In the
general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of
spacetime to the distribution of
matter within it.^{
[1]}
The equations were published by Einstein in 1915 in the form of a
tensor equation^{
[2]} which related the local spacetime
curvature (expressed by the
Einstein tensor) with the local energy,
momentum and stress within that spacetime (expressed by the
stress–energy tensor).^{
[3]}
Analogously to the way that
electromagnetic fields are related to the distribution of
charges and
currents via
Maxwell's equations, the EFE relate the
spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the
metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear
partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The
inertial trajectories of particles and radiation (
geodesics) in the resulting geometry are then calculated using the
geodesic equation.
As well as implying local energy–momentum conservation, the EFE reduce to
Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the
speed of light.^{
[4]}
Exact solutions for the EFE can only be found under simplifying assumptions such as
symmetry. Special classes of
exact solutions are most often studied since they model many gravitational phenomena, such as
rotating black holes and the
expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from
flat spacetime, leading to the
linearized EFE. These equations are used to study phenomena such as
gravitational waves.
where R_{μν} is the
Ricci curvature tensor, and R is the
scalar curvature. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives.
The Einstein gravitational constant is defined as^{
[6]}^{
[7]}
In standard units, each term on the left has units of 1/length^{2}.
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.
The EFE is a tensor equation relating a set of
symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four
Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four
gauge-fixingdegrees of freedom, which correspond to the freedom to choose a coordinate system.
Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions.^{
[9]} The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T_{μν} is everywhere zero) define
Einstein manifolds.
The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor $g_{\mu \nu }$, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic
partial differential equations.^{
[10]}
Sign convention
The above form of the EFE is the standard established by
Misner, Thorne, and Wheeler (MTW).^{
[11]} The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):
With these definitions
Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972)^{
[12]} is (+ − −), Peebles (1980)^{
[13]} and Efstathiou et al. (1990)^{
[14]} are (− + +), Rindler (1977),^{[
citation needed]} Atwater (1974),^{[
citation needed]} Collins Martin & Squires (1989)^{
[15]} and Peacock (1999)^{
[16]} are (− + −).
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative:
The sign of the cosmological term would change in both these versions if the (+ − − −) metric
sign convention is used rather than the MTW (− + + +) metric sign convention adopted here.
Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace $g_{\mu \nu }$ in the expression on the right with the
Minkowski metric without significant loss of accuracy).
the term containing the
cosmological constantΛ was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a
universe that is not expanding or contracting. This effort was unsuccessful because:
any desired steady state solution described by this equation is unstable, and
Einstein then abandoned Λ, remarking to
George Gamow "that the introduction of the cosmological term was the biggest blunder of his life".^{
[17]}
The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent
astronomical observations have shown an
accelerating expansion of the universe, and to explain this a positive value of Λ is needed.^{
[18]}^{
[19]} The cosmological constant is negligible at the scale of a galaxy or smaller.
Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor:
where it is assumed that Λ has SI unit m^{−2} and κ is defined as above.
The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.
Features
Conservation of energy and momentum
General relativity is consistent with the local conservation of energy and momentum expressed as
which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.
Newtonian gravitation can be written as the theory of a scalar field, Φ, which is the gravitational potential in joules per kilogram of the gravitational field g = −∇Φ, see
Gauss's law for gravity
and that the metric and its derivatives are approximately static and that the squares of deviations from the
Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives
which reduces to the Newtonian field equation provided
${\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho \,$
which will occur if
$K={\frac {8\pi G}{c^{4}}}\,.$
Vacuum field equations
A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).
If the energy–momentum tensor T_{μν} is zero in the region under consideration, then the field equations are also referred to as the
vacuum field equations. By setting T_{μν} = 0 in the
trace-reversed field equations, the vacuum equations can be written as
$R_{\mu \nu }=0\,.$
In the case of nonzero cosmological constant, the equations are
is used, then the Einstein field equations are called the Einstein–Maxwell equations (with
cosmological constantΛ, taken to be zero in conventional relativity theory):
where the semicolon represents a
covariant derivative, and the brackets denote
anti-symmetrization. The first equation asserts that the 4-
divergence of the
2-formF is zero, and the second that its
exterior derivative is zero. From the latter, it follows by the
Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential A_{α} such that
in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived.^{
[20]} However, there are global solutions of the equation that may lack a globally defined potential.^{
[21]}
The solutions of the Einstein field equations are
metrics of
spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as
post-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called
exact solutions.^{
[9]}
The study of exact solutions of Einstein's field equations is one of the activities of
cosmology. It leads to the prediction of
black holes and to different models of evolution of the
universe.
One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.^{
[22]} In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,^{
[23]} self-similar solutions to the Einstein field equations are fixed points of the resulting
dynamical system. New solutions have been discovered using these methods by LeBlanc^{
[24]} and Kohli and Haslam.^{
[25]}
The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the
gravitational field is very weak and the
spacetime approximates that of
Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the
Minkowski metric, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of
gravitational radiation.
Polynomial form
Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written
Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of det(g) to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.^{
[26]}
^With the choice of the Einstein gravitational constant as given here, κ = 8πG/c^{4}, the stress–energy tensor on the right side of the equation must be written with each component in units of energy density (i.e., energy per volume, equivalently pressure). In Einstein's original publication, the choice is κ = 8πG/c^{2}, in which case the stress–energy tensor components have units of mass density.