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${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
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The Einstein field equations (EFE; also known as "Einstein's equations") are the set of 10

Similar to the way that

geodesic equation

.

As well as obeying local energy–momentum conservation, the EFE reduce to

Exact solutions for the EFE can only be found under simplifying assumptions such as

gravitational waves

.

Mathematical form

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The Einstein field equations (EFE) may be written in the form:[5]^[1]

where R_μν is the

.

The EFE is a

, 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. 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_μν, 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, hyperbolic-elliptic partial differential equations^{[citation needed]}.

One can write the EFE in a more compact form by defining the Einstein tensor

${\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu },}$

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }.}$

Using

where G = c = 1, this can be rewritten as

${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi T_{\mu \nu }\,.}$

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

.

Sign convention

The above form of the EFE is the standard established by Misner, Thorne, and Wheeler.^[7] The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):

${\displaystyle {\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times (\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma })\\[6pt]G_{\mu \nu }&=[S3]\times {8\pi G \over c^{4}}T_{\mu \nu }\end{aligned}}}$

The third sign above is related to the choice of convention for the Ricci tensor:

${\displaystyle R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }}$

With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972)^[8] is (+ − −), Peebles (1980)^{[citation needed]} and Efstathiou (1990)^{[citation needed]} are (− + +), while Peacock (1994)^{[citation needed]}, Rindler (1977)^{[citation needed]}, Atwater (1974)^{[citation needed]}, Collins Martin & Squires (1989)^{[citation needed]} 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

${\displaystyle R_{\mu \nu }-{1 \over 2}R\,g_{\mu \nu }-\Lambda g_{\mu \nu }=-{8\pi G \over c^{4}}T_{\mu \nu }.}$

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 with respect to the metric of both sides of the EFE one gets

${\displaystyle R-{\frac {D}{2}}R+D\Lambda ={8\pi G \over c^{4}}T\,}$

where D is the spacetime dimension. This expression can be rewritten as

${\displaystyle -R+{\frac {D\Lambda }{({\tfrac {D}{2}}-1)}}={8\pi G \over c^{4}}{\frac {T}{{\tfrac {D}{2}}-1}}\,.}$

If one adds −1/2g_μν times this to the EFE, one gets the following equivalent "trace-reversed" form

${\displaystyle R_{\mu \nu }-{\frac {\Lambda g_{\mu \nu }}{{\tfrac {D}{2}}-1}}={8\pi G \over c^{4}}\left(T_{\mu \nu }-{1 \over {D-2}}T\,g_{\mu \nu }\right).\,}$

For example, in D = 4 dimensions this reduces to

${\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }={8\pi G \over c^{4}}\left(T_{\mu \nu }-{\tfrac {1}{2}}T\,g_{\mu \nu }\right).\,}$

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_μν in the expression on the right with the

without significant loss of accuracy).

The cosmological constant

Einstein modified his original field equations to include a

${\displaystyle R_{\mu \nu }-{1 \over 2}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }\,.}$

Since Λ is constant, the energy conservation law is unaffected.

The cosmological constant term was originally introduced by Einstein to allow for a universe that is not expanding or contracting. This effort was unsuccessful because:

• the universe described by this theory was unstable, and
• observations by
  expanding
  .

So, Einstein abandoned Λ, calling it the "biggest blunder [he] ever made".^[9]

Despite

's motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. For many years the cosmological constant was almost universally considered to be 0. However, recent improved astronomical techniques have found that a positive value of ${\displaystyle \Lambda }$ is needed to explain the

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:

${\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda c^{4}}{8\pi G}}g_{\mu \nu }\,.}$

The resulting vacuum energy is constant and given by

${\displaystyle \rho _{\mathrm {vac} }={\frac {\Lambda c^{2}}{8\pi G}}}$

The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy. Thus, the terms "cosmological constant" and "vacuum energy" 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

${\displaystyle \nabla _{\beta }T^{\alpha \beta }\,=T^{\alpha \beta }{}_{;\beta }\,=0}$.

Derivation of local energy-momentum conservation

Contracting the differential Bianchi identity ${\displaystyle R_{\alpha \beta [\gamma \delta ;\varepsilon ]}=\,0}$ with gαβ gives, using the fact that the metric tensor is covariantly constant, i.e. gαβ;γ = 0, ${\displaystyle R^{\gamma }{}_{\beta \gamma \delta ;\varepsilon }+\,R^{\gamma }{}_{\beta \varepsilon \gamma ;\delta }+\,R^{\gamma }{}_{\beta \delta \varepsilon ;\gamma }=\,0}$ The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten: ${\displaystyle R^{\gamma }{}_{\beta \gamma \delta ;\varepsilon }\,-R^{\gamma }{}_{\beta \gamma \varepsilon ;\delta }\,+R^{\gamma }{}_{\beta \delta \varepsilon ;\gamma }\,=0}$ which is equivalent to ${\displaystyle R_{\beta \delta ;\varepsilon }\,-R_{\beta \varepsilon ;\delta }\,+R^{\gamma }{}_{\beta \delta \varepsilon ;\gamma }\,=0}$ using the definition of the Ricci tensor . Next, contract again with the metric ${\displaystyle g^{\beta \delta }(R_{\beta \delta ;\varepsilon }\,-R_{\beta \varepsilon ;\delta }\,+R^{\gamma }{}_{\beta \delta \varepsilon ;\gamma })\,=0}$ to get ${\displaystyle R^{\delta }{}_{\delta ;\varepsilon }\,-R^{\delta }{}_{\varepsilon ;\delta }\,+R^{\gamma \delta }{}_{\delta \varepsilon ;\gamma }\,=0}$ The definitions of the Ricci curvature tensor and the scalar curvature then show that ${\displaystyle R_{;\varepsilon }\,-2R^{\gamma }{}_{\varepsilon ;\gamma }\,=0}$ which can be rewritten as ${\displaystyle (R^{\gamma }{}_{\varepsilon }\,-{\frac {1}{2}}g^{\gamma }{}_{\varepsilon }R)_{;\gamma }\,=0}$ A final contraction with gεδ gives ${\displaystyle (R^{\gamma \delta }\,-{\frac {1}{2}}g^{\gamma \delta }R)_{;\gamma }\,=0}$ which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices, ${\displaystyle G^{\alpha \beta }{}_{;\beta }\,=0}$ Using the EFE, this immediately gives, ${\displaystyle \nabla _{\beta }T^{\alpha \beta }\,=T^{\alpha \beta }{}_{;\beta }\,=0}$

which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations

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,

.

The correspondence principle

The EFE reduce to

. 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, Φ, which is the gravitational potential in joules per kilogram ${\displaystyle \nabla ^{2}\Phi [{\vec {x}},t]=4\pi G\rho [{\vec {x}},t]}$ where ρ is the mass density. The orbit of a free-falling particle satisfies ${\displaystyle {\ddot {\vec {x}}}[t]=-\nabla \Phi [{\vec {x}}[t],t]\,.}$ In tensor notation, these become ${\displaystyle \Phi _{,ii}=4\pi G\rho \,}$ ${\displaystyle {\frac {d^{2}x^{i}}{{dt}^{2}}}=-\Phi _{,i}\,.}$ In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form ${\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{1 \over 2}Tg_{\mu \nu }\right)}$ for some constant, K, and the geodesic equation ${\displaystyle {\frac {d^{2}x^{\alpha }}{{d\tau }^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.}$ To see how the latter reduce to the former, we assume that the test particle's velocity is approximately zero ${\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)}$ and thus ${\displaystyle {\frac {d}{dt}}\left({\frac {dt}{d\tau }}\right)\approx 0}$ 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 ${\displaystyle {\frac {d^{2}x^{i}}{{dt}^{2}}}\approx -\Gamma _{00}^{i}}$ where two factors of dt/dτ have been divided out. This will reduce to its Newtonian counterpart, provided ${\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={1 \over 2}g^{i\alpha }(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha })\,.}$ Our assumptions force α = i and the time (0) derivatives to be zero. So this simplifies to ${\displaystyle 2\Phi _{,i}\approx g^{ij}(-g_{00,j})\approx -g_{00,i}\,}$ which is satisfied by letting ${\displaystyle g_{00}\approx -c^{2}-2\Phi \,.}$ Turning to the Einstein equations, we only need the time-time component ${\displaystyle R_{00}=K\left(T_{00}-{1 \over 2}Tg_{00}\right)}$ the low speed and static field assumptions imply that ${\displaystyle T_{\mu \nu }\approx \mathrm {diag} (T_{00},0,0,0)\approx \mathrm {diag} (\rho c^{4},0,0,0)\,.}$ So ${\displaystyle T=g^{\alpha \beta }T_{\alpha \beta }\approx g^{00}T_{00}\approx {-1 \over c^{2}}\rho c^{4}=-\rho c^{2}\,}$ and thus ${\displaystyle K\left(T_{00}-{1 \over 2}Tg_{00}\right)\approx K\left(\rho c^{4}-{1 \over 2}(-\rho c^{2})(-c^{2})\right)={1 \over 2}K\rho c^{4}\,.}$ From the definition of the Ricci tensor ${\displaystyle R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.}$ Our simplifying assumptions make the squares of Γ disappear together with the time derivatives ${\displaystyle R_{00}\approx \Gamma _{00,i}^{i}\,.}$ Combining the above equations together ${\displaystyle \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{1 \over 2}Tg_{00}\right)\approx {1 \over 2}K\rho c^{4}\,}$ which reduces to the Newtonian field equation provided ${\displaystyle {1 \over 2}K\rho c^{4}=4\pi G\rho \,}$ which will occur if ${\displaystyle K={\frac {8\pi G}{c^{4}}}\,.}$

Vacuum field equations

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

${\displaystyle R_{\mu \nu }=0\,.}$

In the case of nonzero cosmological constant, the equations are

${\displaystyle R_{\mu \nu }={\frac {\Lambda }{{\tfrac {D}{2}}-1}}g_{\mu \nu }\,.}$

The solutions to the vacuum field equations are called

.

.

Einstein–Maxwell equations

If the energy-momentum tensor T_μν is that of an

${\displaystyle T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left(F^{\alpha }{}^{\psi }F_{\psi }{}^{\beta }+{1 \over 4}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right)}$

is used, then the Einstein field equations are called the Einstein–Maxwell equations (with cosmological constant Λ, taken to be zero in conventional relativity theory):

${\displaystyle R^{\alpha \beta }-{1 \over 2}Rg^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {8\pi G}{c^{4}\mu _{0}}}\left(F^{\alpha }{}^{\psi }F_{\psi }{}^{\beta }+{1 \over 4}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).}$

Additionally, the covariant Maxwell Equations are also applicable in free space:

${\displaystyle F^{\alpha \beta }{}_{;\beta }\,=0}$

${\displaystyle F_{[\alpha \beta ;\gamma ]}={\frac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\frac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\!}$

where the semicolon represents a

that in a coordinate chart it is possible to introduce an electromagnetic field potential A_α such that

${\displaystyle F_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }\!}$

in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived.[12] However, there are global solutions of the equation which may lack a globally defined potential.^[13]

Solutions

The solutions of the Einstein field equations are

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.^[15] In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,^[16] 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 ^[17] and Kohli and Haslam.^[18]

The linearised EFE

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

.

Polynomial form

One might think that EFE are non-polynomial 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:

${\displaystyle \det(g)={\frac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }\,}$

using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:

${\displaystyle g^{\alpha \kappa }={\frac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }/\det(g)\,.}$

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.^[19]

See also

Notes

References

See

General relativity resources

.

• ISBN 0-7167-0344-0{{citation}}: CS1 maint: multiple names: authors list (link
  )
• ISBN 0-471-92567-5

External links

Wikibooks has a book on the topic of: General Relativity

Wikiversity has learning resources about General Relativity

• "Einstein equations", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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
• Video Lecture on Einstein's Field Equations by
  MIT
  Physics Professor Edmund Bertschinger.
• Arch and scaffold: How Einstein found his field equations Physics Today November 2015, History of the Development of the Field Equations
• Black Hole Tech? a computational look at Einstein's Field Equations through programming.