Perihelion Shift Of Mercury's Orbit
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Tests of general relativity serve to establish observational evidence for the
theory of general relativity General relativity, also known as the general theory of relativity, and as 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 phys ...
. The first three tests, proposed by
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
in 1915, concerned the "anomalous"
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...
of the
perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...
of Mercury, the bending of light in
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
s, and the
gravitational redshift In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy correspo ...
. The precession of Mercury was already known; experiments showing light bending in accordance with the predictions of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
were performed in 1919, with increasingly precise measurements made in subsequent tests; and scientists claimed to have measured the gravitational redshift in 1925, although measurements sensitive enough to actually confirm the theory were not made until 1954. A more accurate program starting in 1959 tested general relativity in the weak gravitational field limit, severely limiting possible deviations from the theory. In the 1970s, scientists began to make additional tests, starting with Irwin Shapiro's measurement of the relativistic time delay in radar signal travel time near the Sun. Beginning in 1974, Hulse,
Taylor Taylor, Taylors or Taylor's may refer to: People * Taylor (surname) ** List of people with surname Taylor * Taylor (given name), including Tayla and Taylah * Taylor sept, a branch of Scottish clan Cameron * Justice Taylor (disambiguation) ...
and others studied the behaviour of
binary pulsar A binary pulsar is a pulsar with a binary companion, often a white dwarf or neutron star. (In at least one case, the double pulsar PSR J0737-3039, the companion neutron star is another pulsar as well.) Binary pulsars are one of the few objects ...
s experiencing much stronger gravitational fields than those found in the Solar System. Both in the weak field limit (as in the Solar System) and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been extremely well tested. In February 2016, the
Advanced LIGO The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory designed to detect cosmic gravitational waves and to develop gravitational-wave observations as an astronomical tool. Prior to LIG ...
team announced that they had directly detected gravitational waves from a black hole merger. This discovery, along with additional detections announced in June 2016 and June 2017,Conover, Emily
LIGO snags another set of gravitational waves
''Science News'', June 1, 2017. Retrieved 8 June 2017.
tested general relativity in the very strong field limit, observing to date no deviations from theory.


Classical tests

Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
proposed three tests of general relativity, subsequently called the "classical tests" of general relativity, in 1916: # the perihelion precession of Mercury's orbit # the deflection of light by the
Sun The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...
# the
gravitational redshift In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy correspo ...
of light In the letter to ''
The Times ''The Times'' is a British Newspaper#Daily, daily Newspaper#National, national newspaper based in London. It began in 1785 under the title ''The Daily Universal Register'', adopting its modern name on 1 January 1788. ''The Times'' and its si ...
'' (of London) on November 28, 1919, he described the theory of relativity and thanked his English colleagues for their understanding and testing of his work. He also mentioned three classical tests with comments: : "The chief attraction of the theory lies in its logical completeness. If a single one of the conclusions drawn from it proves wrong, it must be given up; to modify it without destroying the whole structure seems to be impossible."


Perihelion precession of Mercury

Under
Newtonian physics Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved substantial change in the methods ...
, an object in an (isolated) two-body system, consisting of the object orbiting a spherical mass, would trace out an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
with the center of mass of the system at a
focus Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film *Focus (2001 film), ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel *Focus (2015 ...
of the ellipse. The point of closest approach, called the
periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...
(or when the central body is the Sun,
perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...
), is fixed. Hence the major axis of the ellipse remains fixed in space. Both objects orbit around the center of mass of this system, so they each have their own ellipse. However, a number of effects in the Solar System cause the perihelia of planets to precess (rotate) around the Sun in the plane of their orbits, or equivalently, cause the major axis to rotate about the center of mass, hence changing its orientation in space. The principal cause is the presence of other planets which perturb one another's orbit. Another (much less significant) effect is solar
oblateness Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f ...
. Mercury deviates from the precession predicted from these Newtonian effects. This anomalous rate of precession of the perihelion of Mercury's orbit was first recognized in 1859 as a problem in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, by
Urbain Le Verrier Urbain Jean Joseph Le Verrier (; 11 March 1811 – 23 September 1877) was a French astronomer and mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune using only mathematics. ...
. His re-analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848 showed that the actual rate of the precession disagreed from that predicted from Newton's theory by 38″ (
arcseconds A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
) per tropical century (later re-estimated at 43″ by
Simon Newcomb Simon Newcomb (March 12, 1835 – July 11, 1909) was a Canadians, Canadian–Americans, American astronomer, applied mathematician, and autodidactic polymath. He served as Professor of Mathematics in the United States Navy and at Johns Hopkins ...
in 1882). A number of ''ad hoc'' and ultimately unsuccessful solutions were proposed, but they tended to introduce more problems. Le Verrier suggested that another hypothetical planet might exist to account for Mercury's behavior. The previously successful search for Neptune based on its perturbations of the orbit of Uranus led astronomers to place some faith in this possible explanation, and the hypothetical planet was even named Vulcan. Finally, in 1908, W. W. Campbell, Director of the Lick Observatory, after the comprehensive photographic observations by Lick astronomer, Charles D. Perrine, at three solar eclipse expeditions, stated, "In my opinion, Dr. Perrine's work at the three eclipses of 1901, 1905, and 1908 brings the observational side of the famous intramercurial-planet problem definitely to a close." Subsequently, no evidence of Vulcan was found and Einstein's 1915 general theory accounted for Mercury's anomalous precession. Einstein wrote to
Michele Besso Michele Angelo Besso (25May 187315March 1955) was a Swiss-Italian engineer who worked closely with Albert Einstein. Biography Besso was born in Riesbach from a family of Italian Jewish ( Sephardi) descent. He was a close friend of Albert Ei ...
, "Perihelion motions explained quantitatively ... you will be astonished". In general relativity, this remaining
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...
, or change of orientation of the orbital ellipse within its orbital plane, is explained by gravitation being mediated by the curvature of spacetime. Einstein showed that general relativity agrees closely with the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general relativity. Although earlier measurements of planetary orbits were made using conventional telescopes, more accurate measurements are now made with
radar Radar is a system that uses radio waves to determine the distance ('' ranging''), direction ( azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track ...
. The total observed precession of Mercury is (574.10 ± 0.65)″ per century relative to the inertial ICRF. This precession can be attributed to the following causes: The correction by ()″/cy is the prediction of post-Newtonian theory with
parameters A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
\gamma = \beta = 1. Thus the effect can be fully explained by general relativity. More recent calculations based on more precise measurements have not materially changed the situation. In general relativity the perihelion shift ''σ'', expressed in radians per revolution, is approximately given by: : \sigma = \frac \ , where ''L'' is the
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...
, ''T'' is the
orbital period The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...
, ''c'' is the speed of light, and ''e'' is the
orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values be ...
(see:
Two-body problem in general relativity The two-body problem in general relativity (or relativistic two-body problem) is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essen ...
). The other planets experience perihelion shifts as well, but, since they are farther from the Sun and have longer periods, their shifts are lower, and could not be observed accurately until long after Mercury's. For example, the perihelion shift of Earth's orbit due to general relativity is theoretically 3.83868″ per century and experimentally ()″/cy, Venus's is 8.62473″/cy and (8.6247 ± 0.0005)″/cy and Mars' is ()″/cy. Both values have now been measured, with results in good agreement with theory. The
periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...
shift has also now been measured for binary pulsar systems, with
PSR 1913+16 PSR may refer to: Organizations * Pacific School of Religion, Berkeley, California, US * Palestinian Center for Policy and Survey Research * Payment Systems Regulator in the United Kingdom * Physicians for Social Responsibility, US Political par ...
amounting to 4.2° per year. These observations are consistent with general relativity. It is also possible to measure periapsis shift in binary star systems which do not contain ultra-dense stars, but it is more difficult to model the classical effects precisely – for example, the alignment of the stars' spin to their orbital plane needs to be known and is hard to measure directly. A few systems, such as
DI Herculis DI Herculis is an Algol-type eclipsing binary star in the constellation of Hercules. The system has an apparent magnitude of about +8.5 and consists of two young blue stars of spectral type B5 and B4. It is about two thousand light year ...
,Naeye, Robert
"Stellar Mystery Solved, Einstein Safe"
''Sky and Telescope'', September 16, 2009. See als
MIT Press Release
September 17, 2009. Accessed 8 June 2017.
have been measured as test cases for general relativity.


Deflection of light by the Sun

Henry Cavendish Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "inflammable air". He described the density of inflammable a ...
in 1784 (in an unpublished manuscript) and
Johann Georg von Soldner Johann 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. Life He was born in Feuchtwangen in A ...
in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object. The same value as Soldner's was calculated by Einstein in 1911 based on the equivalence principle alone. However, Einstein noted in 1915 in the process of completing general relativity, that his 1911 result (and thus Soldner's 1801 result) is only half of the correct value. Einstein became the first to calculate the correct value for light bending: 1.75
arcsecond A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
s for light that grazes the Sun. The first observation of light deflection was performed by noting the change in position of
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s as they passed near the Sun on the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...
. The observations were performed by
Arthur Eddington Sir Arthur Stanley Eddington, (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician. He was also a philosopher of science and a populariser of science. The Eddington limit, the natural limit to the lu ...
and his collaborators (see
Eddington experiment The Eddington experiment was an observational test of general relativity, organised by the British astronomers Frank Watson Dyson and Arthur Stanley Eddington in 1919. The observations were of the total solar eclipse of 29 May 1919 and were ca ...
) during the total
solar eclipse of May 29, 1919 A total solar eclipse occurred at the Moon's descending node of orbit on Thursday, May 29, 1919, with a magnitude of 1.0719. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of ...
, when the stars near the Sun (at that time in the constellation
Taurus Taurus is Latin for 'bull' and may refer to: * Taurus (astrology), the astrological sign ** Vṛṣabha, in vedic astrology * Taurus (constellation), one of the constellations of the zodiac * Taurus (mythology), one of two Greek mythological ch ...
) could be observed. Observations were made simultaneously in the cities of
Sobral, Ceará Sobral is a municipality in the state of Ceará, Brazil. Sobral is the fifth largest municipality of Ceará, after Fortaleza. Its economy is based on agriculture, services and some manufacturing industries. The city has two public universi ...
, Brazil and in
São Tomé and Príncipe São Tomé and Príncipe, officially the Democratic Republic of São Tomé and Príncipe, is an island country in the Gulf of Guinea, off the western equatorial coast of Central Africa. It consists of two archipelagos around the two main isla ...
on the west coast of Africa. The result was considered spectacular news and made the front page of most major newspapers. It made Einstein and his theory of general relativity world-famous. When asked by his assistant what his reaction would have been if general relativity had not been confirmed by Eddington and Dyson in 1919, Einstein famously made the quip: "Then I would feel sorry for the dear Lord. The theory is correct anyway." The early accuracy, however, was poor and there was doubt that the small number of measured star locations and instrument questions could produce a reliable result. The results were argued by some to have been plagued by
systematic error Observational error (or measurement error) is the difference between a measurement, measured value of a physical quantity, quantity and its unknown true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. Such errors are ...
and possibly
confirmation bias Confirmation bias (also confirmatory bias, myside bias, or congeniality bias) is the tendency to search for, interpret, favor and recall information in a way that confirms or supports one's prior beliefs or Value (ethics and social sciences), val ...
, although modern reanalysis of the dataset suggests that Eddington's analysis was accurate.D. Kennefick, "Testing relativity from the 1919 eclipse- a question of bias", ''Physics Today'', March 2009, pp. 37–42. The measurement was repeated by a team from the
Lick Observatory The Lick Observatory is an astronomical observatory owned and operated by the University of California. It is on the summit of Mount Hamilton (California), Mount Hamilton, in the Diablo Range just east of San Jose, California, United States. The ...
led by the Director W. W. Campbell in the 1922 eclipse as observed in remote Australian station of Wallal, with results based on hundreds of star positions that agreed with the 1919 results and has been repeated several times since, most notably in 1953 by
Yerkes Observatory Yerkes Observatory ( ) is an astronomical observatory located in Williams Bay, Wisconsin, United States. The observatory was operated by the University of Chicago Department of Astronomy and Astrophysics from its founding in 1897 until 2018. O ...
astronomers and in 1973 by a team from the
University of Texas The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas, United States. Founded in 1883, it is the flagship institution of the University of Texas System. With 53,082 students as of fall 2 ...
. Considerable uncertainty remained in these measurements for almost fifty years, until observations started being made at
radio frequencies Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upper ...
. The deflection of starlight by the nearby
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
star Stein 2051 B has also been measured.


Gravitational redshift of light

Einstein predicted the
gravitational redshift In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy correspo ...
of light from the
equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
in 1907, and it was predicted that this effect might be measured in the spectral lines of a
white dwarf star A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place in a white dwarf; wh ...
, which has a very high gravitational field. Initial attempts to measure the gravitational redshift of the spectrum of Sirius-B were done by
Walter Sydney Adams Walter Sydney Adams (December 20, 1876 – May 11, 1956) was an American astronomer. He is renowned for his pioneering work in spectroscopy. Life and work Adams was born in Antioch, Ottoman Empire, to Lucien Harper Adams and Nancy Dorrance ...
in 1925, but the result was criticized as being unusable due to the contamination from light from the (much brighter) primary star,
Sirius Sirius is the brightest star in the night sky. Its name is derived from the Greek word (Latin script: ), meaning 'glowing' or 'scorching'. The star is designated  Canis Majoris, Latinized to Alpha Canis Majoris, and abbr ...
.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.
Holberg, J. B.

''Journal for the History of Astronomy, Vol. 41,'' 1, 2010, p. 41-64. Accessed 6 April 2017.
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 paralla ...
B. The redshift of Sirius B was finally measured by Greenstein ''et al.'' in 1971, obtaining the value for the gravitational redshift of , with more accurate measurements by the Hubble Space Telescope showing .


Tests of special relativity

The general theory of relativity incorporates Einstein's
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, "On the Electrodynamics of Moving Bodies", the theory is presen ...
, and hence tests of special relativity are also testing aspects of general relativity. As a consequence of the
equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
,
Lorentz invariance In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
holds locally in non-rotating, freely falling reference frames. Experiments related to Lorentz invariance special relativity (that is, when gravitational effects can be neglected) are described in
tests of special relativity Special relativity is a physical theory that plays a fundamental role in the description of all physical phenomena, as long as gravitation is not significant. Many experiments played (and still play) an important role in its development and justific ...
.


Modern tests

The modern era of testing general relativity was ushered in largely at the impetus of Dicke and
Schiff Schiff is a Jewish and German language, German surname meaning "ship". The Schiffs are known from "about 1370, the earliest date to which any contemporary Jewish family can be definitely traced". The Schiff family, is a prominent Jewish banking dyna ...
who laid out a framework for testing general relativity. They emphasized the importance not only of the classical tests, but of null experiments, testing for effects which in principle could occur in a theory of gravitation, but do not occur in general relativity. Other important theoretical developments included the inception of alternative theories to general relativity, in particular, scalar–tensor theories such as the
Brans–Dicke theory In physics, the Brans–Dicke theory of gravitation (sometimes called the Jordan–Brans–Dicke theory) is a competitor to Einstein's general theory of relativity. It is an example of a scalar–tensor theory, a gravitational theory in which the ...
; the
parameterized post-Newtonian formalism In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order de ...
in which deviations from general relativity can be quantified; and the framework of the
equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
. Experimentally, new developments in
space exploration Space exploration is the process of utilizing astronomy and space technology to investigate outer space. While the exploration of space is currently carried out mainly by astronomers with telescopes, its physical exploration is conducted bo ...
,
electronics Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...
and
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
have made additional precise experiments possible, such as the Pound–Rebka experiment, laser interferometry and lunar rangefinding.


Post-Newtonian tests of gravity

Early tests of general relativity were hampered by the lack of viable competitors to the theory: it was not clear what sorts of tests would distinguish it from its competitors. General relativity was the only known relativistic theory of gravity compatible with special relativity and observations. This changed in 1960 with the introduction of
Brans–Dicke theory In physics, the Brans–Dicke theory of gravitation (sometimes called the Jordan–Brans–Dicke theory) is a competitor to Einstein's general theory of relativity. It is an example of a scalar–tensor theory, a gravitational theory in which the ...
which provided an alternative theory also in agreement with experimental observations. Ultimately, this led to the development of the parametrized post-Newtonian formalism by Nordtvedt and
Will Will may refer to: Common meanings * Will and testament, instructions for the disposition of one's property after death * Will (philosophy), or willpower * Will (sociology) * Will, volition (psychology) * Will, a modal verb - see Shall and will ...
, which parametrizes, in terms of ten adjustable parameters, all the possible departures from Newton's law of universal gravitation to first order in the velocity of moving objects (''i.e.'' to first order in v/c, where ''v'' is the velocity of an object and ''c'' is the speed of light). This approximation allows the possible deviations from general relativity, for slowly moving objects in weak gravitational fields, to be systematically analyzed. Much effort has been put into constraining the post-Newtonian parameters, and deviations from general relativity are at present severely limited. The experiments testing gravitational lensing and light time delay limits the same post-Newtonian parameter, the so-called Eddington parameter γ, which is a straightforward parametrization of the amount of deflection of light by a gravitational source. It is equal to one for general relativity, and takes different values in other theories (such as Brans–Dicke theory). It is the best constrained of the ten post-Newtonian parameters, but there are other experiments designed to constrain the others. Precise observations of the perihelion shift of Mercury constrain other parameters, as do tests of the strong equivalence principle. One of the goals of the
BepiColombo BepiColombo is a joint mission of the European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA) to the planet Mercury. The mission comprises two satellites launched together: the Mercury Planetary Orbiter (MPO) and ''Mio'' ...
mission to Mercury, is to test the general relativity theory by measuring the parameters gamma and beta of the parametrized post-Newtonian formalism with high accuracy. The experiment is part of the Mercury Orbiter Radio science Experiment (MORE). The spacecraft was launched in October 2018 and is expected to enter orbit around Mercury in December 2025.


Gravitational lensing

One of the most important tests is
gravitational lensing A gravitational lens is matter, such as a galaxy cluster, cluster of galaxies or a point particle, that bends light from a distant source as it travels toward an observer. The amount of gravitational lensing is described by Albert Einstein's Ge ...
. It has been observed in distant astrophysical sources, but these are poorly controlled and it is uncertain how they constrain general relativity. The most precise tests are analogous to Eddington's 1919 experiment: they measure the deflection of radiation from a distant source by the Sun. The sources that can be most precisely analyzed are distant radio sources. In particular, some
quasar A quasar ( ) is an extremely Luminosity, luminous active galactic nucleus (AGN). It is sometimes known as a quasi-stellar object, abbreviated QSO. The emission from an AGN is powered by accretion onto a supermassive black hole with a mass rangi ...
s are very strong radio sources. The directional resolution of any telescope is in principle limited by diffraction; for radio telescopes this is also the practical limit. An important improvement in obtaining positional high accuracies (from milli-arcsecond to micro-arcsecond) was obtained by combining radio telescopes across Earth. The technique is called
very long baseline interferometry Very-long-baseline interferometry (VLBI) is a type of astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescopes on Earth or in space. T ...
(VLBI). With this technique radio observations couple the phase information of the radio signal observed in telescopes separated over large distances. Recently, these telescopes have measured the deflection of radio waves by the Sun to extremely high precision, confirming the amount of deflection predicted by general relativity aspect to the 0.03% level. At this level of precision systematic effects have to be carefully taken into account to determine the precise location of the telescopes on Earth. Some important effects are Earth's
nutation Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference fra ...
, rotation, atmospheric refraction, tectonic displacement and tidal waves. Another important effect is refraction of the radio waves by the
solar corona In astronomy, a corona (: coronas or coronae) is the outermost layer of a star's Stellar atmosphere, atmosphere. It is a hot but relatively luminosity, dim region of Plasma (physics), plasma populated by intermittent coronal structures such as so ...
. Fortunately, this effect has a characteristic
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
, whereas gravitational distortion is independent of wavelength. Thus, careful analysis, using measurements at several frequencies, can subtract this source of error. The entire sky is slightly distorted due to the gravitational deflection of light caused by the Sun (the anti-Sun direction excepted). This effect has been observed by the
European Space Agency The European Space Agency (ESA) is a 23-member International organization, international organization devoted to space exploration. With its headquarters in Paris and a staff of around 2,547 people globally as of 2023, ESA was founded in 1975 ...
astrometric satellite
Hipparcos ''Hipparcos'' was a scientific satellite of the European Space Agency (ESA), launched in 1989 and operated until 1993. It was the first space experiment devoted to precision astrometry, the accurate measurement of the positions and distances of ...
. It measured the positions of about 105 stars. During the full mission about relative positions have been determined, each to an accuracy of typically 3 milliarcseconds (the accuracy for an 8–9 magnitude star). Since the gravitation deflection perpendicular to the Earth–Sun direction is already 4.07 milliarcseconds, corrections are needed for practically all stars. Without systematic effects, the error in an individual observation of 3 milliarcseconds, could be reduced by the square root of the number of positions, leading to a precision of 0.0016 milliarcseconds. Systematic effects, however, limit the accuracy of the determination to 0.3% (Froeschlé, 1997). Launched in 2013, the ''Gaia'' spacecraft will conduct a census of one billion
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s in the
Milky Way The Milky Way or Milky Way Galaxy is the galaxy that includes the Solar System, with the name describing the #Appearance, galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars in other arms of the galax ...
and measure their positions to an accuracy of 24 microarcseconds. Thus it will also provide stringent new tests of gravitational deflection of light caused by the
Sun The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...
which was predicted by General relativity.


Light travel time delay testing

Irwin I. Shapiro Irwin Ira Shapiro is an American astrophysicist and Harvard University Professor, Timken University Professor at Harvard University. He has been a professor at Harvard since 1982. He was the director of the Harvard-Smithsonian Center for Astr ...
proposed another test, beyond the classical tests, which could be performed within the Solar System. It is sometimes called the fourth "classical" test of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. He predicted a relativistic time delay (
Shapiro delay The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic Solar System tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return th ...
) in the round-trip travel time for radar signals reflecting off other planets. The mere curvature of the path of a
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 particles that can ...
passing near the Sun is too small to have an observable delaying effect (when the round-trip time is compared to the time taken if the photon had followed a straight path), but general relativity predicts a time delay that becomes progressively larger when the photon passes nearer to the Sun due to the
time dilation Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unsp ...
in the
gravitational potential In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the ...
of the Sun. Observing radar reflections from Mercury and Venus just before and after they are eclipsed by the Sun agrees with general relativity theory at the 5% level. More recently, the
Cassini probe Cassini may refer to: People * Cassini (surname) * Oleg Cassini (1913–2006), American fashion designer Cassini family: * Giovanni Domenico Cassini (1625–1712), Italian mathematician, astronomer, engineer, and astrologer * Jacques Cassini ...
has undertaken a similar experiment which gave agreement with general relativity at the 0.002% level. However, the following detailed studies revealed that the measured value of the PPN parameter gamma is affected by a gravitomagnetic effect caused by the orbital motion of Sun around the
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
of the solar system. The gravitomagnetic effect in the Cassini radioscience experiment was implicitly postulated by B. Bertotti as having a pure general relativistic origin but its theoretical value has never been tested in the experiment which effectively makes the experimental uncertainty in the measured value of gamma actually larger (by a factor of 10) than 0.002% claimed by B. Bertotti and co-authors in Nature.
Very Long Baseline Interferometry Very-long-baseline interferometry (VLBI) is a type of astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescopes on Earth or in space. T ...
has measured velocity-dependent (gravitomagnetic) corrections to the Shapiro time delay in the field of moving Jupiter and Saturn.


Equivalence principle

The equivalence principle, in its simplest form, asserts that the trajectories of falling bodies in a gravitational field should be independent of their mass and internal structure, provided they are small enough not to disturb the environment or be affected by
tidal forces The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the d ...
. This idea has been tested to extremely high precision by Eötvös torsion balance experiments, which look for a differential acceleration between two test masses. Constraints on this, and on the existence of a composition-dependent fifth force or gravitational
Yukawa interaction In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki Yukawa, is an interaction between particles according to the Yukawa potential. Specifically, it is between a scalar field (or pseudoscalar field) \ \phi\ and a Dira ...
are very strong, and are discussed under
fifth force In physics, a fifth force refers to a hypothetical fundamental interaction (also known as fundamental force) beyond the four known interactions in nature: gravitational, electromagnetic, strong nuclear, and weak nuclear forces. Some speculativ ...
and
weak equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same tr ...
. A version of the equivalence principle, called the
strong equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same tr ...
, asserts that self-gravitation falling bodies, such as stars, planets or black holes (which are all held together by their gravitational attraction) should follow the same trajectories in a gravitational field, provided the same conditions are satisfied. This is called the
Nordtvedt effect In theoretical astrophysics, the Nordtvedt effect refers to the relative motion between the Earth and the Moon that would be observed if the gravitational self-energy of a body contributed differently to its gravitational mass than to its inert ...
and is most precisely tested by the
Lunar Laser Ranging Experiment Lunar Laser Ranging (LLR) is the practice of measuring Lunar distance (astronomy), the distance between the surfaces of the Earth and the Moon using Lidar, laser ranging. The distance can be calculated from the Round-trip delay, round-trip time ...
. Since 1969, it has continuously measured the distance from several rangefinding stations on Earth to reflectors on the Moon to approximately centimeter accuracy. These have provided a strong constraint on several of the other post-Newtonian parameters. Another part of the strong equivalence principle is the requirement that Newton's gravitational constant be constant in time, and have the same value everywhere in the universe. There are many independent observations limiting the possible variation of Newton's
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
, but one of the best comes from lunar rangefinding which suggests that the gravitational constant does not change by more than one part in 1011 per year. The constancy of the other constants is discussed in the Einstein equivalence principle section of the equivalence principle article.


Gravitational redshift and time dilation

The first of the classical tests discussed above, the
gravitational redshift In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy correspo ...
, is a simple consequence of the Einstein equivalence principle and was predicted by Einstein in 1907. As such, it is not a test of general relativity in the same way as the post-Newtonian tests, because any theory of gravity obeying the equivalence principle should also incorporate the gravitational redshift. Nonetheless, confirming the existence of the effect was an important substantiation of relativistic gravity, since the absence of gravitational redshift would have strongly contradicted relativity. The first observation of the gravitational redshift was the measurement of the shift in the spectral lines from the
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
star
Sirius Sirius is the brightest star in the night sky. Its name is derived from the Greek word (Latin script: ), meaning 'glowing' or 'scorching'. The star is designated  Canis Majoris, Latinized to Alpha Canis Majoris, and abbr ...
B by Adams in 1925, discussed above, and follow-on measurements of other white dwarfs. Because of the difficulty of the astrophysical measurement, however, experimental verification using a known terrestrial source was preferable. Experimental verification of gravitational redshift using terrestrial sources took several decades, because it is difficult to find clocks (to measure
time dilation Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unsp ...
) or sources of electromagnetic radiation (to measure redshift) with a frequency that is known well enough that the effect can be accurately measured. It was confirmed experimentally for the first time in 1959 using 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 a ...
, which generates radiation with a very narrow line width. The Pound–Rebka experiment measured the relative redshift of two sources situated at the top and bottom of Harvard University's Jefferson tower. The result was in excellent agreement with general relativity. This was one of the first precision experiments testing general relativity. The experiment was later improved to better than the 1% level by Pound and Snider. The blueshift of a falling photon can be found by assuming it has an equivalent mass based on its frequency (where ''h'' is the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
) along with , a result of special relativity. Such simple derivations ignore the fact that in general relativity the experiment compares clock rates, rather than energies. In other words, the "higher energy" of the photon after it falls can be equivalently ascribed to the slower running of clocks deeper in the gravitational potential well. To fully validate general relativity, it is important to also show that the rate of arrival of the photons is greater than the rate at which they are emitted. A very accurate gravitational redshift experiment, which deals with this issue, was performed in 1976, where a
hydrogen Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
maser A maser is a device that produces coherent electromagnetic waves ( microwaves), through amplification by stimulated emission. The term is an acronym for microwave amplification by stimulated emission of radiation. Nikolay Basov, Alexander Pr ...
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%. Although the
Global Positioning System The Global Positioning System (GPS) is a satellite-based hyperbolic navigation system owned by the United States Space Force and operated by Mission Delta 31. It is one of the global navigation satellite systems (GNSS) that provide ge ...
(GPS) is not designed as a test of fundamental physics, it 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, some engineers resisted the prediction that a noticeable gravitational time dilation would occur, so the first satellite was launched without the clock adjustment that was later built into subsequent satellites. It showed the predicted shift of 38 microseconds per day. This rate of discrepancy is sufficient to substantially impair 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. Other precision tests of general relativity, not discussed here, are the
Gravity Probe A Gravity Probe A (GP-A) was a space-based 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 Adm ...
satellite, launched in 1976, which showed gravity and velocity affect the ability to synchronize the rates of clocks orbiting a central mass and the
Hafele–Keating experiment The Hafele–Keating experiment was a test of the theory of relativity. In 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around th ...
, which used atomic clocks in circumnavigating aircraft to test general relativity and special relativity together.


Frame-dragging tests

Tests of the
Lense–Thirring precession In general relativity, Lense–Thirring precession or the Lense–Thirring effect (; named after Josef Lense and Hans Thirring) is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is ...
, consisting of small secular
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...
s of the orbit of a test particle in motion around a central rotating mass, for example, a planet or a star, have been performed with the
LAGEOS LAGEOS (), Laser Geodynamics Satellite or Laser Geometric Environmental Observation Survey, are a series of two scientific research satellites designed to provide an orbiting laser ranging benchmark for geodynamical studies of the Earth. Each ...
satellites, but many aspects of them remain controversial. The same effect may have been detected in the data of the
Mars Global Surveyor ''Mars Global Surveyor'' (MGS) was an American Robotic spacecraft, robotic space probe developed by NASA's Jet Propulsion Laboratory. It launched November 1996 and collected data from 1997 to 2006. MGS was a global mapping mission that examined ...
(MGS) spacecraft, a former probe in orbit around
Mars Mars is the fourth planet from the Sun. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the atmosph ...
; also such a test raised a debate. First attempts to detect the
Sun The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...
's Lense–Thirring effect on the perihelia of the inner
planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s have been recently reported as well. Frame dragging would cause the orbital plane of stars orbiting near a
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 ...
to precess about the black hole spin axis. This effect should be detectable within the next few years via
astrometric Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, the Milky Way. History ...
monitoring of stars at the center of the
Milky Way The Milky Way or Milky Way Galaxy is the galaxy that includes the Solar System, with the name describing the #Appearance, galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars in other arms of the galax ...
galaxy. By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test the
no-hair theorem The no-hair theorem states that all stationary black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three independent ''externally'' observabl ...
s of general relativity. The
Gravity Probe B Gravity Probe B (GP-B) was a satellite-based experiment whose objective was to test two previously-unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, t ...
satellite, launched in 2004 and operated until 2005, detected frame-dragging and the
geodetic effect The 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 experiment used four quartz spheres the size of ping pong balls coated with a superconductor. Data analysis continued through 2011 due to high noise levels and difficulties in modelling the noise accurately so that a useful signal could be found. Principal investigators at
Stanford University Leland Stanford Junior University, commonly referred to as Stanford University, is a Private university, private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth ...
reported on May 4, 2011, that they had accurately measured the frame dragging effect relative to the distant star IM Pegasi, and the calculations proved to be in line with the prediction of Einstein's theory. The results, published in ''
Physical Review Letters ''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. The journal is considered one of the most prestigious in the field of physics ...
'' measured the
geodetic effect The 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, ...
with an error of about 0.2 percent. The results reported the frame dragging effect (caused by Earth's rotation) added up to 37 milliarcseconds with an error of about 19 percent. Investigator Francis Everitt explained that a milliarcsecond "is the width of a human hair seen at the distance of 10 miles". In January 2012,
LARES Lares ( , ; archaic , singular ) were Tutelary deity#Ancient Rome, guardian deities in ancient Roman religion. Their origin is uncertain; they may have been hero-ancestors, guardians of the hearth, fields, boundaries, or fruitfulness, or an ama ...
satellite was launched on a
Vega Vega is the brightest star in the northern constellation of Lyra. It has the Bayer designation α Lyrae, which is Latinised to Alpha Lyrae and abbreviated Alpha Lyr or α Lyr. This star is relatively close at only from the Sun, and ...
rocket to measure Lense–Thirring effect with an accuracy of about 1%, according to its proponents. This evaluation of the actual accuracy obtainable is a subject of debate.


Tests of the gravitational potential at small distances

It is possible to test whether the gravitational potential continues with the inverse square law at very small distances. Tests so far have focused on a divergence from GR in the form of a
Yukawa potential Yukawa (written: 湯川) is a Japanese surname, but is also applied to proper nouns. People * Diana Yukawa (born 1985), Anglo-Japanese solo violinist. She has had two solo albums with BMG Japan, one of which opened to #1 * Hideki Yukawa (1907–1 ...
V(r) = V_0\left(1 + \alpha e^\right), but no evidence for a potential of this kind has been found. The Yukawa potential with \alpha=1 has been ruled out down to .


Mössbauer rotor experiment

It was conceived as a means to measure the
time dilation Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unsp ...
effect on Earth after being motivated by Einstein's
equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
that implies a rotating observer will be subject to the same transformations as an observer in a gravitational field. Mössbauer rotor experiments hence permit a precise terrestrial test of the
relativistic Doppler effect The relativistic Doppler effect is the change in frequency, wavelength and amplitude of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect, first proposed by Christian Doppler in 1842), when ta ...
. From a radioactive source fixed at the center of a spinning disc or rod,
gamma rays A gamma ray, also known as gamma radiation (symbol ), is a penetrating form of electromagnetic radiation arising from high energy interactions like the radioactive decay of atomic nuclei or astronomical events like solar flares. It consists o ...
travel to an absorber at the rim (in some variations of the experiment this scheme was reversed) and an unabsorbed number of them pass through depending on the rotational speed to arrive at a stationary counter (''i.e.'', detector of gamma quanta resting in the lab frame). In lieu with the
Clock hypothesis Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unsp ...
, Einstein's
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
predicts that the moving absorber's clock at the rim should retard by a specific amount due to time dilation on account of centrifugal binding alone compared to a rest frame absorber. So the transmission of gamma photons through the absorber should increase during rotation, which can be subsequently measured by the stationary counter beyond the absorber. This prediction was actually observed using 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 a ...
, since the equivalence principle, as originally suggested by Einstein, implicitly allows the association of the time dilation due to rotation (calculated as a result of the change in the detector's count rate) with gravitational time dilation. Such experiments were pioneered by Hay ''et al.'' (1960), Champeney ''et al.'' (1965), and Kündig (1963), and all of them had declared confirmation of the prediction of Einstein's theory of relativity. Be that as it may, an early 21st Century re-examination of these endeavors called into question the validity of the past obtained results claiming to have verified time dilation as predicted by Einstein's relativity theory, whereby novel experimentations were carried out that uncovered an ''extra energy shift'' between emitted and absorbed radiation next to the classical relativistic dilation of time. This discovery was first explained as discrediting
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and successfully confirming at the laboratory scale the predictions of an alternative theory of gravity developed by T. Yarman and his colleagues. Against this development, a contentious attempt was made to explain the disclosed extra energy shift as arising from a so-far unknown and allegedly missed ''clock synchronization effect'', which was unusually awarded a prize in 2018 by the
Gravity Research Foundation The Gravity Research Foundation is an organization established in 1948 by businessman Roger Babson (founder of Babson College) to find ways to implement gravitational shielding. Over time, the foundation turned away from trying to block gravity ...
for having secured ''a new proof of general relativity''. However, at the same time period, it was revealed that said author committed several mathematical errors in his calculations, and the supposed contribution of the so-called clock synchronization to the measured time dilation is in fact practically null. As a consequence, a general relativistic explanation for the outcomes of Mössbauer rotor experiments remains open.


Strong field tests

The very strong gravitational fields that are present close to
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s, especially those
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 ...
s which are thought to power
active galactic nuclei An active galactic nucleus (AGN) is a compact region at the center of a galaxy that emits a significant amount of energy across the electromagnetic spectrum, with characteristics indicating that this luminosity is not produced by the stars. Such e ...
and the more active
quasar A quasar ( ) is an extremely Luminosity, luminous active galactic nucleus (AGN). It is sometimes known as a quasi-stellar object, abbreviated QSO. The emission from an AGN is powered by accretion onto a supermassive black hole with a mass rangi ...
s, belong to a field of intense active research. Observations of these quasars and active galactic nuclei are difficult, and interpretation of the observations is heavily dependent upon astrophysical models other than general relativity or competing fundamental theories of gravitation, but they are qualitatively consistent with the black hole concept as modeled in general relativity.


Binary pulsars

Pulsars A pulsar (''pulsating star, on the model of quasar'') is a highly magnetized rotating neutron star that emits beams of electromagnetic radiation out of its magnetic poles. This radiation can be observed only when a beam of emission is pointin ...
are rapidly rotating
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s which emit regular radio pulses as they rotate. As such they act as clocks which allow very precise monitoring of their orbital motions. Observations of pulsars in orbit around other stars have all demonstrated substantial
periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...
precessions that cannot be accounted for classically but can be accounted for by using general relativity. For example, the Hulse–Taylor
binary pulsar A binary pulsar is a pulsar with a binary companion, often a white dwarf or neutron star. (In at least one case, the double pulsar PSR J0737-3039, the companion neutron star is another pulsar as well.) Binary pulsars are one of the few objects ...
PSR B1913+16 PSR may refer to: Organizations * Pacific School of Religion, Berkeley, California, US * Palestinian Center for Policy and Survey Research * Payment Systems Regulator in the United Kingdom * Physicians for Social Responsibility, US Political p ...
(a pair of neutron stars in which one is detected as a pulsar) has an observed precession of over 4° of arc per year (periastron shift per orbit only about 10−6). This precession has been used to compute the masses of the components. Similarly to the way in which atoms and molecules emit electromagnetic radiation, a gravitating mass that is in
quadrupole A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure re ...
type or higher order vibration, or is asymmetric and in rotation, can emit gravitational waves. These
gravitational waves Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by H ...
are predicted to travel at the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. For example, planets orbiting the Sun constantly lose energy via gravitational radiation, but this effect is so small that it is unlikely it will be observed in the near future (Earth radiates about 200 watts of
gravitational radiation Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by ...
). The radiation of gravitational waves has been inferred from the Hulse–Taylor binary (and other binary pulsars). Precise timing of the pulses shows that the stars orbit only approximately according to
Kepler's Laws In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replaced circular orbits and epicycles in ...
: over time they gradually spiral towards each other, demonstrating an
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
loss in close agreement with the predicted energy radiated by gravitational waves. For their discovery of the first binary pulsar and measuring its orbital decay due to gravitational-wave emission, Hulse and
Taylor Taylor, Taylors or Taylor's may refer to: People * Taylor (surname) ** List of people with surname Taylor * Taylor (given name), including Tayla and Taylah * Taylor sept, a branch of Scottish clan Cameron * Justice Taylor (disambiguation) ...
won the 1993
Nobel Prize in Physics The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
. A "double pulsar" discovered in 2003,
PSR J0737-3039 PSR may refer to: Organizations * Pacific School of Religion, Berkeley, California, US * Palestinian Center for Policy and Survey Research * Payment Systems Regulator in the United Kingdom * Physicians for Social Responsibility, US Political p ...
, has a periastron precession of 16.90° per year; unlike the Hulse–Taylor binary, both
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s are detected as pulsars, allowing precision timing of both members of the system. Due to this, the tight orbit, the fact that the system is almost edge-on, and the very low transverse velocity of the system as seen from Earth, J0737−3039 provides by far the best system for strong-field tests of general relativity known so far. Several distinct relativistic effects are observed, including orbital decay as in the Hulse–Taylor system. After observing the system for two and a half years, four independent tests of general relativity were possible, the most precise (the Shapiro delay) confirming the general relativity prediction within 0.05% (nevertheless the periastron shift per orbit is only about 0.0013% of a circle and thus it is not a higher-order relativity test). In 2013, an international team of astronomers reported new data from observing a pulsar-white dwarf system
PSR J0348+0432 PSR may refer to: Organizations * Pacific School of Religion, Berkeley, California, US * Palestinian Center for Policy and Survey Research * Payment Systems Regulator in the United Kingdom * Physicians for Social Responsibility, US Political pa ...
, in which they have been able to measure a change in the orbital period of 8 millionths of a second per year, and confirmed GR predictions in a regime of extreme gravitational fields never probed before; but there are still some competing theories that would agree with these data.


Direct detection of gravitational waves

A number of gravitational-wave detectors have been built with the intent of directly detecting the gravitational waves emanating from such astronomical events as the merger of two
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s or
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s. In February 2016, the
Advanced LIGO The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory designed to detect cosmic gravitational waves and to develop gravitational-wave observations as an astronomical tool. Prior to LIG ...
team announced that they had directly detected gravitational waves from a stellar
binary black hole A binary black hole (BBH), or black hole binary, is an astronomical object consisting of two black holes in close orbit around each other. Like black holes themselves, binary black holes are often divided into binary stellar black holes, formed e ...
merger, with additional detections announced in June 2016, June 2017, and August 2017. General relativity predicts gravitational waves, as does any theory of gravitation in which changes in the gravitational field propagate at a finite speed. Then, the LIGO response function could discriminate among the various theories. Since gravitational waves can be directly detected, it is possible to use them to learn about the Universe. This is
gravitational-wave astronomy Gravitational-wave astronomy is a subfield of astronomy concerned with the detection and study of gravitational waves emitted by astrophysical sources. Gravitational waves are minute distortions or ripples in spacetime caused by the acceleration ...
. Gravitational-wave astronomy can test general relativity by verifying that the observed waves are of the form predicted (for example, that they only have two transverse polarizations), and by checking that
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s are the objects described by solutions of the
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
. Gravitational-wave astronomy can also test Maxwell-Einstein field equations. This version of the field equations predicts that spinning magnetars (i.e., neutron stars with extremely strong magnetic dipole field) should emit gravitational waves. "These amazing observations are the confirmation of a lot of theoretical work, including Einstein's general theory of relativity, which predicts gravitational waves", said Stephen Hawking.


Direct observation of black holes

The galaxy M87 was the subject of observation by the
Event Horizon Telescope The Event Horizon Telescope (EHT) is a Astronomical interferometer, telescope array consisting of a global network of radio telescopes. The EHT project combines data from several very-long-baseline interferometry (VLBI) stations around Earth, wh ...
(EHT) in 2017; the 10 April 2019 issue of ''
Astrophysical Journal Letters ''The Astrophysical Journal'' (''ApJ'') is a peer-reviewed scientific journal of astrophysics and astronomy, established in 1895 by American astronomers George Ellery Hale and James Edward Keeler. The journal discontinued its print edition and b ...
'' (vol. 875, No. 1) was dedicated to the EHT results, publishing six
open-access Open access (OA) is a set of principles and a range of practices through which nominally copyrightable publications are delivered to readers free of access charges or other barriers. With open access strictly defined (according to the 2001 de ...
papers. The
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside 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 c ...
of the black hole at the center of M87 was directly imaged at the wavelength of radio waves by the EHT; the image was revealed in a press conference on 10 April 2019, the first image of a black hole's event horizon. In May 2022, the EHT provided the first image of the super massive black hole Sagittarius A* in the center of our own Milky Way galaxy.


Gravitational redshift and orbit precession of star in strong gravity field

Gravitational redshift in light from the S2 star orbiting the supermassive black hole
Sagittarius A* Sagittarius A*, abbreviated as Sgr A* ( ), is the supermassive black hole at the Galactic Center of the Milky Way. Viewed from Earth, it is located near the border of the constellations Sagittarius and Scorpius, about 5.6° south o ...
in the center of the Milky Way has been measured with the
Very Large Telescope The Very Large Telescope (VLT) is an astronomical facility operated since 1998 by the European Southern Observatory, located on Cerro Paranal in the Atacama Desert of northern Chile. It consists of four individual telescopes, each equipped with ...
using GRAVITY, NACO and SIFONI instruments. Additionally, there has now been detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole.


Strong equivalence principle

The strong equivalence principle of general relativity requires universality of free fall to apply even to bodies with strong self-gravity. Direct tests of this principle using Solar System bodies are limited by the weak self-gravity of the bodies, and tests using pulsar–white-dwarf binaries have been limited by the weak gravitational pull of the Milky Way. With the discovery of a triple star system called
PSR J0337+1715 PSR may refer to: Organizations * Pacific School of Religion, Berkeley, California, US * Palestinian Center for Policy and Survey Research * Payment Systems Regulator in the United Kingdom * Physicians for Social Responsibility, US Political pa ...
, located about 4,200 light-years from Earth, the strong equivalence principle can be tested with a high accuracy. This system contains a
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
in a 1.6-day orbit with a
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
star, and the pair in a 327-day orbit with another white dwarf further away. This system permits a test that compares how the gravitational pull of the outer white dwarf affects the pulsar, which has strong self-gravity, and the inner white dwarf. The result shows that the accelerations of the pulsar and its nearby white-dwarf companion differ fractionally by no more than 2.6 (95% confidence level).


X-ray spectroscopy

This technique is based on the idea that photon trajectories are modified in the presence of a gravitational body. A very common astrophysical system in the universe is a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
surrounded by an
accretion disk An accretion disk is a structure (often a circumstellar disk) formed by diffuse material in orbital motion around a massive central body. The central body is most frequently a star. Friction, uneven irradiance, magnetohydrodynamic effects, and ...
. The radiation from the general neighborhood, including the accretion disk, is affected by the nature of the central black hole. Assuming Einstein's theory is correct, astrophysical black holes are described by the Kerr metric. (A consequence of the
no-hair theorem The no-hair theorem states that all stationary black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three independent ''externally'' observabl ...
s.) Thus, by analyzing the radiation from such systems, it is possible to test Einstein's theory. Most of the radiation from these black hole – accretion disk systems (e.g., black hole binaries and
active galactic nuclei An active galactic nucleus (AGN) is a compact region at the center of a galaxy that emits a significant amount of energy across the electromagnetic spectrum, with characteristics indicating that this luminosity is not produced by the stars. Such e ...
) arrives in the form of X-rays. When modeled, the radiation is decomposed into several components. Tests of Einstein's theory are possible with the thermal spectrum (only for black hole binaries) and the reflection spectrum (for both black hole binaries and active galactic nuclei). The former is not expected to provide strong constraints, while the latter is much more promising. In both cases, systematic uncertainties might make such tests more challenging.


Cosmological tests

Tests of general relativity on the largest scales are not nearly so stringent as Solar System tests. The earliest such test was the prediction and discovery of the
expansion of the universe The expansion of the universe is the increase in proper length, distance between Gravitational binding energy, gravitationally unbound parts of the observable universe with time. It is an intrinsic and extrinsic properties (philosophy), intrins ...
. In 1922,
Alexander Friedmann Alexander Alexandrovich Friedmann (also spelled Friedman or Fridman; ; ; – September 16, 1925) was a Russian and Soviet physicist and mathematician. He originated the pioneering theory that the universe is expanding, governed by a set of eq ...
found that the Einstein equations have non-stationary solutions (even in the presence of the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is a coefficient that Albert Einstein initially added to his field equations of general rel ...
). In 1927,
Georges Lemaître Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, and mathematician who made major contributions to cosmology and astrophysics. He was the first to argue that the ...
showed that static solutions of the Einstein equations, which are possible in the presence of the cosmological constant, are unstable, and therefore the static universe envisioned by Einstein could not exist (it must either expand or contract). W.Pauli, 1958, pp. 219–220 Lemaître made an explicit prediction that the universe should expand. Kragh, 2003, p. 153 He also derived a redshift-distance relationship, which is now known as the
Hubble Law Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther a galaxy is from the Earth, the faster ...
. Later, in 1931, Einstein himself agreed with the results of Friedmann and Lemaître. The expansion of the universe discovered by
Edwin Hubble Edwin Powell Hubble (November 20, 1889 – September 28, 1953) was an American astronomer. He played a crucial role in establishing the fields of extragalactic astronomy and observational cosmology. Hubble proved that many objects previously ...
in 1929 was then considered by many (and continues to be considered by some now) as a direct confirmation of general relativity. In the 1930s, largely due to the work of E. A. Milne, it was realised that the linear relationship between redshift and distance derives from the general assumption of uniformity and isotropy rather than specifically from general relativity. Rudnicki, 1991, p. 28. ''The Hubble Law was viewed by many as an observational confirmation of General Relativity in the early years'' However the prediction of a non-static universe was non-trivial, indeed dramatic, and primarily motivated by general relativity. Some other cosmological tests include searches for primordial gravitational waves generated during
cosmic inflation In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the very early universe. Following the inflationary period, the universe continued to expand, but at a slower ...
, which may be detected in the
cosmic microwave background The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
or by a proposed space-based gravitational-wave interferometer called the Big Bang Observer. Other tests at high redshift are constraints on other theories of gravity, and the variation of the gravitational constant since
Big Bang nucleosynthesis In physical cosmology, Big Bang nucleosynthesis (also known as primordial nucleosynthesis, and abbreviated as BBN) is a model for the production of light nuclei, deuterium, 3He, 4He, 7Li, between 0.01s and 200s in the lifetime of the universe ...
(it varied by no more than 40% since then). In August 2017, the findings of tests conducted by astronomers using the
European Southern Observatory The European Organisation for Astronomical Research in the Southern Hemisphere, commonly referred to as the European Southern Observatory (ESO), is an intergovernmental organization, intergovernmental research organisation made up of 16 m ...
's
Very Large Telescope The Very Large Telescope (VLT) is an astronomical facility operated since 1998 by the European Southern Observatory, located on Cerro Paranal in the Atacama Desert of northern Chile. It consists of four individual telescopes, each equipped with ...
(VLT), among other instruments, were released, and positively demonstrated gravitational effects predicted by Albert Einstein. One of these tests observed the orbit of the stars circling around
Sagittarius A* Sagittarius A*, abbreviated as Sgr A* ( ), is the supermassive black hole at the Galactic Center of the Milky Way. Viewed from Earth, it is located near the border of the constellations Sagittarius and Scorpius, about 5.6° south o ...
, a black hole about 4 million times as massive as the sun. Einstein's theory suggested that large objects bend the space around them, causing other objects to diverge from the straight lines they would otherwise follow. Although previous studies have validated Einstein's theory, this was the first time his theory had been tested on such a gigantic object. The findings were published in ''
The Astrophysical Journal ''The Astrophysical Journal'' (''ApJ'') is a peer-reviewed scientific journal of astrophysics and astronomy, established in 1895 by American astronomers George Ellery Hale and James Edward Keeler. The journal discontinued its print edition and ...
''.


Gravitational lensing

Astronomers using the Hubble Space Telescope and the Very Large Telescope have made precise tests of general relativity on galactic scales. The nearby galaxy ESO 325-G004 acts as a strong gravitational lens, distorting light from a distant galaxy behind it to create an
Einstein ring An Einstein ring, also known as an Einstein–Chwolson ring or Chwolson ring (named for Orest Chwolson), is created when light from a galaxy or star passes by a massive object en route to the Earth. Due to gravitational lensing, the light is di ...
around its centre. By comparing the mass of ESO 325-G004 (from measurements of the motions of stars inside this galaxy) with the curvature of space around it, astronomers found that gravity behaves as predicted by general relativity on these astronomical length-scales.


See also

*
General relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
*
Tests of special relativity Special relativity is a physical theory that plays a fundamental role in the description of all physical phenomena, as long as gravitation is not significant. Many experiments played (and still play) an important role in its development and justific ...


References


Notes


Other research papers

* * * * A. Einstein, "Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen", ''Jahrbuch der Radioaktivitaet und Elektronik'' 4 (1907); translated "On the relativity principle and the conclusions drawn from it", in ''The collected papers of Albert Einstein. Vol. 2 : The Swiss years: writings, 1900–1909'' (Princeton University Press, Princeton, New Jersey, 1989), Anna Beck translator. Einstein proposes the gravitational redshift of light in this paper, discussed online a
The Genesis of General Relativity
* A. Einstein, "Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes", ''Annalen der Physik'' 35 (1911); translated "On the Influence of Gravitation on the Propagation of Light" in ''The collected papers of Albert Einstein. Vol. 3 : The Swiss years: writings, 1909–1911'' (Princeton University Press, Princeton, New Jersey, 1994), Anna Beck translator, and in ''The Principle of Relativity'', (Dover, 1924), pp 99–108, W. Perrett and G. B. Jeffery translators, . The deflection of light by the sun is predicted from the principle of equivalence. Einstein's result is half the full value found using the general theory of relativity. * * M. Froeschlé, F. Mignard and F. Arenou,
Determination of the PPN parameter γ with the Hipparcos data
Hipparcos Venice '97, ESA-SP-402 (1997). * * * *


Textbooks

* S. M. Carroll,
Spacetime and Geometry: an Introduction to General Relativity
', Addison-Wesley, 2003. A graduate-level general relativity textbook. * A. S. Eddington,
Space, Time and Gravitation
', Cambridge University Press, reprint of 1920 ed. * A. Gefter, "Putting Einstein to the Test", ''Sky and Telescope'' July 2005, p. 38. A popular discussion of tests of general relativity. * H. Ohanian and R. Ruffini, ''Gravitation and Spacetime, 2nd Edition'' Norton, New York, 1994, . A general relativity textbook. * * C. M. Will, ''Theory and Experiment in Gravitational Physics'', Cambridge University Press, Cambridge (1993). A standard technical reference. * C. M. Will, ''Was Einstein Right?: Putting General Relativity to the Test'', Basic Books (1993). This is a popular account of tests of general relativity.


Living Reviews papers

* * An online, technical review, covering much of the material in ''Theory and experiment in gravitational physics.'' It is less comprehensive but more up to date.


External links





(for amount of observed and GR shifts). {{DEFAULTSORT:Tests Of General Relativity Mercury (planet)