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 Einstein's theory of general relativity. It is a dynamical equation which describes how matter changes the geometry of spacetime, this curved geometry being interpreted as the gravitational field of the matter source. The motion of objects (with a mass much smaller than the matter source) in this gravitational field is described by the geodesic equation. Physics (from the Greek, φυσικός (phusikos), natural, and φύσις (phusis), nature) is the science of nature in the broadest sense. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... Two-dimensional visualisation of space-time distortion. ... World line of the orbit of the Earth depicted as a circle in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, Z, making the circle appear as a helix. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ...

Mathematical form of the Einstein field equation

The Einstein field equation is usually written 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 stress-energy tensor and the constant is given in terms of π (pi), c (the speed of light) and G (the gravitational constant). In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds and which is defined in index-free notation as, where is the Ricci tensor, is the metric tensor and is the Ricci scalar (or scalar curvature). ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... The stress-energy tensor is a tensor quantity in relativity. ... The minuscule, or lower-case, pi The mathematical constant π is commonly used in mathematics and physics. ... Cherenkov effect in a swimming pool nuclear reactor. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

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: Accelerating universe is a term for the idea that our universe is undergoing divergent rapid expansion. ...

where R_ab are the components of the Ricci tensor, R is the Ricci scalar and Λ is the cosmological constant. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) occurs in Einsteins theory of general relativity. ...

Using the definition of the Einstein tensor, the previous equation now reads: In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds and which is defined in index-free notation as, where is the Ricci tensor, is the metric tensor and is the Ricci scalar (or scalar curvature). ...

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. A tensor A, with components Aij, is said to be symmetric if Aij = Aji for all i, j. ...

These equations, together with the geodesic equation, form the core of the mathematical formulation of general relativity. In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... Two-dimensional visualisation of space-time distortion. ...

See also Einstein-Hilbert action In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen...

(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:

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 metrics (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 physics, he Kerr metric describes the geometry of spacetime around 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. ... 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. ...

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