Einstein field equations

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Mathematical form[edit]

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


where R_{\mu \nu}\, is the Ricci curvature tensor, R\, is the scalar curvature, g_{\mu \nu}\, is the metric tensor, \Lambda\, is the cosmological constant, G\, is Newton's gravitational constant, c\, is the speed of light in vacuum, and T_{\mu \nu}\, is the stress–energy tensor.



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_{\mu \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, hyperbolic-elliptic partial differential equations[citation needed].


G_{\mu \nu} = R_{\mu \nu} - {1 \over 2}R g_{\mu \nu},


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


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


These equations, together with the geodesic equation,[5] which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.

Sign convention[edit]


{R^\mu}_{\alpha \beta \gamma} & = [S2] \times (\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}) \\[6pt]
G_{\mu \nu} & = [S3] \times {8 \pi G \over c^4} T_{\mu \nu}


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) is (+--)\,, Peebles (1980) and Efstathiou (1990) are (-++)\, while Peacock (1994), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are (-+-)\,.


R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R - g_{\mu \nu} \Lambda = -{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[edit]

Taking the trace of both sides of the EFE one gets

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

which simplifies to

-R + 4 \Lambda = {8 \pi G \over c^4} T \,.

If one adds - {1 \over 2} g_{\mu \nu} \, times this to the EFE, one gets the following equivalent "trace-reversed" form

R_{\mu \nu} - g_{\mu \nu} \Lambda = {8 \pi G \over c^4} \left(T_{\mu \nu} - {1 \over 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_{\mu\nu} \, in the expression on the right with the Minkowski metric without significant loss of accuracy).

The cosmological constant[edit]

Main article: Cosmological constant

Einstein modified his original field equations to include a cosmological constant term \Lambda proportional to the metric


Since \Lambda is constant, the energy conservation law is unaffected.


  • the universe described by this theory was unstable, and
  • observations of Hubble confirmed that our universe is expanding.

So, Einstein abandoned \Lambda, calling it the "biggest blunder [he] ever made".[6]

Despite Einstein'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. Indeed, recent improved astronomical techniques have found that a positive value of \Lambda is needed to explain the accelerating universe.[7][8]


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

The resulting vacuum energy is constant and given by

\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 term vacuum energy is now used interchangeably in general relativity.


Conservation of energy and momentum[edit]


\nabla_\beta T^{\alpha\beta}  \,  = T^{\alpha\beta}{}_{;\beta}  \, = 0.




The correspondence principle[edit]


Vacuum field equations[edit]

A Swiss commemorative coin showing the vacuum field equations with zero cosmological constant (top).

If the energy-momentum tensor T_{\mu \nu} is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting T_{\mu \nu} = 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

R_{\mu \nu} = \Lambda g_{\mu \nu} \,.


Manifolds with a vanishing Ricci tensor,  R_{\mu \nu}=0 , are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.

Einstein–Maxwell equations[edit]

If the energy-momentum tensor T_{\mu \nu} is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor

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)


R^{\alpha \beta} - {1 \over 2}R g^{\alpha \beta} + g^{\alpha \beta} \Lambda = \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:

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. \!


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.[9] However, there are global solutions of the equation which may lack a globally defined potential.[10]




One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.[12] In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,[13] 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 [14] and Kohli and Haslam.[15]

The linearised EFE[edit]


Polynomial form[edit]


\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}


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)


See also[edit]


See General relativity resources.

  1. ^ a b Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik 354 (7): 769. Bibcode:1916AnP...354..769E. doi:10.1002/andp.19163540702. 
  2. ^ Einstein, Albert (November 25, 1915). "Die Feldgleichungen der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847. Retrieved 2006-09-12. 
  4. ^ Carroll, Sean (2004). Spacetime and Geometry - An Introduction to General Relativity. pp. 151–159. ISBN 0-8053-8732-3. 
  8. ^ Turner, Michael S. (May 2001). "Making Sense of the New Cosmology". Int.J.Mod.Phys. A17S1 17: 180–196. arXiv:astro-ph/0202008. Bibcode:2002IJMPA..17S.180T. doi:10.1142/S0217751X02013113. 
  10. ^ Trautman, Andrzej (1977). "Solutions of the Maxwell and Yang-Mills equations associated with hopf fibrings". International Journal of Theoretical Physics 16 (9): 561–565. Bibcode:1977IJTP...16..561T. doi:10.1007/BF01811088. .
  11. ^ Stephani, Hans; D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt (2003). Exact Solutions of Einstein's Field Equations. Cambridge University Press. ISBN 0-521-46136-7. 
  12. ^ Ellis, GFR and MacCallum, M, "A class of homogeneous cosmological models", Comm. Math. Phys. Volume 12, Number 2 (1969), 108-141.
  13. ^ Hsu, L and Wainwright, J, "Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions", Class. Quantum Grav. 3 (1986) 1105-1124"
  14. ^ LeBlanc, V.G, "Asymptotic states of magnetic Bianchi I cosmologies", 1997 Class. Quantum Grav. 14 2281
  16. ^ Einstein's Field Equations in Polynomial Form|http://arxiv.org/pdf/gr-qc/0507026.pdf

External links[edit]