Time dilation is the difference in elapsedtime as measured by twoclocks, either because of a relativevelocity between them (special relativity), or a difference ingravitational potential between their locations (general relativity). When unspecified, "time dilation" usually refers to the effect due to velocity.
After compensating for varying signal delays resulting from the changing distance between anobserver and a moving clock (i.e.Doppler effect), the observer will measure the moving clock as ticking more slowly than a clock at rest in the observer's ownreference frame. There is a difference between observed and measured relativistic time dilation - the observer does not visually perceive time dilation in the same way that they measure it.[1] In addition, a clock that is close to a massive body (and which therefore is at lower gravitational potential) will record less elapsed time than a clock situated farther from the same massive body (and which is at a higher gravitational potential).
These predictions of thetheory of relativity have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation ofsatellite navigation systems such asGPS andGalileo.[2]
Time dilation by theLorentz factor was predicted by several authors at the turn of the 20th century.[3][4]Joseph Larmor (1897) wrote that, at least for those orbiting a nucleus, individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio:.[5]Emil Cohn (1904) specifically related this formula to the rate of clocks.[6] In the context ofspecial relativity it was shown byAlbert Einstein (1905) that this effect concerns the nature of time itself, and he was also the first to point out its reciprocity or symmetry.[7] Subsequently,Hermann Minkowski (1907) introduced the concept ofproper time which further clarified the meaning of time dilation.[8]
From the local frame of reference of the blue clock, the red clock, being in motion, is measured as ticking slower.[9]
Special relativity indicates that, for an observer in aninertial frame of reference, a clock that is moving relative to the observer will be measured to tick more slowly than a clock at rest in the observer's frame of reference. This is sometimes called special relativistic time dilation. The faster therelative velocity, the greater the time dilation between them, with time slowing to a stop as one clock approaches thespeed of light (299,792,458 m/s).
In theory, time dilation would make it possible for passengers in a fast-moving vehicle to advance into the future in a short period of their own time. With sufficiently high speeds, the effect would be dramatic. For example, one year of travel might correspond to ten years on Earth. Indeed, a constant 1 g acceleration would permit humans to travel throughthe entire known Universe in one human lifetime.[10]
With current technology severely limiting the velocity of space travel, the differences experienced in practice are minuscule. After 6 months on theInternational Space Station (ISS), orbiting Earth at a speed of about 7,700 m/s, an astronaut would have aged about 0.005 seconds less than he would have on Earth.[11] The cosmonautsSergei Krikalev andSergey Avdeev both experienced time dilation of about 20 milliseconds compared to time that passed on Earth.[12][13]
Left: Observer at rest measures time 2L/c between co-local events of light signal generation at A and arrival at A. Right: Events according to an observer moving to the left of the setup: bottom mirror A when signal is generated at timet'=0, top mirror B when signal gets reflected at timet'=D/c, bottom mirror A when signal returns at timet'=2D/c
Time dilation can be inferred from the observed constancy of the speed of light in all reference frames dictated by thesecond postulate of special relativity. This constancy of the speed of light means that, counter to intuition, the speeds of material objects and light are not additive. It is not possible to make the speed of light appear greater by moving towards or away from the light source.[14][15][16][17]
Consider then, a simple vertical clock consisting of two mirrorsA andB, between which a light pulse is bouncing. The separation of the mirrors isL and the clock ticks once each time the light pulse hits mirrorA.
In the frame in which the clock is at rest (see left part of the diagram), the light pulse traces out a path of length2L and the time period between the ticks of the clock is equal to2L divided by the speed of lightc:
From the frame of reference of a moving observer traveling at the speedv relative to the resting frame of the clock (right part of diagram), the light pulse is seen as tracing out a longer, angled path2D. Keeping the speed of light constant for all inertial observers requires a lengthening (that is dilation) of the time period between the ticks of this clock from the moving observer's perspective. That is to say, as measured in a frame moving relative to the local clock, this clock will be running (that is ticking) more slowly, since tick rate equals one over the time period between ticks 1/.
Straightforward application of thePythagorean theorem leads to the well-known prediction of special relativity:
The total time for the light pulse to trace its path is given by:
The length of the half path can be calculated as a function of known quantities as:
Elimination of the variablesD andL from these three equations results in:
Time dilation equation
which expresses the fact that the moving observer's period of the clock is longer than the period in the frame of the clock itself. TheLorentz factor gamma (γ) is defined as[18]
Because all clocks that have a common period in the resting frame should have a common period when observed from the moving frame, all other clocks—mechanical, electronic, optical (such as an identical horizontal version of the clock in the example)—should exhibit the same velocity-dependent time dilation.[19]
Transversal time dilation. The blue dots represent a pulse of light. Each pair of dots with light "bouncing" between them is a clock. In the frame of each group of clocks, the other group is measured to tick more slowly, because the moving clock's light pulse has to travel a larger distance than the stationary clock's light pulse. That is so, even though the clocks are identical and their relative motion is perfectly reciprocal.
Given a certain frame of reference, and the "stationary" observer described earlier, if a second observer accompanied the "moving" clock, each of the observers would measure the other's clock as ticking at aslower rate than their own local clock, due to them both measure the other to be the one that is in motion relative to their own stationary frame of reference.
Common sense would dictate that, if the passage of time has slowed for a moving object, said object would observe the external world's time to be correspondingly sped up. Counterintuitively, special relativity predicts the opposite. When two observers are in motion relative to each other, each will measure the other's clock slowing down, in concordance with them being in motion relative to the observer's frame of reference.
Time UV of a clock in S is shorter compared to Ux′ in S′, and time UW of a clock in S′ is shorter compared to Ux in S.
While this seems self-contradictory, a similar oddity occurs in everyday life. If two persons A and B observe each other from a distance, B will appear small to A, but at the same time, A will appear small to B. Being familiar with the effects ofperspective, there is no contradiction or paradox in this situation.[20]
The reciprocity of the phenomenon also leads to the so-calledtwin paradox where the aging of twins, one staying on Earth and the other embarking on space travel, is compared, and where the reciprocity suggests that both persons should have the same age when they reunite. On the contrary, at the end of the round-trip, the traveling twin will be younger than the sibling on Earth. The dilemma posed by the paradox can be explained by the fact that situation is not symmetric. The twin staying on Earth is in a single inertial frame, and the traveling twin is in two different inertial frames: one on the way out and another on the way back. See alsoTwin paradox § Role of acceleration.
A comparison ofmuon lifetimes at different speeds is possible. In the laboratory, slow muons are produced; and in the atmosphere, very fast-moving muons are introduced by cosmic rays. Taking the muon lifetime at rest as the laboratory value of 2.197 μs, the lifetime of a cosmic-ray-produced muon traveling at 98% of the speed of light is about five times longer, in agreement with observations. An example is Rossi and Hall (1941), who compared the population of cosmic-ray-produced muons at the top of a mountain to that observed at sea level.[21]
The lifetime of particles produced in particle accelerators are longer due to time dilation. In such experiments, the "clock" is the time taken by processes leading to muon decay, and these processes take place in the moving muon at its own "clock rate", which is much slower than the laboratory clock. This is routinely taken into account in particle physics, and many dedicated measurements have been performed. For instance, in the muon storage ring at CERN the lifetime of muons circulating with γ = 29.327 was found to be dilated to 64.378 μs, confirming time dilation to an accuracy of 0.9 ± 0.4 parts per thousand.[22]
The stated purpose by Ives and Stilwell (1938, 1941) of these experiments was to verify the time dilation effect, predicted by Larmor–Lorentz ether theory, due to motion through the ether using Einstein's suggestion that Doppler effect incanal rays would provide a suitable experiment. These experiments measured theDoppler shift of the radiation emitted fromcathode rays, when viewed from directly in front and from directly behind. The high and low frequencies detected were not the classically predicted values:The high and low frequencies of the radiation from the moving sources were measured as:[23]as deduced by Einstein (1905) from theLorentz transformation, when the source is running slow by the Lorentz factor.
Hasselkamp, Mondry, and Scharmann[24] (1979) measured the Doppler shift from a source moving at right angles to the line of sight. The most general relationship between frequencies of the radiation from the moving sources is given by:as deduced by Einstein (1905).[25] Forϕ = 90° (cosϕ = 0) this reduces tofdetected =frestγ. This lower frequency from the moving source can be attributed to the time dilation effect and is often called thetransverse Doppler effect and was predicted by relativity.
In 2010 time dilation was observed at speeds of less than 10 metres per second using optical atomic clocks connected by 75 metres of optical fiber.[26]
Clock C in relative motion between two synchronized clocks A and B. C meets A atd, and B atf.
Twin paradox. One twin has to change frames, leading to differentproper times in the twin's world lines.
In theMinkowski diagram from the first image on the right, clock C resting in inertial frame S′ meets clock A atd and clock B atf (both resting in S). All three clocks simultaneously start to tick in S. The worldline of A is the ct-axis, the worldline of B intersectingf is parallel to the ct-axis, and the worldline of C is the ct′-axis. All events simultaneous withd in S are on the x-axis, in S′ on the x′-axis.
Theproper time between two events is indicated by a clock present at both events.[27] It is invariant, i.e., in all inertial frames it is agreed that this time is indicated by that clock. Intervaldf is, therefore, the proper time of clock C, and is shorter with respect to the coordinate timesef=dg of clocks B and A in S. Conversely, also proper timeef of B is shorter with respect to timeif in S′, because evente was measured in S′ already at timei due to relativity of simultaneity, long before C started to tick.
From that it can be seen, that the proper time between two events indicated by an unaccelerated clock present at both events, compared with the synchronized coordinate time measured in all other inertial frames, is always theminimal time interval between those events. However, the interval between two events can also correspond to the proper time of accelerated clocks present at both events. Under all possible proper times between two events, the proper time of the unaccelerated clock ismaximal, which is the solution to thetwin paradox.[27]
Lorentz factor as a function of speed (in natural units wherec = 1). Notice that for small speeds (as v tends to zero), γ is approximately 1.
In addition to the light clock used above, the formula for time dilation can be more generally derived from the temporal part of theLorentz transformation.[28] Let there be two events at which the moving clock indicates and, thus:
Since the clock remains at rest in its inertial frame, it follows, thus the interval is given by:
where Δt is the time interval betweentwo co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on their clock), known as theproper time, Δt′ is the time interval between those same events, as measured by another observer, inertially moving with velocityv with respect to the former observer,v is the relative velocity between the observer and the moving clock,c is the speed of light, and theLorentz factor (conventionally denoted by the Greek lettergamma or γ) is:
Thus the duration of the clock cycle of a moving clock is found to be increased: it is measured to be "running slow". The range of such variances in ordinary life, wherev ≪c, even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored for most purposes. As an approximate threshold, time dilation of 0.5% may become important when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light).[29]
In special relativity, time dilation is most simply described in circumstances where relative velocity is unchanging. Nevertheless, the Lorentz equations allow one to calculateproper time and movement in space for the simple case of a spaceship which is applied with a force per unit mass, relative to some reference object in uniform (i.e. constant velocity) motion, equal tog throughout the period of measurement.
Lett be the time in an inertial frame subsequently called the rest frame. Letx be a spatial coordinate, and let the direction of the constant acceleration as well as the spaceship's velocity (relative to the rest frame) be parallel to thex-axis. Assuming the spaceship's position at timet = 0 beingx = 0 and the velocity beingv0 and defining the following abbreviation:
In the case wherev(0) =v0 = 0 andτ(0) =τ0 = 0 the integral can be expressed as a logarithmic function or, equivalently, as aninverse hyperbolic function:
As functions of the proper time of the ship, the following formulae hold:[31]
Theclock hypothesis is the assumption that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity. This is equivalent to stating that a clock moving along a path measures theproper time, defined by:
The clock hypothesis was implicitly (but not explicitly) included in Einstein's original 1905 formulation of special relativity. Since then, it has become a standard assumption and is usually included in the axioms of special relativity, especially in light of experimental verification up to very high accelerations inparticle accelerators.[32][33]
Time dilation explains why two working clocks will report different times after different accelerations. For example, time goes slower at theISS, lagging approximately 0.01 seconds for every 12 Earth months passed. ForGPS satellites to work, they must adjust for similar bending ofspacetime to coordinate properly with systems on Earth.[2]Time passes more quickly further from a center of gravity, as is witnessed with massive objects (like the Earth).
Gravitational time dilation is experienced by an observer that, at a certain altitude within a gravitational potential well, finds that their local clocks measure less elapsed time than identical clocks situated at higher altitude (and which are therefore at higher gravitational potential).
Gravitational time dilation is at play e.g. for ISS astronauts. While the astronauts'relative velocity slows down their time, the reduced gravitational influence at their location speeds it up, although to a lesser degree. Also, a climber's time is theoretically passing slightly faster at the top of a mountain compared to people at sea level. It has also been calculated that due to time dilation, thecore of the Earth is 2.5 years younger than thecrust.[34] "A clock used to time a full rotation of the Earth will measure the day to be approximately an extra 10 ns/day longer for every km of altitude above the reference geoid."[35] Travel to regions of space where extreme gravitational time dilation is taking place, such as near (but not beyond theevent horizon of) ablack hole, could yield time-shifting results analogous to those of near-lightspeed space travel.
Contrarily to velocity time dilation, in which both observers measure the other as aging slower (a reciprocal effect), gravitational time dilation is not reciprocal. This means that with gravitational time dilation both observers agree that the clock nearer the center of the gravitational field is slower in rate, and they agree on the ratio of the difference.
In 1959,Robert Pound andGlen Rebka measured the very slightgravitational redshift in the frequency of light emitted at a lower height, where Earth's gravitational field is relatively more intense. The results were within 10% of the predictions of general relativity. In 1964, Pound and J. L. Snider measured a result within 1% of the value predicted by gravitational time dilation.[36] (SeePound–Rebka experiment)
In 2010, gravitational time dilation was measured at the Earth's surface with a height difference of only one meter, using optical atomic clocks.[26]
Combined effect of velocity and gravitational time dilation
Daily time dilation (gain or loss if negative) in microseconds as a function of (circular) orbit radiusr =rs/re, wherers is satellite orbit radius andre is the equatorial Earth radius, calculated using the Schwarzschild metric. Atr ≈ 1.497[Note 1] there is no time dilation. Here the effects of motion and reduced gravity cancel. ISS astronauts fly below, whereas GPS and geostationary satellites fly above.[2]
High-accuracy timekeeping, low-Earth-orbit satellite tracking, andpulsar timing are applications that require the consideration of the combined effects of mass and motion in producing time dilation. Practical examples include theInternational Atomic Time standard and its relationship with theBarycentric Coordinate Time standard used for interplanetary objects.
Relativistic time dilation effects for theSolar System and the Earth can be modeled very precisely by theSchwarzschild solution to the Einstein field equations. In the Schwarzschild metric, the interval is given by:[38][39]
where:
is a small increment of proper time (an interval that could be recorded on an atomic clock),
are small increments in the three coordinates of the clock's position,
represents the sum of the Newtonian gravitational potentials due to the masses in the neighborhood, based on their distances from the clock. This sum includes any tidal potentials.
The coordinate time is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses (), and stationary in the system of coordinates (). The exact relation between the rate of proper time and the rate of coordinate time for a clock with a radial component of velocity is:
where:
is the radial velocity,
is the escape speed,
, and are velocities as a percentage of speed of lightc,
is the Newtonian potential; hence equals half the square of the escape speed.
The above equation is exact under the assumptions of the Schwarzschild solution. It reduces to velocity time dilation equation in the presence of motion and absence of gravity, i.e.. It reduces to gravitational time dilation equation in the absence of motion and presence of gravity, i.e..
Daily time dilation over circular orbit height split into its components. On this chart, onlyGravity Probe A was launched specifically totest general relativity. The other spacecraft on this chart (except for the ISS, whose range of points is marked "theory") carry atomic clocks whose proper operationdepend on the validity of general relativity.
Hafele and Keating, in 1971, flewcaesium atomic clocks east and west around the Earth in commercial airliners, to compare the elapsed time against that of a clock that remained at theU.S. Naval Observatory. Two opposite effects came into play. The clocks were expected to age more quickly (show a larger elapsed time) than the reference clock since they were in a higher (weaker) gravitational potential for most of the trip (c.f.Pound–Rebka experiment). But also, contrastingly, the moving clocks were expected to age more slowly because of the speed of their travel. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40±23 nanoseconds during the eastward trip and should have gained 275±21 nanoseconds during the westward trip. Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59±10 nanoseconds during the eastward trip and gained 273±7 nanoseconds during the westward trip (where the error bars represent standard deviation).[40] In 2005, theNational Physical Laboratory in the United Kingdom reported their limited replication of this experiment.[41] The NPL experiment differed from the original in that the caesium clocks were sent on a shorter trip (London–Washington, D.C. return), but the clocks were more accurate. The reported results are within 4% of the predictions of relativity, within the uncertainty of the measurements.
TheGlobal Positioning System can be considered a continuously operating experiment in both special and general relativity. The in-orbit clocks are corrected for both special and general relativistic time dilation effectsas described above, so that (as observed from the Earth's surface) they run at the same rate as clocks on the surface of the Earth.[42]
Velocity and gravitational time dilation have been the subject of science fiction works in a variety of media. Some examples in film are the moviesInterstellar andPlanet of the Apes.[43] InInterstellar, a key plot point involves a planet, which is close to arotating black hole and on the surface of which one hour is equivalent to seven years on Earth due to time dilation.[44] PhysicistKip Thorne collaborated in making the film and explained its scientific concepts in the bookThe Science of Interstellar.[45][46]
TheQueen song'39 was written by astrophysicist as well as musicianBrian May, and is centred around the time dilation effect on spacefarers searching for a new home for mankind, as we gradually ruin planet Earth. They return successful, only to find that all and everything they knew has long since passed away.
Time dilation was used in theDoctor Who episodes "World Enough and Time" and "The Doctor Falls", which take place on a spaceship in the vicinity of a black hole. Due to the immense gravitational pull of the black hole and the ship's length (400 miles), time moves faster at one end than the other. When The Doctor's companion, Bill, gets taken away to the other end of the ship, she waits years for him to rescue her; in his time, only minutes pass.[47] Furthermore, the dilation allows theCybermen to evolve at a "faster" rate than previously seen in the show.
Tau Zero, a novel byPoul Anderson, is an early example of the concept in science fiction literature. In the novel, a spacecraft uses aBussard ramjet to accelerate to high enough speeds that the crew spends five years on board, but thirty-three years pass on the Earth before they arrive at their destination. The velocity time dilation is explained by Anderson in terms of thetau factor which decreases closer and closer to zero as the ship approaches the speed of light—hence the title of the novel.[48] Due to an accident, the crew is unable to stop accelerating the spacecraft, causing such extreme time dilation that the crew experiences theBig Crunch at the end of the universe.[49] Other examples in literature, such asRocannon's World,Hyperion andThe Forever War, similarly make use of relativistic time dilation as a scientifically plausible literary device to have certain characters age slower than the rest of the universe.[50][51]
^Average time dilation has a weak dependence on the orbital inclination angle (Ashby 2003, p.32). Ther ≈ 1.497 result corresponds to[37] the orbital inclination of modern GPS satellites, which is 55 degrees.
^Bailey, H.; Borer, K.; Combley, F.; Drumm, H.; Krienen, F.; Lange, F.; Picasso, E.; von Ruden, W.; Farley F. J. M.; Field J. H.; Flegel W. & Hattersley P. M. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit".Nature.268 (5618):301–305.Bibcode:1977Natur.268..301B.doi:10.1038/268301a0.S2CID4173884.
^Roos, C. E.; Marraffino, J.; Reucroft, S.; Waters, J.; Webster, M. S.; Williams, E. G. H. (1980). "σ+/- lifetimes and longitudinal acceleration".Nature.286 (5770):244–245.Bibcode:1980Natur.286..244R.doi:10.1038/286244a0.S2CID4280317.
^Burns, M. Shane; Leveille, Michael D.; Dominguez, Armand R.; Gebhard, Brian B.; Huestis, Samuel E.; Steele, Jeffrey; Patterson, Brian; Sell, Jerry F.; Serna, Mario; Gearba, M. Alina; Olesen, Robert; O'Shea, Patrick; Schiller, Jonathan (18 September 2017). "Measurement of gravitational time dilation: An undergraduate research project".American Journal of Physics.85 (10):757–762.arXiv:1707.00171.Bibcode:2017AmJPh..85..757B.doi:10.1119/1.5000802.S2CID119503665.
