Editing General relativity (section)

=== 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>