einstein's field equation

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 space-time is curved by matter, and (the other way round) how matter is influenced by the curvature of space-time (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):

G_{ab} = {8 \pi G \over c^4} T_{ab}

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 stress-energy tensor and the constant is given in terms of

\pi

(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:

G_{ab} = R_{ab} - {R \over 2} g_{ab}  + \Lambda g_{ab}

where

R_{ab}

are the components of the Ricci tensor,

R

is the Ricci scalar and

\Lambda

is the cosmological constant. Using the definition of the Einstein tensor, the previous equation now reads:

R_{ab} - {R \over 2} g_{ab} + \Lambda g_{ab} = {8 \pi G \over c^4} T_{ab}

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 Einstein-Hilbert 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 energy-momentum tensor (by the field equations). An exact solution is a metric which corresponds to a physically realizable energy-momentum tensor. Exact solutions are sometimes termed 'metrics'. Some well-known 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)
Reissner-Nordstrom metric (which describes the geometry around a charged spherical mass)
Kerr-Newman metric (which describes the geometry around a charged-rotating spherical mass)
Friedmann-Robertson-Walker (FRW) metric (which is an important model of an expanding universe)
pp-wave 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 weak-field approximation and the slow-motion 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

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