Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The higher the gravitational potential (the farther the clock is from the source of gravitation), the faster time passes. Albert Einstein originally predicted this effect in his theory of relativity and it has since been confirmed by tests of general relativity.^{[1]}
This has been demonstrated by noting that atomic clocks at differing altitudes (and thus different gravitational potential) will eventually show different times. The effects detected in such Earthbound experiments are extremely small, with differences being measured in nanoseconds. Relative to Earth's age in billions of years, Earth's core is effectively 2.5 years younger than its surface.^{[2]} Demonstrating larger effects would require greater distances from the Earth or a larger gravitational source.
Gravitational time dilation was first described by Albert Einstein in 1907^{[3]} as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of proper time at different positions as described by a metric tensor of spacetime. The existence of gravitational time dilation was first confirmed directly by the Pound–Rebka experiment in 1959.
Clocks that are far from massive bodies (or at higher gravitational potentials) run more quickly, and clocks close to massive bodies (or at lower gravitational potentials) run more slowly. For example, considered over the total timespan of Earth (4.6 billion years), a clock set at the peak of Mount Everest would be about 39 hours ahead of a clock set at sea level.^{[4]}^{[5]} This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects.^{[6]}
According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.^{[7]}
Consider a family of observers along a straight "vertical" line, each of whom experiences a distinct constant gforce directed along this line (e.g., a long accelerating spacecraft^{[8]}^{[9]}, a skyscraper, a shaft on a planet). Let be the dependence of gforce on "height", a coordinate along the aforementioned line. The equation with respect to a base observer at is
where is the total time dilation at a distant position , is the dependence of gforce on "height" , is the speed of light, and denotes exponentiation by e.
For simplicity, in a Rindler's family of observers in a flat spacetime, the dependence would be
with constant , which yields
On the other hand, when is nearly constant and is much smaller than , the linear "weak field" approximation can also be used.
See Ehrenfest paradox for application of the same formula to a rotating reference frame in flat spacetime.
A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes spacetime in the vicinity of a nonrotating massive spherically symmetric object. The equation is
where
To illustrate then, without accounting for the effects of rotation, proximity to Earth's gravitational well will cause a clock on the planet's surface to accumulate around 0.0219 fewer seconds over a period of one year than would a distant observer's clock. In comparison, a clock on the surface of the sun will accumulate around 66.4 fewer seconds in one year.
In the Schwarzschild metric, freefalling objects can be in circular orbits if the orbital radius is larger than (the radius of the photon sphere). The formula for a clock at rest is given above; the formula below gives the gravitational time dilation for a clock in a circular orbit but it does not include the opposing time dilation caused by the clock's motion. (Both dilations are shown in the figure below).
Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. The clocks aboard the airplanes were slightly faster than clocks on the ground. The effect is significant enough that the Global Positioning System's artificial satellites need to have their clocks corrected.^{[10]}
Additionally, time dilations due to height differences of less than one metre have been experimentally verified in the laboratory.^{[11]}
Gravitational time dilation has also been confirmed by the Pound–Rebka experiment, observations of the spectra of the white dwarf Sirius B, and experiments with time signals sent to and from Viking 1 Mars lander.
The BSSN formalism is a formalism of general relativity that was developed by Thomas W. Baumgarte, Stuart L. Shapiro, Masaru Shibata and Takashi Nakamura between 1987 and 1999. It is a modification of the ADM formalism developed during the 1950s.
The ADM formalism is a Hamiltonian formalism that does not permit stable and longterm numerical simulations. In the BSSN formalism, the ADM equations are modified by introducing auxiliary variables. The formalism has been tested for a longterm evolution of linear gravitational waves and used for a variety of purposes such as simulating the nonlinear evolution of gravitational waves or the evolution and collision of black holes.
Barycentric Coordinate TimeBarycentric Coordinate Time (TCB, from the French Tempscoordonnée barycentrique) 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. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame comoving with the barycenter of the Solar system: that is, a clock that performs exactly the same movements as the Solar system but is outside the system's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Sun and the rest of the system.
TCB was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the problematic 1976 definition of Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCB is defined in the context of the general theory of relativity. The relationships between TCB and other relativistic time scales are defined with fully general relativistic metrics.
Because the reference frame for TCB is not influenced by the gravitational potential caused by the Solar system, TCB ticks faster than clocks on the surface of the Earth by 1.550505 × 10−8 (about 490 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCB differ from the traditional values of physical constants (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales). Adapting the large body of existing software to change from TDB to TCB is an ongoing task, and as of 2002 many calculations continue to use TDB in some form.
Time coordinates on the TCB scale are conventionally specified using traditional means of specifying days, carried over from nonuniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCB was set to match ET at around Julian Date 2443144.5 (19770101T00Z). More precisely, it was defined that TCB instant 19770101T00:00:32.184 exactly corresponds to the International Atomic Time (TAI) instant 19770101T00:00:00.000 exactly, at the geocenter. This is also the instant at which TAI introduced corrections for gravitational time dilation.
Coordinate 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 (but not all) coordinate systems, an event is specified by one time coordinate and three spatial coordinates. The time specified by the time coordinate is referred to as coordinate time to distinguish it from proper time.
In the special case of an inertial observer in special relativity, by convention the coordinate time at an event is the same as the proper time measured by a clock that is at the same location as the event, that is stationary relative to the observer and that has been synchronised to the observer's clock using the Einstein synchronisation convention.
General relativityGeneral relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and supersedes Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay. The predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and selfconsistent theory of quantum gravity.
Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an endstate for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes. For example, microquasars and active galactic nuclei result from the presence of stellar black holes and supermassive black holes, respectively. The bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.
Geocentric Coordinate TimeGeocentric Coordinate Time (TCG  Tempscoordonnée géocentrique) is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to precession, nutation, the Moon, and artificial satellites of the Earth. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame comoving with the center of the Earth: that is, a clock that performs exactly the same movements as the Earth but is outside the Earth's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Earth.
TCG was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the illdefined Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCG is defined in the context of the general theory of relativity. The relationships between TCG and other relativistic time scales are defined with fully general relativistic metrics.
Because the reference frame for TCG is not rotating with the surface of the Earth and not in the gravitational potential of the Earth, TCG ticks faster than clocks on the surface of the Earth by a factor of about 7.0 × 10−10 (about 22 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCG differ from the traditional values of physical constants. (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales.) Adapting the large body of existing software to change from TDB to TCG is a formidable task, and as of 2002 many calculations continue to use TDB in some form.
Time coordinates on the TCG scale are conventionally specified using traditional means of specifying days, carried over from nonuniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCG was set to match ET at around Julian Date 2443144.5 (19770101T00Z). More precisely, it was defined that TCG instant 19770101T00:00:32.184 exactly corresponds to TAI instant 19770101T00:00:00.000 exactly. This is also the instant at which TAI introduced corrections for gravitational time dilation.
TCG is a Platonic time scale: a theoretical ideal, not dependent on a particular realisation. For practical purposes, TCG must be realised by actual clocks in the Earth system. Because of the linear relationship between Terrestrial Time (TT) and TCG, the same clocks that realise TT also serve for TCG. See the article on TT for details of the relationship and how TT is realised.
Barycentric Coordinate Time (TCB) is the analog of TCG, used for calculations relating to the solar system beyond Earth orbit. TCG is defined by a different reference frame from TCB, such that they are not linearly related. Over the long term, TCG ticks more slowly than TCB by about 1.6 × 10−8 (about 0.5 seconds per year). In addition there are periodic variations, as Earth moves within the Solar system. When the Earth is at perihelion in January, TCG ticks even more slowly than it does on average, due to gravitational time dilation from being deeper in the Sun's gravity well and also velocity time dilation from moving faster relative to the Sun. At aphelion in July the opposite holds, with TCG ticking faster than it does on average.
Geodetic effectThe geodetic effect (also known as geodetic precession, de Sitter precession or de Sitter effect) represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment. The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker. It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.The term geodetic effect has two slightly different meanings as the moving body may be spinning or nonspinning. Nonspinning bodies move in geodesics, whereas spinning bodies move in slightly different orbits.The difference between de Sitter precession and Lense–Thirring precession (frame dragging) is that the de Sitter effect is due simply to the presence of a central mass, whereas Lense–Thirring precession is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession.
Gravity Probe AGravity Probe A (GPA) was a spacebased experiment to test the equivalence principle, a feature of Einstein's theory of relativity. It was performed jointly by the Smithsonian Astrophysical Observatory and the National Aeronautics and Space Administration. The experiment sent a hydrogen maser, a highly accurate frequency standard, into space to measure with high precision the rate at which time passes in a weaker gravitational field. Masses cause distortions in spacetime, which leads to the effects of length contraction and time dilation, both predicted results of Albert Einstein's theory of general relativity. Because of the bending of spacetime, an observer on Earth (in a lower gravitational potential) should measure a different rate at which time passes than an observer that is sufficiently high up in Earth's atmosphere (at higher gravitational potential). This effect is known as gravitational time dilation.
The experiment was a test of a major fallout of Einstein's general relativity, the equivalence principle. The equivalence principle states that a reference frame in a uniform gravitational field is indistinguishable from a reference frame that is under uniform acceleration. Further, the equivalence principle predicts that phenomenon of different time flow rates, present in a uniformly accelerating reference frame, will also be present in a stationary reference frame that is in a uniform gravitational field.
The probe was launched on June 18, 1976 from the NASAWallops Flight Center in Wallops Island, Virginia. The probe was carried via a Scout rocket, and attained a height of 10,000 km (6,200 mi), while remaining in space for 1 hour and 55 minutes, as intended. It returned to Earth by splashing down into the Atlantic Ocean.
Hafele–Keating 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. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity.
History of general relativityGeneral relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the observed gravitational attraction between masses results from the warping of space and time by those masses.
Before the advent of general relativity, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses, even though Newton himself did not regard the theory as the final word on the nature of gravity. Within a century of Newton's formulation, careful astronomical observation revealed unexplainable variations between the theory and the observations. Under Newton's model, gravity was the result of an attractive force between massive objects. Although even Newton was bothered by the unknown nature of that force, the basic framework was extremely successful at describing motion.
However, experiments and observations show that Einstein's description accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, while others are the subject of ongoing research.
General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where gravitational attraction is so strong that not even light can escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.
Introduction to general relativityGeneral relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915. According to general relativity, the observed gravitational effect between masses results from their warping of spacetime.
By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion.
Experiments and observations show that Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, most recently gravitational waves.
General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where the gravitational effect is strong enough that even light cannot escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.
Although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data. Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of quantum physics to produce a complete and selfconsistent theory of quantum gravity.
Johann Georg von SoldnerJohann Georg von Soldner (16 July 1776 in Feuchtwangen, Ansbach – 13 May 1833 in Bogenhausen, Munich) was a German physicist, mathematician and astronomer, first in Berlin and later in 1808 in Munich.
Quantum clockA quantum clock is a type of atomic clock with laser cooled single ions confined together in an electromagnetic ion trap. Developed in 2010 by National Institute of Standards and Technology physicists, the clock was 37 times more precise than the thenexisting international standard. The quantum logic clock is based on an aluminium spectroscopy ion with a logic atom.
Both the aluminiumbased quantum clock and the mercurybased optical atomic clock track time by the ion vibration at an optical frequency using a UV laser, that is 100,000 times higher than the microwave frequencies used in NISTF1 and other similar time standards around the world. Quantum clocks like this are able to be far more precise than microwave standards.
Reissner–Nordström metricIn physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, nonrotating, spherically symmetric body of mass M. The analogous solution for a charged, rotating body is given by the KerrNewman metric.
The metric was discovered between 1916 and 1921 by Hans Reissner, Hermann Weyl, Gunnar Nordström and George Barker Jeffery.
Schwarzschild radiusThe Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a physical parameter that shows up in the Schwarzschild solution to Einstein's field equations, corresponding to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with every quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.
The Schwarzschild radius is given as
where G is the gravitational constant, M is the object mass, and c is the speed of light.
Tempo (astronomy)Tempo is a computer program to analyze radio observations of pulsars. Once enough observations are available, Tempo can deduce the pulsar rotation rate and phase, astrometric position and rates of change, and parameters of binary systems, by fitting models to pulse times of arrival measured at one or more terrestrial observatories. This is a nontrivial procedure because much larger effects must be removed before the detailed fit can be performed. These include:
Dispersion of the pulses in the Interstellar medium, the solar system, and the ionosphere
Observatory motion (including Earth rotation, precession, nutation, polar motion and orbital motion)
Tropospheric propagation delay
Gravitational time dilation due to binary companions and Solar system bodies.Tempo is maintained and distributed on SourceForge. There is a reference manual available, but no general documentation.
Tempo is a relatively old program, and is being replaced by Tempo2. The main advantages of Tempo2, from the abstract, are:
We have developed tempo2, a new pulsar timing package that contains propagation and other relevant effects implemented at the 1ns level of precision (a factor of ~100 more precise than previously obtainable). In contrast with earlier timing packages, tempo2 is compliant with the general relativistic framework of the IAU 1991 and 2000 resolutions and hence uses the International Celestial Reference System, Barycentric Coordinate Time and uptodate precession, nutation and polar motion models.
Terrestrial TimeTerrestrial Time (TT) is a modern astronomical time standard defined by the International Astronomical Union, primarily for timemeasurements of astronomical observations made from the surface of Earth.
For example, the Astronomical Almanac uses TT for its tables of positions (ephemerides) of the Sun, Moon and planets as seen from Earth. In this role, TT continues Terrestrial Dynamical Time (TDT or TD), which in turn succeeded ephemeris time (ET). TT shares the original purpose for which ET was designed, to be free of the irregularities in the rotation of Earth.
The unit of TT is the SI second, the definition of which is currently based on the caesium atomic clock, but TT is not itself defined by atomic clocks. It is a theoretical ideal, and real clocks can only approximate it.
TT is distinct from the time scale often used as a basis for civil purposes, Coordinated Universal Time (UTC). TT indirectly underlies UTC, via International Atomic Time (TAI). Because of the historical difference between TAI and ET when TT was introduced, TT is approximately 32.184 s ahead of TAI.
Theory of relativityThe theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity. Special relativity applies to elementary particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of gravitation and its relation to other forces of nature. It applies to the cosmological and astrophysical realm, including astronomy.The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200yearold theory of mechanics created primarily by Isaac Newton. It introduced concepts including spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves.
Time dilationAccording to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field. As a result of the nature of spacetime, a clock that is moving relative to an observer will be measured to tick slower than a clock that is at rest in the observer's own frame of reference. A clock that is under the influence of a stronger gravitational field than an observer's will also be measured to tick slower than the observer's own clock.
Such time dilation has been repeatedly demonstrated, for instance by small disparities in a pair of atomic clocks after one of them is sent on a space trip, or by clocks on the Space Shuttle running slightly slower than reference clocks on Earth, or clocks on GPS and Galileo satellites running slightly faster. Time dilation has also been the subject of science fiction works, as it technically provides the means for forward time travel.
Twin paradoxIn physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a highspeed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, according to an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not a paradox in the sense of a logical contradiction.
Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason". Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration. General relativity is not necessary to explain the twin paradox; special relativity alone can explain the phenomenon.Time dilation has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment. It was also confirmed in particle accelerators by measuring the time dilation of circulating particle beams.
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