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Einstein-Hilbert action

In general relativity, Einstein's field equations can be derived from an action principle starting from the Einstein-Hilbert action:

$S[g]=k\int d^nx \sqrt{|\det(g)|} R$

where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen (see below). In Brans-Dicke theory, k is replaced by a scalar field.

In general relativity, the action is assumed to be a functional of the metric only, i.e. the connection is given by the Levi-Civita connection. Some extensions of general relativity assume the metric and connection to be independent however and vary with respect to both independently.

The Einstein-Hilbert action is said to have been written down first by the German mathematician David Hilbert.

 Contents

Derivation of Einstein's field equations

The Einstein-Hilbert action as stated above will actually yield the vacuum Einstein equations. So as starting point a matter Lagrangian LM should be added:

$S = \int d^nx \sqrt{|\det(g)|} \left[ k\, R + L_\mathrm{M} \right]$

The variation with respect to the metric yields

$\delta S = \int d^nx \sqrt{|\det(g)|} \left[ k \left( \delta R + R \frac{1}{\sqrt{|\det(g)|}} \delta \sqrt{|\det(g)|} \right) + \frac{1}{\sqrt{|\det(g)|}} \frac{\delta \sqrt{|\det(g)|} L_\mathrm{M}}{\delta g^{mn}} \right] \delta g^{mn}$

the last term of which is by definition called the stress-energy tensor Tmn. See Belinfante-Rosenfeld tensor

$-\frac{1}{2} T_{mn} := \frac{1}{\sqrt{|\det(g)|}} \frac{\delta}{\delta g^{mn}} \sqrt{|\det(g)|} L_\mathrm{M}$

Note that this is the conventional definition in general relativity, although there are several inequivalent definitions, in particular the canonical stress-energy tensor .

The following are standard text book calculations which have in part been taken from Carroll (see References).

Variation of the Ricci scalar

The variation of the Riemann curvature tensor with respect to the metric is

$\delta R^r{}_{mln} = \nabla_l (\delta \Gamma^r_{nm}) - \nabla_n (\delta \Gamma^r_{lm})$

where δΓ is the variation of the Levi-Civita connection (which is not written down explicitly as it is not required subsequently).

Due to R=gmnRmn and Rmn=Rrmr n the variation of the scalar curvature is

$\delta R = R_{mn} \delta g^{mn} + \nabla_s ( g^{mn} \delta\Gamma^s_{nm} - g^{ms}\delta\Gamma^r_{rm} )$

where the second term yields a surface term by Stokes' theorem as long as k is a constant and does not contribute when the variation δgmn is supposed to vanish at infinity.

Variation of the determinant

We use the following property of a determinant

$\,\! \det(g) = \exp \mathrm{tr} \ln g$

to determine the variation

$\,\! \delta \det(g)=\det(g) g^{mn} \delta g_{mn}$
$\delta\sqrt{|\det(g)|}=\frac{1}{2} \sqrt{|\det(g)|} (-g_{mn} \delta g^{mn})$

where δ(gmngmn)=0 has been used.

Equation of motion

From

$\delta S=\int d^nx \sqrt{|\det(g)|} \left[ k ( R_{mn} - \frac{1}{2} g_{mn} R) - \frac{1}{2} T_{mn} \right] \delta g^{mn}$

$R_{mn} - \frac{1}{2} g_{mn} R = \frac{8 \pi G}{c^4} T_{mn}$

which is Einstein's field equation and

$k = \frac{c^4}{16 \pi G}$

has been chosen such that the non-relativistic limit yields the usual form of Newtons gravity law, where G is the gravitational constant.

The stress-energy tensor may be written as

$T_{mn} = g_{mn} L_\mathrm{M} - 2 \frac{\delta L_\mathrm{M}}{\delta g^{mn}}$

where the functional derivative can be replaced by a partial derivative if the matter Lagrangean does not depend on derivatives of the metric as is common in general relativity.