Time Dilation

Time dilation is a scientific concept related to Relativity which states, basically, that for an observer onboard a spaceship travelling at a certain speed any speed at all (though only noticeable at appreciable fractions of the speed of light) with respect to Earth (or any inertial reference frame of your choice), time passes more slowly than it would for an observer on Earth. When near-lightspeed travel becomes involved, the effects become quite drastic: A person might go on a space journey that seems to him to last one year and, on returning, find that 10 years have passed on Earth. This is sometimes extrapolated by science fiction authors to apply to FTL Travel as well, though this does not make much sense physically.

In general relativity, an additional time dilation effect is caused by gravity. Time passes more slowly nearer to the bottom of a gravitational potential well (e.g. on the surface of a planet) than higher up in one (e.g. in an airplane). This dilation, in addition to the dilation due to differences in velocity, needs to be compensated for by clocks on satellites.

In fiction, this effect is often used to facilitate a variation on Mayfly-December Romance, with the earth-bound partner as the "short-lived" one compared to the space traveller. Such plots can also involve a familial relationship instead of a romance -- in this case, the earth-bound character is usually the space traveller's twin or child.

Note that though many writers extend this so traveling faster than light means aging backwards, that isn't how the math says it works. The time scale factor for speeds faster than c is imaginary, not negative. However, if an object is travelling faster-than-light that means there is always some slower-than-light frame reference that sees the object travelling backwards in time -- or possibly moving in the opposite direction, with events on the object occurring backwards. The Other Wiki has an article on tachyons, theoretical objects that move faster than the speed of light, and explains how this works.

Rip Van Winkle is a related trope in more fantastical works. Year Inside, Hour Outside is the inverse, and tends to show up in softer works or those using magic rather than physics.

There are a couple of quirks about real-world time dilation that also make it somewhat different from what you see in fiction:

 * You have to get pretty darned close to the speed of light to see any significant time dilation effects at all. At 90% of the speed of light, your "gamma factor" -- how much slower your clock will seem to a stationary observer -- will only be about 2. To get a gamma factor of five million (roughly what you'd need for one minute of your time to equal a decade in "rest time") would require moving at 99.999999999998% of the speed of light.
 * Time dilation works both ways. To the guy standing on Earth, you're whizzing past him at 90% of the speed of light, so to him your clock seems to only be running at half normal speed. But to you, you're standing still and the Earth is whizzing past you at 90% of the speed of light, so to you the clocks on Earth seem to be running at half normal speed. To you, the people back on Earth are the ones aging more slowly. This seeming paradox is only resolved upon carefully examining the path taken by both observers -- or, in this example, by you and your friend back on Earth -- and realizing that you had to decelerate and turn back, tracing a path in 4-dimensional spacetime that is not a straight line. According the odd notion of distance defined in 4-d spacetime, a straight line actually has the longest possible distance, corresponding to the longest possible time. Then you, the traveler, will experience a smaller subjective amount of time than your buddy on Earth.
 * GPS has a small correction for time dilation between the surface of the earth and up in space. A satellite in space experiences 0.6 nanoseconds more for every second on Earth.

Anime & Manga

 * Gun Buster, the original giant robot time dilation story. They even call it the Urashima effect (though it was change to the "Rip Van Winkle effect" on the DVD to avoid telling an enormous story.).
 * Soukou no Strain, since there's no FTL travel in the series.
 * In The World of Narue, Narue's elder sister Kanaka came to stay with them on Earth. Unfortunately, Narue and her father had access to FTL travel while Kanaka snuck aboard a ship traveling normal lightspeed. The result is that when Kanaka reaches earth, enough time has passed there that Narue, the younger sister, is now physically older.

Comic Books

 * The Alan Moore comic The Ballad of Halo Jones deals with this in the war period of Halo's life. The planet Moab has such an extreme gravitational field that all the soldiers have to wear high pressure armor to fight. When the Amazon Brigade marches towards combat, the fight is frozen but gradually speeds up as they get closer to it, being at normal speed by the time they arrive. Every time Halo goes out on a mission for an afternoon, she misses another birthday. She ends up getting promoted this way, and she returns one last time to find that war ended months ago.
 * The Marvel/Epic comic series The Alien Legion had the members of Force Nomad fighting a battle in the event horizon of a black hole. There they met a race of aliens that had been there for decades, but all their star charts were millions of years out of date. They helped our guys get out, but their system of measurements was incomprehensible to all but one of them (This was because it was amazingly ancient, and so was the translator's race). They were able to leave after a short time, but discovered that fifteen years had passed on the outside. During that time, they had been declared dead and Tamara's infant daughter was a teenager in her own platoon.
 * The origin story of X-O Manowar: a Visigoth soldier is enslaved by aliens on a slower-than-light ship. Seven years passed on the ship, but 1600 years passed on Earth by the time he returned.

Film -- Live Action

 * Planet of the Apes. Not very important in the book, more so in The Movie.
 * Used for a cheap gag towards the end of The Ice Pirates.

Literature

 * The short story Shore Leave Blacks by Nancy Etchemendy is about the pilot of near-lightspeed spacecraft dealing with the problems this causes her, as she ends up leaving behind not only a husband/lover (I forget which) but a son who will be her age or older by the time she returns.
 * Flatterland, unofficial sequel to Flatland, illustrates this concept with the story of the Paradox Twins. One of them travels at near-lightspeed to the moon and back (or something like that) and returns to find his twin older than he is.
 * Time For The Stars by Robert A. Heinlein is about a pair of telepathically linked twins, one of whom goes on a near-lightspeed colonization trip and ends up substantially younger than the other one.
 * The sequels to Ender's Game take place 3,000 years after it, but involve many of the same characters as they've spent most of their life on spaceships traveling at relativistic speeds. However, he got the bit about Time Dilation working both ways wrong. Relativity and don't mix well.
 * The IF in Ender's Game relied on this trope specifically.
 * The Forever War deals with quite a bit of time dilation. In order to travel faster than light a warship has to accelerate to relativistic speeds towards a collapsar. Every combat mission results in a 20 to 100+ leap forward for the crew.
 * Alastair Reynolds plays with time dilation a lot in his novels. The most notable example is probably Pushing Ice, which involves a near-future spaceship encountering an advanced alien artifact which accelerates to such a degree that they end up millions of years in the future.
 * This is The Gentle Giants of Ganymede's premise, when humanity runs into a shipload of Ancient Astronauts who had been forced to circle around the solar system for the last several million years due to an emergency hyperspace jump gone wrong, until their engines finally ran out of fuel.
 * In the Strugatski Brothers' Noon Universe, FTL travel is impossible. All ships travel at near-light speeds and return centuries later (objective time). This all changes when the crew of a starship decides to try something new. Normally, constant or slowly-increasing acceleration is maintained for most of the voyage. The captain of the ship decides to try high acceleration for most of the trip (about 10g) in order to reverse the effects of time dilation, as this would fall outside of the Special Theory of Relativity. According to the captain, General Theory of Relativity allows for this. This works, and the ship returns to Earth six months later; however, several years pass for the crew.
 * In Captain French, or the Quest for Paradise by Mikhail Akhmanov and Christopher Nicholas Gilmore, the only way to travel between the stars is with the use of a relativistic drive system. It does not require any acceleration (i.e. works like the lightspeed in Star Wars, only with STL) and takes only seconds for the crew. However, decades, if not centuries, pass in the outside universe. It still takes months of travel to and from the edges of star systems in order to minimize the risk of Critical Existence Failure. For this reason, space travel is only done by colonists and space traders and no interstellar government is possible.
 * The titular protagonist mentions once intercepting a message sent out from one world about a scientist claiming to have proven Fermats Last Theorem. While French admits that this is probably a big deal for a mathematician, the message has no commercial value to him as a space trader. In fact, in all his millennia of travel, he has only intercepted about two dozen interstellar messages, as sending them requires putting up a powerful relay satellite in orbit as well as maintaining it for little or no return. As such, the various human worlds are isolated with an occasional space trader passing through every half-a-century or so. Of course, given that aging has been eliminated in this 'verse, it's entirely possible for a planetbound person to meet the same space trader who comes back 200 years later.
 * In Sergey Lukyanenko's Line of Delirium trilogy (very loosely based on Master of Orion), all ships must decelerate before exiting hyperspace. Failure to do so would result in the ship exiting at near-light speeds and experiencing extreme time dilation. There are also cases of warships escaping from battle using their sublight engines, being forced to accelerate to near-light speeds when their hyperdrive is damaged. The crew of one such human warship commits suicide when they find themselves in a post-war galaxy over 100 years after they left.
 * This is the premise behind the short story prequel Shadows of Dreams, when a Psilon battleship decelerates from near-light speeds near a small human colony, believing that Psilons and humans are still at war.
 * The Honor Harrington books occasionally mention time dilation, though generally in the context of reducing perceived travel time in hyperspace by a few days or as an extra wrinkle of a few minutes during combat rather than as a significant alteration to characters' lives.
 * The biggest impact that its had on the story so far is that it nearly torpedoed the title character's career before it got started. Dealing with the math involved nearly caused her to flunk out of a compulsory astrogation course in the Academy.
 * In Frederik Pohl's Gateway, a major shocker in the ending is that
 * Time dilation is one of the quirks of the distorted planet in The Inverted World, which is in the shape of a rotating hyperboloid. North of optimum, where the circumference of the world grows exponentially smaller and the speed of its rotation correspondingly slower, time moves faster. South of optimum, where the circumference of the world grows exponentially larger and the speed of its rotation correspondingly greater, time moves slower.
 * This effect is present in The Left Hand of Darkness; the Ekumenical diplomats consider it a good thing, since if their mission on a particular planet fails, they can hop over to the nearest planet, then turn around and come right back and have a whole new generation of leaders with which to try again.
 * In Poul Anderson's Tau Zero, the runaway Bussard ramjet Leonora Christine is going so close to the speed of light that the crew witness the final end of the universe. (They also manage to survive it and end up in the next universe born after ours dies.)
 * And they then take their time about slowing down... so that after the Big Crunch and Big Bang, there is time for stars and planets to form and life to evolve to a reasonably advanced level before they finally finish their journey.
 * There is a novel where the main character is an astronaut on a ship equipped with an antimatter drive, allowing it to rapidly accelerate to relativistic speeds. Their goal - go to the nearest galaxy and return. Realizing she can't live without him, his girlfriend decides to put herself into an induced coma, a form of extreme cryogenics. Against all odds, her pod is recovered from the ruins of the lab thousands of years later, and she is revived. By this time, FTL travel is discovered, so the humans travel to the other galaxy and rescue her boyfriend, who has been trapped in a stasis field.
 * Inverted in The Clockwork Rocket by Greg Egan, which takes place in a universe where the laws of physics are different from ours, and traveling quickly means more time passes. The protagonists take a trip in a high-speed Generation Ship in order to have enough time to develop the technology they need to avert a coming disaster.

Live Action Television

 * In an episode of Stargate Atlantis, the crew encountered a group of actual Ancients flying at 99,9% of the speed of light (their hyperdrive did not function), which is the reason why they were alive in the present time, despite the rest of their race being wiped out/ascending long ago.
 * Battlestar Galactica Reimagined:  went on a sublight journey that took 2000 years from an outside perspective, but a lot less to them.
 * Starhunter has a couple of examples.
 * At the end of the first season the protagonists end up being trapped in Hyperspace, and at the beginning of the second season one of them comes out 15 years later the same age, and much of the season is her searching for her uncle, who is still trapped there. At one point it's suggested that by the time they find him his son maybe be twice his age.
 * One episode features a middle aged man who hires the bounty hunter protagonists to rescue his kidnapped parents, who's kidnappers had been traveling at very fast speeds for 50 years causing them to age only 8 months, meaning he's now twice their age.
 * Carl Sagan laid it out, along with the other bizarre effects that come with approaching the speed of light, in an episode of Cosmos, Travels in Space and Time.
 * In Andromeda, the titular ship and its captain spend three centuries at the edge of the event horizon of a black hole where only a few seconds pass. Enough time for a massive civil war and the subsequent collapse of civilization.
 * The Bellerophon, Earth's earliest starship, traveled half the speed of light across the galaxy for 3000 years, but far less passed on the ship.
 * In fact, the crew still looks fairly young, implying that they were traveling at much higher speeds than mere .5c (you wouldn't experience much Time Dilation at that speed). The ship can, apparently, accelerate ridiculously fast using its massive fusion engine (lacking weapons, the ship actively uses the exhaust as a weapon), although they often need to replenish their fuel.
 * One episode of the Dirty Pair TV anime had a space travel magnate try to separate his son from a lover he disapproved of by launching her on the prototype of a slower-than-light "Time Dilation Tour" ship he had handy; the plan was that she'd only return after the son had aged the fifty years of the trip. The Lovely Angels can't stop the launch, but free the son in time for him to follow his love on another of the ships. The father finally gets on the final ship because he can't live without his son.

Music

 * A musical example: Queen's "'39" is about a crewman on a spaceship who travels to a distant planet and returns after a year, only to discover that a hundred years have passed back home and only the descendants of his loved ones remain. It ends on quite a down note.
 * Julia Ecklar and Anne Prather's "Pushin' the Speed of Light" is a classic filk on the subject, and a Tear Jerker: "And you've left behind you the world of men with no way in space to go home again."

Radio Show

 * The BBC Radio series Earthsearch uses time dilation as a major plot point. The children of a starship's second-generation crew, the sole survivors of a disaster that killed the crew and erased a huge amount of scientific data, return the ship to their home solar system 115 years after it set out, only to find that a million years have passed outside. Oh, and the Earth is missing.

Tabletop RPG

 * GURPS: Spaceships goes so far as to provide the equation for relativistic time dilation to be used for ships with sufficiently powerful engines.

Video Game

 * According to her backstory from Super Mario Galaxy, Rosalina was whisked into outer space by several Lumas, in which she then befriended, but even though she spent a few days traveling through space on the Lumas' spaceship, by the time she returned to Earth,
 * In the X-Universe series, every ship can mount a "Singularity Engine Time Accelerator" which can speed up the flow of time up to 10x, depending on the game settings. Its primary use is to speed up travel time in-system for the player's benefit. Activating the device at high settings is heavy on the CPU and tends to cause Artificial Stupidity.
 * In in-game lore, malfunctioning SETA drives can supposedly crank up the effect to several thousand times normal speed: back in the days of X2 and X3 when stations had bulletin boards that featured news articles, one story covered a pilot who lost a year's worth of time when his SETA device went haywire and took several hours for him to shut down.

Web Original

 * Orions Arm has relativistic space travel at speeds in excess of .9c, but it's usually requires a Transapient tech Reactionless Drive. But it's not much of an issue because people can have essentially unlimited lifespans (average Nearbaseline lifespan is about 3000 years). Also the setting's Portal Network requires the two mouths of the wormhole to be towed into position at sub-relativistic speeds (~.77c).
 * In Homestuck, at one point two characters travel between two windows in the fourth wall, taking what is explicitly stated as three nanoseconds (the time it takes light to travel one yard) to do so from an outside perspective. It is stated to take three years from their own perspective, which is actually not how time dilation works.
 * Andrew then claimed that it was 'plot ditation'. Since the characters had started their quest the previous day, but from our side of the fourth wall it had taken three years, they would have to wait until the same amount of time had passed for them to re-join.
 * Problem Sleuth is not much better at depicting time dilation either: as a character approaches the event horizon of a black hole, the events for him start to become faster until "all the events in history happen at once", which Andrew himself admitted was unrealistic but more dramatic.

Western Animation

 * Reversed in My Life As a Teenage Robot. Sheldon was sent to space by accident, and returned some days later as a 60-year-old man. Dr. Wakeman then tried to use a machine to de-age him, but he ended up as a baby, and was sent to space again to return to his normal age.
 * Which, to a physics student, sounds more impossible than traveling faster than the speed of light.
 * In Generator Rex, César's mobile laboratory was thrown into space, traveling so fast that where 15 minutes passed for him, five years passed on Earth.

Literature

 * Jump drive in CJ Cherryh's Alliance Union universe causes some time dilation, jumps take a week or less for the crew and a month or two for planetsiders and stationers. Of course, humans have to be sedated for jump and most other oxygen-breathing species are knocked unconscious.
 * Of course, ships have to accelerate to a large fraction of c to jump.
 * Some levels of hyperspace in the Uplift series.
 * Home Fires by Gene Wolfe uses this along with a Mayfly-December Romance. The traveler is a female soldier, and her husband ages a couple of decades to her two years.
 * Stanisław Lem's Return From the Stars is about an astronaut who tries to cope with the changed world after returning from a 127-year mission (which lasted 10 years for him).
 * In Hyperion humanity has a way of circumventing this with portals. The problem is that you have to send a few dozen (or hundred, depending on the route) lightspeed or so ships out to build a portal at the other end. Now, there is time dilation to begin with, and since that version of Hyperspace Is a Scary Place, they also travel as Human Popsicles, so the people sent to build them leave their whole life behind.
 * In the short story "Remembering Siri" (included in Hyperion as "The Consul's Tale"), a woman on a backwater planet falls in love with a dashing Space Marine who stops by for shore leave once a decade (in her timeframe). When she dies of old age, he has only aged 5 years, but their son is 43.
 * In the fourth book, it helps the protagonist to cut a few years off the age difference between him and his love.

Tabletop RPG

 * One more scary thing about the Warp is that you can never tell when you will emerge. More than once a fleet has arrived centuries after (or before)departing.

Webcomic
"QQ: "Wait, a Mark IV? But the Mark III isn't due out for another year!
 * The D.A.V.E. (Dangerous And Very Expensive) drive in Freefall works by reversing time dilation, weeks pass outside while decades pass on board. So passengers spend the trip frozen.
 * In Quentyn Quinn Space Ranger the titular character missed three years without realizing it while on a long mission through largely uncharted space.

Delivery guy: "Oh no. The Mark III came out last year. The Mark IV just hit the shelves last month!""