a solution and the solution will be unique. The equations for the value of the field at \(x, y\), and \(z\) are:

Solving these equations for \(z\) yields

which gives a unique value for \(z\) in the generic case. But looking at the space of all possible dynamics for this lattice (i.e., the space of all possible weighting factors), we find a singularity in the case where \(1-W_{zx}W_{xz} - W_{zy}W_{yz} = 0\). If we choose weighting factors in just this way, then arbitrary data at \(v\) and \(w\) cannot be continued to a global solution. Indeed, if the scalar field is everywhere non-negative, then this particular choice of dynamics puts ironclad constraints on the value of the field at \(v\) and \(w\): the field there must be zero (assuming \(W_{vx}\) and \(W_{wy}\) to be non-zero), and similarly all nodes in their past must have field value zero. If the field can take negative values, then the values at \(v\) and \(w\) must be so chosen that \(vW_{vx}W_{xz} = -wW_{wy}W_{yz}\). In either case, the field values at \(v\) and \(w\) are severely constrained by the existence of the CTC region even though these nodes lie completely to the past of that region. It is this sort of constraint which we find to be unlike anything which appears in standard physics.

Our toy models suggest three things. The first is that it may be impossible to prove in complete generality that arbitrary data on a partial Cauchy surface can always be continued to a global solution: our artificial case provides an example where it cannot. The second is that such odd constraints are not likely to be generic: we had to delicately fine-tune the dynamics to get a problem. The third is that the opposite problem, namely data on a partial Cauchy surface being consistent with many different global solutions, is likely to be generic: we did not have to do any fine-tuning to get this result.

This third point leads to a peculiar sort of indeterminism, illustrated by the case of the unwritten book: the entire state on \(S\) does not determine what will happen in the future even though the local dynamics is deterministic and there are no other “edges” to space-time from which data could influence the result. What happens in the time travel region is constrained but not determined by what happens on \(S\), and the dynamics does not even supply any probabilities for the various possibilities. The example of the photographic negative discussed in section 2, then, seems likely to be unusual, for in that case there is a unique fixed point for the dynamics, and the set-up plus the dynamical laws determine the outcome. In the generic case one would rather expect multiple fixed points, with no room for anything to influence, even probabilistically, which would be realized. (See the supplement on

Remarks and Limitations on the Toy Models .

It is ironic that time travel should lead generically not to contradictions or to constraints (in the normal region) but to underdetermination of what happens in the time travel region by what happens everywhere else (an underdetermination tied neither to a probabilistic dynamics nor to a free edge to space-time). The traditional objection to time travel is that it leads to contradictions: there is no consistent way to complete an arbitrarily constructed story about how the time traveler intends to act. Instead, though, it appears that the more significant problem is underdetermination: the story can be consistently completed in many different ways.

Echeverria, Klinkhammer, and Thorne (1991) considered the case of 3-dimensional single hard spherical ball that can go through a single time travel wormhole so as to collide with its younger self.

Figure 9 [An extended description of figure 9 is in the supplement.]

The threat of paradox in this case arises in the following form. Consider the initial trajectory of a ball as it approaches the time travel region. For some initial trajectories, the ball does not undergo a collision before reaching mouth 1, but upon exiting mouth 2 it will collide with its earlier self. This leads to a contradiction if the collision is strong enough to knock the ball off its trajectory and deflect it from entering mouth 1. Of course, the Wheeler-Feynman strategy is to look for a “glancing blow” solution: a collision which will produce exactly the (small) deviation in trajectory of the earlier ball that produces exactly that collision. Are there always such solutions? [ 4 ]

Echeverria, Klinkhammer & Thorne found a large class of initial trajectories that have consistent “glancing blow” continuations, and found none that do not (but their search was not completely general). They did not produce a rigorous proof that every initial trajectory has a consistent continuation, but suggested that it is very plausible that every initial trajectory has a consistent continuation. That is to say, they have made it very plausible that, in the billiard ball wormhole case, the time travel structure of such a wormhole space-time does not result in constraints on states on spacelike surfaces in the non-time travel region.

In fact, as one might expect from our discussion in the previous section, they found the opposite problem from that of inconsistency: they found underdetermination. For a large class of initial trajectories there are multiple different consistent “glancing blow” continuations of that trajectory (many of which involve multiple wormhole traversals). For example, if one initially has a ball that is traveling on a trajectory aimed straight between the two mouths, then one obvious solution is that the ball passes between the two mouths and never time travels. But another solution is that the younger ball gets knocked into mouth 1 exactly so as to come out of mouth 2 and produce that collision. Echeverria et al. do not note the possibility (which we pointed out in the previous section) of the existence of additional balls in the time travel region. We conjecture (but have no proof) that for every initial trajectory of \(A\) there are some, and generically many, multiple-ball continuations.

Friedman, Morris, et al. (1990) examined the case of source-free non-self-interacting scalar fields traveling through such a time travel wormhole and found that no constraints on initial conditions in the non-time travel region are imposed by the existence of such time travel wormholes. In general there appear to be no known counter examples to the claim that in “somewhat realistic” time-travel space-times with a partial Cauchy surface there are no constraints imposed on the state on such a partial Cauchy surface by the existence of CTC’s. (See, e.g., Friedman & Morris 1991; Thorne 1994; Earman 1995; Earman, Smeenk, & Wüthrich 2009; and Dowe 2007.)

How about the issue of constraints in the time travel region \(T\)? Prima facie , constraints in such a region would not appear to be surprising. But one might still expect that there should be no constraints on states on a spacelike surface, provided one keeps the surface “small enough”. In the physics literature the following question has been asked: for any point \(p\) in \(T\), and any space-like surface \(S\) that includes \(p\) is there a neighborhood \(E\) of \(p\) in \(S\) such that any solution on \(E\) can be extended to a solution on the whole space-time? With respect to this question, there are some simple models in which one has this kind of extendability of local solutions to global ones, and some simple models in which one does not have such extendability, with no clear general pattern. The technical mathematical problems are amplified by the more conceptual problem of what it might mean to say that one could create a situation which forces the creation of closed timelike curves. (See, e.g., Yurtsever 1990; Friedman, Morris, et al. 1990; Novikov 1992; Earman 1995; and Earman, Smeenk, & Wüthrich 2009). What are we to think of all of this?

The toy models above all treat billiard balls, fields, and other objects propagating through a background spacetime with CTCs. Even if we can show that a consistent solution exists, there is a further question: what kind of matter and dynamics could generate CTCs to begin with? There are various solutions of Einstein’s equations with CTCs, but how do these exotic spacetimes relate to the models actually used in describing the world? In other words, what positive reasons might we have to take CTCs seriously as a feature of the actual universe, rather than an exotic possibility of primarily mathematical interest?

We should distinguish two different kinds of “possibility” that we might have in mind in posing such questions (following Stein 1970). First, we can consider a solution as a candidate cosmological model, describing the (large-scale gravitational degrees of freedom of the) entire universe. The case for ruling out spacetimes with CTCs as potential cosmological models strikes us as, surprisingly, fairly weak. Physicists used to simply rule out solutions with CTCs as unreasonable by fiat, due to the threat of paradoxes, which we have dismantled above. But it is also challenging to make an observational case. Observations tell us very little about global features, such as the existence of CTCs, because signals can only reach an observer from a limited region of spacetime, called the past light cone. Our past light cone—and indeed the collection of all the past light cones for possible observers in a given spacetime—can be embedded in spacetimes with quite different global features (Malament 1977, Manchak 2009). This undercuts the possibility of using observations to constrain global topology, including (among other things) ruling out the existence of CTCs.

Yet the case in favor of taking cosmological models with CTCs seriously is also not particularly strong. Some solutions used to describe black holes, which are clearly relevant in a variety of astrophysical contexts, include CTCs. But the question of whether the CTCs themselves play an essential representational role is subtle: the CTCs arise in the maximal extensions of these solutions, and can plausibly be regarded as extraneous to successful applications. Furthermore, many of the known solutions with CTCs have symmetries, raising the possibility that CTCs are not a stable or robust feature. Slight departures from symmetry may lead to a solution without CTCs, suggesting that the CTCs may be an artifact of an idealized model.

The second sense of possibility regards whether “reasonable” initial conditions can be shown to lead to, or not to lead to, the formation of CTCs. As with the toy models above, suppose that we have a partial Cauchy surface \(S\), such that all the temporal funny business lies to the future. Rather than simply assuming that there is a region with CTCs to the future, we can ask instead whether it is possible to create CTCs by manipulating matter in the initial, well-behaved region—that is, whether it is possible to build a time machine. Several physicists have pursued “chronology protection theorems” aiming to show that the dynamics of general relativity (or some other aspects of physics) rules this out, and to clarify why this is the case. The proof of such a theorem would justify neglecting solutions with CTCs as a source of insight into the nature of time in the actual world. But as of yet there are several partial results that do not fully settle the question. One further intriguing possibility is that even if general relativity by itself does protect chronology, it may not be possible to formulate a sensible theory describing matter and fields in solutions with CTCs. (See SEP entry on Time Machines; Smeenk and Wüthrich 2011 for more.)

There is a different question regarding the limitations of these toy models. The toy models and related examples show that there are consistent solutions for simple systems in the presence of CTCs. As usual we have made the analysis tractable by building toy models, selecting only a few dynamical degrees of freedom and tracking their evolution. But there is a large gap between the systems we have described and the time travel stories they evoke, with Kurt traveling along a CTC with murderous intentions. In particular, many features of the manifest image of time are tied to the thermodynamical properties of macroscopic systems. Rovelli (unpublished) considers a extremely simple system to illustrate the problem: can a clock move along a CTC? A clock consists of something in periodic motion, such as a pendulum bob, and something that counts the oscillations, such as an escapement mechanism. The escapement mechanism cannot work without friction; this requires dissipation and increasing entropy. For a clock that counts oscillations as it moves along a time-like trajectory, the entropy must be a monotonically increasing function. But that is obviously incompatible with the clock returning to precisely the same state at some future time as it completes a loop. The point generalizes, obviously, to imply that anything like a human, with memory and agency, cannot move along a CTC.

Since it is not obvious that one can rid oneself of all constraints in realistic models, let us examine the argument that time travel is implausible, and we should think it unlikely to exist in our world, in so far as it implies such constraints. The argument goes something like the following. In order to satisfy such constraints one needs some pre-established divine harmony between the global (time travel) structure of space-time and the distribution of particles and fields on space-like surfaces in it. But it is not plausible that the actual world, or any world even remotely like ours, is constructed with divine harmony as part of the plan. In fact, one might argue, we have empirical evidence that conditions in any spatial region can vary quite arbitrarily. So we have evidence that such constraints, whatever they are, do not in fact exist in our world. So we have evidence that there are no closed time-like lines in our world or one remotely like it. We will now examine this argument in more detail by presenting four possible responses, with counterresponses, to this argument.

Response 1. There is nothing implausible or new about such constraints. For instance, if the universe is spatially closed, there has to be enough matter to produce the needed curvature, and this puts constraints on the matter distribution on a space-like hypersurface. Thus global space-time structure can quite unproblematically constrain matter distributions on space-like hypersurfaces in it. Moreover we have no realistic idea what these constraints look like, so we hardly can be said to have evidence that they do not obtain.

Counterresponse 1. Of course there are constraining relations between the global structure of space-time and the matter in it. The Einstein equations relate curvature of the manifold to the matter distribution in it. But what is so strange and implausible about the constraints imposed by the existence of closed time-like curves is that these constraints in essence have nothing to do with the Einstein equations. When investigating such constraints one typically treats the particles and/or field in question as test particles and/or fields in a given space-time, i.e., they are assumed not to affect the metric of space-time in any way. In typical space-times without closed time-like curves this means that one has, in essence, complete freedom of matter distribution on a space-like hypersurface. (See response 2 for some more discussion of this issue). The constraints imposed by the possibility of time travel have a quite different origin and are implausible. In the ordinary case there is a causal interaction between matter and space-time that results in relations between global structure of space-time and the matter distribution in it. In the time travel case there is no such causal story to be told: there simply has to be some pre-established harmony between the global space-time structure and the matter distribution on some space-like surfaces. This is implausible.

Response 2. Constraints upon matter distributions are nothing new. For instance, Maxwell’s equations constrain electric fields \(\boldsymbol{E}\) on an initial surface to be related to the (simultaneous) charge density distribution \(\varrho\) by the equation \(\varrho = \text{div}(\boldsymbol{E})\). (If we assume that the \(E\) field is generated solely by the charge distribution, this conditions amounts to requiring that the \(E\) field at any point in space simply be the one generated by the charge distribution according to Coulomb’s inverse square law of electrostatics.) This is not implausible divine harmony. Such constraints can hold as a matter of physical law. Moreover, if we had inferred from the apparent free variation of conditions on spatial regions that there could be no such constraints we would have mistakenly inferred that \(\varrho = \text{div}(\boldsymbol{E})\) could not be a law of nature.

Counterresponse 2. The constraints imposed by the existence of closed time-like lines are of quite a different character from the constraint imposed by \(\varrho = \text{div}(\boldsymbol{E})\). The constraints imposed by \(\varrho = \text{div}(\boldsymbol{E})\) on the state on a space-like hypersurface are:

• local constraints (i.e., to check whether the constraint holds in a region you just need to see whether it holds at each point in the region),
• quite independent of the global space-time structure,
• quite independent of how the space-like surface in question is embedded in a given space-time, and
• very simply and generally stateable.

On the other hand, the consistency constraints imposed by the existence of closed time-like curves (i) are not local, (ii) are dependent on the global structure of space-time, (iii) depend on the location of the space-like surface in question in a given space-time, and (iv) appear not to be simply stateable other than as the demand that the state on that space-like surface embedded in such and such a way in a given space-time, do not lead to inconsistency. On some views of laws (e.g., David Lewis’ view) this plausibly implies that such constraints, even if they hold, could not possibly be laws. But even if one does not accept such a view of laws, one could claim that the bizarre features of such constraints imply that it is implausible that such constraints hold in our world or in any world remotely like ours.

Response 3. It would be strange if there are constraints in the non-time travel region. It is not strange if there are constraints in the time travel region. They should be explained in terms of the strange, self-interactive, character of time travel regions. In this region there are time-like trajectories from points to themselves. Thus the state at such a point, in such a region, will, in a sense, interact with itself. It is a well-known fact that systems that interact with themselves will develop into an equilibrium state, if there is such an equilibrium state, or else will develop towards some singularity. Normally, of course, self-interaction isn’t true instantaneous self-interaction, but consists of a feed-back mechanism that takes time. But in time travel regions something like true instantaneous self-interaction occurs. This explains why constraints on states occur in such time travel regions: the states “ ab initio ” have to be “equilibrium states”. Indeed in a way this also provides some picture of why indeterminism occurs in time travel regions: at the onset of self-interaction states can fork into different equi-possible equilibrium states.

Counterresponse 3. This is explanation by woolly analogy. It all goes to show that time travel leads to such bizarre consequences that it is unlikely that it occurs in a world remotely like ours.

Response 4. All of the previous discussion completely misses the point. So far we have been taking the space-time structure as given, and asked the question whether a given time travel space-time structure imposes constraints on states on (parts of) space-like surfaces. However, space-time and matter interact. Suppose that one is in a space-time with closed time-like lines, such that certain counterfactual distributions of matter on some neighborhood of a point \(p\) are ruled out if one holds that space-time structure fixed. One might then ask

Why does the actual state near \(p\) in fact satisfy these constraints? By what divine luck or plan is this local state compatible with the global space-time structure? What if conditions near \(p\) had been slightly different?

And one might take it that the lack of normal answers to these questions indicates that it is very implausible that our world, or any remotely like it, is such a time travel universe. However the proper response to these question is the following. There are no constraints in any significant sense. If they hold they hold as a matter of accidental fact, not of law. There is no more explanation of them possible than there is of any contingent fact. Had conditions in a neighborhood of \(p\) been otherwise, the global structure of space-time would have been different. So what? The only question relevant to the issue of constraints is whether an arbitrary state on an arbitrary spatial surface \(S\) can always be embedded into a space-time such that that state on \(S\) consistently extends to a solution on the entire space-time.

But we know the answer to that question. A well-known theorem in general relativity says the following: any initial data set on a three dimensional manifold \(S\) with positive definite metric has a unique embedding into a maximal space-time in which \(S\) is a Cauchy surface (see, e.g., Geroch & Horowitz 1979: 284 for more detail), i.e., there is a unique largest space-time which has \(S\) as a Cauchy surface and contains a consistent evolution of the initial value data on \(S\). Now since \(S\) is a Cauchy surface this space-time does not have closed time like curves. But it may have extensions (in which \(S\) is not a Cauchy surface) which include closed timelike curves, indeed it may be that any maximal extension of it would include closed timelike curves. (This appears to be the case for extensions of states on certain surfaces of Taub-NUT space-times. See Earman, Smeenk, & Wüthrich 2009). But these extensions, of course, will be consistent. So properly speaking, there are no constraints on states on space-like surfaces. Nonetheless the space-time in which these are embedded may or may not include closed time-like curves.

Counterresponse 4. This, in essence, is the stonewalling answer which we indicated in section 1. However, whether or not you call the constraints imposed by a given space-time on distributions of matter on certain space-like surfaces “genuine constraints”, whether or not they can be considered lawlike, and whether or not they need to be explained, the existence of such constraints can still be used to argue that time travel worlds are so bizarre that it is implausible that our world or any world remotely like ours is a time travel world.

Suppose that one is in a time travel world. Suppose that given the global space-time structure of this world, there are constraints imposed upon, say, the state of motion of a ball on some space-like surface when it is treated as a test particle, i.e., when it is assumed that the ball does not affect the metric properties of the space-time it is in. (There is lots of other matter that, via the Einstein equation, corresponds exactly to the curvature that there is everywhere in this time travel worlds.) Now a real ball of course does have some effect on the metric of the space-time it is in. But let us consider a ball that is so small that its effect on the metric is negligible. Presumably it will still be the case that certain states of this ball on that space-like surface are not compatible with the global time travel structure of this universe.

This means that the actual distribution of matter on such a space-like surface can be extended into a space-time with closed time-like lines, but that certain counterfactual distributions of matter on this space-like surface can not be extended into the same space-time. But note that the changes made in the matter distribution (when going from the actual to the counterfactual distribution) do not in any non-negligible way affect the metric properties of the space-time. (Recall that the changes only effect test particles.) Thus the reason why the global time travel properties of the counterfactual space-time have to be significantly different from the actual space-time is not that there are problems with metric singularities or alterations in the metric that force significant global changes when we go to the counterfactual matter distribution. The reason that the counterfactual space-time has to be different is that in the counterfactual world the ball’s initial state of motion starting on the space-like surface, could not “meet up” in a consistent way with its earlier self (could not be consistently extended) if we were to let the global structure of the counterfactual space-time be the same as that of the actual space-time. Now, it is not bizarre or implausible that there is a counterfactual dependence of manifold structure, even of its topology, on matter distributions on spacelike surfaces. For instance, certain matter distributions may lead to singularities, others may not. We may indeed in some sense have causal power over the topology of the space-time we live in. But this power normally comes via the Einstein equations. But it is bizarre to think that there could be a counterfactual dependence of global space-time structure on the arrangement of certain tiny bits of matter on some space-like surface, where changes in that arrangement by assumption do not affect the metric anywhere in space-time in any significant way . It is implausible that we live in such a world, or that a world even remotely like ours is like that.

Let us illustrate this argument in a different way by assuming that wormhole time travel imposes constraints upon the states of people prior to such time travel, where the people have so little mass/energy that they have negligible effect, via the Einstein equation, on the local metric properties of space-time. Do you think it more plausible that we live in a world where wormhole time travel occurs but it only occurs when people’s states are such that these local states happen to combine with time travel in such a way that nobody ever succeeds in killing their younger self, or do you think it more plausible that we are not in a wormhole time travel world? [ 5 ]

An alternative approach to time travel (initiated by Deutsch 1991) abstracts away from the idealized toy models described above. [ 6 ] This computational approach considers instead the evolution of bits (simple physical systems with two discrete states) through a network of interactions, which can be represented by a circuit diagram with gates corresponding to the interactions. Motivated by the possibility of CTCs, Deutsch proposed adding a new kind of channel that connects the output of a given gate back to its input —in essence, a backwards-time step. More concretely, given a gate that takes \(n\) bits as input, we can imagine taking some number \(i \lt n\) of these bits through a channel that loops back and then do double-duty as inputs. Consistency requires that the state of these \(i\) bits is the same for output and input. (We will consider an illustration of this kind of system in the next section.) Working through examples of circuit diagrams with a CTC channel leads to similar treatments of Consistency and Underdetermination as the discussion above (see, e.g., Wallace 2012: § 10.6). But the approach offers two new insights (both originally due to Deutsch): the Easy Knowledge paradox, and a particularly clear extension to time travel in quantum mechanics.

A computer equipped with a CTC channel can exploit the need to find consistent evolution to solve remarkably hard problems. (This is quite different than the first idea that comes to mind to enhance computational power: namely to just devote more time to a computation, and then send the result back on the CTC to an earlier state.) The gate in a circuit incorporating a CTC implements a function from the input bits to the output bits, under the constraint that the output and input match the i bits going through the CTC channel. This requires, in effect, finding the fixed point of the relevant function. Given the generality of the model, there are few limits on the functions that could be implemented on the CTC circuit. Nature has to solve a hard computational problem just to ensure consistent evolution. This can then be extended to other complex computational problems—leading, more precisely, to solutions of NP -complete problems in polynomial time (see Aaronson 2013: Chapter 20 for an overview and further references). The limits imposed by computational complexity are an essential part of our epistemic situation, and computers with CTCs would radically change this.

We now turn to the application of the computational approach to the quantum physics of time travel (see Deutsch 1991; Deutsch & Lockwood 1994). By contrast with the earlier discussions of constraints in classical systems, they claim to show that time travel never imposes any constraints on the pre-time travel state of quantum systems. The essence of this account is as follows. [ 7 ]

A quantum system starts in state \(S_1\), interacts with its older self, after the interaction is in state \(S_2\), time travels while developing into state \(S_3\), then interacts with its younger self, and ends in state \(S_4\) (see figure 10 ).

Figure 10 [An extended description of figure 10 is in the supplement.]

Deutsch assumes that the set of possible states of this system are the mixed states, i.e., are represented by the density matrices over the Hilbert space of that system. Deutsch then shows that for any initial state \(S_1\), any unitary interaction between the older and younger self, and any unitary development during time travel, there is a consistent solution, i.e., there is at least one pair of states \(S_2\) and \(S_3\) such that when \(S_1\) interacts with \(S_3\) it will change to state \(S_2\) and \(S_2\) will then develop into \(S_3\). The states \(S_2, S_3\) and \(S_4\) will typically be not be pure states, i.e., will be non-trivial mixed states, even if \(S_1\) is pure. In order to understand how this leads to interpretational problems let us give an example. Consider a system that has a two dimensional Hilbert space with as a basis the states \(\vc{+}\) and \(\vc{-}\). Let us suppose that when state \(\vc{+}\) of the young system encounters state \(\vc{+}\) of the older system, they interact and the young system develops into state \(\vc{-}\) and the old system remains in state \(\vc{+}\). In obvious notation:

Similarly, suppose that:

Let us furthermore assume that there is no development of the state of the system during time travel, i.e., that \(\vc{+}_2\) develops into \(\vc{+}_3\), and that \(\vc{-}_2\) develops into \(\vc{-}_3\).

Now, if the only possible states of the system were \(\vc{+}\) and \(\vc{-}\) (i.e., if there were no superpositions or mixtures of these states), then there is a constraint on initial states: initial state \(\vc{+}_1\) is impossible. For if \(\vc{+}_1\) interacts with \(\vc{+}_3\) then it will develop into \(\vc{-}_2\), which, during time travel, will develop into \(\vc{-}_3\), which inconsistent with the assumed state \(\vc{+}_3\). Similarly if \(\vc{+}_1\) interacts with \(\vc{-}_3\) it will develop into \(\vc{+}_2\), which will then develop into \(\vc{+}_3\) which is also inconsistent. Thus the system can not start in state \(\vc{+}_1\).

But, says Deutsch, in quantum mechanics such a system can also be in any mixture of the states \(\vc{+}\) and \(\vc{-}\). Suppose that the older system, prior to the interaction, is in a state \(S_3\) which is an equal mixture of 50% \(\vc{+}_3\) and 50% \(\vc{-}_3\). Then the younger system during the interaction will develop into a mixture of 50% \(\vc{+}_2\) and 50% \(\vc{-}_2\), which will then develop into a mixture of 50% \(\vc{+}_3\) and 50% \(\vc{-}_3\), which is consistent! More generally Deutsch uses a fixed point theorem to show that no matter what the unitary development during interaction is, and no matter what the unitary development during time travel is, for any state \(S_1\) there is always a state \(S_3\) (which typically is not a pure state) which causes \(S_1\) to develop into a state \(S_2\) which develops into that state \(S_3\). Thus quantum mechanics comes to the rescue: it shows in all generality that no constraints on initial states are needed!

One might wonder why Deutsch appeals to mixed states: will superpositions of states \(\vc{+}\) and \(\vc{-}\) not suffice? Unfortunately such an idea does not work. Suppose again that the initial state is \(\vc{+}_1\). One might suggest that that if state \(S_3\) is

one will obtain a consistent development. For one might think that when initial state \(\vc{+}_1\) encounters the superposition

it will develop into superposition

and that this in turn will develop into

as desired. However this is not correct. For initial state \(\vc{+}_1\) when it encounters

will develop into the entangled state

In so far as one can speak of the state of the young system after this interaction, it is in the mixture of 50% \(\vc{+}_2\) and 50% \(\vc{-}_2\), not in the superposition

So Deutsch does need his recourse to mixed states.

This clarification of why Deutsch needs his mixtures does however indicate a serious worry about the simplifications that are part of Deutsch’s account. After the interaction the old and young system will (typically) be in an entangled state. Although for purposes of a measurement on one of the two systems one can say that this system is in a mixed state, one can not represent the full state of the two systems by specifying the mixed state of each separate part, as there are correlations between observables of the two systems that are not represented by these two mixed states, but are represented in the joint entangled state. But if there really is an entangled state of the old and young systems directly after the interaction, how is one to represent the subsequent development of this entangled state? Will the state of the younger system remain entangled with the state of the older system as the younger system time travels and the older system moves on into the future? On what space-like surfaces are we to imagine this total entangled state to be? At this point it becomes clear that there is no obvious and simple way to extend elementary non-relativistic quantum mechanics to space-times with closed time-like curves: we apparently need to characterize not just the entanglement between two systems, but entanglement relative to specific spacetime descriptions.

How does Deutsch avoid these complications? Deutsch assumes a mixed state \(S_3\) of the older system prior to the interaction with the younger system. He lets it interact with an arbitrary pure state \(S_1\) younger system. After this interaction there is an entangled state \(S'\) of the two systems. Deutsch computes the mixed state \(S_2\) of the younger system which is implied by this entangled state \(S'\). His demand for consistency then is just that this mixed state \(S_2\) develops into the mixed state \(S_3\). Now it is not at all clear that this is a legitimate way to simplify the problem of time travel in quantum mechanics. But even if we grant him this simplification there is a problem: how are we to understand these mixtures?

If we take an ignorance interpretation of mixtures we run into trouble. For suppose that we assume that in each individual case each older system is either in state \(\vc{+}_3\) or in state \(\vc{-}_3\) prior to the interaction. Then we regain our paradox. Deutsch instead recommends the following, many worlds, picture of mixtures. Suppose we start with state \(\vc{+}_1\) in all worlds. In some of the many worlds the older system will be in the \(\vc{+}_3\) state, let us call them A -worlds, and in some worlds, B -worlds, it will be in the \(\vc{-}_3\) state. Thus in A -worlds after interaction we will have state \(\vc{-}_2\) , and in B -worlds we will have state \(\vc{+}_2\). During time travel the \(\vc{-}_2\) state will remain the same, i.e., turn into state \(\vc{-}_3\), but the systems in question will travel from A -worlds to B -worlds. Similarly the \(\vc{+}\) \(_2\) states will travel from the B -worlds to the A -worlds, thus preserving consistency.

Now whatever one thinks of the merits of many worlds interpretations, and of this understanding of it applied to mixtures, in the end one does not obtain genuine time travel in Deutsch’s account. The systems in question travel from one time in one world to another time in another world, but no system travels to an earlier time in the same world. (This is so at least in the normal sense of the word “world”, the sense that one means when, for instance, one says “there was, and will be, only one Elvis Presley in this world.”) Thus, even if it were a reasonable view, it is not quite as interesting as it may have initially seemed. (See Wallace 2012 for a more sympathetic treatment, that explores several further implications of accepting time travel in conjunction with the many worlds interpretation.)

We close by acknowledging that Deutsch’s starting point—the claim that this computational model captures the essential features of quantum systems in a spacetime with CTCs—has been the subject of some debate. Several physicists have pursued a quite different treatment of evolution of quantum systems through CTC’s, based on considering the “post-selected” state (see Lloyd et al. 2011). Their motivations for implementing the consistency condition in terms of the post-selected state reflects a different stance towards quantum foundations. A different line of argument aims to determine whether Deutsch’s treatment holds as an appropriate limiting case of a more rigorous treatment, such as quantum field theory in curved spacetimes. For example, Verch (2020) establishes several results challenging the assumption that Deutsch’s treatment is tied to the presence of CTC’s, or that it is compatible with the entanglement structure of quantum fields.

What remains of the grandfather paradox in general relativistic time travel worlds is the fact that in some cases the states on edgeless spacelike surfaces are “overconstrained”, so that one has less than the usual freedom in specifying conditions on such a surface, given the time-travel structure, and in some cases such states are “underconstrained”, so that states on edgeless space-like surfaces do not determine what happens elsewhere in the way that they usually do, given the time travel structure. There can also be mixtures of those two types of cases. The extent to which states are overconstrained and/or underconstrained in realistic models is as yet unclear, though it would be very surprising if neither obtained. The extant literature has primarily focused on the problem of overconstraint, since that, often, either is regarded as a metaphysical obstacle to the possibility time travel, or as an epistemological obstacle to the plausibility of time travel in our world. While it is true that our world would be quite different from the way we normally think it is if states were overconstrained, underconstraint seems at least as bizarre as overconstraint. Nonetheless, neither directly rules out the possibility of time travel.

If time travel entailed contradictions then the issue would be settled. And indeed, most of the stories employing time travel in popular culture are logically incoherent: one cannot “change” the past to be different from what it was, since the past (like the present and the future) only occurs once. But if the only requirement demanded is logical coherence, then it seems all too easy. A clever author can devise a coherent time-travel scenario in which everything happens just once and in a consistent way. This is just too cheap: logical coherence is a very weak condition, and many things we take to be metaphysically impossible are logically coherent. For example, it involves no logical contradiction to suppose that water is not molecular, but if both chemistry and Kripke are right it is a metaphysical impossibility. We have been interested not in logical possibility but in physical possibility. But even so, our conditions have been relatively weak: we have asked only whether time-travel is consistent with the universal validity of certain fundamental physical laws and with the notion that the physical state on a surface prior to the time travel region be unconstrained. It is perfectly possible that the physical laws obey this condition, but still that time travel is not metaphysically possible because of the nature of time itself. Consider an analogy. Aristotle believed that water is homoiomerous and infinitely divisible: any bit of water could be subdivided, in principle, into smaller bits of water. Aristotle’s view contains no logical contradiction. It was certainly consistent with Aristotle’s conception of water that it be homoiomerous, so this was, for him, a conceptual possibility. But if chemistry is right, Aristotle was wrong both about what water is like and what is possible for it. It can’t be infinitely divided, even though no logical or conceptual analysis would reveal that.

Similarly, even if all of our consistency conditions can be met, it does not follow that time travel is physically possible, only that some specific physical considerations cannot rule it out. The only serious proof of the possibility of time travel would be a demonstration of its actuality. For if we agree that there is no actual time travel in our universe, the supposition that there might have been involves postulating a substantial difference from actuality, a difference unlike in kind from anything we could know if firsthand. It is unclear to us exactly what the content of possible would be if one were to either maintain or deny the possibility of time travel in these circumstances, unless one merely meant that the possibility is not ruled out by some delineated set of constraints. As the example of Aristotle’s theory of water shows, conceptual and logical “possibility” do not entail possibility in a full-blooded sense. What exactly such a full-blooded sense would be in case of time travel, and whether one could have reason to believe it to obtain, remain to us obscure.

• Aaronson, Scott, 2013, Quantum Computing since Democritus , Cambridge: Cambridge University Press. doi:10.1017/CBO9780511979309
• Arntzenius, Frank, 2006, “Time Travel: Double Your Fun”, Philosophy Compass , 1(6): 599–616. doi:10.1111/j.1747-9991.2006.00045.x
• Clarke, C.J.S., 1977, “Time in General Relativity” in Foundations of Space-Time Theory , Minnesota Studies in the Philosophy of Science , Vol VIII, Earman, J., Glymour, C., and Stachel, J. (eds), pp. 94–108. Minneapolis: University of Minnesota Press.
• Deutsch, David, 1991, “Quantum Mechanics near Closed Timelike Lines”, Physical Review D , 44(10): 3197–3217. doi:10.1103/PhysRevD.44.3197
• Deutsch, David and Michael Lockwood, 1994, “The Quantum Physics of Time Travel”, Scientific American , 270(3): 68–74. doi:10.1038/scientificamerican0394-68
• Dowe, Phil, 2007, “Constraints on Data in Worlds with Closed Timelike Curves”, Philosophy of Science , 74(5): 724–735. doi:10.1086/525617
• Earman, John, 1972, “Implications of Causal Propagation Outside the Null Cone”, Australasian Journal of Philosophy , 50(3): 222–237. doi:10.1080/00048407212341281
• Earman, John, 1995, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes , New York: Oxford University Press.
• Earman, John, Christopher Smeenk, and Christian Wüthrich, 2009, “Do the Laws of Physics Forbid the Operation of Time Machines?”, Synthese , 169(1): 91–124. doi:10.1007/s11229-008-9338-2
• Echeverria, Fernando, Gunnar Klinkhammer, and Kip S. Thorne, 1991, “Billiard Balls in Wormhole Spacetimes with Closed Timelike Curves: Classical Theory”, Physical Review D , 44(4): 1077–1099. doi:10.1103/PhysRevD.44.1077
• Effingham, Nikk, 2020, Time Travel: Probability and Impossibility , Oxford: Oxford University Press. doi:10.1093/oso/9780198842507.001.0001
• Fletcher, Samuel C., 2020, “The Principle of Stability”, Philosopher’s Imprint , 20: article 3. [ Fletcher 2020 available online ]
• Friedman, John and Michael Morris, 1991, “The Cauchy Problem for the Scalar Wave Equation Is Well Defined on a Class of Spacetimes with Closed Timelike Curves”, Physical Review Letters , 66(4): 401–404. doi:10.1103/PhysRevLett.66.401
• Friedman, John, Michael S. Morris, Igor D. Novikov, Fernando Echeverria, Gunnar Klinkhammer, Kip S. Thorne, and Ulvi Yurtsever, 1990, “Cauchy Problem in Spacetimes with Closed Timelike Curves”, Physical Review D , 42(6): 1915–1930. doi:10.1103/PhysRevD.42.1915
• Geroch, Robert and Gary Horowitz, 1979, “Global Structures of Spacetimes”, in General Relativity: An Einstein Centenary Survey , Stephen Hawking and W. Israel (eds.), Cambridge/New York: Cambridge University Press, Chapter 5, pp. 212–293.
• Gödel, Kurt, 1949, “A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy”, in Albert Einstein, Philosopher-Scientist , Paul Arthur Schilpp (ed.), Evanston, IL: Library of Living Philosophers, 557–562.
• Hocking, John G. and Gail S. Young, 1961, Topology , (Addison-Wesley Series in Mathematics), Reading, MA: Addison-Wesley.
• Horwich, Paul, 1987, “Time Travel”, in his Asymmetries in Time: Problems in the Philosophy of Science , , Cambridge, MA: MIT Press, 111–128.
• Kutach, Douglas N., 2003, “Time Travel and Consistency Constraints”, Philosophy of Science , 70(5): 1098–1113. doi:10.1086/377392
• Lewis, David, 1976, “The Paradoxes of Time Travel”, American Philosophical Quarterly , 13(2): 145–152.
• Lloyd, Seth, Lorenzo Maccone, Raul Garcia-Patron, Vittorio Giovannetti, and Yutaka Shikano, 2011, “Quantum Mechanics of Time Travel through Post-Selected Teleportation”, Physical Review D , 84(2): 025007. doi:10.1103/PhysRevD.84.025007
• Malament, David B., 1977, “Observationally Indistinguishable Spacetimes: Comments on Glymour’s Paper”, in Foundations of Space-Time Theories , John Earman, Clark N. Glymour, and John J. Stachel (eds.), (Minnesota Studies in the Philosophy of Science 8), Minneapolis, MN: University of Minnesota Press, 61–80.
• –––, 1984, “‘Time Travel’ in the Gödel Universe”, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , 1984(2): 91–100. doi:10.1086/psaprocbienmeetp.1984.2.192497
• –––, 1985, “Minimal Acceleration Requirements for ‘Time Travel’, in Gödel Space‐time”, Journal of Mathematical Physics , 26(4): 774–777. doi:10.1063/1.526566
• Manchak, John Byron, 2009, “Can We Know the Global Structure of Spacetime?”, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics , 40(1): 53–56. doi:10.1016/j.shpsb.2008.07.004
• –––, 2011, “On Efficient ‘Time Travel’ in Gödel Spacetime”, General Relativity and Gravitation , 43(1): 51–60. doi:10.1007/s10714-010-1068-3
• Maudlin, Tim, 1990, “Time-Travel and Topology”, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , 1990(1): 303–315. doi:10.1086/psaprocbienmeetp.1990.1.192712
• Novikov, I. D., 1992, “Time Machine and Self-Consistent Evolution in Problems with Self-Interaction”, Physical Review D , 45(6): 1989–1994. doi:10.1103/PhysRevD.45.1989
• Smeenk, Chris and Christian Wüthrich, 2011, “Time Travel and Time Machines”, in the Oxford Handbook on Time , Craig Callender (ed.), Oxford: Oxford University Press, 577–630..
• Stein, Howard, 1970, “On the Paradoxical Time-Structures of Gödel”, Philosophy of Science , 37(4): 589–601. doi:10.1086/288328
• Thorne, Kip S., 1994, Black Holes and Time Warps: Einstein’s Outrageous Legacy , (Commonwealth Fund Book Program), New York: W.W. Norton.
• Verch, Rainer, 2020, “The D-CTC Condition in Quantum Field Theory”, in Progress and Visions in Quantum Theory in View of Gravity , Felix Finster, Domenico Giulini, Johannes Kleiner, and Jürgen Tolksdorf (eds.), Cham: Springer International Publishing, 221–232. doi:10.1007/978-3-030-38941-3_9
• Wallace, David, 2012, The Emergent Multiverse: Quantum Theory According to the Everett Interpretation , Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780199546961.001.0001
• Wasserman, Ryan, 2018, Paradoxes of Time Travel , Oxford: Oxford University Press. doi:10.1093/oso/9780198793335.001.0001
• Weyl, Hermann, 1918/1920 [1922/1952], Raum, Zeit, Materie , Berlin: Springer; fourth edition 1920. Translated as Space—Time—Matter , Henry Leopold Brose (trans.), New York: Dutton, 1922. Reprinted 1952, New York: Dover Publications.
• Wheeler, John Archibald and Richard Phillips Feynman, 1949, “Classical Electrodynamics in Terms of Direct Interparticle Action”, Reviews of Modern Physics , 21(3): 425–433. doi:10.1103/RevModPhys.21.425
• Yurtsever, Ulvi, 1990, “Test Fields on Compact Space‐times”, Journal of Mathematical Physics , 31(12): 3064–3078. doi:10.1063/1.528960

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Adlam, Emily, unpublished, “ Is There Causation in Fundamental Physics? New Insights from Process Matrices and Quantum Causal Modelling ”, 2022, arXiv: 2208.02721. doi:10.48550/ARXIV.2208.02721
• Rovelli, Carlo, unpublished, “ Can We Travel to the Past? Irreversible Physics along Closed Timelike Curves ”, arXiv: 1912.04702. doi:10.48550/ARXIV.1912.04702

causation: backward | determinism: causal | quantum mechanics | quantum mechanics: retrocausality | space and time: being and becoming in modern physics | time machines | time travel

Copyright © 2023 by Christopher Smeenk < csmeenk2 @ uwo . ca > Frank Arntzenius Tim Maudlin

• Accessibility

Support SEP

Mirror sites.

View this site from another server:

• Info about mirror sites

The Stanford Encyclopedia of Philosophy is copyright © 2023 by The Metaphysics Research Lab , Department of Philosophy, Stanford University

Library of Congress Catalog Data: ISSN 1095-5054

A beginner's guide to time travel

Learn exactly how Einstein's theory of relativity works, and discover how there's nothing in science that says time travel is impossible.

Everyone can travel in time . You do it whether you want to or not, at a steady rate of one second per second. You may think there's no similarity to traveling in one of the three spatial dimensions at, say, one foot per second. But according to Einstein 's theory of relativity , we live in a four-dimensional continuum — space-time — in which space and time are interchangeable.

Einstein found that the faster you move through space, the slower you move through time — you age more slowly, in other words. One of the key ideas in relativity is that nothing can travel faster than the speed of light — about 186,000 miles per second (300,000 kilometers per second), or one light-year per year). But you can get very close to it. If a spaceship were to fly at 99% of the speed of light, you'd see it travel a light-year of distance in just over a year of time. 

That's obvious enough, but now comes the weird part. For astronauts onboard that spaceship, the journey would take a mere seven weeks. It's a consequence of relativity called time dilation , and in effect, it means the astronauts have jumped about 10 months into the future. 

Traveling at high speed isn't the only way to produce time dilation. Einstein showed that gravitational fields produce a similar effect — even the relatively weak field here on the surface of Earth . We don't notice it, because we spend all our lives here, but more than 12,400 miles (20,000 kilometers) higher up gravity is measurably weaker— and time passes more quickly, by about 45 microseconds per day. That's more significant than you might think, because it's the altitude at which GPS satellites orbit Earth, and their clocks need to be precisely synchronized with ground-based ones for the system to work properly. 

The satellites have to compensate for time dilation effects due both to their higher altitude and their faster speed. So whenever you use the GPS feature on your smartphone or your car's satnav, there's a tiny element of time travel involved. You and the satellites are traveling into the future at very slightly different rates.

But for more dramatic effects, we need to look at much stronger gravitational fields, such as those around black holes , which can distort space-time so much that it folds back on itself. The result is a so-called wormhole, a concept that's familiar from sci-fi movies, but actually originates in Einstein's theory of relativity. In effect, a wormhole is a shortcut from one point in space-time to another. You enter one black hole, and emerge from another one somewhere else. Unfortunately, it's not as practical a means of transport as Hollywood makes it look. That's because the black hole's gravity would tear you to pieces as you approached it, but it really is possible in theory. And because we're talking about space-time, not just space, the wormhole's exit could be at an earlier time than its entrance; that means you would end up in the past rather than the future.

Trajectories in space-time that loop back into the past are given the technical name "closed timelike curves." If you search through serious academic journals, you'll find plenty of references to them — far more than you'll find to "time travel." But in effect, that's exactly what closed timelike curves are all about — time travel

This article is brought to you by  How It Works .

How It Works is the action-packed magazine that's bursting with exciting information about the latest advances in science and technology, featuring everything you need to know about how the world around you — and the universe — works.

There's another way to produce a closed timelike curve that doesn't involve anything quite so exotic as a black hole or wormhole: You just need a simple rotating cylinder made of super-dense material. This so-called Tipler cylinder is the closest that real-world physics can get to an actual, genuine time machine. But it will likely never be built in the real world, so like a wormhole, it's more of an academic curiosity than a viable engineering design.

Yet as far-fetched as these things are in practical terms, there's no fundamental scientific reason — that we currently know of — that says they are impossible. That's a thought-provoking situation, because as the physicist Michio Kaku is fond of saying, "Everything not forbidden is compulsory" (borrowed from T.H. White's novel, "The Once And Future King"). He doesn't mean time travel has to happen everywhere all the time, but Kaku is suggesting that the universe is so vast it ought to happen somewhere at least occasionally. Maybe some super-advanced civilization in another galaxy knows how to build a working time machine, or perhaps closed timelike curves can even occur naturally under certain rare conditions.

This raises problems of a different kind — not in science or engineering, but in basic logic. If time travel is allowed by the laws of physics, then it's possible to envision a whole range of paradoxical scenarios . Some of these appear so illogical that it's difficult to imagine that they could ever occur. But if they can't, what's stopping them? 

Thoughts like these prompted Stephen Hawking , who was always skeptical about the idea of time travel into the past, to come up with his "chronology protection conjecture" — the notion that some as-yet-unknown law of physics prevents closed timelike curves from happening. But that conjecture is only an educated guess, and until it is supported by hard evidence, we can come to only one conclusion: Time travel is possible.

A party for time travelers 

Hawking was skeptical about the feasibility of time travel into the past, not because he had disproved it, but because he was bothered by the logical paradoxes it created. In his chronology protection conjecture, he surmised that physicists would eventually discover a flaw in the theory of closed timelike curves that made them impossible. 

In 2009, he came up with an amusing way to test this conjecture. Hawking held a champagne party (shown in his Discovery Channel program), but he only advertised it after it had happened. His reasoning was that, if time machines eventually become practical, someone in the future might read about the party and travel back to attend it. But no one did — Hawking sat through the whole evening on his own. This doesn't prove time travel is impossible, but it does suggest that it never becomes a commonplace occurrence here on Earth.

The arrow of time 

One of the distinctive things about time is that it has a direction — from past to future. A cup of hot coffee left at room temperature always cools down; it never heats up. Your cellphone loses battery charge when you use it; it never gains charge. These are examples of entropy , essentially a measure of the amount of "useless" as opposed to "useful" energy. The entropy of a closed system always increases, and it's the key factor determining the arrow of time.

It turns out that entropy is the only thing that makes a distinction between past and future. In other branches of physics, like relativity or quantum theory, time doesn't have a preferred direction. No one knows where time's arrow comes from. It may be that it only applies to large, complex systems, in which case subatomic particles may not experience the arrow of time.

Time travel paradox 

If it's possible to travel back into the past — even theoretically — it raises a number of brain-twisting paradoxes — such as the grandfather paradox — that even scientists and philosophers find extremely perplexing.

Killing Hitler

A time traveler might decide to go back and kill him in his infancy. If they succeeded, future history books wouldn't even mention Hitler — so what motivation would the time traveler have for going back in time and killing him?

Killing your grandfather

Instead of killing a young Hitler, you might, by accident, kill one of your own ancestors when they were very young. But then you would never be born, so you couldn't travel back in time to kill them, so you would be born after all, and so on … 

A closed loop

Suppose the plans for a time machine suddenly appear from thin air on your desk. You spend a few days building it, then use it to send the plans back to your earlier self. But where did those plans originate? Nowhere — they are just looping round and round in time.

Sign up for the Live Science daily newsletter now

Get the world’s most fascinating discoveries delivered straight to your inbox.

Andrew May holds a Ph.D. in astrophysics from Manchester University, U.K. For 30 years, he worked in the academic, government and private sectors, before becoming a science writer where he has written for Fortean Times, How It Works, All About Space, BBC Science Focus, among others. He has also written a selection of books including Cosmic Impact and Astrobiology: The Search for Life Elsewhere in the Universe, published by Icon Books.

Something 'kicked' this hypervelocity star racing through the Milky Way at 1.3 million miles per hour

James Webb telescope reveals long-studied baby star is actually 'twins' — and they're throwing identical tantrums

'It's pretty incredible, the guy's got three legs': Watch lion looking for sex make record-breaking swim across treacherous river filled with crocs and hippos

Most Popular

• 2 Mysterious 4,000-year-old 'palace' with maze-like walls found on Greek island of Crete
• 3 GPT-4 has passed the Turing test, researchers claim
• 4 Planet Nine: Is the search for this elusive world nearly over?
• 5 NASA engineers finally fix Voyager 1 spacecraft — from 15 billion miles away
• 2 The sun's magnetic field is about to flip. Here's what to expect.
• 3 X-ray vision chip gives phones 'Superman' power to view objects through walls
• 4 Newly deciphered papyrus describes 'miracle' performed by 5-year-old Jesus
• 5 The 1st 'major lunar standstill' in more than 18 years is about to occur. Here's how to see it.

A Student Just Proved Paradox-Free Time Travel Is Possible

Now we can all go back to 2019.

• This follows recent research observing that the present is not changed by a time-traveling qubit.
• It's still not very nice to step on butterflies, though.

In a peer-reviewed paper, an honors undergraduate student says he has mathematically proven the physical feasibility of a specific kind of time travel. The paper appears in Classical and Quantum Gravity .

University of Queensland student Germain Tobar, who the university’s press release calls “prodigious,” worked with UQ physics professor Fabio Costa on this paper . In “ Reversible dynamics with closed time-like curves and freedom of choice ,” Tobar and Costa say they’ve found a middle ground in mathematics that solves a major logical paradox in one model of time travel. Let’s dig in.

The math itself is complex, but it boils down to something fairly simple. Time travel discussion focuses on closed time-like curves (CTCs), something Albert Einstein first posited. And Tobar and Costa say that as long as just two pieces of an entire scenario within a CTC are still in “causal order” when you leave, the rest is subject to local free will.

“Our results show that CTCs are not only compatible with determinism and with the local 'free choice' of operations, but also with a rich and diverse range of scenarios and dynamical processes,” their paper concludes.

In a university statement, Costa illustrates the science with an analogy:

“Say you travelled in time, in an attempt to stop COVID-19's patient zero from being exposed to the virus. However if you stopped that individual from becoming infected, that would eliminate the motivation for you to go back and stop the pandemic in the first place. This is a paradox, an inconsistency that often leads people to think that time travel cannot occur in our universe. [L]ogically it's hard to accept because that would affect our freedom to make any arbitrary action. It would mean you can time travel, but you cannot do anything that would cause a paradox to occur."

Some outcomes of this are grouped as the “ butterfly effect ,” which refers to unintended large consequences of small actions. But the real truth, in terms of the mathematical outcomes, is more like another classic parable: the monkey’s paw. Be careful what you wish for, and be careful what you time travel for. Tobar explains in the statement:

“In the coronavirus patient zero example, you might try and stop patient zero from becoming infected, but in doing so you would catch the virus and become patient zero, or someone else would. No matter what you did, the salient events would just recalibrate around you. Try as you might to create a paradox, the events will always adjust themselves, to avoid any inconsistency.”

While that sounds frustrating for the person trying to prevent a pandemic or kill Hitler, for mathematicians, it helps to smooth a fundamental speed bump in the way we think about time. It also fits with quantum findings from Los Alamos , for example, and the way random walk mathematics behave in one and two dimensions.

At the very least, this research suggests that anyone eventually designing a way to meaningfully travel in time could do so and experiment without an underlying fear of ruining the world—at least not right away.

Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all. 

.css-cuqpxl:before{padding-right:0.3125rem;content:'//';display:inline;} Pop Mech Pro .css-xtujxj:before{padding-left:0.3125rem;content:'//';display:inline;}

In a Show of Force, Russian Navy Visits Cuba

The Real WWII Fighter That Killed Godzilla

The CIA’s Plan to Deploy an Army of Super Spies

She Was Pronounced Dead—Then Found Gasping for Air

A B-1B Lancer Was Seen With a New Super Bomb

All About China’s Terrifying New Aircraft Carrier

Navy Fighter Seen Packing Huge, Ship-Fired Missile

Hieroglyphics Unveil Ramses II’s Lost Sarcophagus

This Navy Warship Program Is Kind of a Disaster

Every Single Cell in Your Body Could Be Conscious

How to Live Forever, or Die Trying

Articles on Time travel

Displaying 1 - 20 of 25 articles.

Is time travel even possible? An astrophysicist explains the science behind the science fiction

Adi Foord , University of Maryland, Baltimore County

Are black holes time machines? Yes, but there’s a catch

Sam Baron , Australian Catholic University

What are wormholes? An astrophysicist explains these shortcuts through  space-time

Dejan Stojkovic , University at Buffalo

Curious Kids: is it possible to see what is happening in distant solar systems now?

Jacco van Loon , Keele University

Can we time travel? A theoretical physicist provides some answers

Peter Watson , Carleton University

Curious Kids: what would happen if someone moved at twice the speed of light?

Time travel could be possible, but only with parallel timelines

Barak Shoshany , Brock University

Why does gravity pull us down and not up?

Mario Borunda , Oklahoma State University

New warp drive research dashes faster than light travel dreams – but reveals stranger possibilities

Curious Kids: is time travel possible for humans?

Lucy Strang , The University of Melbourne and Jacqueline Bondell , Swinburne University of Technology

Rotating black holes may serve as gentle portals for hyperspace travel

Gaurav Khanna , UMass Dartmouth

The great movie scenes: Back to the Future

Bruce Isaacs , University of Sydney

Time travel is possible – but only if you have an object with infinite mass

Stephen Hawking’s final book suggests time travel may one day be possible – here’s what to make of it

Peter Millington , University of Nottingham

Like a TARDIS in your head, memory helps you travel through time

Alice Mason , The University of Western Australia

Time travel: a conversation between a scientist and a literature professor

Richard Bower , Durham University and Simon John James , Durham University

Star Trek’s version of time travel is more realistic than most sci fi

Lloyd Strickland , Manchester Metropolitan University

Anthill 1: About time

Annabel Bligh , The Conversation and Gemma Ware , The Conversation

How to build a time machine

Steve Humble , Newcastle University

It’s Back to the Future Day today – so what are the next future predictions?

Michael Cowling , CQUniversity Australia ; Hamza Bendemra , Australian National University ; Justin Zobel , The University of Melbourne ; Philip Branch , Swinburne University of Technology ; Robert Merkel , Monash University ; Thas Ampalavanapillai Nirmalathas , The University of Melbourne , and Toby Walsh , Data61

Related Topics

• Albert Einstein
• Astrophysics
• Curious Kids
• Curious Kids US
• General Relativity
• Stephen Hawking

Top contributors

Associate Professor, Philosophy of Science, The University of Melbourne

Senior Lecturer in Philosophy, The University of Edinburgh

Professor of Physics, University of Rhode Island

Professor, University of Sydney

Honorary Research Associate, Centre for Time

Associate Professor in Telecommunications Engineering, Swinburne University of Technology

Emeritus Fellow, Trinity College, University of Cambridge

Former Academic Research Fellow, Khalifa University

Associate Professor, Film Studies, University of Sydney

Associate Professor – Information & Communication Technology (ICT), CQUniversity Australia

Lecturer in Software Engineering, Monash University

Professor in Physics, Durham University

Pro Vice-Chancellor, Graduate & International Research, The University of Melbourne

Professor of Philosophy and Intellectual History, Manchester Metropolitan University

Professor of Electrical and Electronic Engineering and Deputy Dean Research at Faculty of Engineering and Information Technology, The University of Melbourne

• X (Twitter)
• Unfollow topic Follow topic

What is the future of travel?

All aboard! After the pandemic upended life and leisure as we know it, travel is roaring back. The industry is set to make a full recovery by the end of 2024, after losing 75 percent of its value in 2020. Much of this has been so-called “revenge travel,” or people embarking on international or bucket list trips that were delayed by the pandemic. But domestic travel is recovering quickly too and is set to represent 70 percent of travel spending by 2030.

Get to know and directly engage with senior McKinsey experts on travel and tourism

Margaux Constantin is a partner in McKinsey’s Dubai office, Matteo Pacca is a senior partner in the Paris office, and Vik Krishnan is a senior partner in the Bay Area office.

We’ve done a deep dive into the latest travel trends and how industry players can adjust accordingly in The state of travel and hospitality 2024 report. Check out the highlights below, as well as McKinsey’s insights on AI in travel, mass tourism, and much more.

Learn more about McKinsey’s Travel, Logistics, and Infrastructure Practice .

Who are today’s travelers, and what do they want?

In February and March 2024, McKinsey surveyed  more than 5,000 people in China, Germany, the United Arab Emirates (UAE), the United Kingdom, and the United States who had taken at least one leisure trip in the past two years. Here are six highlights from the results of that survey:

• Travel is a top priority, especially for younger generations. Sixty-six percent of travelers we surveyed said they are more interested in travel now than before the COVID-19 pandemic. And millennials and Gen Zers  are traveling more and spending a higher share of their income on travel than their older counterparts.
• Younger travelers are keen to travel abroad. Gen Zers and millennials who responded to our survey are planning nearly an equal number of international and domestic trips in 2024. Older generations are planning to take twice as many domestic trips.
• Baby boomers are willing to spend if they see value. Baby boomers still account for 20 percent of overall travel spending. They are willing to spend on comforts such as nonstop flights. On the other hand, they are more willing to forego experiences to save money while traveling, unlike Gen Zers who will cut all other expense categories before they trim experiences.
• Travel is a collective story, with destinations as the backdrop. Travelers both want to hear other travelers’ stories and share their own. Ninety-two percent of younger travelers were inspired by social media in some shape or form for their last trip.
• What travelers want depends on where they’re from. Sixty-nine percent of Chinese respondents said they plan to visit a famous sight on their next trip, versus the 20 percent of European and North American travelers who said the same. Respondents living in the UAE also favor iconic destinations, as well as shopping and outdoor activities.

Learn more about McKinsey’s  Travel, Logistics, and Infrastructure Practice .

What are the top three travel industry trends today?

Travel is back, but traveler flows are shifting. McKinsey has isolated three major themes for industry stakeholders to consider as they look ahead.

• The bulk of travel spending is close to home. Seventy-five percent of travel spend is domestic. The United States is currently the world’s largest domestic travel market, but China is set to overtake it in the coming years. Stakeholders should make sure they capture the full potential of domestic travelers before turning their attention abroad.
• New markets such as India, Southeast Asia, and Eastern Europe are growing sources of outbound tourism. Indians’ travel spending is expected to grow 9 percent per year between now and 2030; annual growth projections for Southeast Asians and Eastern Europeans are both around 7 percent.
• Unexpected destinations are finding new ways to lure travelers and establish themselves alongside enduring favorites. Rwanda, for example, has capitalized on sustainable tourism by limiting gorilla trekking permits and directing revenue toward conservation.

Introducing McKinsey Explainers : Direct answers to complex questions

For a more in-depth look at these trends, check out McKinsey’s State of travel and hospitality 2024   report .

How will AI change how people travel?

In the 1950s, the introduction of the jet engine dramatically reduced travel times, changing the way people traveled forever. Now AI is upending the industry  in a similarly fundamental way. Industry players down to individual travelers are using advances in generative AI (gen AI) , machine learning , and deep learning  to reimagine what it means to plan, book, and experience travel. “It’s quite clear,” says McKinsey partner Vik Krishnan , “that gen AI significantly eases  the process of travel discovery.”

For travel companies, the task now is to rethink how they interact with customers, develop products and services, and manage operations in the age of AI. According to estimates by McKinsey Digital, companies that holistically address digital and analytics opportunities have the potential to see an earnings improvement of up to 25 percent .

McKinsey and Skift Research interviewed executives from 17 companies across five types of travel business. Here are three key findings on how travel companies can reckon with emerging technologies, drawn from the resulting report The promise of travel in the age of AI :

• Segmentation. Companies can use AI to create hyperspecific customer segments to guide how they interact with and serve customers. Segmentation can be based on a single macro characteristic (such as business versus leisure), or it can be so specific as to relate to just one customer.
• Surprise and delight. In the travel context, gen AI could take the form of digital assistants that interact with customers throughout their journeys, providing personalized trip itineraries and tailored recommendations and helping to resolve unexpected disruptions.
• Equipping workers better. AI tools can free up frontline workers’ time, allowing them to focus more on personal customer interactions. These tools can also shorten the training time for new hires and quickly upskill  the existing workforce.

AI is important, yes. But, according to Ella Alkalay Schreiber, general manager (GM) of fintech at Hopper, “The actual challenge is to understand the data, ask the right questions, read prediction versus actual, and do this in a timely manner. The actual challenge is the human thinking, the common sense .”

How is mass tourism changing travel?

More people are traveling than ever before. The most visited destinations are experiencing more concentrated flows of tourists ; 80 percent of travelers visit just 10 percent of the world’s tourist destinations. Mass tourism can encumber infrastructure, frustrate locals, and even harm the attractions that visitors came to see in the first place.

Tourism stakeholders can collectively look for better ways to handle visitor flows before they become overwhelming. Destinations should remain alert to early warning signs about high tourism concentration and work to maximize the benefits of tourism, while minimizing its negative impacts.

For one thing, destinations should understand their carrying capacity of tourists—that means the specific number of visitors a destination can accommodate before harm is caused to its physical, economic, or sociocultural environment. Shutting down tourism once the carrying capacity is reached isn’t always possible—or advisable. Rather, destinations should focus on increasing carrying capacity to enable more growth.

Next, destinations should assess their readiness to handle mass tourism and choose funding sources and mechanisms that can address its impacts. Implementing permitting systems for individual attractions can help manage capacity and mitigate harm. Proceeds from tourism can be reinvested into local communities to ensure that residents are not solely responsible for repairing the wear and tear caused by visitors.

After risks and funding sources have been identified, destinations can prepare for growing tourist volumes in the following ways:

• Build and equip a tourism-ready workforce to deliver positive tourism experiences.
• Use data (gathered from governments, businesses, social media platforms, and other sources) to manage visitor flows.
• Be deliberate about which tourist segments to attract (business travelers, sports fans, party groups, et cetera), and tailor offerings and communications accordingly.
• Distribute visitor footfall across different areas, nudging tourists to visit less-trafficked locations, and during different times, promoting off-season travel.
• Be prepared for sudden, unexpected fluctuations triggered by viral social media and cultural trends.
• Preserve cultural and natural heritage. Engage locals, especially indigenous people, to find the balance between preservation and tourism.

How can the travel sector accelerate the net-zero transition?

Global warming is getting worse, and the travel sector contributes up to 11 percent of total carbon emissions. Many consumers are aware that travel is part of the problem, but they’re reticent to give up their trips: travel activity is expected to soar by 85 percent  from 2016 to 2030. Instead, they’re increasing pressure on companies in the travel sector to achieve net zero . It’s a tall order: the range of decarbonization technologies in the market is limited, and what’s available is expensive.

But decarbonization doesn’t have to be a loss-leading proposition. Here are four steps  travel companies can take toward decarbonization that can potentially create value:

• Identify and sequence decarbonization initiatives. Awareness of decarbonization levers is one thing; implementation is quite another. One useful tool to help develop an implementation plan is the marginal abatement cost curve pathway framework, which provides a cost-benefit analysis of individual decarbonization levers and phasing plans.
• Partner to accelerate decarbonization of business travel. Many organizations will reduce their business travel, which accounts for 30 percent of all travel spend. This represents an opportunity for travel companies to partner with corporate clients on decarbonization. Travel companies can support their partners in achieving their decarbonization goals by nudging corporate users to make more sustainable choices, while making reservations and providing data to help partners track their emissions.
• Close the ‘say–do’ gap among leisure travelers. One McKinsey survey indicates that 40 percent of travelers globally say they are willing to pay at least 2 percent more for carbon-neutral flights. But Skift’s latest consumer survey reveals that only 14 percent  of travelers said they actually paid more for sustainable travel options. Travel companies can help close this gap by making sustainable options more visible during booking and using behavioral science to encourage travelers to make sustainable purchases.
• Build new sustainable travel options for the future. The travel sector can proactively pioneer sustainable new products and services. Green business building will require companies to create special initiatives, led by teams empowered to experiment without the pressure of being immediately profitable.

What’s the future of air travel?

Air travel is becoming more seasonal, as leisure travel’s increasing share of the market creates more pronounced summer peaks. Airlines have responded by shifting their schedules to operate more routes at greater frequency during peak periods. But airlines have run into turbulence when adjusting to the new reality. Meeting summer demand means buying more aircraft and hiring more crew; come winter, these resources go unutilized, which lowers productivity . But when airlines don’t run more flights in the summer, they leave a lot of money on the table.

How can airlines respond to seasonality? Here are three approaches :

• Mitigate winter weakness by employing conventional pricing and revenue management techniques, as well as creative pricing approaches (including, for example, monitoring and quickly seizing on sudden travel demand spikes, such as those created by a period of unexpectedly sunny weather).
• Adapt to seasonality by moving crew training sessions to off-peak periods, encouraging employee holiday taking during trough months, and offering workers seasonal contracts. Airlines can also explore outsourcing of crew, aircraft, maintenance, and even insurance.
• Leverage summer strengths, ensuring that commercial contracts reflect summer’s higher margins.

How is the luxury travel space evolving?

Quickly. Luxury travelers are not who you might expect: many are under the age of 60 and not necessarily from Europe or the United States. Perhaps even more surprisingly, they are not all millionaires: 35 percent of luxury-travel spending is by travelers with net worths between $100,000 and $1 million. Members of this group are known as aspirational luxury travelers, and they have their own set of preferences. They might be willing to spend big on one aspect of their trip—a special meal or a single flight upgrade—but not on every travel component. They prefer visibly branded luxury and pay close attention to loyalty program points and benefits .

The luxury-hospitality space is projected to grow faster than any other segment, at 6 percent per year  through 2025. And competition for luxury hotels is intensifying too: customers now have the option of renting luxurious villas with staff, or booking nonluxury hotels with luxury accoutrements such as rainfall showerheads and mattress toppers.

Another critical evolution is that the modern consumer, in the luxury space and elsewhere, values experiences over tangible things (exhibit).

Luxury properties may see more return from investing in a culture of excellence—powered by staff who anticipate customer needs, exceed expectations, create cherished memories, and make it all feel seamless—than in marble floors and gold-plated bath fixtures. Here are a few ways luxury properties can foster a culture of excellence :

• Leaders should assume the role of chief culture officer. GMs of luxury properties should lead by example, to help nurture a healthy and happy staff culture, and listen and respond to staff concerns.
• Hire for personalities, not resumes. “You can teach someone how to set a table,” said one GM we interviewed, “but you can’t teach a positive disposition.”
• Celebrate and reward employees. Best-in-class service is about treating customers with generosity and care. Leaders in the service sector can model this behavior by treating employees similarly.
• Create a truly distinctive customer experience . McKinsey research has shown that the top factor influencing customer loyalty in the lodging sector is “an experience worth paying more for”—not the product. Train staff to focus on tiny details as well as major needs to deliver true personalization.

What’s the latest in travel loyalty programs?

Loyalty programs are big business . They’ve evolved past being simply ways to boost sales or strengthen customer relationships; now, for many travel companies, they are profit centers in their own right. One major development was that travel companies realized they could sell loyalty points in bulk to corporate partners, who in turn offered the points to their customers as rewards. In 2019, United’s MileagePlus loyalty program sold $3.8 billion worth of miles to third parties, which accounted for 12 percent of the airline’s total revenue for that year. In 2022, American Airlines’ loyalty program brought in $3.1 billion in revenue, and Marriott’s brought in $2.7 billion.

But as this transition has happened, travel players have shifted focus away from the original purpose of these programs. Travel companies are seeing these loyalty programs primarily as revenue generators, rather than ways to improve customer experiences . As a result, loyalty program members have become increasingly disloyal. Recent loyalty surveys conducted by McKinsey revealed a steep decline in the likelihood that a customer would recommend airline, hotel, and cruise line loyalty programs to a friend. The same surveys also found that airline loyalty programs are driving fewer customer behavior changes than they used to.

So how can travel brands win customers’ loyalty back? Here are three steps to consider:

• Put experience at the core of loyalty programs. According to our 2023 McKinsey Travel Loyalty Survey , American respondents said they feel more loyal to Amazon than to the top six travel players combined, despite the absence of any traditional loyalty program. One of the reasons for Amazon’s success may be the frictionless experience it provides customers. Companies should strive to design loyalty programs around experiential benefits that make travelers feel special and seamlessly integrate customer experiences between desktop, mobile, and physical locations.
• Use data to offer personalization  to members. Travel brands have had access to customer data for a long time. But many have yet to deploy it for maximum value. Companies can use personalization to tailor both experiences and offers for loyalty members; our research has shown that 78 percent  of consumers are more likely to make a repeat purchase when offered a personalized experience.
• Rethink partnerships. Traditionally, travel companies have partnered with banks to offer cobranded credit cards. But many credit card brands now offer their own, self-branded travel rewards ecosystems. These types of partnerships may have diminishing returns in the future. When rethinking partnerships, travel brands should seek to build richer connections with customers, while boosting engagement. Uber’s partnership with Marriott, for example, gives users the option to link the brands’ loyalty programs, tapping into two large customer bases and providing more convenient travel experiences.

In a changing travel ecosystem, travel brands will need to ask themselves some hard questions if they want to earn back their customers’ loyalty.

Learn more about McKinsey’s Travel, Logistics, and Infrastructure Practice . And check out travel-related job opportunities if you’re interested in working at McKinsey.

Articles referenced include:

• “ Updating perceptions about today’s luxury traveler ,” May 29, 2024, Caroline Tufft , Margaux Constantin , Matteo Pacca , and Ryan Mann
• “ The way we travel now ,” May 29, 2024, Caroline Tufft , Margaux Constantin , Matteo Pacca , and Ryan Mann
• “ Destination readiness: Preparing for the tourist flows of tomorrow ,” May 29, 2024, Caroline Tufft , Margaux Constantin , Matteo Pacca , and Ryan Mann
• “ How the world’s best hotels deliver exceptional customer experience ,” March 18, 2024, Ryan Mann , Ellen Scully, Matthew Straus, and Jillian Tellez Holub
• “ How airlines can handle busier summers—and comparatively quiet winters ,” January 8, 2024, Jaap Bouwer, Ludwig Hausmann , Nina Lind , Christophe Verstreken, and Stavros Xanthopoulos
• “ Travel invented loyalty as we know it. Now it’s time for reinvention. ,” November 15, 2023, Lidiya Chapple, Clay Cowan, Ellen Scully, and Jillian Tellez Holub
• “ What AI means for travel—now and in the future ,” November 2, 2023, Alex Cosmas  and Vik Krishnan
• “ The promise of travel in the age of AI ,” September 27, 2023, Susann Almasi, Alex Cosmas , Sam Cowan, and Ben Ellencweig
• “ The future of tourism: Bridging the labor gap enhancing customer experience ,” August 1, 2023, Urs Binggeli, Zi Chen, Steffen Köpke, and Jackey Yu
• “ Hotels in the 2030s: Perspectives from Accor’s C-suite ,” July 27, 2023, Aurélia Bettati
• “ Tourism in the metaverse: Can travel go virtual? ,” May 4, 2023, Margaux Constantin , Giuseppe Genovese, Kashiff Munawar, and Rebecca Stone
• “ Three innovations to solve hotel staffing shortages ,” April 3, 2023, Ryan Mann , Esteban Ramirez, and Matthew Straus
• “ Accelerating the transition to net-zero travel ,” September 20, 2022, Danielle Bozarth , Olivier Cheret, Vik Krishnan , Mackenzie Murphy, and Jules Seeley
• “ The six secrets of profitable airlines ,” June 28, 2022, Jaap Bouwer, Alex Dichter , Vik Krishnan , and Steve Saxon
• “ How to ‘ACE’ hospitality recruitment ,” June 23, 2022, Margaux Constantin , Steffen Köpke, and Joost Krämer
• “ Opportunities for industry leaders as new travelers take to the skies ,” April 5, 2022, Mishal Ahmad, Frederik Franz, Tomas Nauclér, and Daniel Riefer
• “ Rebooting customer experience to bring back the magic of travel ,” September 21, 2021, Vik Krishnan , Kevin Neher, Maurice Obeid , Ellen Scully, and Jules Seeley

Want to know more about the future of travel?

Related articles.

The promise of travel in the age of AI

Travel Disruptors: Bringing fintech to travel booking

‘We Are the World Power.’ How Joe Biden Leads

You can read the transcript of the interview here and the fact-check here .

Joe Biden makes his way through the West Wing telling stories. In the Cabinet Room, with sun pouring through French doors from the Rose Garden outside, he remembers the first time he sat around the long mahogany table, its high-backed leather chairs ordered by seniority. It was more than 50 years ago, Biden says, and Richard Nixon told National Security Adviser Henry Kissinger to brief the 30-year-old first-term Delaware Senator on the still secret timing of the U.S. withdrawal from Vietnam. Walking slowly through the halls, the President unspools anecdotes about heads of state: Vladimir Putin, Xi Jinping , Emmanuel Macron. In the Oval Office, he talks about his childhood home in Scranton, Pa., and the 2008 phone call from Barack Obama asking Biden to be his running mate.

Biden recounts these memories over the course of more than 90 minutes on a warm spring day, speaking in a quiet, sometimes scattershot way. The impression he gives is one of advancing age and broad experience, of a man who has lived history. Biden leads the U.S. as the American century is fading into an uncertain future, a changing world of threats, opportunities, and power shifts. At 81, he holds fast to a vision that has reigned since World War II, in which a rich and powerful America leads an alliance of democracies to safeguard the globe from tyranny.

On June 6, Biden will travel to Normandy, France, to memorialize an event that has served for eight decades as a focal point of this vision. He will arrive as the 12th—and certainly the last—American President who was alive on that day in 1944, when 73,000 American troops led the largest amphibious invasion in human history, accelerating Nazi Germany’s defeat and Europe’s liberation. For generations, D-Day has been a hallowed anniversary. The President says commemorating it is as much about the future as the past. “We’re playing [that role] even more,” Biden says. “We are the world power.”

Whether this view of America’s role in the world will outlast Biden’s presidency is an open question. Voters face a clear choice this November. Biden calls America’s democratic values the “grounding wire of our global power” and its alliances “our greatest asset.” His presumptive opponent, former President Donald Trump, called for withdrawing American forces in Europe and Asia and has promised, most recently in his April 12 interview with TIME, to cut loose even our closest allies if they don’t do as he tells them. By his own account, Trump sees all countries as unreliable, the relations between them transactional. That sentiment has spread throughout a Republican Party that once championed America’s values abroad. J.D. Vance, the Ohio Senator in contention to become Trump’s Vice President, tells TIME that the D-Day story has become a sepia-toned distraction. “The foreign policy establishment is obsessed with World War II historical analogies,” says Vance, “and everything is some fairy tale they tell themselves from the 1930s and 1940s.”

Buy TIME's President Biden 'If He Wins' cover here

During his 40 months in office, events have tested Biden’s vision of American world leadership. Alliances haven’t been enough to win a new European war in Ukraine. U.S. power and leverage haven’t prevented a humanitarian catastrophe in the Middle East, marked by alleged war crimes. Putin is trying to assemble an axis of autocrats from Tehran to Beijing. In China, the U.S. faces an adversary potentially its equal in economic and military power that is intent on tearing down the American global order. President Xi has told his military to be ready to invade Taiwan by 2027, U.S. officials say, raising the possibility of a dark analogue to Normandy in Asia. Biden doesn’t rule out sending troops to defend Taiwan if China attacked, saying, “It would depend on the circumstances.”

Biden’s record in facing these tests is more than just nostalgic talk. He has added two powerful European militaries to NATO, and will soon announce the doubling of the number of countries in the Atlantic alliance that are paying more than the target 2% of their GDP toward defense, the White House says. His Administration has worked to prevent the war in Gaza from igniting a broader regional conflict. He brokered the first trilateral summit with long-distrustful regional partners South Korea and Japan, and coaxed the Philippines to move away from Beijing’s orbit and provide the U.S. new access to four military bases. He has rallied European and Asian countries to curtail China’s economic sway. “We have put together the strongest alliance in the history of the world,” Biden says, so that “we are able to move in a way that recognizes how much the world has changed and still lead.”

But American Presidents must earn a mandate from their fellow citizens, and it’s far from clear that Biden can. In surveys, large majorities say that he is too old to lead. As he walked TIME through the West Wing and sat for a 35-minute interview on May 28, the President, with his stiff gait, muffled voice, and fitful syntax, cut a striking contrast with the intense, loquacious figure who served as Senator and Vice President. Biden bristles at the suggestion that he is aging out of his job. Asked whether he could handle its rigors though the end of a second term, when he will be 86, he shot back, “I can do it better than anybody you know.” Age aside, Biden’s handling of foreign affairs gets poor marks from voters, and not just for the bungled withdrawal of U.S. forces from Afghanistan or the ongoing war in Gaza. While 65% of Americans still believe that the U.S. should take a leading or major role in the world, that number is down 14 points from 2003 and is at its lowest level since Gallup began polling the issue two years earlier.

Biden, who is the most experienced foreign policy President in a generation, believes that role is in America’s interest. “When we strengthen our alliances, we amplify our power as well as our ability to disrupt threats before they can reach our shores,” he said soon after taking office. To judge the merit of Biden’s plan to sustain American world leadership, voters can look to his record: what he has accomplished, where he has fallen short, and how he intends to build on his work in a second term.

Around 3 p.m. on Dec. 13, 2021, the White House Situation Room put through a call from Biden to his Finnish counterpart, Sauli Niinisto. Putin’s invasion of Ukraine was still more than two months away, and Finland, with its 830-mile border with Russia and tense history with Moscow, had long declined to join NATO. Less than a quarter of Finns supported entering the alliance at the time. But Biden had decided, aides say, that if Russia invaded, the West’s response should be not just to defend NATO, but to strengthen it.

On March 4, days after the invasion, Biden met with the newly enthusiastic Niinisto in the Oval Office. Together they called Swedish Prime Minister Magdalena Andersson, who had resisted joining the alliance, to try to persuade her. After both countries applied for membership in May 2022, Biden turned to getting the rest of NATO to accept them. In June, he called Turkey’s leader Recep Tayyip Erdogan from Air Force One on the way to a summit in Madrid, in hopes of getting Erdogan’s support for expanding the alliance. Dangling a one-on-one meeting, Biden said of Turkey’s long-sought access to America’s F-16 fighter jets, “Let’s find a way to get that done,” according to the White House. By March 2024, Sweden and Finland were in. “Everybody thought, including you guys, thought I was crazy,” Biden says. “Guess what? I did it.”

The accession of Finland and Sweden was part of Biden’s broader efforts to respond to Russia’s invasion of Ukraine by rallying the free world. Starting in October 2021, Biden held a series of meetings with European and NATO leaders, discussing postinvasion support for Ukraine, including military assistance, sanctions, diplomacy, and economic support. Biden also brought Asian allies into the effort. South Korea and Japan have imposed sanctions on Russia and its arms suppliers. The result, Biden advisers say, is a strengthened alliance of shared democratic values worldwide. “He has connected Europe and Asia in a way no previous President has,” says National Security Adviser Jake Sullivan.

Others view all the investment in Ukraine as a distraction from the bigger challenge America faces in East Asia. “Who doesn’t think that $200 billion spent in Europe would’ve been incredibly useful in the Pacific?” says Elbridge Colby, a former Trump Administration Pentagon official and lead architect of the 2018 National Defense Strategy. “Great nations fail,” says Lieut. General Keith Kellogg, Trump’s former National Security Adviser, when “you fix somebody else’s potholes instead of fixing your potholes.” 

Biden says he remains committed to Ukrainian victory. Asked about the war’s endgame, Biden says, “Peace looks like making sure Russia never, never, never, never occupies Ukraine.” But last year’s Ukrainian counteroffensive was a failure. Russia recently has made its largest advances since the opening months of the invasion. Alliance building may have reached its limit, along with Americans’ appetite for funding a war of attrition. Biden’s allies in Kyiv complain he has been too cautious, giving Ukraine enough weapons to survive the war but not to win it. “It’s not a decisive stance,” says a senior official in President Volodymyr Zelensky’s government. “It’s not the way to victory.”

On balance, however, even longtime critics are impressed with Biden’s efforts in Ukraine. Former Defense Secretary and CIA director Robert Gates wrote in 2014 that Biden had “been wrong on nearly every major foreign policy and national-security issue over the past four decades.” But on May 19, Gates said that Biden’s response to Russia’s invasion has gone a long way toward repairing the damage of the disastrous Afghanistan withdrawal. “He gained a lot of credibility with the speed with which he assembled the coalition of partner countries, allies, and friends before, during, and after the Russian invasion of Ukraine,” Gates told CBS’s  Face the Nation.

Biden says his response has been part of a broader deterrence strategy. “If we ever let Ukraine go down, mark my words, you’ll see Poland go, and you’ll see all those nations along the actual border of Russia [fall],” he tells TIME. But in other theaters, the high-minded Normandy vision has given way to a different kind of diplomacy.

Halfway through our interview, Biden responds to a question about America’s relationship with Saudi Arabia by saying that the U.S. has two kinds of alliances: “There are values-based, and there are practical-based.” During the campaign, Biden had sworn to make Saudi Arabia a “pariah.” One of his first moves in office was to cut off certain arms supplies over the kingdom’s war in Yemen, which has displaced 4.5 million people and killed 377,000, including 11,000 children, according to the U.N. Soon after, the de facto Saudi ruler, Crown Prince Mohammed bin Salman, known as MBS, met with China’s Foreign Minister and proposed greater cooperation on nuclear energy and security with Beijing, already the kingdom’s largest economic partner.

The Biden Administration quietly pivoted. A new “great game” was afoot, with the world dividing between competing Chinese and American spheres of influence. For all Biden’s efforts to stimulate a green transition, Saudi Arabia was still providing much of the world’s energy. Moreover, the Saudis had expressed willingness during the Trump presidency to normalize relations with Israel, which would tilt the regional balance of power against Iran and in the U.S.’s favor. On Sept. 27, 2021, Sullivan traveled to Saudi Arabia with instructions from Biden to explore the possibility of a peace deal between the kingdom and Israel.

Biden himself traveled to Saudi Arabia in July 2022, bucking a flurry of criticism for meeting with MBS, who has led a widespread crackdown on clerics, academics, and human-rights advocates critical of his regime, according to Human Rights Watch, and who the U.S. says ordered the murder of Washington Post journalist Jamal Khashoggi . But the visit helped stabilize relations. Over the course of the next year, it began to look as if Biden’s moral climb down with MBS had brought the Saudis back on the U.S. side, and restarted a possible bargain with Israel. The outlines of that deal, Biden now says, were “overwhelmingly in our interest.”

Hamas, the terrorist group that controls Gaza, was determined not to allow it. Days after its Oct. 7 attack against Israel, which killed some 1,200 people, Hamas spokesman Ghazi Hamad told TIME, “We planned for this because Israel thinks it can make peace with anyone, it can make normalization with any country, it can oppress the Palestinians, so we decided to shock the Israelis in order to wake up others.” Eight Americans were among the estimated 240 taken hostage in the massacre. The Biden Administration has sought to secure their release, but it is not clear how many of the American hostages have survived; three reportedly have been killed. “We believe there are those that are still alive,” Biden tells TIME. “I met with all the families. But we don’t have final proof on exactly who’s alive.”

Biden’s reaction to Oct. 7 was to provide rock-solid support to Israel. Within a week he had deployed two aircraft carriers to the region. Quietly, he tried to rally Egypt and Saudi Arabia to resist expansion of the conflict into a war between Israel and Iran. Biden’s “practical-based” alliance building appeared to pay off on April 13, when Iran responded to an Israeli attack on a satellite diplomatic office by launching more than 300 missiles and drones in its first-ever direct attack on Israel. The Saudis and Jordanians reportedly provided intelligence assistance and opened their airspace to U.S. and other jets. With Israel leading the way, the ad hoc alliance managed to shoot down all but four of the projectiles, with no fatalities. More important, the episode helped avert a region-wide war.

But Prime Minister Benjamin Netanyahu has upped the cost of Biden’s commitment to Israel at every turn. Nearly eight months after the conflict started, the death toll in Gaza, according to the local Hamas-led Ministry of Health, has climbed to more than 36,000 people, including an unknown number of Hamas fighters. More than 1.7 million have been displaced by Israeli attacks that have destroyed much of the enclave. On May 20, the prosecutor for the International Criminal Court requested a war-crimes indictment for Netanyahu, his Defense Minister, and three leaders of Hamas. Four days later, in a largely symbolic move, the International Court of Justice ordered Israel to halt operations in Gaza . Human Rights Watch says Israel has “imposed collective punishments on the civilian population, deprived the civilian population of objects indispensable to its survival, and used starvation of civilians as a weapon of war.”

Asked if Israeli forces have committed war crimes in Gaza, Biden says, “It’s uncertain.” From the start, the Administration knew Israel was pushing the limits of legal warfare, the Washington Post and others have reported. The conflict is driving a wedge between the U.S. and its allies. On May 31, Biden laid out a phased cease-fire plan that would end the war and secure the release of hostages. He has continued to pursue the complicated regional deal with Saudi Arabia. Some close to Biden say the only holdout to the broader pact is Netanyahu. The President declines to say as much, but when asked by TIME if Netanyahu is prolonging the war for his own political reasons, Biden admits, “There is every reason for people to draw that conclusion.”

As aides try to bring the interview to a close, Biden turns to China. Hawks say Beijing is in a sprint to match American economic and military production. By some measures, it is catching up on GDP and defense manufacturing, and already has a larger navy. But Biden takes a bullish view of the competition with the rising Asian power. “Everybody talks about how, how strong China is and how powerful they are,” Biden says. “You’ve got an economy that’s on the brink there. The idea that their economy is booming, give me a break.” That doesn’t mean they can’t pose a threat. Asked if China is using AI or other means to meddle in the upcoming U.S. election, Biden says someone is, but declines to say who. Pressed, he adds, “I think China would have an interest in meddling.”

What Biden describes as China’s economic weakness could make confrontation more, not less, likely—another argument, as he sees it, for expanding America’s alliances in East Asia. And in that arena, the President has pursued a mix of “values-based” and “practical” approaches.

Biden was on Air Force One on his way to a fundraiser in Illinois on May 11, 2022, when the results of the Philippine presidential elections were announced, showing that Ferdinand “Bongbong” Marcos Jr. had won. Biden “had the instinct to just pick up the phone and said, ‘Hey, let’s get together soon and start building a relationship,’” Sullivan says. It was a long shot. Marcos has a pending $2 billion judgment against him in a U.S. court relating to his parents’ human-rights record during their more than 20-year dictatorship, which ended in 1986. The Philippines are now rated “partly free” by Freedom House, and the outgoing President, Rodrigo Duterte, had courted China even as Beijing claimed nearby islands and territorial waters. Marcos had sent cold signals to the U.S. during his campaign.

Biden’s call was the first Marcos had received from a foreign leader. As U.S. officials followed up, they briefed the new President on the parallels between Putin’s invasion of Ukraine and Xi’s declared goals in the South China Sea. Biden dispatched Vice President Kamala Harris and his Secretaries of State and Defense to woo Marcos. Deputy Secretary of State Wendy Sherman made clear Marcos had diplomatic immunity and would be welcome in the U.S. Less than a year after Biden’s congratulatory call, Marcos made a visit to the White House. More significantly, he provided the U.S. military new access to four bases in the Philippines. In April and May, the two countries engaged in their largest military exercises together, simulating an effort to repulse an amphibious landing. “The President got engaged early in a very personal way,” says Sullivan, “and then kind of showed both respect for him and a vision for where the relationship would go.”

Biden has pursued this brand of personal realpolitik across Asia. He elevated the communist autocracy in Vietnam to the highest diplomatic status, comprehensive strategic partner, and has moved to embrace the increasingly repressive regime of Narendra Modi in India. He has tried to boost the “Quad” alliance with India, Japan, and Australia, upgrading it from a meeting of Foreign Ministers to one of heads of state. In April 2023, Biden convened a Camp David summit with the South Korean President Yoon Suk-yeol and Japanese Prime Minister Fumio Kishida. Overcoming long-fraught relations between Seoul and Tokyo, the three countries criticized China’s behavior in the South China Sea and declared “a hinge point of history, when geopolitical competition, the climate crisis, Russia’s war of aggression against Ukraine, and nuclear provocations test us.”

Critics say the problem is too much friend making and not enough deterrence. The U.K. recently said China may be preparing to provide lethal aid to Russia, a move that Biden said in March 2022 would put Xi “in significant jeopardy” of harsh U.S. sanctions. “The single biggest problem with the Biden team is their failure to grasp what it takes to achieve effective deterrence against aggressors,” says Matt Pottinger, who was Deputy National Security Adviser under Trump. “They failed against the Taliban, then Putin, and then Iran and its proxies. And now Beijing is making moves that could prove fateful for the world.” Former Trump official Colby says Biden’s diplomatic work is a weak substitute for the one thing that can deter China’s rise. “These high-profile photo ops,” says Colby, “are not a substitute for raw military power.” He points to recent statements by senior U.S. military officials that China is outpacing the U.S. on missile- and shipbuilding, and war games showing the U.S. losing badly in a contest over Taiwan, and says the U.S. should put all its efforts into defending the “first island chain” of Japan, Taiwan, and the Philippines.

Biden believes that withdrawing from Europe and the Middle East to focus on East Asia would backfire, aides say. If America abandons its allies elsewhere, they argue, its Asian allies will abandon it, in turn. The U.S. needs European and Middle Eastern countries to increase its economic and military advantages over China. And ultimately, failing to confront instability now—in Ukraine, Gaza or elsewhere—will only make doing so later a more costly distraction from the competition with Beijing.

Back in the Cabinet Room after the interview, the sun is lower, and Biden has more stories. He turns to a sideboard with a commendation from the Kosovo government to his son Beau Biden, who died of cancer nine years ago. The President relates with evident pride his son’s work supporting its judicial system. A mention of diplomat Richard Holbrooke, who brokered the Dayton accords for the Balkans, elicits a story about Afghanistan and an argument Biden had with Holbrooke over the search for peace there. 

On a matching sideboard on the other side of the door, the President opens an album with travel pictures, launching a series of anecdotes about the Popes he has known, including John Paul II and Benedict, whom Biden calls “the Rottweiler.” Recounting an exchange with one over abortion, he casts an eye toward the cracked door to the Oval Office and asks an aide, “Are they in there?” Turning back to his visitors, he says, “Let me show you one more picture.”

This avuncular politicking remains a Biden trademark, one that has helped with allies overseas but failed to unite Americans at home, as Biden pledged when running for President. Not that he has stopped trying. Biden ultimately persuaded Republican House Speaker Mike Johnson to move a roughly $95 billion supplemental aid package for Ukraine, the Middle East, and Taiwan. To build support for his Middle East peace package, he has worked both sides of the aisle. On Nov. 8, 2023, Biden sat for two hours in the windowless Roosevelt Room with a bipartisan group of nine Senators who had just returned from the region, asking for impressions from the trip and moderating a conversation between them, Sullivan, and Middle East coordinator Brett McGurk. At the end, he pulled Democratic Senator Chris Coons and Republican Senator Lindsey Graham into the Oval Office for separate 10-minute conversations about next steps in the effort, says Coons.

Biden may be right that despite the partisanship, a consensus exists for a values-based, pragmatic role for America in the world. His challenge is to get Americans to focus on that rather than on other issues driven by foreign affairs, like inflation or immigration. Biden denies that his expansion of Trump’s trade war with China will increase prices, and says his only regret about lifting Trump’s anti-immigration measures is that he didn’t do it sooner. His goal in a second term, he says, is “to finish what he started.”

At stake is the direction of the world for the coming century. At Normandy, Biden will make the case for what historian Hal Brands says is “the 80-year tradition of internationalism that has been quite good for America and the world.” The alternative, says Brands, would be a “more vicious and chaotic” world where Americans ultimately would be less safe, prosperous, and free, but only after everyone else suffered first.

Wrapping up his conversation with TIME, Biden offers cookies from a tray in the outer Oval. “They’re homemade,” he says. Turning to leave, he offers a final salutation: “Keep the faith.” But then he pauses and turns back, as the phrase triggers one last story. It’s about a relative who had his own response to that admonition. And here Biden taps one of his visitors on the chest and says, “Spread the faith.”

— With reporting by Simon Shuster/Kyiv; Leslie Dickstein, Simmone Shah, and Julia Zorthian/New York; and Melissa August, Brian Bennett, Vera Bergengruen, Eric Cortellessa, and Sam Jacobs/Washington

Order your copy of the President Biden 'If He Wins' issue here

Correction appended, June 5: The original version of this story misstated the terms of an agreement the Biden Administration struck with the Philippines. The Philippines provided the U.S. new access to four existing military bases, not permission for four new bases.

More Must-Reads from TIME

• Melinda French Gates Is Going It Alone
• Lai Ching-te Is Standing His Ground
• Do Less. It’s Good for You
• There's Something Different About Will Smith
• What Animal Studies Are Revealing About Their Minds—and Ours
• What a Hospice Nurse Wants You to Know About Death
• 15 LGBTQ+ Books to Read for Pride
• Want Weekly Recs on What to Watch, Read, and More? Sign Up for Worth Your Time

Contact us at [email protected]

• Search Please fill out this field.
• Manage Your Subscription
• Give a Gift Subscription
• Newsletters
• Sweepstakes

Italy's Deepest Lake Is Also Its Most Glamorous — With Luxury Hotels, Stunning Mountain Views, and Fairy-tale Towns

Lake Como is one of Italy's dreamiest lake destinations — here's how to plan the perfect visit.

Elizabeth Heath is a writer and editor living on a hill in Umbria, from where she writes about travel in Italy, the rest of Europe, and farther afield.

© Marco Bottigelli/Getty Images

It’s hard to think of Lake Como without thinking of its famous residents: Clooney, Versace, Messi, Madonna. At one point or another, they’ve all staked their claim to a piece of lakeside real estate here. And one can hardly blame them for coveting a sliver of Como’s fabulous shoreline. No place in Italy has quite the cache of Lake Como, and its reputation as the epitome of the good life is well-deserved.

One of a cluster of northern Italian lakes that includes Garda, Maggiore, Lugano, and several smaller siblings, Lake Como, or Lago di Como, is Italy’s deepest lake. Its distinctive shape — an upside down Y — was formed by a massive glacier, and the lake is still fed from waters descending from the nearby Alps. That sets the scene for some high drama, landscape-wise, with steep mountains cloaked with dense forests and deep blue waters ringed by charming lakeside towns .

The Romans were the first to build grand villas on Lake Como, and ever since, generations of la dolce vita seekers have followed suit, using towns such as Bellagio, Como, Menaggio, and Tremezzo as the base for lakeside leisure. We asked several travel experts about their longtime love affairs with the lake, and got their advice on the best places to stay, dine, and experience Lake Como.

Top 5 Can’t Miss

• Lake Como’s five-star hotels are numerous and legendary, including relative newcomer Passalacqua, which opened in 2022 and occupies an 18th-century aristocratic villa.
• All of our experts agree that getting out on the water is a must, whether via a lake ferry, a privately chartered Riva boat, or a self-driven motorboat.
• Whether you dine on the water — say at Harry’s Bar in Cernobbio — or journey uphill to Al Veluu for its sweeping views, make sure at least one of your meals takes in the scenery.
• Yes, you can buy luxury fashion goods during your visit, but consider some Italian-made items, like the wine, olive oil, and vinegars from Enoteca da Gigi in Como.
• Join a guided or self-guided hike or bike ride to discover the many scenic trails around the lake, where the gobsmacking views are the reward for all that elevation change.

Best Hotels

Courtesy of Il Sereno Lago di Como

Grand Hotel Tremezzo

“I always say the arrival to Grand Hotel Tremezzo on Lake Como is cinematic,” says luxury travel expert Annie Fitzsimmons, whose new book, " 100 Hotels of a Lifetime ," features the art nouveau facade of this palatial resort on the cover. “I love everything about the view … the town of Bellagio across the lake, the swimming pool seemingly suspended over the water,” she shares, adding that owner Valentina de Santis and her team are “the best on the planet.”

Grand Hotel Victoria

TravelLustre travel consultant Nicole Bono calls this belle époque beauty her favorite hotel on the lake. “There's a private beach club for guests steps away from the hotel where you can go for a dip, as well as a stunning pool,” she says. Samy Ghachem, general manager of La Dolce Vita Orient Express , says Victoria has “the best spa on the lake.”

Passalacqua

“Eighteenth century Passalacqua is a dream from another time,” says travel writer Nicole Trilivas. “It’s like walking onto a vintage film set, with cascading gardens, jewel-box sitting rooms, and pastel-hued suites as spoiling, decadent, and lavish as wedding cakes.” Trilivas is also a fan of their Bellini cocktail, made with white peach, pink pepper, and Champagne.

A sexy ambience permeates this 40-suite hotel, where striking modern architecture stands out on a lakeshore dotted with centuries-old villas and art nouveau edifices. And on a lake known for its grand dames, it says a lot that Il Sereno was T+L readers’ favorite resort in Europe in 2023 . Expect a swim-up suite , Riva boats for guest use, and Michelin-star dining.

Palace Hotel

For a palatial vibe right in the center of Como, Ghachem calls Palace Hotel , run by the Villa d’Este group, one of his favorites. “Cross the road and you're on the water. The hotel is located within walking distance of most restaurants, shopping, and most activities, including the ferry terminal.”

See more top hotels in Lake Como here.

Best Things to Do

Amanda Blackard/Travel + Leisure

Go on a boat ride.

“I always advise every traveler going to Lake Como that the real beauty of the region is best seen from the water,” says Ghachem. “You absolutely need to get out on a boat. Most hotels can make arrangements for private Venetian-style boats or the more exciting and elegant Riva boats. There are, of course, less pricey options, including the numerous ferry routes.” Como Lake Artists offers reasonably priced self-drive boats.

Hire a photographer.

Eli Wagner , a Travel + Leisure A-list advisor and founder of Wagner Bespoke Travel , says the best way to preserve your Lake Como memories is to hire a pro. “Hire a photographer to join you and your family on the boat day. Instead of the traditional holiday cards, having Lake Como as the backdrop will be memorable for your holiday photos for many years to come.”

Hit the trail.

When you’re not on the water, head out on one of Lake Como’s numerous hiking trails. From Torno, Ghachem recommends the trail up to Montepiatto “for a stunning view over the northern and southern parts of the lake.” From Como town, it’s just an hour walk up to the village of Brunate, where Ghachem says “you’ll be rewarded with a number of bars and local restaurants before catching the funicular back down.”

Embrace la dolce far niente.

The sweetness of doing nothing — that’s a favorite pastime on Lake Como. “It's a great destination to lounge with your spritz and do a whole lot of nothing,” says Bono. “People can just relax and take it easy, strolling, shopping, and enjoying the scenery.”

Best Shopping

Enoteca da gigi, como.

“This is one of my favorite wine bars in the old town,” says Ghachem. “You can taste before you buy.” In addition to wines from all over Italy, he says the olive oil and vinegar selection here is top-notch.

Riflessi di Gusto

This old-school boutique for foodie gifts carries carefully and locally sourced, made-in-Italy products, including many hard-to-find items that slip easily into a carry-on. Ghachem says the selection of dry pastas is the best on the lake.

La Bottega del Legno 

“There are a lot of super-touristy stores all over the lake, but you can also find some wonderful boutiques,” says Bono. Bellagio’s compact old town is her favorite for shopping, and the historic Bottega del Legno is a great stop for wood-carved items, including real wooden toys.

Best Restaurants

chekyfoto/Getty Images

Bono calls this hilltop restaurant “a true showstopper,” while Wagner says it “embodies the essence of Lake Como.” Fresh, locally sourced ingredients, including mushrooms delivered daily, earn high marks. “Go at sunset and take in the amazing panorama,” adds Bono.

Harry's Bar

Ghachem appreciates the familiar comforts of this longtime restaurant on the Cernobbio waterfront. “You can't go wrong here,” he says. “There’s great ambience — it’s just upscale enough.” Grab an outdoor seat overlooking the lake and take advantage of the terrific wine list.

L'Orologio

“Most concierges would not recommend this place as it's almost too local,” says Ghachem, adding that this Como restaurant is “for sure the least touristy place in town.” Tagliolini with porcini mushrooms and truffle is his go-to order here.

Best Time to Visit

Our experts agree that shoulder season (September and October) is the best time to visit Lake Como. “The entire lake is lush — everything is so stunning this time of year,” says Bono. Wagner votes for September. “[This is] when the summer crowds have dissipated and the sun is still shining,” he shares. Ghachem also suggests May, before the summer crowds arrive. But note that if you do visit in these “in-between” seasons, you’ll find cooler air and water temperatures that may dissuade you from taking a dip in the lake.

June, July, and August are peak travel months for the lake. However, if you reserve early and prepare for the crowds, you’ll find a quintessential Italian lake experience awaits.

Many hotels and restaurants close for the winter, but enough remain open that if you want a quiet — albeit chilly — vacation, this might be the time to visit. At Christmas, the villages along the lake take on a magical atmosphere.  

How to Get There

Como town, the starting point for many vacations on the lake, is about one hour from Milan’s Malpensa Airport, and about a 40-minute train ride from Milano Centrale, Milan’s main train station. Bellagio, Menaggio, and other small towns along the lake are served by ferries (less frequent in the offseason) or can be reached by car.

Served by several intra-Europe carriers, Milan Bergamo Airport is about one hour and 40 minutes from Como by car.

How to Get Around

“If you're happy to self-drive, having a car can be very useful,” says Bono. This is especially true for reaching destinations not directly on the lake. But if you don’t want to rent a car, Bono says, “private transfers and car services, as well as ferries and water taxis are the way to go.” Some hotels rent Fiat 500s, which make for fun driving and great photo ops, though Ghachem warns that the roads around the lake are very narrow and often quite curvy. 

Related Articles

Every product is independently selected by (obsessive) editors. Things you buy through our links may earn us a commission.

How Often Should You Replace Your Pillows?

As with mattresses , pillows don’t have a hard-and-fast expiration date, so the only truly honest answer for when to replace them is a resounding “It depends.” The basic rule of thumb — and what I’ve heard over and over from experts — is to replace your pillows every two years, especially if you have asthma, a dust-mite allergy, or sensitive skin. But a well-maintained pillow can certainly last longer if it’s still supportive enough that you aren’t waking up with aches and pains. Whether it has been two years or not, here are the main things to consider before buying a new pillow .

Above all, make sure your pillow is supporting your head and neck

Beyond comfort, the main reason for sleeping with a pillow at all is to keep your head and neck in a neutral alignment with your spine. And because you sleep on it for hours every night, your pillow will inevitably wear out over time. If you’re dealing with any kind of neck pain , it may be a sign that your pillow isn’t doing its job, especially if it has gotten too flat or compressed after years of use, says chiropractor Dr. Jordan Duncan .

To test the supportiveness of your pillow, you can try what Dr. Joshua Tal , a psychologist who specializes in sleep disorders, calls the “shoe test,” which was created by Dr. Michael Breus , a clinical psychologist and founder of the Sleep Doctor wellness company. “What you do is you fold your pillow in half, put a shoe on the back side of the pillow, and then let go of the pillow,” explains Tal. “If the pillow folds back into shape and flings the shoe off of it, you’re good. If it doesn’t, it’s kind of lost its ability to hold your head up properly.” (Be sure to use a substantial shoe and not, say, a flip-flop.)

Tal also mentioned another test he uses with his patients: “I advise clients to stand up against the wall as if you were sleeping — so a back sleeper would stand with their back to the wall; a side sleeper would stand with their side to the wall — and then rest your head on the wall and notice how far it has to move to do that,” he says. “Then put your pillow in between where you’re resting your head and see if you’re standing up straight. That’s the key: You should be standing up straight if you have a good pillow.”

If your pillow looks or feels very lumpy — a possibility for down , down-alternative, or shredded-foam pillows — that may also be a sign it’s losing some of its structural integrity.

Pillows can get pretty gross over time

Because you’re essentially smashing your face and hair into them every night, and because they’re not always washing machine–friendly, pillows tend to need replacing more often than other bedding, solely from a hygiene perspective. The main concerns are asthma , allergies, and skin: According to Melanie Carver, chief mission officer of the Asthma and Allergy Foundation of America (AAFA), the fabric of a pillow is permeable to dust mites, and after two years of use, your pillow can be 10 percent dead dust mites and their droppings. Icky, yes, as well as particularly bad for people with dust-mite allergies . Other potential allergy triggers like pet dander and mold can also be absorbed by your pillow and cause symptoms to flare. If you have asthma, allergens in your pillow can make your symptoms worse.

Pillows can also trap dirt, oil, and dead skin cells, and according to board-certified dermatologist Dr. Annie Chiu , all of these can be irritants for your skin — even if you are diligently washing your pillowcase . Chiu explains that a pillowcase won’t act as a complete barrier, so irritants in your pillow can still cause trouble, especially if you have sensitive skin or persistent acne .

Yellowing or stains can be an indicator that your pillow has absorbed lots of moisture — from sweat, drool, skin-care or hair-care products, or going to sleep with wet hair — and needs to be replaced.

How important is the “every two years” rule, really?

If you’re balking at the idea of replacing your pillow every two years, you’re not alone. Two years is what AAFA officially recommends — and just about every expert I spoke to echoed it. But my own informal polling of friends and colleagues and a deep dive into Reddit discussions on how often to replace pillows suggest many people aren’t replacing their pillows as frequently as every two years, no matter what the experts say. Ultimately, while you may need to replace basic fiber pillows even sooner than two years because they may flatten more quickly, pillows made of sturdier materials can last much longer. You should mainly be mindful of whether issues like asthma, allergies, or neck pain are getting worse the longer you have your pillow.

Keeping your pillow as clean as possible will help it last as long as possible

Carver told me that washing your pillow once a month in hot water (130 degrees Fahrenheit) can help remove allergens like dust mites. However, this advice only really applies to pillows that are totally washable — and in my experience, the majority of them are not. Synthetic-fill down-alternative pillows and down pillows are the most likely to be washing machine–friendly, so if you have allergies or asthma and want to follow Carver’s guidance, you may want to seek out those fill types.

Most memory-foam and latex pillows have only a washable cover since the foam itself usually cannot be washed. And a 2013 paper from the American Academy of Allergy, Asthma & Immunology (AAAAI) found that foam pillows are just as susceptible to dust mites as feather pillows, so it’s still important to wash your pillow covers regularly. Memory-foam pillows are often recommended as allergy-safe since the foam itself isn’t a food source to dust mites the way natural materials like feathers are, but like other types of synthetic pillows, foam can still collect your dead skin, which feeds the mites.

You can sometimes clean memory foam by sprinkling it with baking soda and vacuuming it, as recommended by cleaning expert Jolie Kerr, to help remove dust mites or dead-skin buildup. And while some experts (including Kerr) say it’s possible to clean memory foam in water as long as you hand-wash it, I have attempted this and forewarn you that it will take a long time for the foam to dry. (This means you may be risking mold or mildew, so proceed with caution and make sure you’ve got a well-ventilated space and a fan to speed up the process.) For latex pillows, you can follow a similar process as with memory foam. Latex manufacturer Turmerry recommends vacuuming the latex and then spot-cleaning with a damp cloth and mild detergent.

Consider a pillow protector

In addition to cleaning, Carver also recommends using an Asthma & Allergy Friendly–certified pillow protector to keep out dust mites and other allergens in the first place — they’re like mattress protectors for your pillow that go around the pillow before you add your actual pillowcase. But even with these precautions, AAFA still recommends replacing your pillow every two years.

Additional reporting by Hilary Reid and Chloe Anello .

The Strategist  is designed to surface useful, expert recommendations for things to buy across the vast e-commerce landscape. Every product is independently selected by our team of editors, whom you can read about  here . We update links when possible, but note that deals can expire and all prices are subject to change.

• the strategist
• strategist explains

Every product is independently selected by (obsessive) editors. Things you buy through our links may earn us a commission.

Deal of the Day

Micro sales, greatest hits, most viewed stories.

• What Julianne Nicholson Can’t Live Without
• Some Really Good Deals at Quince’s First-Ever Sale
• The Best Fitness Trackers for All Types of Activities
• What’s the Difference Between BB Creams and CC Creams?
• The 11 Very Best Shampoos
• The 22 Very Best Sunscreens for Your Face

Today’s Top Clicked

Things you buy through our links may earn Vox Media a commission

How to Survive a Long Flight

Air travel has become an increasingly miserable experience for most people in recent years. Before you even get on the plane itself, you must brave the hordes at airport security and hope that, on the other side, your flight isn’t on a Boeing 737 Max and doesn’t get delayed or canceled. Food and drinks inside the airport are comically bad and overpriced, while finding a seat near your gate feels like The Hunger Games . By the time you finally cram yourself into a tiny seat with little leg room, you may already be uncomfortable and cranky — and you’re not even halfway to your destination .

If, like me, you’re not a big fan of the experience in the first place, long-haul travel can be particularly challenging — especially if you’re stuck in economy class with no hope of a free upgrade. So how exactly do you survive a long flight? Here are eight suggestions from the Cut staff for strategies and travel essentials that will help you pass the time, get comfortable, and even catch some sleep.

1. Try to Game the System

My husband and I always book the window and aisle seats of a three-seat row, and often that means the middle seat will be empty. (Not always, but it’s worth a try for the extra space!) I also bring inflatable seat cushions and pillows to try to make the experience more comfortable. — Chantal Fernandez, features writer

2. Shift Your Schedule Beforehand

I’ve done a lot of long-haul international flights — I’m talking 12-plus hours — and I start changing my schedule in the days leading up to a trip. I use the app Timeshifter : I enter my flight date and number into the app, and it gives a game plan for the days leading up to the trip and after. It tells me when to sleep, when to avoid caffeine, when to get some sun — all things that can help minimize jet lag! — Tariro Mzezewa, morning blogger

3. Download Movies and Podcasts Ahead of Time

If you leave me to my own devices, I am one of those weirdos who will stare at the map all flight long . So as to not raise alarms from my seatmates, I’ve started to give myself some distractions. Preparation starts days in advance, when I hoard podcasts and documentary films that I want to consume. I download them and start the flight with those. Then I always carry my own eye mask, ear plugs, and melatonin gummies (and reserve a window seat). I pop one of those gummies a couple hours in and sleep.. — Joanna Nikas, deputy style editor 

4. Keep Busy

I’m a nervous flier, so I like to keep busy with a variety of low-effort activities outside of constantly checking the flight tracker. Keeping noise canceling headphones on hand is a must for listening to the playlists I’ve downloaded (Beyoncé’s entire discography and calming sounds) and trying to get some sleep. I also keep my iPad on hand with books to read and apps to keep me busy, like Happy Color, which is just paint by numbers. If all else fails, use some of the extra time to free up space on your phone by deleting apps and old screenshots. Just bring a charger. — Chinea Rodriguez, shopping writer

5. Knock Yourself Out

My best advice isn’t original: Spend as much of the flight unconscious as possible. Pull an all-nighter beforehand, pop some melatonin or an edible — whatever you need. Then have a nice big drink of water, put in your noise-canceling headphones, and go to sleep. I also always wear a hoodie, hood up. Planes are gross and I don’t want to get lice. Plus it’s added privacy. — Rachel Bashein, managing editor

6. Practice the Local Language

Sleeping on planes is nearly impossible for me. If I have to be awake for six to ten hours on a tube hurtling through the sky, I’m going to watch exactly one in-flight movie and then I’ll spend the rest of the time brushing up on my language skills. If I’m traveling to a Spanish-speaking country, I like to listen to Radio Ambulante , an excellent podcast that tells stories from all over Latin America. For anywhere else, there’s Duolingo. I may not be refreshed when I step off the plane, but at least I’m mentally ready to make small talk with my cab driver. — Catherine Thompson, features editor

7. Bring Plenty of Treats

I do not eat plane food. It’s not a bougie thing, it’s an IBS thing. So it’s always important that I have plenty of snacks when I’m getting on a long flight and that I’ve had a good meal before I’ve boarded. If I don’t have time to eat beforehand, I’ll grab a sandwich from my favorite bakery to take along or grab something at the airport. But I always shop for snacks before heading out the door. I tend to grab something salty, something sweet and then a fun drink I’ve been wanting to try so that I have those to look forward to while I’m rotting away in my seat. — Brooke LaMantia, editorial assistant

8. Do Some Self-care

Long flights are hell, and my friends who are rich enough to fly first class or bold enough to Xanax themselves into unconsciousness really have it figured out. I’m too paranoid to mess around with sedatives thousands of feet in the air, so I try to make things more tolerable by caring for my body the best I can. I use Colgate Wisps so that I don’t have to spend three minutes in a plane bathroom brushing my teeth, pack some nice-smelling travel-size hand sanitizer and lotion, and apply deodorant every few hours. It’s hard to feel like a human when you have approximately two feet of leg room. This helps. — Danielle Cohen, staff writer

• summer travel
• recommendations

The Cut Shop

Most viewed stories.

• And You Thought You Had Roommate Drama
• My Job Was My Life. Then I Got Fired.
• I Have a Terrible Memory. Am I Better Off That Way?
• Riding the Subway With Anna Delvey
• Bill Belichick Is How Many Years Older Than His Girlfriend?
• The Stolen Kids of Sarah Lawrence
• Why Did These YouTubers Give Away Their Son?

Editor’s Picks

Most Popular

What is your email.

This email will be used to sign into all New York sites. By submitting your email, you agree to our Terms and Privacy Policy and to receive email correspondence from us.

Sign In To Continue Reading

Create your free account.

Password must be at least 8 characters and contain:

• Lower case letters (a-z)
• Upper case letters (A-Z)
• Numbers (0-9)
• Special Characters (!@#$%^&*)

As part of your account, you’ll receive occasional updates and offers from New York , which you can opt out of anytime.

We've detected unusual activity from your computer network

To continue, please click the box below to let us know you're not a robot.

Why did this happen?

Please make sure your browser supports JavaScript and cookies and that you are not blocking them from loading. For more information you can review our Terms of Service and Cookie Policy .

For inquiries related to this message please contact our support team and provide the reference ID below.