Einstein equation
$G\_\{\backslash mu\; \backslash nu\}\; +\; \backslash Lambda\; g\_\{\backslash mu\; \backslash nu\}=\; \{8\backslash pi\; G\backslash over\; c^4\}\; T\_\{\backslash mu\; \backslash nu\}$ 

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The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.^{[1]} First published by Einstein in 1915^{[2]} as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).^{[3]}
Similar to the way that electromagnetic fields are determined using charges and currents via Maxwell's equations, the EFE are used to determine the spacetime geometry resulting from the presence of massenergy and linear momentum, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy 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 obeying local energymomentum conservation, the EFE reduce to Newton's law of gravitation where the gravitational field is weak and velocities 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 as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as flat spacetime with a small deviation, leading to the linearised EFE. These equations are used to study phenomena such as gravitational waves.
Contents
Mathematical form
The Einstein field equations (EFE) may be written in the form:^{[1]}
Template:Equation box 1
where $R\_\{\backslash mu\; \backslash nu\}\backslash ,$ is the Ricci curvature tensor, $R\backslash ,$ the scalar curvature, $g\_\{\backslash mu\; \backslash nu\}\backslash ,$ the metric tensor, $\backslash Lambda\backslash ,$ is the cosmological constant, $G\backslash ,$ is Newton's gravitational constant, $c\backslash ,$ the speed of light in vacuum, and $T\_\{\backslash mu\; \backslash nu\}\backslash ,$ the stress–energy tensor.
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 fixing degrees 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 fourdimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define Einstein manifolds.
Despite the simple appearance of the equations they are actually quite complicated. 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 $g\_\{\backslash mu\; \backslash nu\}$, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolicelliptic partial differential equations.
One can write the EFE in a more compact form by defining the Einstein tensor
 $G\_\{\backslash mu\; \backslash nu\}\; =\; R\_\{\backslash mu\; \backslash nu\}\; \; \{1\; \backslash over\; 2\}R\; g\_\{\backslash mu\; \backslash nu\},$
which is a symmetric secondrank tensor that is a function of the metric. The EFE can then be written as
 $G\_\{\backslash mu\; \backslash nu\}\; +\; g\_\{\backslash mu\; \backslash nu\}\; \backslash Lambda\; =\; \{8\; \backslash pi\; G\; \backslash over\; c^4\}\; T\_\{\backslash mu\; \backslash nu\}.$
Using geometrized units where G = c = 1, this can be rewritten as
 $G\_\{\backslash mu\; \backslash nu\}\; +\; g\_\{\backslash mu\; \backslash nu\}\; \backslash Lambda\; =\; 8\; \backslash pi\; T\_\{\backslash mu\; \backslash nu\}\backslash ,.$
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.
These equations, together with the geodesic equation,^{[5]} which dictates how freelyfalling matter moves through spacetime, form the core of the mathematical formulation of general relativity.
Sign convention
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
 $$
\begin{align} g_{\mu \nu} & = [S1] \times \operatorname{diag}(1,+1,+1,+1) \\[6pt] {R^\mu}_{a \beta \gamma} & = [S2] \times (\Gamma^\mu_{a \gamma,\beta}\Gamma^\mu_{a \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma a}\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta a}) \\[6pt] G_{\mu \nu} & = [S3] \times {8 \pi G \over c^4} T_{\mu \nu} \end{align}
The third sign above is related to the choice of convention for the Ricci tensor:
 $R\_\{\backslash mu\; \backslash nu\}=[S2]\backslash times\; [S3]\; \backslash times\; \{R^a\}\_\{\backslash mu\; a\; \backslash nu\}$
With these definitions Misner, Thorne, and Wheeler classify themselves as $(+++)\backslash ,$, whereas Weinberg (1972) is $(+)\backslash ,$, Peebles (1980) and Efstathiou (1990) are $(++)\backslash ,$ while Peacock (1994), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are $(+)\backslash ,$.
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
 $R\_\{\backslash mu\; \backslash nu\}\; \; \{1\; \backslash over\; 2\}g\_\{\backslash mu\; \backslash nu\}\backslash ,R\; \; g\_\{\backslash mu\; \backslash nu\}\; \backslash Lambda\; =\; \{8\; \backslash pi\; G\; \backslash over\; c^4\}\; T\_\{\backslash mu\; \backslash nu\}.$
The sign of the (very small) cosmological term would change in both these versions, if the +−−− metric sign convention is used rather than the MTW −+++ metric sign convention adopted here.
Equivalent formulations
Taking the trace of both sides of the EFE one gets
 $R\; \; 2\; R\; +\; 4\; \backslash Lambda\; =\; \{8\; \backslash pi\; G\; \backslash over\; c^4\}\; T\; \backslash ,$
which simplifies to
 $R\; +\; 4\; \backslash Lambda\; =\; \{8\; \backslash pi\; G\; \backslash over\; c^4\}\; T\; \backslash ,.$
If one adds $\; \{1\; \backslash over\; 2\}\; g\_\{\backslash mu\; \backslash nu\}\; \backslash ,$ times this to the EFE, one gets the following equivalent "tracereversed" form
 $R\_\{\backslash mu\; \backslash nu\}\; \; g\_\{\backslash mu\; \backslash nu\}\; \backslash Lambda\; =\; \{8\; \backslash pi\; G\; \backslash over\; c^4\}\; \backslash left(T\_\{\backslash mu\; \backslash nu\}\; \; \{1\; \backslash over\; 2\}T\backslash ,g\_\{\backslash mu\; \backslash nu\}\backslash right)\; \backslash ,.$
Reversing the trace again would restore the original EFE. The tracereversed form may be more convenient in some cases (for example, when one is interested in weakfield limit and can replace $g\_\{\backslash mu\backslash nu\}\; \backslash ,$ in the expression on the right with the Minkowski metric without significant loss of accuracy).
The cosmological constant
Einstein modified his original field equations to include a cosmological term proportional to the metric
 $R\_\{\backslash mu\; \backslash nu\}\; \; \{1\; \backslash over\; 2\}g\_\{\backslash mu\; \backslash nu\}\backslash ,R\; +\; g\_\{\backslash mu\; \backslash nu\}\; \backslash Lambda\; =\; \{8\; \backslash pi\; G\; \backslash over\; c^4\}\; T\_\{\backslash mu\; \backslash nu\}\; \backslash ,.$
The constant $\backslash Lambda$ is the cosmological constant. Since $\backslash Lambda$ is constant, the energy conservation law is unaffected.
The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations of distant galaxies by Hubble a decade later confirmed that our universe is, in fact, not static but expanding. So $\backslash Lambda$ was abandoned, with Einstein calling it the "biggest blunder [he] ever made".^{[6]} For many years the cosmological constant was almost universally considered to be 0.
Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. Indeed, recent improved astronomical techniques have found that a positive value of $\backslash Lambda$ is needed to explain the accelerating universe.^{[7]}^{[8]}
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, written as part of the stress–energy tensor:
 $T\_\{\backslash mu\; \backslash nu\}^\{\backslash mathrm\{(vac)\}\}\; =\; \; \backslash frac\{\backslash Lambda\; c^4\}\{8\; \backslash pi\; G\}\; g\_\{\backslash mu\; \backslash nu\}\; \backslash ,.$
The resulting vacuum energy is constant and given by
 $\backslash rho\_\{\backslash mathrm\{vac\}\}\; =\; \backslash frac\{\backslash Lambda\; c^2\}\{8\; \backslash pi\; G\}$
The existence of a cosmological constant is thus equivalent to the existence of a nonzero vacuum energy. The terms are now 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
 $\backslash nabla\_\backslash beta\; T^\{\backslash alpha\backslash beta\}\; \backslash ,\; =\; T^\{\backslash alpha\backslash beta\}\{\}\_\{;\backslash beta\}\; \backslash ,\; =\; 0$.
Derivation of local energymomentum conservation Contracting the differential Bianchi identity
 $R\_\{\backslash alpha\backslash beta[\backslash gamma\backslash delta;\backslash varepsilon]\}\; =\; \backslash ,\; 0$
with $g^\{\backslash alpha\backslash gamma\}$ gives, using the fact that the metric tensor is covariantly constant, i.e. $g^\{\backslash alpha\backslash beta\}\{\}\_\{;\backslash gamma\}=0$,
 $R^\backslash gamma\{\}\_\{\backslash beta\backslash gamma\backslash delta;\backslash varepsilon\}\; +\; \backslash ,\; R^\backslash gamma\{\}\_\{\backslash beta\backslash varepsilon\backslash gamma;\backslash delta\}\; +\; \backslash ,\; R^\backslash gamma\{\}\_\{\backslash beta\backslash delta\backslash varepsilon;\backslash gamma\}\; =\; \backslash ,\; 0$
The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
 $R^\backslash gamma\{\}\_\{\backslash beta\backslash gamma\backslash delta;\backslash varepsilon\}\; \backslash ,\; \; R^\backslash gamma\{\}\_\{\backslash beta\backslash gamma\backslash varepsilon;\backslash delta\}\; \backslash ,\; +\; R^\backslash gamma\{\}\_\{\backslash beta\backslash delta\backslash varepsilon;\backslash gamma\}\; \backslash ,\; =\; 0$
which is equivalent to
 $R\_\{\backslash beta\backslash delta;\backslash varepsilon\}\; \backslash ,\; \; R\_\{\backslash beta\backslash varepsilon;\backslash delta\}\; \backslash ,\; +\; R^\backslash gamma\{\}\_\{\backslash beta\backslash delta\backslash varepsilon;\backslash gamma\}\; \backslash ,\; =\; 0$
using the definition of the Ricci tensor.
Next, contract again with the metric
 $g^\{\backslash beta\backslash delta\}(R\_\{\backslash beta\backslash delta;\backslash varepsilon\}\; \backslash ,\; \; R\_\{\backslash beta\backslash varepsilon;\backslash delta\}\; \backslash ,\; +\; R^\backslash gamma\{\}\_\{\backslash beta\backslash delta\backslash varepsilon;\backslash gamma\})\; \backslash ,\; =\; 0$
to get
 $R^\backslash delta\{\}\_\{\backslash delta;\backslash varepsilon\}\; \backslash ,\; \; R^\backslash delta\{\}\_\{\backslash varepsilon;\backslash delta\}\; \backslash ,\; +\; R^\{\backslash gamma\backslash delta\}\{\}\_\{\backslash delta\backslash varepsilon;\backslash gamma\}\; \backslash ,\; =\; 0$
The definitions of the Ricci curvature tensor and the scalar curvature then show that
 $R\_\{;\backslash varepsilon\}\; \backslash ,\; \; 2R^\backslash gamma\{\}\_\{\backslash varepsilon;\backslash gamma\}\; \backslash ,\; =\; 0$
which can be rewritten as
 $(R^\backslash gamma\{\}\_\{\backslash varepsilon\}\; \backslash ,\; \; \backslash frac\{1\}\{2\}g^\backslash gamma\{\}\_\{\backslash varepsilon\}R)\_\{;\backslash gamma\}\; \backslash ,\; =\; 0$
A final contraction with $g^\{\backslash varepsilon\backslash delta\}$ gives
 $(R^\{\backslash gamma\backslash delta\}\; \backslash ,\; \; \backslash frac\{1\}\{2\}g^\{\backslash gamma\backslash delta\}R)\_\{;\backslash gamma\}\; \backslash ,\; =\; 0$
which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices,
 $G^\{\backslash alpha\backslash beta\}\{\}\_\{;\backslash beta\}\; \backslash ,\; =\; 0$
Using the EFE, this immediately gives,
 $\backslash nabla\_\backslash beta\; T^\{\backslash alpha\backslash beta\}\; \backslash ,\; =\; T^\{\backslash alpha\backslash beta\}\{\}\_\{;\backslash beta\}\; \backslash ,\; =\; 0$
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.
Nonlinearity
The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics which is linear in the wavefunction.
The correspondence principle
The EFE reduce to Newton's law of gravity by using both the weakfield approximation and the slowmotion approximation. In fact, the constant G appearing in the EFE is determined by making these two approximations.
Derivation of Newton's law of gravity Newtonian gravitation can be written as the theory of a scalar field, $\backslash Phi\; \backslash !$, which is the gravitational potential in joules per kilogram
 $\backslash nabla^2\; \backslash Phi\; [\backslash vec\{x\},t]\; =\; 4\; \backslash pi\; G\; \backslash rho\; [\backslash vec\{x\},t]$
where $\backslash rho\; \backslash !$ is the mass density. The orbit of a freefalling particle satisfies
 $\backslash ddot\{\backslash vec\{x\}\}[t]\; =\; \; \backslash nabla\; \backslash Phi\; [\backslash vec\{x\}\; [t],t]\; \backslash ,.$
In tensor notation, these become
 $\backslash Phi\_\{,i\; i\}\; =\; 4\; \backslash pi\; G\; \backslash rho\; \backslash ,$
 $\backslash frac\{d^2\; x^i\}=\backslash frac\{1\}\{3\}\backslash left(F\_\{\backslash alpha\backslash beta;\backslash gamma\}\; +\; F\_\{\backslash beta\backslash gamma;\backslash alpha\}+F\_\{\backslash gamma\backslash alpha;\backslash beta\}\backslash right)=\backslash frac\{1\}\{3\}\backslash left(F\_\{\backslash alpha\backslash beta,\backslash gamma\}\; +\; F\_\{\backslash beta\backslash gamma,\backslash alpha\}+F\_\{\backslash gamma\backslash alpha,\backslash beta\}\backslash right)=\; 0.\; \backslash !$
where the semicolon represents a covariant derivative, and the brackets denote antisymmetrization. The first equation asserts that the 4divergence of the twoform F 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
 $F\_\{\backslash alpha\backslash beta\}\; =\; A\_\{\backslash alpha;\backslash beta\}\; \; A\_\{\backslash beta;\backslash alpha\}\; =\; A\_\{\backslash alpha,\backslash beta\}\; \; A\_\{\backslash beta,\backslash alpha\}\backslash !$
in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived.^{[9]} However, there are global solutions of the equation which may lack a globally defined potential.^{[10]}
Solutions
Main article: Solutions of the Einstein field equationsThe 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 nonlinear, 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 postNewtonian approximations. Even so, there are numerous cases where the field equations have been solved completely, and those are called exact solutions.^{[11]}
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.
The linearised EFE
Main articles: Linearized Einstein field equations and Linearized gravityThe 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, with terms that are quadratic in or higher powers of the deviation being ignored. This linearisation procedure can be used to investigate the phenomena of gravitational radiation.
Polynomial form
One might think that EFE are nonpolynomial since they contain the inverse of the metric tensor. However, the equations can be arranged so that they contain only the metric tensor and not its inverse. First, the determinant of the metric in 4 dimensions can be written:
 $$
\det(g) = \frac{1}{24} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\alpha\kappa} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu} \,
using the LeviCivita symbol; and the inverse of the metric in 4 dimensions can be written as:
 $$
g^{\alpha\kappa} = \frac{1}{6} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu} / \det(g) \,.
Substituting this definition of the inverse of the metric into the equations then multiplying both sides by det(g) until there are none left in 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.^{[12]}
See also
References
See General relativity resources.
 Aczel, Amir D., 1999. God's Equation: Einstein, Relativity, and the Expanding Universe. Delta Science. A popular account.
 Charles Misner, Kip Thorne, and John Wheeler, 1973. Gravitation. W H Freeman.
External links
 Template:Springer
 Caltech Tutorial on Relativity — A simple introduction to Einstein's Field Equations.
 The Meaning of Einstein's Equation — An explanation of Einstein's field equation, its derivation, and some of its consequences
 MIT Physics Professor Edmund Bertschinger.
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