Editing General relativity (section)

=== 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:
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|equation=<math>G_{\mu\nu}\equiv R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} = \kappa T_{\mu\nu}\,</math>
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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.