Editing General relativity (section)

== From classical mechanics to general relativity ==
General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a [[heuristic]] derivation of general relativity.<ref>The following exposition re-traces that of {{Harvnb|Ehlers|1973|loc=sec. 1}}</ref><ref>{{Cite web|last=Al-Khalili|first=Jim|date=26 March 2021|title=Gravity and Me: The force that shapes our lives|url=https://www.bbc.co.uk/programmes/b08kgv7f|access-date=9 April 2021|website=www.bbc.co.uk}}</ref>

=== Geometry of Newtonian gravity<!--'Einstein's elevator experiment' redirects here--> ===
[[File:Elevator gravity.svg|thumb|According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8&nbsp;m/s<sup>2</sup> (the acceleration due to gravity on the surface of the Earth).]]
At the base of [[classical mechanics]] is the notion that a [[physical body|body]]'s motion can be described as a combination of free (or [[inertia]]l) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second [[Newton's laws of motion|law of motion]], which states that the net [[force]] acting on a body is equal to that body's (inertial) [[mass]] multiplied by its [[acceleration]].<ref>{{Harvnb|Arnold|1989|loc=ch. 1}}</ref> The preferred inertial motions are related to the geometry of space and time: in the standard [[frame of reference|reference frames]] of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are [[geodesic]]s, straight [[world lines]] in [[curved spacetime]].<ref>{{Harvnb|Ehlers|1973|pp=5f}}</ref>

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as [[electromagnetism]] or [[friction]]), can be used to define the geometry of space, as well as a time [[coordinate]]. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of [[Loránd Eötvös|Eötvös]] and its successors (see [[Eötvös experiment]]), there is a universality of free fall (also known as the weak [[equivalence principle]], or the universal equality of inertial and passive-gravitational mass): the trajectory of a [[test body]] in free fall depends only on its position and initial speed, but not on any of its material properties.<ref>{{Harvnb|Will|1993|loc=sec. 2.4}}, {{Harvnb|Will|2006|loc=sec. 2}}</ref> A simplified version of this is embodied in '''Einstein's elevator experiment'''<!--boldface per WP:R#PLA-->, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.<ref>{{Harvnb|Wheeler|1990|loc=ch. 2}}</ref>

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific [[connection (mathematics)|connection]] which depends on the [[gradient]] of the [[gravitational potential]]. Space, in this construction, still has the ordinary [[Euclidean geometry]]. However, space''time'' as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not [[integrable systems|integrable]]. From this, one can deduce that spacetime is curved. The resulting [[Newton–Cartan theory]] is a geometric formulation of Newtonian gravity using only [[Covariance and contravariance of vectors#Informal usage|covariant]] concepts, i.e. a description which is valid in any desired coordinate system.<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.2}}, {{Harvnb|Havas|1964}}, {{Harvnb|Künzle|1972}}. The simple thought experiment in question was first described in {{Harvnb|Heckmann|Schücking|1959}}</ref> In this geometric description, [[tidal effect]]s—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.<ref>{{Harvnb|Ehlers|1973|pp=10f}}</ref>

=== Relativistic generalization ===
[[File:Light cone.svg|thumb|left|upright|[[Light cone]]]]
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a [[limiting case (philosophy of science)|limiting case]] of (special) relativistic mechanics.<ref>Good introductions are, in order of increasing presupposed knowledge of mathematics, {{Harvnb|Giulini|2005}}, {{Harvnb|Mermin|2005}}, and {{Harvnb|Rindler|1991}}; for accounts of precision experiments, cf. part IV of {{Harvnb|Ehlers|Lämmerzahl|2006}}</ref> In the language of [[symmetry]]: where gravity can be neglected, physics is [[Lorentz invariance|Lorentz invariant]] as in special relativity rather than [[Galilean invariance|Galilei invariant]] as in classical mechanics. (The defining symmetry of special relativity is the [[Poincaré group]], which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the [[speed of light]], and with high-energy phenomena.<ref>An in-depth comparison between the two symmetry groups can be found in {{Harvnb|Giulini|2006}}</ref>

With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each [[event (relativity)|event]] {{math|A}}, there is a set of events that can, in principle, either influence or be influenced by {{math|A}} via signals or interactions that do not need to travel faster than light (such as event {{math|B}} in the image), and a set of events for which such an influence is impossible (such as event {{math|C}} in the image). These sets are [[frame of reference|observer]]-independent.<ref>{{Harvnb|Rindler|1991|loc=sec. 22}}, {{Harvnb|Synge|1972|loc=ch. 1 and 2}}</ref> In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a [[conformal structure]]<ref>{{Harvnb|Ehlers|1973|loc=sec. 2.3}}</ref> or conformal geometry.

Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global [[inertial frame]]s. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight [[time-like]] lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.4}}, {{Harvnb|Schutz|1985|loc=sec. 5.1}}</ref>

A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of [[preferred frame]]s. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. [[#Gravitational time dilation and frequency shift|below]]). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.<ref>{{Harvnb|Ehlers|1973|pp=17ff}}; a derivation can be found in {{Harvnb|Mermin|2005|loc=ch. 12}}. For the experimental evidence, cf. the section [[#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]], below</ref> The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the [[Equivalence Principle#The Einstein equivalence principle|Einstein equivalence principle]], a crucial guiding principle for generalizing special-relativistic physics to include gravity.<ref>{{Harvnb|Rindler|2001|loc=sec. 1.13}}; for an elementary account, see {{Harvnb|Wheeler|1990|loc=ch. 2}}; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. {{Harvnb|Norton|1985}}</ref>

The same experimental data shows that time as measured by clocks in a gravitational field—[[proper time]], to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the [[Minkowski metric]]. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The [[metric tensor (general relativity)|metric tensor]] that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or [[pseudo-Riemannian]] metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the [[Levi-Civita connection]], and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable [[Local reference frame|locally inertial coordinates]], the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.4}} for the experimental evidence, see once more section [[#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]]. Choosing a different connection with non-zero [[torsion tensor|torsion]] leads to a modified theory known as [[Einstein–Cartan theory]]</ref>

=== Einstein's equations ===
{{Main|Einstein field equations|Mathematics of general relativity}}
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the [[energy–momentum tensor]], which includes both [[energy density|energy]] and momentum [[density|densities]] as well as [[stress (physics)|stress]]: [[pressure]] and shear.<ref>{{Harvnb|Ehlers|1973|p=16}}, {{Harvnb|Kenyon|1990|loc=sec. 7.2}}, {{Harvnb|Weinberg|1972|loc=sec. 2.8}}</ref> Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the [[field equation]] for gravity relates this tensor and the [[Ricci curvature|Ricci tensor]], which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, [[conservation of energy]]–momentum corresponds to the statement that the energy–momentum tensor is [[divergence]]-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-[[manifold]] counterparts, [[covariant derivative]]s studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations:
{{Equation box 1
|indent=:
|title='''Einstein's field equations'''
|equation=<math>G_{\mu\nu}\equiv R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} = \kappa T_{\mu\nu}\,</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

On the left-hand side is the [[Einstein tensor]], <math>G_{\mu\nu}</math>, which is symmetric and a specific divergence-free combination of the Ricci tensor <math>R_{\mu\nu}</math> and the metric. In particular,
: <math>R=g^{\mu\nu}R_{\mu\nu}</math>
is the curvature scalar. The Ricci tensor itself is related to the more general [[Riemann curvature tensor]] as
: <math>R_{\mu\nu}={R^\alpha}_{\mu\alpha\nu}.</math>

On the right-hand side, <math>\kappa</math> is a constant and <math>T_{\mu\nu}</math> is the energy–momentum tensor. All tensors are written in [[abstract index notation]].<ref>{{Harvnb|Ehlers|1973|pp=19–22}}; for similar derivations, see sections 1 and 2 of ch. 7 in {{Harvnb|Weinberg|1972}}. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. {{Harvnb|Lovelock|1972}}. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as [[Bianchi identities]], the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. {{Harvnb|Schutz|1985|loc=sec. 8.3}}</ref> Matching the theory's prediction to observational results for [[planet]]ary [[orbit]]s or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant <math>\kappa</math> is found to be <math display="inline">\kappa=\frac{8\pi G}{c^4}</math>, where <math>G</math> is the [[Newtonian constant of gravitation]] and <math>c</math> the speed of light in vacuum.<ref>{{Harvnb|Kenyon|1990|loc=sec. 7.4}}</ref> When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,
: <math>R_{\mu\nu}=0.</math>

In general relativity, the [[world line]] of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.

The [[Geodesics in general relativity|geodesic equation]] is:
: <math> {d^2 x^\mu \over ds^2}+\Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}=0,</math>
where <math>s</math> is a scalar parameter of motion (e.g. the [[proper time]]), and <math> \Gamma^\mu {}_{\alpha \beta}</math> are [[Christoffel symbols]] (sometimes called the [[affine connection]] coefficients or [[Levi-Civita connection]] coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the [[summation convention]] is used for repeated indices <math>\alpha</math> and <math>\beta</math>. The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to [[Newton's laws of motion]] which likewise provide formulae for the acceleration of a particle. This equation of motion employs the [[Einstein notation]], meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a [[test particle]] whose motion is described by the geodesic equation.

=== Total force in general relativity ===
{{See also|Two-body problem in general relativity}}
In general relativity, the effective [[gravitational potential energy]] of an object of mass ''m'' revolving around a massive central body ''M'' is given by<ref>{{cite book |title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity|last=Weinberg, Steven|publisher=John Wiley|year=1972|isbn=978-0-471-92567-5}}</ref><ref>{{cite book |title=Relativity, Gravitation and Cosmology: a Basic Introduction|last=Cheng, Ta-Pei|publisher=Oxford and New York: Oxford University Press|year=2005|isbn=978-0-19-852957-6}}</ref>

:<math>U_f(r) =-\frac{GMm}{r}+\frac{L^{2}}{2mr^{2}}-\frac{GML^{2}}{mc^{2}r^{3}}</math>

A conservative total [[force]] can then be obtained as its [[Force#Potential_energy|negative gradient]]

:<math>F_f(r)=-\frac{GMm}{r^{2}}+\frac{L^{2}}{mr^{3}}-\frac{3GML^{2}}{mc^{2}r^{4}}</math>

where ''L'' is the [[angular momentum]]. The first term represents the [[Newton's law of universal gravitation|force of Newtonian gravity]], which is described by the inverse-square law. The second term represents the [[centrifugal force]] in the circular motion. The third term represents the relativistic effect.

=== Alternatives to general relativity ===
{{Main|Alternatives to general relativity}}
There are [[alternatives to general relativity]] built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are [[Whitehead's theory of gravitation|Whitehead's theory]], [[Brans–Dicke theory]], [[teleparallelism]], [[f(R) gravity|''f''(''R'') gravity]] and [[Einstein–Cartan theory]].<ref>{{Harvnb|Brans|Dicke|1961}}, {{Harvnb|Weinberg|1972|loc=sec. 3 in ch. 7}}, {{Harvnb|Goenner|2004|loc=sec. 7.2}}, and {{Harvnb|Trautman|2006}}, respectively</ref>