Time dilation
Encyclopedia
In the theory of relativity
, time dilation is an observed difference of elapsed time between two events
as measured by observers
either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer's own equally accurate clocks. This effect arises not from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the nature of spacetime itself.
). The point of view of the other observer will be that again the local clock (this time the other clock) is correct and it is the distant moving one that is slow. From a local perspective, time registered by clocks that are at rest with respect to the local frame of reference (and far from any gravitational mass) always appears to pass at the same rate.
", appears to go slower than the clock that is more distant from the mass (or higher in altitude away from the center of the gravitational mass). That does not mean that the two observers fully agree: each still makes the local clock to be correct; the observer more distant from the mass (higher in altitude) measures the other clock (closer to the mass, lower in altitude) to be slower than the local correct rate, and the observer situated closer to the mass (lower in altitude) measures the other clock (farther from the mass, higher in altitude) to be faster than the local correct rate. They agree at least that the clock nearer the mass is slower in rate and on the ratio of the difference.
's theories of relativity
, time dilation in these two circumstances can be summarized:
Special and general relativistic effects can combine, for example in some timescale applications mentioned below.
In special relativity, the time dilation effect is reciprocal: as observed from the point of view of either of two clocks which are in motion with respect to each other, it will be the other clock that is time dilated. (This presumes that the relative motion of both parties is uniform; that is, they do not accelerate with respect to one another during the course of the observations.)
In contrast, gravitational time dilation (as treated in general relativity) is not reciprocal: an observer at the top of a tower will observe that clocks at ground level tick slower, and observers on the ground will agree about the direction and the ratio of the difference. There is not full agreement, as all the observers make their own local clocks out to be correct, but the direction and ratio of gravitational time dilation is agreed by all observers, independent of their altitude.
This constancy of the speed of light means, counter to intuition, that speeds of material objects and light are not additive. It is not possible to make the speed of light appear faster by approaching at speed towards the material source that is emitting light. It is not possible to make the speed of light appear slower by receding from the source at speed. From one point of view, it is the implications of this unexpected constancy that take away from constancies expected elsewhere.
Consider a simple clock consisting of two mirrors A and B, between which a light pulse is bouncing. The separation of the mirrors is L and the clock ticks once each time it hits a given mirror.
In the frame where the clock is at rest (diagram at right), the light pulse traces out a path of length 2L and the period of the clock is 2L divided by the speed of light:
From the frame of reference of a moving observer traveling at the speed v (diagram at lower right), the light pulse traces out a longer, angled path. The second postulate of special relativity states that the speed of light is constant in all frames, which implies a lengthening of the period of this clock from the moving observer's perspective. That is to say, in a frame moving relative to the clock, the clock appears to be running more slowly. Straightforward application of the Pythagorean theorem
leads to the wellknown 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
Substituting D from this equation into the previous and solving for gives:
and thus, with the definition of :
which expresses the fact that for the moving observer the period of the clock is longer than in the frame of the clock itself.
A similar oddity occurs in everyday life. If Sam sees Abigail at a distance she appears small to him and at the same time Sam appears small to Abigail. Being very familiar with the effects of perspective
, we see no mystery or a hint of a paradox in this situation.
One is accustomed to the notion of relativity with respect to distance: the distance from Los Angeles to New York is by convention the same as the distance from New York to Los Angeles. On the other hand, when speeds are considered, one thinks of an object as "actually" moving, overlooking that its motion is always relative to something else — to the stars, the ground or to oneself. If one object is moving with respect to another, the latter is moving with respect to the former and with equal relative speed.
In the special theory of relativity, a moving clock is found to be ticking slowly with respect to the observer's clock. If Sam and Abigail are on different trains in nearlightspeed relative motion, Sam measures (by all methods of measurement) clocks on Abigail's train to be running slowly and similarly, Abigail measures clocks on Sam's train to be running slowly.
Note that in all such attempts to establish "synchronization" within the reference system, the question of whether something happening at one location is in fact happening simultaneously with something happening elsewhere, is of key importance. Calculations are ultimately based on determining which events
are simultaneous. Furthermore, establishing simultaneity of events separated in space necessarily requires transmission of information between locations, which by itself is an indication that the speed of light will enter the determination of simultaneity.
It is a natural and legitimate question to ask how, in detail, special relativity can be selfconsistent if clock A is timedilated with respect to clock B and clock B is also timedilated with respect to clock A. It is by challenging the assumptions built into the common notion of simultaneity that logical consistency can be restored. Simultaneity is a relationship between an observer in a particular frame of reference and a set of events. By analogy, left and right are accepted to vary with the position of the observer, because they apply to a relationship. In a similar vein, Plato
explained that up and down describe a relationship to the earth and one would not fall off at the antipodes
.
Within the framework of the theory and its terminology there is a relativity of simultaneity
that affects how the specified events are aligned with respect to each other by observers in relative motion. Because the pairs of putatively simultaneous moments are identified differently by different observers (as illustrated in the twin paradox
article), each can treat the other clock as being the slow one without relativity being selfcontradictory. This can be explained in many ways, some of which follow.
). It is now usually called the Einstein synchronization procedure, since it appeared in his 1905 paper.
An observer with a clock sends a light signal out at time t_{1} according to his clock. At a distant event, that light signal is reflected back, and arrives back at the observer at time t_{2} according to his clock. Since the light travels the same path at the same rate going both out and back for the observer in this scenario, the coordinate time of the event of the light signal being reflected for the observer t_{E} is t_{E} = (t_{1} + t_{2}) / 2. In this way, a single observer's clock can be used to define temporal coordinates which are good anywhere in the universe.
Symmetric time dilation occurs with respect to temporal coordinate systems set up in this manner. It is an effect where another clock is being viewed as running slowly by an observer. Observers do not consider their own clock time to be timedilated, but may find that it is observed to be timedilated in another coordinate system.
where Δt is the time interval between two colocal events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock) – this is known as the proper time
, Δt' is the time interval between those same events, as measured by another observer, inertially moving with velocity v 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
is the Lorentz factor
. 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, where even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored. It is only when an object approaches speeds on the order of 30,000 km/s
(1/10 the speed of light
) that time dilation becomes important.
Time dilation by the Lorentz factor was predicted by Joseph Larmor
(1897), at least for electrons orbiting a nucleus. Thus "... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio :" (Larmor 1897). Time dilation of magnitude corresponding to this (Lorentz) factor has been experimentally confirmed, as described below.
standard and its relationship with the Barycentric Coordinate Time
standard used for interplanetary objects.
Relativistic time dilation effects for the solar system and the Earth can be modeled very precisely by the Schwarzschild solution to the Einstein field equations. In the Schwarzschild metric, the interval dt_{E} is given by:
where:
The coordinate velocity of the clock is
The coordinate time is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses (U=0), and stationary in the system of coordinates (v=0). 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, and
The above equation is exact under the assumptions of the Schwarzschild solution.
, is a continuously running test of the time dilation of special relativity. The specific experiments include:
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 2.22 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. In this experiment 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.
acceleration would permit humans to travel through the entire known Universe in one human lifetime. The space travellers could return to Earth billions of years in the future. A scenario based on this idea was presented in the novel Planet of the Apes by Pierre Boulle
.
A more likely use of this effect would be to enable humans to travel to nearby stars without spending their entire lives aboard the ship. However, any such application of time dilation during interstellar travel
would require the use of some new, advanced method of propulsion
. The Orion Project
has been the only major attempt toward this idea.
Current space flight technology has fundamental theoretical limits based on the practical problem that an increasing amount of energy is required for propulsion as a craft approaches the speed of light
. The likelihood of collision with small space debris
and other particulate material is another practical limitation. At the velocities presently attained, however, time dilation is not a factor in space travel. Travel to regions of spacetime where gravitational time dilation is taking place, such as within the gravitational field of a black hole but outside the event horizon
(perhaps on a hyperbolic trajectory exiting the field), could also yield results consistent with present theory.
and movement in space for the simple case of a spaceship whose acceleration
, relative to some referent object in uniform (i.e. constant velocity) motion, equals g throughout the period of measurement.
Let t be the time in an inertial frame subsequently called the rest frame. Let x 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 the xaxis. Assuming the spaceship's position at time t = 0 being x = 0 and the velocity being v_{0} and defining the following abbreviation
the following formulas hold:
Position:
Velocity:
Proper time:
In the case where v(0) = v_{0} = 0 and τ(0) = τ_{0} = 0 the integral can be expressed as a logarithmic function or, equivalently, as an inverse hyperbolic function:
The blue dots represent pulses of light. One cycle of lightpulses between two green ships takes two seconds of "green time", one second for each leg.
As seen from the perspective of the reds, the transit time of the light pulses they exchange among each other is one second of "red time" for each leg. As seen from the perspective of the greens, the red ships' cycle of exchanging light pulses travels a diagonal path that is two lightseconds long. (As seen from the green perspective the reds travel 1.73 () lightseconds of distance for every two seconds of green time.)
One of the red ships emits a light pulse towards the greens every second of red time. These pulses are received by ships of the green fleet with twosecond intervals as measured in green time. Not shown in the animation is that all aspects of physics are proportionally involved. The light pulses that are emitted by the reds at a particular frequency as measured in red time are received at a lower frequency as measured by the detectors of the green fleet that measure against green time, and vice versa.
The animation cycles between the green perspective and the red perspective, to emphasize the symmetry. As there is no such thing as absolute motion in relativity (as is also the case for Newtonian mechanics), both the green and the red fleet are entitled to consider themselves motionless in their own frame of reference.
Again, it is vital to understand that the results of these interactions and calculations reflect the real state of the ships as it emerges from their situation of relative motion. It is not a mere quirk of the method of measurement or communication.
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
, time dilation is an observed difference of elapsed time between two events
Event (relativity)
In physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference.Strictly speaking, the...
as measured by observers
Observer (special relativity)
The term observer in special relativity refers most commonly to an inertial reference frame. Less often it may refer to an arbitrary noninertial reference frame; in particular, a Rindler frame is sometimes called an "accelerating observer". In such cases an inertial reference frame may be called...
either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer's own equally accurate clocks. This effect arises not from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the nature of spacetime itself.
Overview
Time dilation can arise from: the relative velocity of motion between two observers, or
 the difference in their distance from a gravitational mass.
Relative velocity time dilation
When two observers are in relative uniform motion and uninfluenced by any gravitational mass, the point of view of each will be that the other's (moving) clock is ticking at a slower rate than the local clock. The faster the relative velocity, the greater the magnitude of time dilation. This case is sometimes called special relativistic time dilation. It is often interpreted as time "slowing down" for the other (moving) clock. But that is only true from the physical point of view of the local observer, and of others at relative rest (i.e. in the local observer's frame of referenceFrame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
). The point of view of the other observer will be that again the local clock (this time the other clock) is correct and it is the distant moving one that is slow. From a local perspective, time registered by clocks that are at rest with respect to the local frame of reference (and far from any gravitational mass) always appears to pass at the same rate.
Gravitational time dilation
There is another case of time dilation, where both observers are differently situated in their distance from a significant gravitational mass, such as (for terrestrial observers) the Earth or the Sun. One may suppose for simplicity that the observers are at relative rest (which is not the case of two observers both rotating with the Earth—an extra factor described below). In the simplified case, the general theory of relativity describes how, for both observers, the clock that is closer to the gravitational mass, i.e. deeper in its "gravity wellGravity well
A gravity well or gravitational well is a conceptual model of the gravitational field surrounding a body in space. The more massive the body the deeper and more extensive the gravity well associated with it. The Sun has a farreaching and deep gravity well. Asteroids and small moons have much...
", appears to go slower than the clock that is more distant from the mass (or higher in altitude away from the center of the gravitational mass). That does not mean that the two observers fully agree: each still makes the local clock to be correct; the observer more distant from the mass (higher in altitude) measures the other clock (closer to the mass, lower in altitude) to be slower than the local correct rate, and the observer situated closer to the mass (lower in altitude) measures the other clock (farther from the mass, higher in altitude) to be faster than the local correct rate. They agree at least that the clock nearer the mass is slower in rate and on the ratio of the difference.
Time dilation: special vs. general theories of relativity
In Albert EinsteinAlbert Einstein
Albert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theories of relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
, time dilation in these two circumstances can be summarized:
 In special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
(or, hypothetically far from all gravitational mass), clocks that are moving with respect to an inertial system of observation are measured to be running slower. This effect is described precisely by the Lorentz transformationLorentz transformationIn physics, the Lorentz transformation or LorentzFitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
.
 In general relativityGeneral relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, clocks at lower potentials in a gravitational fieldGravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
—such as in closer proximity to a planet—are found to be running slower. The articles on gravitational time dilationGravitational time dilationGravitational time dilation is the effect of time passing at different rates in regions of different gravitational potential; the lower the gravitational potential, the more slowly time passes...
and gravitational red shift give a more detailed discussion.
Special and general relativistic effects can combine, for example in some timescale applications mentioned below.
In special relativity, the time dilation effect is reciprocal: as observed from the point of view of either of two clocks which are in motion with respect to each other, it will be the other clock that is time dilated. (This presumes that the relative motion of both parties is uniform; that is, they do not accelerate with respect to one another during the course of the observations.)
In contrast, gravitational time dilation (as treated in general relativity) is not reciprocal: an observer at the top of a tower will observe that clocks at ground level tick slower, and observers on the ground will agree about the direction and the ratio of the difference. There is not full agreement, as all the observers make their own local clocks out to be correct, but the direction and ratio of gravitational time dilation is agreed by all observers, independent of their altitude.
Simple inference of time dilation due to relative velocity
Time dilation can be inferred from the observed fact of the constancy of the speed of light in all reference frames.This constancy of the speed of light means, counter to intuition, that speeds of material objects and light are not additive. It is not possible to make the speed of light appear faster by approaching at speed towards the material source that is emitting light. It is not possible to make the speed of light appear slower by receding from the source at speed. From one point of view, it is the implications of this unexpected constancy that take away from constancies expected elsewhere.
Consider a simple clock consisting of two mirrors A and B, between which a light pulse is bouncing. The separation of the mirrors is L and the clock ticks once each time it hits a given mirror.
In the frame where the clock is at rest (diagram at right), the light pulse traces out a path of length 2L and the period of the clock is 2L divided by the speed of light:
From the frame of reference of a moving observer traveling at the speed v (diagram at lower right), the light pulse traces out a longer, angled path. The second postulate of special relativity states that the speed of light is constant in all frames, which implies a lengthening of the period of this clock from the moving observer's perspective. That is to say, in a frame moving relative to the clock, the clock appears to be running more slowly. Straightforward application of the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
leads to the wellknown 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
Substituting D from this equation into the previous and solving for gives:
and thus, with the definition of :
which expresses the fact that for the moving observer the period of the clock is longer than in the frame of the clock itself.
Time dilation due to relative velocity symmetric between observers
Common sense would dictate that if time passage has slowed for a moving object, the moving object would observe the external world to be correspondingly "sped up". Counterintuitively, special relativity predicts the opposite.A similar oddity occurs in everyday life. If Sam sees Abigail at a distance she appears small to him and at the same time Sam appears small to Abigail. Being very familiar with the effects of perspective
Perspective (visual)
Perspective, in context of vision and visual perception, is the way in which objects appear to the eye based on their spatial attributes; or their dimensions and the position of the eye relative to the objects...
, we see no mystery or a hint of a paradox in this situation.
One is accustomed to the notion of relativity with respect to distance: the distance from Los Angeles to New York is by convention the same as the distance from New York to Los Angeles. On the other hand, when speeds are considered, one thinks of an object as "actually" moving, overlooking that its motion is always relative to something else — to the stars, the ground or to oneself. If one object is moving with respect to another, the latter is moving with respect to the former and with equal relative speed.
In the special theory of relativity, a moving clock is found to be ticking slowly with respect to the observer's clock. If Sam and Abigail are on different trains in nearlightspeed relative motion, Sam measures (by all methods of measurement) clocks on Abigail's train to be running slowly and similarly, Abigail measures clocks on Sam's train to be running slowly.
Note that in all such attempts to establish "synchronization" within the reference system, the question of whether something happening at one location is in fact happening simultaneously with something happening elsewhere, is of key importance. Calculations are ultimately based on determining which events
Event (relativity)
In physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference.Strictly speaking, the...
are simultaneous. Furthermore, establishing simultaneity of events separated in space necessarily requires transmission of information between locations, which by itself is an indication that the speed of light will enter the determination of simultaneity.
It is a natural and legitimate question to ask how, in detail, special relativity can be selfconsistent if clock A is timedilated with respect to clock B and clock B is also timedilated with respect to clock A. It is by challenging the assumptions built into the common notion of simultaneity that logical consistency can be restored. Simultaneity is a relationship between an observer in a particular frame of reference and a set of events. By analogy, left and right are accepted to vary with the position of the observer, because they apply to a relationship. In a similar vein, Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
explained that up and down describe a relationship to the earth and one would not fall off at the antipodes
Antipodes
In geography, the antipodes of any place on Earth is the point on the Earth's surface which is diametrically opposite to it. Two points that are antipodal to one another are connected by a straight line running through the centre of the Earth....
.
Within the framework of the theory and its terminology there is a relativity of simultaneity
Relativity of simultaneity
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...
that affects how the specified events are aligned with respect to each other by observers in relative motion. Because the pairs of putatively simultaneous moments are identified differently by different observers (as illustrated in the twin paradox
Twin paradox
In physics, the twin paradox is a thought experiment in special relativity, in which a twin makes a journey into space in a highspeed rocket and returns home to find he has aged less than his identical twin who stayed on Earth...
article), each can treat the other clock as being the slow one without relativity being selfcontradictory. This can be explained in many ways, some of which follow.
Temporal coordinate systems and clock synchronization
In Relativity, temporal coordinate systems are set up using a procedure for synchronizing clocks, discussed by Poincaré (1900) in relation to Lorentz's local time (see relativity of simultaneityRelativity of simultaneity
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...
). It is now usually called the Einstein synchronization procedure, since it appeared in his 1905 paper.
An observer with a clock sends a light signal out at time t_{1} according to his clock. At a distant event, that light signal is reflected back, and arrives back at the observer at time t_{2} according to his clock. Since the light travels the same path at the same rate going both out and back for the observer in this scenario, the coordinate time of the event of the light signal being reflected for the observer t_{E} is t_{E} = (t_{1} + t_{2}) / 2. In this way, a single observer's clock can be used to define temporal coordinates which are good anywhere in the universe.
Symmetric time dilation occurs with respect to temporal coordinate systems set up in this manner. It is an effect where another clock is being viewed as running slowly by an observer. Observers do not consider their own clock time to be timedilated, but may find that it is observed to be timedilated in another coordinate system.
Time dilation due to relative velocity
The formula for determining time dilation in special relativity is:where Δt is the time interval between two colocal events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock) – this is known as the proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
, Δt
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, and
is the Lorentz factor
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
. 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, where even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored. It is only when an object approaches speeds on the order of 30,000 km/s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
(1/10 the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
) that time dilation becomes important.
Time dilation by the Lorentz factor was predicted by Joseph Larmor
Joseph Larmor
Sir Joseph Larmor , a physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter...
(1897), at least for electrons orbiting a nucleus. Thus "... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio :" (Larmor 1897). Time dilation of magnitude corresponding to this (Lorentz) factor has been experimentally confirmed, as described below.
Time dilation due to gravitation and motion together
High accuracy time keeping, low earth orbit satellite tracking, and pulsar timing are applications that require the consideration of the combined effects of mass and motion in producing time dilation. Practical examples include the International Atomic TimeInternational Atomic Time
International Atomic Time is a highprecision atomic coordinate time standard based on the notional passage of proper time on Earth's geoid...
standard and its relationship with the Barycentric Coordinate Time
Barycentric Coordinate Time
Barycentric Coordinate Time is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to orbits of planets, asteroids, comets, and interplanetary spacecraft in the Solar system...
standard used for interplanetary objects.
Relativistic time dilation effects for the solar system and the Earth can be modeled very precisely by the Schwarzschild solution to the Einstein field equations. In the Schwarzschild metric, the interval dt_{E} is given by:
where:
 dt_{E} is a small increment of proper time t_{E} (an interval that could be recorded on an atomic clock);
 dt_{c} is a small increment in the coordinate t_{c} (coordinate timeCoordinate timeIn the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many coordinate systems, an event is specified by one time coordinate and three spatial coordinates...
);  dx, dy and dz are small increments in the three coordinates x, y, z of the clock's position; and
 GM_{i}/r_{i} represents the sum of the Newtonian gravitational potentials due to the masses in the neighborhood, based on their distances r_{i} from the clock. This sum GM_{i}/r_{i} includes any tidal potentials, and is represented as U (using the positive astronomical sign convention for gravitational potentials).
The coordinate velocity of the clock is
The coordinate time is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses (U=0), and stationary in the system of coordinates (v=0). 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, and
 U = GM_{i}/r_{i} is the Newtonian potential, equivalent to half of the escape velocity squared.
The above equation is exact under the assumptions of the Schwarzschild solution.
Experimental confirmation
Time dilation has been tested a number of times. The routine work carried on in particle accelerators since the 1950s, such as those at CERNCERN
The European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...
, is a continuously running test of the time dilation of special relativity. The specific experiments include:
Velocity time dilation tests
 Ives and Stilwell (1938, 1941). The stated purpose of these experiments was to verify the time dilation effect, predicted by LamorLorentz ether theory, due to motion through the ether using Einstein's suggestion that Doppler effect in canal rays would provide a suitable experiment. These experiments measured the Doppler shift of the radiation emitted from cathode rayCathode rayCathode rays are streams of electrons observed in vacuum tubes. If an evacuated glass tube is equipped with two electrodes and a voltage is applied, the glass opposite of the negative electrode is observed to glow, due to electrons emitted from and travelling perpendicular to the cathode Cathode...
s, when viewed from directly in front and from directly behind. The high and low frequencies detected were not the classical values predicted.

 and and
 i.e. for sources with invariant frequencies The high and low frequencies of the radiation from the moving sources were measured as
 and
 as deduced by Einstein (1905) from the Lorentz transformation, when the source is running slow by the Lorentz factor.
 Rossi and Hall (1941) compared the population of cosmicrayproduced muonMuonThe muon mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...
s at the top of a mountain to that observed at sea level. Although the travel time for the muons from the top of the mountain to the base is several muon halflives, the muon sample at the base was only moderately reduced. This is explained by the time dilation attributed to their high speed relative to the experimenters. That is to say, the muons were decaying about 10 times slower than if they were at rest with respect to the experimenters.
 Hasselkamp, Mondry, and Scharmann (1979) measured the Doppler shift from a source moving at right angles to the line of sight (the transverse Doppler shift). The most general relationship between frequencies of the radiation from the moving sources is given by:

 as deduced by Einstein (1905)http://www.fourmilab.ch/etexts/einstein/specrel/www/. For () this reduces to . Thus there is no transverse Doppler shift, and the lower frequency of the moving source can be attributed to the time dilation effect alone.
 In 2010 time dilation was observed at speeds of less than 10 meters per second using optical atomic clocks connected by 75 meters of optical fiber.
Gravitational time dilation tests
 In 1959 Robert PoundRobert PoundRobert Vivian Pound was an American physicist who helped discover nuclear magnetic resonance and who devised the famous PoundRebka experiment supporting general relativity .Pound was born in Ridgeway, Ontario....
and Glen A. Rebka measured the very slight gravitational red shift 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. Later Pound and Snider (in 1964) derived an even closer result of 1%. This effect is as predicted by gravitational time dilation. (See Pound–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.
Velocity and gravitational time dilation combinedeffect tests
 Hafele and KeatingHafeleKeating experimentThe Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesiumbeam atomic clocks aboard commercial airliners and flew twice around the world, first eastward, then westward, and compared...
, in 1971, flew caesiumCaesiumCaesium or cesium is the chemical element with the symbol Cs and atomic number 55. It is a soft, silverygold alkali metal with a melting point of 28 °C , which makes it one of only five elemental metals that are liquid at room temperature...
atomic clocks east and west around the Earth in commercial airliners, to compare the elapsed time against that of a clock that remained at the US 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). 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). In 2005, the National Physical LaboratoryNational Physical Laboratory, UKThe National Physical Laboratory is the national measurement standards laboratory for the United Kingdom, based at Bushy Park in Teddington, London, England. It is the largest applied physics organisation in the UK.Description:...
in the United KingdomUnited KingdomThe United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
reported their limited replication of this experiment. 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.
 The Global Positioning SystemGlobal Positioning SystemThe Global Positioning System is a spacebased global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...
can be considered a continuously operating experiment in both special and general relativity. The inorbit clocks are corrected for both special and general relativistic time dilation effects as 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.
Muon lifetime
A comparison of muonMuon
The muon mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...
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 2.22 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. In this experiment 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.
Time dilation and space flight
Time dilation would make it possible for passengers in a fastmoving vehicle to travel further into the future while aging very little, in that their great speed slows down the rate of passage of onboard time. That is, the ship's clock (and according to relativity, any human travelling with it) shows less elapsed time than the clocks of observers on Earth. For sufficiently high speeds the effect is dramatic. For example, one year of travel might correspond to ten years at home. Indeed, a constant 1 gStandard gravity
Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...
acceleration would permit humans to travel through the entire known Universe in one human lifetime. The space travellers could return to Earth billions of years in the future. A scenario based on this idea was presented in the novel Planet of the Apes by Pierre Boulle
Pierre Boulle
Pierre Boulle was a French novelist largely known for two famous works, The Bridge over the River Kwai and Planet of the Apes .Biography:...
.
A more likely use of this effect would be to enable humans to travel to nearby stars without spending their entire lives aboard the ship. However, any such application of time dilation during interstellar travel
Interstellar travel
Interstellar space travel is manned or unmanned travel between stars. The concept of interstellar travel in starships is a staple of science fiction. Interstellar travel is much more difficult than interplanetary travel. Intergalactic travel, or travel between different galaxies, is even more...
would require the use of some new, advanced method of propulsion
Spacecraft propulsion
Spacecraft propulsion is any method used to accelerate spacecraft and artificial satellites. There are many different methods. Each method has drawbacks and advantages, and spacecraft propulsion is an active area of research. However, most spacecraft today are propelled by forcing a gas from the...
. The Orion Project
Project Orion (nuclear propulsion)
Project Orion was a study of a spacecraft intended to be directly propelled by a series of explosions of atomic bombs behind the craft...
has been the only major attempt toward this idea.
Current space flight technology has fundamental theoretical limits based on the practical problem that an increasing amount of energy is required for propulsion as a craft approaches the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. The likelihood of collision with small space debris
Space debris
Space debris, also known as orbital debris, space junk, and space waste, is the collection of objects in orbit around Earth that were created by humans but no longer serve any useful purpose. These objects consist of everything from spent rocket stages and defunct satellites to erosion, explosion...
and other particulate material is another practical limitation. At the velocities presently attained, however, time dilation is not a factor in space travel. Travel to regions of spacetime where gravitational time dilation is taking place, such as within the gravitational field of a black hole but outside the event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
(perhaps on a hyperbolic trajectory exiting the field), could also yield results consistent with present theory.
Time dilation at constant acceleration
In special relativity, time dilation is most simply described in circumstances where relative velocity is unchanging. Nevertheless, the Lorentz equations allow one to calculate proper timeProper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
and movement in space for the simple case of a spaceship whose acceleration
Proper acceleration
In relativity theory, proper acceleration is the physical acceleration experienced by an object. It is acceleration relative to a freefall, or inertial, observer who is momentarily at rest relative to the object being measured...
, relative to some referent object in uniform (i.e. constant velocity) motion, equals g throughout the period of measurement.
Let t be the time in an inertial frame subsequently called the rest frame. Let x 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 the xaxis. Assuming the spaceship's position at time t = 0 being x = 0 and the velocity being v_{0} and defining the following abbreviation
the following formulas hold:
Position:
Velocity:
Proper time:
In the case where v(0) = v_{0} = 0 and τ(0) = τ_{0} = 0 the integral can be expressed as a logarithmic function or, equivalently, as an inverse hyperbolic function:
Spacetime geometry of velocity time dilation
The green dots and red dots in the animation represent spaceships. The ships of the green fleet have no velocity relative to each other, so for the clocks onboard the individual ships the same amount of time elapses relative to each other, and they can set up a procedure to maintain a synchronized standard fleet time. The ships of the "red fleet" are moving with a velocity of 0.866 of the speed of light with respect to the green fleet.The blue dots represent pulses of light. One cycle of lightpulses between two green ships takes two seconds of "green time", one second for each leg.
As seen from the perspective of the reds, the transit time of the light pulses they exchange among each other is one second of "red time" for each leg. As seen from the perspective of the greens, the red ships' cycle of exchanging light pulses travels a diagonal path that is two lightseconds long. (As seen from the green perspective the reds travel 1.73 () lightseconds of distance for every two seconds of green time.)
One of the red ships emits a light pulse towards the greens every second of red time. These pulses are received by ships of the green fleet with twosecond intervals as measured in green time. Not shown in the animation is that all aspects of physics are proportionally involved. The light pulses that are emitted by the reds at a particular frequency as measured in red time are received at a lower frequency as measured by the detectors of the green fleet that measure against green time, and vice versa.
The animation cycles between the green perspective and the red perspective, to emphasize the symmetry. As there is no such thing as absolute motion in relativity (as is also the case for Newtonian mechanics), both the green and the red fleet are entitled to consider themselves motionless in their own frame of reference.
Again, it is vital to understand that the results of these interactions and calculations reflect the real state of the ships as it emerges from their situation of relative motion. It is not a mere quirk of the method of measurement or communication.