HOME

TheInfoList



OR:

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s travelling out of a gravitational well (seem to) lose
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
. This loss of energy corresponds to a decrease in the wave
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
and increase in the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
, known more generally as a '' redshift''. The opposite effect, in which photons (seem to) gain energy when travelling into a gravitational well, is known as a gravitational blueshift (a type of ''
blueshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
''). The effect was first described by Einstein in 1907, eight years before his publication of the full theory of relativity. Gravitational redshift can be interpreted as a consequence of the equivalence principle (that gravity and acceleration are equivalent and the redshift is caused by the Doppler effect) or as a consequence of the mass-energy equivalence and conservation of energy ('falling' photons gain energy), though there are numerous subtleties that complicate a rigorous derivation. A gravitational redshift can also equivalently be interpreted as
gravitational time dilation 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 lower the gravitational potential (the closer ...
at the source of the radiation: if two oscillators (attached to
transmitter In electronics and telecommunications, a radio transmitter or just transmitter is an electronic device which produces radio waves with an antenna. The transmitter itself generates a radio frequency alternating current, which is applied to the ...
s producing electromagnetic radiation) are operating at different
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
s, the oscillator at the higher gravitational potential (farther from the attracting body) will seem to ‘tick’ faster; that is, when observed from the same location, it will have a higher measured frequency than the oscillator at the lower gravitational potential (closer to the attracting body). To first approximation, gravitational redshift is proportional to the difference in
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
divided by the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
squared, z = \Delta U / c^2, thus resulting in a very small effect. Light escaping from the surface of the sun was predicted by Einstein in 1911 to be redshifted by roughly 2 ppm or 2 × 10−6. Navigational signals from
GPS satellites GPS satellite blocks are the various production generations of the Global Positioning System (GPS) used for satellite navigation. The first satellite in the system, Navstar 1, was launched on 22 February 1978. The GPS satellite constellatio ...
orbiting at 20,000 km altitude are perceived blueshifted by approximately 0.5 ppb or 5 × 10−10, corresponding to a (negligible) increase of less than 1 Hz in the frequency of a 1.5 GHz GPS radio signal (however, the accompanying
gravitational time dilation 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 lower the gravitational potential (the closer ...
affecting the atomic clock in the satellite ''is'' crucially important for accurate navigation). On the surface of the Earth the gravitational potential is proportional to height, \Delta U = g \Delta h, and the corresponding redshift is roughly 10−16 (0.1 part per
quadrillion Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-E ...
) per meter of change in
elevation The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Ver ...
and/or
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
. In
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, the magnitude of a gravitational redshift is often expressed as the velocity that would create an equivalent shift through the
relativistic Doppler effect The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special the ...
. In such units, the 2 ppm sunlight redshift corresponds to a 633 m/s receding velocity, roughly of the same magnitude as convective motions in the sun, thus complicating the measurement. The GPS satellite gravitational blueshift velocity equivalent is less than 0.2 m/s, which is negligible compared to the actual Doppler shift resulting from its orbital velocity. In astronomical objects with strong gravitational fields the redshift can be much greater; for example, light from the surface of a
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
is gravitationally redshifted on average by around 50 km/s/c (around 170 ppm). Observing the gravitational redshift in the solar system is one of the
classical tests of general relativity Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the bendin ...
. Measuring the gravitational redshift to high precision with
atomic clock An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwe ...
s can serve as a test of
Lorentz symmetry In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
and guide searches for
dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
.


Prediction by the equivalence principle and general relativity


Uniform gravitational field or acceleration

Einstein's theory of general relativity incorporates the equivalence principle, which can be stated in various different ways. One such statement is that gravitational effects are locally undetectable for a free-falling observer. Therefore, in a laboratory experiment at the surface of the earth, all gravitational effects should be equivalent to the effects that would have been observed if the laboratory had been accelerating through outer space at ''g''. One consequence is a gravitational Doppler effect. If a light pulse is emitted at the floor of the laboratory, then a free-falling observer says that by the time it reaches the ceiling, the ceiling has accelerated away from it, and therefore when observed by a detector fixed to the ceiling, it will be observed to have been Doppler shifted toward the red end of the spectrum. This shift, which the free-falling observer considers to be a kinematical Doppler shift, is thought of by the laboratory observer as a gravitational redshift. Such an effect was verified in the 1959
Pound–Rebka experiment The Pound–Rebka experiment was an experiment in which gamma rays were emitted from the top of a tower and measured by a receiver at the bottom of the tower. The purpose of the experiment was to test Albert Einstein's theory of general relativit ...
. In a case such as this, where the gravitational field is uniform, the change in wavelength is given by : z = \frac\approx \frac, where \Delta y is the change in height. Since this prediction arises directly from the equivalence principle, it does not require any of the mathematical apparatus of general relativity, and its verification does not specifically support general relativity over any other theory that incorporates the equivalence principle. On Earth's surface (or in a spaceship accelerating at 1g), the gravitational redshift is approximately 1.1 × 10−16, the equivalent of a 3.3 × 10−8 m/s Doppler shift, for every meter of height differential.


Spherically symmetric gravitational field

When the field is not uniform, the simplest and most useful case to consider is that of a spherically symmetric field. By Birkhoff's theorem, such a field is described in general relativity by the Schwarzschild metric, d\tau^2 = \left(1 - r_\text/R\right)dt^2 + \ldots, where d\tau is the clock time of an observer at distance ''R'' from the center, dt is the time measured by an observer at infinity, r_\text is the Schwarzschild radius 2GM/c^2, "..." represents terms that vanish if the observer is at rest, G is Newton's gravitational constant, M the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
of the gravitating body, and c the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. The result is that frequencies and wavelengths are shifted according to the ratio : 1 + z = \frac = \left(1 - \frac\right)^ where * \lambda_\infty\,is the wavelength of the light as measured by the observer at infinity, * \lambda_\text\, is the wavelength measured at the source of emission, and * R_\text is the radius at which the photon is emitted. This can be related to the redshift parameter conventionally defined as z = \lambda_\infty/\lambda_\text - 1. In the case where neither the emitter nor the observer is at infinity, the transitivity of Doppler shifts allows us to generalize the result to \lambda_1/\lambda_2 = \left left(1 - r_\text/R_1\right)/\left(1 - r_\text/R_2\right)\right. The redshift formula for the frequency \nu = c/\lambda is \nu_o/\nu_\text = \lambda_\text/\lambda_o. When R_1 - R_2 is small, these results are consistent with the equation given above based on the equivalence principle. The redshift ratio may also be expressed in terms of a (Newtonian) escape velocity v_\text at R_\text = 2GM/v_\text^2, resulting in the corresponding
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
: :1 + z = \gamma_\text = \frac . For an object compact enough to have an
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
, the redshift is not defined for photons emitted inside the Schwarzschild radius, both because signals cannot escape from inside the horizon and because an object such as the emitter cannot be stationary inside the horizon, as was assumed above. Therefore, this formula only applies when R_\text is larger than r_\text. When the photon is emitted at a distance equal to the Schwarzschild radius, the redshift will be ''infinitely'' large, and it will not escape to ''any'' finite distance from the Schwarzschild sphere. When the photon is emitted at an infinitely large distance, there is no redshift.


Newtonian limit

In the Newtonian limit, i.e. when R_\text is sufficiently large compared to the Schwarzschild radius r_\text, the redshift can be approximated as : z = \frac \approx \frac\frac = \frac = \frac where g is the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
at R_\text. For Earth's surface with respect to infinity, ''z'' is approximately 7 × 10−10 (the equivalent of a 0.2 m/s radial Doppler shift); for the Moon it is approximately 3 × 10−11 (about 1 cm/s). The value for the surface of the sun is about 2 × 10−6, corresponding to 0.64 km/s. (For non-relativisitc velocities, the radial Doppler equivalent velocity can be approximated by multiplying ''z'' with the speed of light.) The z-value can be expressed succinctly in terms of the
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
at R_\text, since the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
is equal to half the square of the
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
, thus: :z \approx \frac\left( \frac \right)^2 where v_\text is the escape velocity at R_\text. It can also be related to the circular orbit velocity v_\text at R_\text, which equals v_\text/\sqrt, thus :z \approx \left( \frac \right)^2. For example, the gravitational blueshift of distant starlight due to the sun's gravity, which the Earth is orbiting at about 30 km/s, would be approximately 1 × 10−8 or the equivalent of a 3 m/s radial Doppler shift. However, the Earth is in free-fall around the sun, and is thus an inertial observer, so the effect is not visible. For an object in a (circular) orbit, the gravitational redshift is of comparable magnitude as the transverse Doppler effect, z \approx \tfrac \beta^2 where ''β''=''v''/''c'', while both are much smaller than the radial Doppler effect, for which z \approx \beta.


Experimental verification


Astronomical observations

A number of experimenters initially claimed to have identified the effect using astronomical measurements, and the effect was considered to have been finally identified in the spectral lines of the star
Sirius B Sirius is the brightest star in the night sky. Its name is derived from the Greek word , or , meaning 'glowing' or 'scorching'. The star is designated α Canis Majoris, Latinized to Alpha Canis Majoris, and abbreviated Alpha CMa ...
by W.S. Adams in 1925.Hetherington, N. S.
"Sirius B and the gravitational redshift - an historical review"
''Quarterly Journal Royal Astronomical Society, vol. 21,'' Sept. 1980, p. 246-252. Accessed 6 April 2017.
However, measurements by Adams have been criticized as being too lowHolberg, J. B.

''Journal for the History of Astronomy, Vol. 41,'' 1, 2010, p. 41-64. Accessed 6 April 2017.
and these observations are now considered to be measurements of spectra that are unusable because of scattered light from the primary, Sirius A. The first accurate measurement of the gravitational redshift of a white dwarf was done by Popper in 1954, measuring a 21 km/s gravitational redshift of
40 Eridani 40 Eridani is a triple star system in the constellation of Eridanus, abbreviated 40 Eri. It has the Bayer designation Omicron2 Eridani, which is Latinized from ο2 Eridani and abbreviated Omicron2 Eri or ο2 Eri. Based on parallax ...
B. The redshift of
Sirius B Sirius is the brightest star in the night sky. Its name is derived from the Greek word , or , meaning 'glowing' or 'scorching'. The star is designated α Canis Majoris, Latinized to Alpha Canis Majoris, and abbreviated Alpha CMa ...
was finally measured by Greenstein ''et al.'' in 1971, obtaining the value for the gravitational redshift of 89±16 km/s, with more accurate measurements by the Hubble Space Telescope, showing 80.4±4.8 km/s. James W. Brault, a graduate student of
Robert Dicke Robert Henry Dicke (; May 6, 1916 – March 4, 1997) was an American astronomer and physicist who made important contributions to the fields of astrophysics, atomic physics, cosmology and gravity. He was the Albert Einstein Professor in Scienc ...
at
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
, measured the gravitational redshift of the sun using optical methods in 1962. In 2020, a team of scientists published the most accurate measurement of the solar gravitational redshift so far, made by analyzing Fe spectral lines in sunlight reflected by the moon; their measurement of a mean global 638 ± 6m/s lineshift is in agreement with the theoretical value of 633.1 m/s. Measuring the solar redshift is complicated by the Doppler shift caused by the motion of the sun's surface, which is of similar magnitude as the gravitational effect. In 2011 the group of Radek Wojtak of the Niels Bohr Institute at the University of Copenhagen collected data from 8000 galaxy clusters and found that the light coming from the cluster centers tended to be red-shifted compared to the cluster edges, confirming the energy loss due to gravity. In 2018, the star S2 made its closest approach to
Sgr A* Sagittarius A* ( ), abbreviated Sgr A* ( ), is the supermassive black hole at the Galactic Center of the Milky Way. It is located near the border of the constellations Sagittarius and Scorpius, about 5.6° south of the ecliptic, vi ...
, the 4-million solar mass
supermassive black hole A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions of times the mass of the Sun (). Black holes are a class of astronomical ob ...
at the centre of the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
, reaching 7650 km/s or about 2.5% of
the speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
while passing the black hole at a distance of just 120 AU, or 1400 Schwarzschild radii. Independent analyses by the GRAVITY collaboration (led by
Reinhard Genzel Reinhard Genzel http://royalsociety.org/people/reinhard-genzel/ Professor Reinhard Genzel ForMemRS (; born 24 March 1952) is a German astrophysicist, co-director of the Max Planck Institute for Extraterrestrial Physics, a professor at LMU and a ...
) and the KECK/UCLA Galactic Center Group (led by
Andrea Ghez Andrea is a given name which is common worldwide for both males and females, cognate to Andreas, Andrej and Andrew. Origin of the name The name derives from the Greek word ἀνήρ (''anēr''), genitive ἀνδρός (''andrós''), that re ...
) revealed a combined transverse Doppler and gravitational redshift up to 200 km/s/c, in agreement with general relativity predictions. In 2021, Mediavilla ( IAC, Spain) & Jiménez-Vicente ( UGR, Spain) were able to use measurements of the gravitational redshift in quasars up to cosmological redshift of z~3 to confirm the predictions of Einstein's Equivalence Principle and the lack of cosmological evolution within 13%.


Terrestrial tests

The effect is now considered to have been definitively verified by the experiments of Pound, Rebka and Snider between 1959 and 1965. The
Pound–Rebka experiment The Pound–Rebka experiment was an experiment in which gamma rays were emitted from the top of a tower and measured by a receiver at the bottom of the tower. The purpose of the experiment was to test Albert Einstein's theory of general relativit ...
of 1959 measured the gravitational redshift in spectral lines using a terrestrial 57Fe gamma source over a vertical height of 22.5 metres. This paper was the first determination of the gravitational redshift which used measurements of the change in wavelength of gamma-ray photons generated with the
Mössbauer effect The Mössbauer effect, or recoilless nuclear resonance fluorescence, is a physical phenomenon discovered by Rudolf Mössbauer in 1958. It involves the resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in ...
, which generates radiation with a very narrow line width. The accuracy of the gamma-ray measurements was typically 1%. An improved experiment was done by Pound and Snider in 1965, with an accuracy better than the 1% level. A very accurate gravitational redshift experiment was performed in 1976, where a
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
maser A maser (, an acronym for microwave amplification by stimulated emission of radiation) is a device that produces coherent electromagnetic waves through amplification by stimulated emission. The first maser was built by Charles H. Townes, Ja ...
clock on a rocket was launched to a height of 10,000 km, and its rate compared with an identical clock on the ground. It tested the gravitational redshift to 0.007%. Later tests can be done with the
Global Positioning System The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
(GPS), which must account for the gravitational redshift in its timing system, and physicists have analyzed timing data from the GPS to confirm other tests. When the first satellite was launched, it showed the predicted shift of 38 microseconds per day. This rate of the discrepancy is sufficient to substantially impair the function of GPS within hours if not accounted for. An excellent account of the role played by general relativity in the design of GPS can be found in Ashby 2003. In 2020 a group at the
University of Tokyo , abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project b ...
measured the gravitational redshift of two strontium-87
optical lattice An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congrega ...
clocks. The measurement took place at
Tokyo Tower is a communications and observation tower in the Shiba-koen district of Minato, Tokyo, Japan, built in 1958. At , it is the second- tallest structure in Japan. The structure is an Eiffel Tower-inspired lattice tower that is painted white and ...
where the clocks were separated by approximately 450 m and connected by telecom fibers. The gravitational redshift can be expressed as : z = \frac = (1+\alpha)\frac , where \Delta\nu=\nu_-\nu_ is the gravitational redshift, \nu_ is the optical clock transition frequency, \Delta U=\Delta U_-\Delta U_ is the difference in gravitational potential, and \alpha denotes the violation from general relativity. By Ramsey spectroscopy of the strontium-87 optical clock transition (429 THz, 698 nm) the group determined the gravitational redshift between the two optical clocks to be 21.18 Hz, corresponding to a ''z''-value of approximately 5 × 10−14. Their measured value of \alpha, (1.4 \pm 9.1)\times 10^ , is an agreement with recent measurements made with hydrogen masers in elliptical orbits.


Early historical development of the theory

The gravitational weakening of light from high-gravity stars was predicted by
John Michell John Michell (; 25 December 1724 – 21 April 1793) was an English natural philosopher and clergyman who provided pioneering insights into a wide range of scientific fields including astronomy, geology, optics, and gravitation. Considered "o ...
in 1783 and
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
in 1796, using
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
's concept of light corpuscles (see:
emission theory Emission theory, also called emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. Emission theories obey the principle of rel ...
) and who predicted that some stars would have a gravity so strong that light would not be able to escape. The effect of gravity on light was then explored by Johann Georg von Soldner (1801), who calculated the amount of deflection of a light ray by the sun, arriving at the Newtonian answer which is half the value predicted by
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. All of this early work assumed that light could slow down and fall, which is inconsistent with the modern understanding of light waves. Once it became accepted that light was an electromagnetic wave, it was clear that the frequency of light should not change from place to place, since waves from a source with a fixed frequency keep the same frequency everywhere. One way around this conclusion would be if time itself were altered—if clocks at different points had different rates. This was precisely Einstein's conclusion in 1911. He considered an accelerating box, and noted that according to the
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
, the clock rate at the "bottom" of the box (the side away from the direction of acceleration) was slower than the clock rate at the "top" (the side toward the direction of acceleration). Indeed, in a frame moving (in x direction) with velocity v relative to the rest frame, the clocks at a nearby position dx are ahead by (dx/c)(v/c) (to the first order); so an acceleration g (that changes speed by g/dt per time dt) makes clocks at the position dx to be ahead by (dx/c)(g/c)dt, that is, tick at a rate : R=1+(g/c^2)dx The equivalence principle implies that this change in clock rate is the same whether the acceleration g is that of an accelerated frame without gravitational effects, or caused by a gravitational field in a stationary frame. Since acceleration due to gravitational potential V is -dV/dx, we get : = g/c^2 = - \, so –- in weak fields –- the change \Delta R in the clock rate is equal to -\Delta V/c^2. Since the light would be slowed down by gravitational time dilation (as seen by outside observer), the regions with lower gravitational potential would act like a medium with higher refractive index causing light to
deflect Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflection ...
. This reasoning allowed Einstein in 1911 to reproduce the incorrect Newtonian value for the deflection of light. At the time he only considered the time-dilating manifestation of gravity, which is the dominating contribution at non-relativistic speeds; however relativistic objects travel through space a comparable amount as they do though time, so purely spatial curvature becomes just as important. After constructing the full theory of general relativity, Einstein solved in 1915 the full
post-Newtonian approximation In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor (general relativity), metric tensor. The approximations are expanded in small ...
for the Sun's gravity and calculated the correct amount of light deflection – double the Newtonian value. Einstein's prediction was confirmed by many experiments, starting with Arthur Eddington's 1919 solar eclipse expedition. The changing rates of clocks allowed Einstein to conclude that light waves change frequency as they move, and the frequency/energy relationship for photons allowed him to see that this was best interpreted as the effect of the gravitational field on the mass–energy of the photon. To calculate the changes in frequency in a nearly static gravitational field, only the time component of the metric tensor is important, and the lowest order approximation is accurate enough for ordinary stars and planets, which are much bigger than their
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...
.


See also

*
Tests of general relativity Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the ben ...
* Equivalence principle *
Gravitational time dilation 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 lower the gravitational potential (the closer ...
* Redshift * Gravitational wave#Redshifting (Redshifting of Gravitational waves due to speed or cosmic expansion)


Citations


References


Primary sources

* * * * Albert Einstein, "Relativity: the Special and General Theory". (@Project Gutenberg). * * *


Other sources

* {{Cite book , last1 = Misner , first1 = Charles W. , last2 = Thorne , first2 = Kip S. , last3 = Wheeler , first3 = John Archibald , title = Gravitation , publisher =
W. H. Freeman W. H. Freeman and Company is an imprint of Macmillan Higher Education, a division of Macmillan Publishers. Macmillan publishes monographs and textbooks for the sciences under the imprint. History The company was founded in 1946 by William H. ...
, location = San Francisco , date=1973-09-15 , isbn = 978-0-7167-0344-0 Albert Einstein Effects of gravitation