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). Although no generally agreed definition exists, an exact solution of the Einstein field equation is usually taken to mean a Lorentz metric which solves the field equation given an energy-momentum tensor. Often the requirement of symmetry conditions on the metric, initial and boundary conditions on the spacetime or algebraic restrictions on the Riemann tensor are imposed in obtaining exact solutions. For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

Some well-known and popular exact solutions include:

1. Schwarzschild metric (which describes the spacetime geometry around a spherical mass)
2. Kerr metric (which describes the geometry around a rotating spherical mass)
3. Reissner-Nordstrom metric (which describes the geometry around a charged spherical mass)
4. Kerr-Newman metric (which describes the geometry around a charged-rotating spherical mass)
5. Friedmann-Robertson-Walker (FRW) metric (which is an important model of an expanding universe)
6. pp-wave spacetimes (which describe various types of gravitational waves)
7. wormhole metrics (which serve as theoretical models for time travel)
8. Alcubierre metric (which serves as a theoretical model of space travel)

Solutions (1), (2), (3) and (4) also include black holes. This article needs cleanup. ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... In physics and astronomy, a Reissner-Nordstrøm black hole is a black hole that carries electric charge , no angular momentum, and mass . ... The Kerr-Newman metric is a solution of Einsteins field equations that describes the spacetime geometry around a charged (), rotating () black hole of mass m. ... The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ... Introduction In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einsteins field equation. ... The Alcubierre metric, sometimes known as the Alcubierre Drive (metric) or the Warp Drive spacetime, is a solution of the field equations of general relativity that is used to model faster than light travel. ... An artists impression of a black hole with a closely orbiting companion star that exceeds its Roche limit. ...

Techniques

Exact solutions may be found by using a variety of techniques, many of them being algebraic ones. Certain classifications of tensors are useful in studying exact solutions, for example the Petrov classification of the Weyl tensor and the Segre classification of the energy-momentum tensor. There is also a method of classifying the Riemann tensor algebraically. Other techniques involve the study of symmetries in general relativity, where, for example, certain vector fields are imposed on the spacetime and the resulting metrics are to be found. The Petrov classsification provides a means of algebraically classifying the Weyl tensor. ... In differential geometry, the Weyl curvature tensor is the traceless component of the Riemann curvature tensor. ... The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

In the study of exact solutions, and in particular, of symmetries in general relativity, it is often useful to decompose the Riemann tensor into its trace and trace-free parts. This is accomplished by taking the definition of the Weyl tensor in terms of the Riemann tensor and the Ricci tensor and making the Riemann tensor the subject of the formula. In four dimensions, this gives: In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...

where the Weyl tensor is the trace-free part (as it satisfies C^a_bad = 0) and the tensors G and E have the following components:

G_abcd = g_a[cg_d]b

where are the components of the trace-free Ricci tensor. The Petrov and Segre classifications may be used in conjunction with this expression for the Riemann tensor to write a list of all possible Riemann tensors.

Exact solutions for which the energy-momentum tensor is identically zero in the region under consideration are termed vacuum solutions and represent the gravitational field in a region of spacetime where there are no material gravitational sources. Strictly speaking, as the gravitational field can do work (in moving planets around the Sun, for example), the field possesses energy (although determining the precise location of this energy in the field is still a problem) and therefore by E = mc² has an effective mass which thereby creates another gravitational field - this feedback effect has the consequence that the gravitational field produced by two bodies is not the sum of the gravitational fields of the individual bodies. This is one of the major difficulties in finding exact solutions of the field equations, and quite often simplifying assumptions such as linearising the field equations are made. Vacuum solutions of the Einstein field equations are those for which the energy-momentum tensor is zero in a given region of spacetime, i. ...

References

• Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. and Herlt, E. Exact Solutions of Einstein's Field Equations (2nd edn.) (2003) CUP ISBN 0521461367

• Adler, R., Bazin, M. and Schiffer, M. Introduction to General Relativity (2nd edn.) (1975) McGraw-Hill New York ISBN 0-07-000423-4