Web Element: Math

Einstein field equations

[math]R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} [/math]

[math]R_{\mu \nu} [/math] Ricci curvature tensor

Mathematical form

Template:Spacetime The Einstein field equations (EFE) may be written in the form:

Template:Equation box 1 EFE on a wall in Leiden where Template:Mvar is the Ricci curvature tensor, Template:Mvar is the scalar curvature, Template:Mvar is the metric tensor, Template:Mvar is the cosmological constant, Template:Mvar is Newton's gravitational constant, Template:Mvar is the speed of light in vacuum, and Template:Mvar is 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 four-dimensional theory, some theorists have explored their consequences in Template:Mvar dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when Template:Mvar 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 Template:Mvar, 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 ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.

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

[math]G_{\mu \nu} = R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu},[/math]

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

[math]G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.[/math]

In standard units, each term on the left has units of 1/length². With this choice of Einstein constant as 8πG/c⁴, then 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 = pressure).

Using geometrized units where Template:Math, this can be rewritten as

[math]G_{\mu \nu} + \Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu}\,.[/math]

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, which dictates how freely-falling matter moves through space-time, 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.Template:Sfnp The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):

[math] \begin{align} g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt] {R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\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}\right) \\[6pt] G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu} \end{align} [/math]

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

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

With these definitions Misner, Thorne, and Wheeler classify themselves as Template:Math, whereas Weinberg (1972)Template:Sfnp and Peacock (1994)Template:Sfnp are Template:Math, Peebles (1980) and Efstathiou et al. (1990) are Template:Math, Rindler (1977)Template:Citation needed, Atwater (1974)Template:Citation needed, Collins Martin & Squires (1989) are Template:Math.

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

[math]R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\frac{8 \pi G}{c^4} T_{\mu \nu}.[/math]

The sign of the (very small) cosmological term would change in both these versions, if the Template:Math metric sign convention is used rather than the MTW Template:Math metric sign convention adopted here.

Equivalent formulations

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

[math]R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,[/math]

where Template:Mvar is the spacetime dimension. This expression can be rewritten as

[math]-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.[/math]

If one adds Template:Math times this to the EFE, one gets the following equivalent "trace-reversed" form

[math]R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,[/math]

For example, in Template:Math dimensions this reduces to

[math]R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,[/math]

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 Template:Mvar in the expression on the right with the Minkowski metric without significant loss of accuracy).