# Einstein-Hilbert action

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

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

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.

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## 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:

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

The variation with respect to the metric yields

[itex]\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} [itex]

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

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

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

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

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

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

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

[itex]\,\! \det(g) = \exp \mathrm{tr} \ln g[itex]

to determine the variation

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

where δ(gmngmn)=0 has been used.

### Equation of motion

From

[itex]\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}[itex]

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

which is Einstein's field equation and

[itex]k = \frac{c^4}{16 \pi G}[itex]

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

The stress-energy tensor may be written as

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

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.

## References

• Carroll, Sean M. (Dec, 1997). Lecture Notes on General Relativity, NSF-ITP-97-147, 231pp, arXiv:gr-qc/9712019

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