Einstein's equations
Main articles: Einstein field equations and 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 and momentum densities as well as stress: pressure and shear.[38] 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 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 derivatives 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 set of equations are what are called Einstein's (field) equations:
Einstein's field equations
G μ ν ≡ R μ ν − 1 2 R g μ ν = 8 π G c 4 T μ ν {\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }\,} G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }\,
On the left-hand side is the Einstein tensor, G μ ν {\displaystyle G_{\mu \nu }} G_{\mu \nu }, which is symmetric and a specific divergence-free combination of the Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} R_{\mu \nu } and the metric. In particular,
R = g μ ν R μ ν {\displaystyle R=g^{\mu \nu }R_{\mu \nu }\,} R=g^{\mu \nu }R_{\mu \nu }\,
is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as
R μ ν = R α μ α ν . {\displaystyle R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.\,} R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.\,
On the right-hand side, T μ ν {\displaystyle T_{\mu \nu }} T_{\mu \nu } is the energy–momentum tensor. All tensors are written in abstract index notation.[39] Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant is found to be 8 π G c 4 {\textstyle {\frac {8\pi G}{c^{4}}}} {\textstyle {\frac {8\pi G}{c^{4}}}}, where G {\displaystyle G} G is the gravitational constant and c {\displaystyle c} c the speed of light in vacuum.[40] When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,
R μ ν = 0. {\displaystyle R_{\mu \nu }=0.\,} R_{\mu \nu }=0.\,
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 geodesic equation is:
d 2 x μ d s 2 + Γ μ α β d x α d s d x β d s = 0 , {\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,} {\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,}
where s {\displaystyle s} s is a scalar parameter of motion (e.g. the proper time), and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} 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 α {\displaystyle \alpha } \alpha and β {\displaystyle \beta } \beta . 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.