For other topics related to Einstein see Einstein (disambig)
Introduction
In physics, the Einstein field equation or Einstein equation is a tensor equation in the theory of gravitation. It is the dynamical equation of the physical theory called general relativity, describing how matter creates gravity (the curvature of spacetime) and, conversely, how gravity affects matter.
Mathematical form of the Einstein field equation
The Einstein field equation describes how spacetime is curved by matter, and (the other way round) how matter is influenced by the curvature of spacetime (i.e. how the curvature influences masses).
As it is a tensor equation, the Einstein field equation is usually written out in terms of its components. The resulting set of equations are then called the Einstein field equations (EFE's):
where G_{ab} are the components of the Einstein tensor, which is composed of derivatives of the metric tensor with components g_{ab}, and T_{ab} are the components of the stressenergy tensor and the constant is given in terms of π (pi), c (the speed of light) and G (the gravitational constant).
One of the solutions of the EFE's represents an expanding universe. In Einstein's time, nobody actually believed that the universe was expanding (even Einstein). To eliminate such a solution from arising, Einstein changed the equation to:
where R_{ab} are the components of the Ricci tensor, R is the Ricci scalar and Λ is the cosmological constant.
Using the definition of the Einstein tensor, the previous equation now reads:
The metric, with components g_{ab}, is a symmetric 4 x 4 tensor, so it has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6 in number.
These equations, together with the geodesic equation, form the core of the mathematical formulation of general relativity.
See also EinsteinHilbert action
(Exact) solutions of the field equation
Strictly speaking, any Lorentz metric is a solution of the Einstein field equation, as this amounts to nothing more than a mathematical definition of the energymomentum tensor (by the field equations). An exact solution is a metric which corresponds to a physically realizable energymomentum tensor. Exact solutions are sometimes termed 'metrics'.
Some wellknown and popular metrics include:

Schwarzschild metric (which describes the spacetime geometry around a spherical mass)

Kerr metric (which describes the geometry around a rotating spherical mass)

ReissnerNordstrom metric (which describes the geometry around a charged spherical mass)

KerrNewman metric (which describes the geometry around a chargedrotating spherical mass)

FriedmannRobertsonWalker (FRW) metric (which is an important model of an expanding universe)

ppwave metrics (which describe various types of gravitational waves)

wormhole metrics (which serve as theoretical models for time travel)

Alcubierre metric (which serves as a theoretical model of space travel)
Solutions (1), (2), (3) and (4) also include black holes.
The correspondence principle
Einstein's equation reduces to Newton's law of gravity by using both the weakfield approximation and the slowmotion approximation . In fact, the constant appearing in the EFE's is determined by making these two approximations.
Initial value and Cauchy problems
References
 Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972) ISBN 0471925675
 Stephani, H., Kramer, D., MacCallum, M., Hoenselaers C. and Herlt, E. Exact Solutions of Einstein's Field Equations (2nd edn.) (2003) CUP ISBN 0521461367
Last updated: 05112005 00:37:45