Editing General relativity (section)

== Consequences of Einstein's theory ==
General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication.

=== Gravitational time dilation and frequency shift ===
{{Main|Gravitational time dilation}}
[[File:Gravitational red-shifting.png|thumb|Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body]]
Assuming that the equivalence principle holds,<ref>{{Harvnb|Rindler|2001|pp=24–26 vs. pp. 236–237}} and {{Harvnb|Ohanian|Ruffini|1994|pp=164–172}}. Einstein derived these effects using the equivalence principle as early as 1907, cf. {{Harvnb|Einstein|1907}} and the description in {{Harvnb|Pais|1982|pp=196–198}}</ref> gravity influences the passage of time. Light sent down into a [[gravity well]] is [[blueshift]]ed, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is [[redshift]]ed; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.<ref>{{Harvnb|Rindler|2001|pp=24–26}}; {{Harvnb|Misner|Thorne|Wheeler|1973 |loc=§&nbsp;38.5}}</ref>

Gravitational redshift has been measured in the laboratory<ref>[[Pound–Rebka experiment]], see {{Harvnb|Pound|Rebka|1959}}, {{Harvnb|Pound|Rebka|1960}}; {{Harvnb|Pound|Snider|1964}}; a list of further experiments is given in {{Harvnb|Ohanian|Ruffini|1994|loc=table 4.1 on p. 186}}</ref> and using astronomical observations.<ref>{{Harvnb|Greenstein|Oke|Shipman|1971}}; the most recent and most accurate Sirius B measurements are published in {{Harvnb|Barstow, Bond et al.|2005}}.</ref> Gravitational time dilation in the Earth's gravitational field has been measured numerous times using [[atomic clocks]],<ref>Starting with the [[Hafele–Keating experiment]], {{Harvnb|Hafele|Keating|1972a}} and {{Harvnb|Hafele|Keating|1972b}}, and culminating in the [[Gravity Probe A]] experiment; an overview of experiments can be found in {{Harvnb|Ohanian|Ruffini|1994|loc=table 4.1 on p. 186}}</ref> while ongoing validation is provided as a side effect of the operation of the [[Global Positioning System]] (GPS).<ref>GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see {{Harvnb|Ashby|2002}} and {{Harvnb|Ashby|2003}}</ref> Tests in stronger gravitational fields are provided by the observation of [[binary pulsar]]s.<ref>{{Harvnb|Stairs|2003}} and {{Harvnb|Kramer|2004}}</ref> All results are in agreement with general relativity.<ref>General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; {{Harvnb|Ohanian|Ruffini|1994|loc=sec. 4.2}}</ref> However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.<ref>{{Harvnb|Ohanian|Ruffini|1994|pp=164–172}}</ref>

=== Light deflection and gravitational time delay ===
{{Main|Schwarzschild geodesics|Kepler problem in general relativity|Gravitational lens|Shapiro delay}}
[[File:Light deflection.png|thumb|left|upright|Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)]]
General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a massive object. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes the [[Sun]].<ref>Cf. {{Harvnb|Kennefick|2005}} for the classic early measurements by Arthur Eddington's expeditions. For an overview of more recent measurements, see {{Harvnb|Ohanian|Ruffini|1994|loc=ch. 4.3}}. For the most precise direct modern observations using quasars, cf. {{Harvnb|Shapiro|Davis|Lebach|Gregory|2004}}</ref>

This and related predictions follow from the fact that light follows what is called a light-like or [[Geodesic (general relativity)|null geodesic]]—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the [[Invariant (mathematics)|invariance]] of lightspeed in special relativity.<ref>This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell [[Lagrangian (field theory)|Lagrangian]] using a [[WKB approximation]], cf. {{Harvnb|Ehlers|1973|loc=sec. 5}}</ref> As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),<ref>{{Harvnb|Blanchet|2006|loc=sec. 1.3}}</ref> several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,<ref>{{Harvnb|Rindler|2001|loc=sec. 1.16}}; for the historical examples, {{Harvnb|Israel|1987|pp=202–204}}; in fact, Einstein published one such derivation as {{Harvnb|Einstein|1907}}. Such calculations tacitly assume that the geometry of space is [[Euclidean space|Euclidean]], cf. {{Harvnb|Ehlers|Rindler|1997}}</ref> the angle of deflection resulting from such calculations is only half the value given by general relativity.<ref>From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. {{Harvnb|Rindler|2001|loc=sec. 11.11}}</ref>

Closely related to light deflection is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.<ref>For the Sun's gravitational field using radar signals reflected from planets such as [[Venus]] and Mercury, cf. {{Harvnb|Shapiro|1964}}, {{Harvnb|Weinberg|1972|loc=ch. 8, sec. 7}}; for signals actively sent back by space probes ([[transponder]] measurements), cf. {{Harvnb|Bertotti|Iess|Tortora|2003}}; for an overview, see {{Harvnb|Ohanian|Ruffini|1994|loc=table 4.4 on p. 200}}; for more recent measurements using signals received from a [[pulsar]] that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. {{Harvnb|Stairs|2003|loc=sec. 4.4}}</ref> In the [[parameterized post-Newtonian formalism]] (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.<ref>{{Harvnb|Will|1993|loc=sec. 7.1 and 7.2}}</ref>
{{clear}}

=== Gravitational waves ===
{{Main|Gravitational wave}}
[[File:Gravwav.gif|thumb|Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave]]
Predicted in 1916<ref>{{cite journal|author=Einstein, A|title=Näherungsweise Integration der Feldgleichungen der Gravitation|date=22 June 1916|url=http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|journal=[[Prussian Academy of Sciences|Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin]]|issue=part&nbsp;1|pages=688–696|bibcode=1916SPAW.......688E|access-date=12 February 2016|archive-url=https://web.archive.org/web/20190321062928/http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|archive-date=21 March 2019}}</ref><ref>{{cite journal|author=Einstein, A|title=Über Gravitationswellen|date=31 January 1918|url=http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin|issue=part&nbsp;1|pages=154–167|bibcode=1918SPAW.......154E|access-date=12 February 2016|archive-url=https://web.archive.org/web/20190321062928/http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|archive-date=21 March 2019}}</ref> by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to [[electromagnetic wave]]s. On 11 February 2016, the Advanced LIGO team announced that they had [[Gravitational wave observation|directly detected gravitational waves]] from a [[Binary black hole|pair]] of black holes [[Stellar collision|merging]].<ref name="Discovery 2016">{{cite journal |title=Einstein's gravitational waves found at last |journal=Nature News| url=http://www.nature.com/news/einstein-s-gravitational-waves-found-at-last-1.19361 |date=11 February 2016 |last1= Castelvecchi |first1=Davide |last2=Witze |first2=Witze |doi=10.1038/nature.2016.19361 |s2cid=182916902|access-date= 11 February 2016 }}</ref><ref name="Abbot">{{cite journal |title=Observation of Gravitational Waves from a Binary Black Hole Merger| author1=B. P. Abbott |collaboration=LIGO Scientific Collaboration and Virgo Collaboration| journal=Physical Review Letters| year=2016| volume=116|issue=6| doi= 10.1103/PhysRevLett.116.061102 | pmid=26918975| page=061102|arxiv = 1602.03837 |bibcode = 2016PhRvL.116f1102A | s2cid=124959784 }}</ref><ref name="NSF">{{cite web|title = Gravitational waves detected 100 years after Einstein's prediction |website= NSF – National Science Foundation|url = https://www.nsf.gov/news/news_summ.jsp?cntn_id=137628 |date = 11 February 2016}}</ref>

The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right).<ref>Most advanced textbooks on general relativity contain a description of these properties, e.g. {{Harvnb|Schutz|1985|loc=ch. 9}}</ref> Since Einstein's equations are [[non-linear]], arbitrarily strong gravitational waves do not obey [[linear superposition]], making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by <math>10^{-21}</math> or less. Data analysis methods routinely make use of the fact that these linearized waves can be [[Fourier decomposition|Fourier decomposed]].<ref>For example {{Harvnb|Jaranowski|Królak|2005}}</ref>

Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space<ref>{{Harvnb|Rindler|2001|loc=ch. 13}}</ref> or [[Gowdy universe]]s, varieties of an expanding cosmos filled with gravitational waves.<ref>{{Harvnb|Gowdy|1971}}, {{Harvnb|Gowdy|1974}}</ref> But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.<ref>See {{Harvnb|Lehner|2002}} for a brief introduction to the methods of numerical relativity, and {{Harvnb|Seidel|1998}} for the connection with gravitational wave astronomy</ref>

=== Orbital effects and the relativity of direction ===
{{Main|Two-body problem in general relativity}}
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation ([[precession]]) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.

==== Precession of apsides ====
[[File:Relativistic precession.svg|thumb|upright=1.05|Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star. The influence of other planets is ignored.]]
{{Main|Apsidal precession}}

In general relativity, the [[apsis|apsides]] of any orbit (the point of the orbiting body's closest approach to the system's [[center of mass]]) will [[apsidal precession|precess]]; the orbit is not an [[ellipse]], but akin to an ellipse that rotates on its focus, resulting in a [[rose (mathematics)|rose curve]]-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a [[test particle]]. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by [[Urbain Le Verrier]] in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.<ref>{{Harvnb|Schutz|2003|pp=48–49}}, {{Harvnb|Pais|1982|pp=253–254}}</ref>

The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)<ref>{{Harvnb|Rindler|2001|loc=sec. 11.9}}</ref> or the much more general [[post-Newtonian formalism]].<ref>{{Harvnb|Will|1993|pp=177–181}}</ref> It is due to the influence of gravity on the geometry of space and to the contribution of [[self-energy]] to a body's gravity (encoded in the [[nonlinearity]] of Einstein's equations).<ref>In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. {{Harvnb|Will|2006|loc=sec. 3.5}} and {{Harvnb|Will|1993|loc=sec. 7.3}}</ref> Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),<ref>The most precise measurements are [[VLBI]] measurements of planetary positions; see {{Harvnb|Will|1993|loc=ch. 5}}, {{Harvnb|Will|2006|loc=sec. 3.5}}, {{Harvnb|Anderson|Campbell|Jurgens|Lau|1992}}; for an overview, {{Harvnb|Ohanian|Ruffini|1994|pp=406–407}}</ref> as well as in binary pulsar systems, where it is larger by five [[order of magnitude|orders of magnitude]].<ref>{{Harvnb|Kramer|Stairs|Manchester|McLaughlin|2006}}</ref>

In general relativity the perihelion shift <math>\sigma</math>, expressed in radians per revolution, is approximately given by:{{sfn|Dediu|Magdalena|Martín-Vide|2015|p=[https://books.google.com/books?id=XmwiCwAAQBAJ&pg=PA141 141]}}

:<math>\sigma=\frac {24\pi^3L^2} {T^2c^2(1-e^2)} \ ,</math>

where:
*<math>L</math> is the [[semi-major axis]]
*<math>T</math> is the [[orbital period]]
*<math>c</math> is the speed of light in vacuum
*<math>e</math> is the [[orbital eccentricity]]

==== Orbital decay ====
<!--This subsection is linked to from the subsection Gravitational Waves in Astrophysical Applications, please do not change its title -->

[[File:PSRJ0737−3039shift2021.png|thumb|upright=0.8|Orbital decay for PSR J0737−3039: time shift (in [[second|s]]), tracked over 16 years (2021).<ref name=":1">{{Cite journal|last1=Kramer|first1=M.|last2=Stairs|first2=I. H.|last3=Manchester|first3=R. N.|last4=Wex|first4=N.|last5=Deller|first5=A. T.|last6=Coles|first6=W. A.|last7=Ali|first7=M.|last8=Burgay|first8=M.|last9=Camilo|first9=F.|last10=Cognard|first10=I.|last11=Damour|first11=T.|date=13 December 2021|title=Strong-Field Gravity Tests with the Double Pulsar|url=https://link.aps.org/doi/10.1103/PhysRevX.11.041050|journal=Physical Review X|language=en|volume=11|issue=4|page=041050|doi=10.1103/PhysRevX.11.041050|arxiv=2112.06795|bibcode=2021PhRvX..11d1050K|s2cid=245124502|issn=2160-3308}}</ref>]]

According to general relativity, a [[Binary system (astronomy)|binary system]] will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the [[Solar System]] or for ordinary [[double star]]s, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting [[neutron star]]s, one of which is a [[pulsar]]: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.<ref>{{Harvnb|Stairs|2003}}, {{Harvnb|Schutz|2003|pp=317–321}}, {{Harvnb|Bartusiak|2000|pp=70–86}}</ref>

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by [[Russell Alan Hulse|Hulse]] and [[Joseph Hooton Taylor, Jr.|Taylor]], using the binary pulsar [[PSR1913+16]] they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 [[Nobel Prize]] in physics.<ref>{{Harvnb|Weisberg|Taylor|2003}}; for the pulsar discovery, see {{Harvnb|Hulse|Taylor|1975}}; for the initial evidence for gravitational radiation, see {{Harvnb|Taylor|1994}}</ref> Since then, several other binary pulsars have been found, in particular the double pulsar [[PSR J0737−3039]], where both stars are pulsars<ref>{{Harvnb|Kramer|2004}}</ref> and which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations.<ref name=":1" />

==== Geodetic precession and frame-dragging ====
{{Main|Geodetic precession|Frame dragging}}
Several relativistic effects are directly related to the relativity of direction.<ref>{{Harvnb|Penrose|2004|loc=§&nbsp;14.5}}, {{Harvnb|Misner|Thorne|Wheeler|1973|loc=§&nbsp;11.4}}</ref> One is [[geodetic effect|geodetic precession]]: the axis direction of a [[gyroscope]] in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("[[parallel transport]]").<ref>{{Harvnb|Weinberg|1972|loc=sec. 9.6}}, {{Harvnb|Ohanian|Ruffini|1994|loc=sec. 7.8}}</ref> For the Moon–Earth system, this effect has been measured with the help of [[lunar laser ranging]].<ref>{{Harvnb|Bertotti|Ciufolini|Bender|1987}}, {{Harvnb|Nordtvedt|2003}}</ref> More recently, it has been measured for test masses aboard the satellite [[Gravity Probe B]] to a precision of better than 0.3%.<ref>{{Harvnb|Kahn|2007}}</ref><ref>A mission description can be found in {{Harvnb|Everitt|Buchman|DeBra|Keiser|2001}}; a first post-flight evaluation is given in {{Harvnb|Everitt|Parkinson|Kahn|2007}}; further updates will be available on the mission website {{Harvnb|Kahn|1996–2012}}.</ref>

Near a rotating mass, there are gravitomagnetic or [[frame-dragging]] effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for [[Kerr solution|rotating black holes]] where, for any object entering a zone known as the [[ergosphere]], rotation is inevitable.<ref>{{Harvnb|Townsend|1997|loc=sec. 4.2.1}}, {{Harvnb|Ohanian|Ruffini|1994|pp=469–471}}</ref> Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.<ref>{{Harvnb|Ohanian|Ruffini|1994|loc=sec. 4.7}}, {{Harvnb|Weinberg|1972|loc=sec. 9.7}}; for a more recent review, see {{Harvnb|Schäfer|2004}}</ref> Somewhat controversial tests have been performed using the [[LAGEOS]] satellites, confirming the relativistic prediction.<ref>{{Harvnb|Ciufolini|Pavlis|2004}}, {{Harvnb|Ciufolini|Pavlis|Peron|2006}}, {{Harvnb|Iorio|2009}}</ref> Also the [[Mars Global Surveyor]] probe around Mars has been used.<ref>{{Harvnb|Iorio|2006}}, {{Harvnb|Iorio|2010}}</ref>