Einstein-hilbert Action

$Sg=k\backslash int\; d^nx\; \backslash sqrt$>

/dd> 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. 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: $S\; =\; \backslash int\; d^nx\; \backslash sqrt$R + L_\mathrm{M} \right> /dd> The variation with respect to the metric yields $\backslash delta\; S\; =\; \backslash int\; d^nx\; \backslash sqrt$\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} > /dd> the last term of which is by definition called the stress-energy tensor Tmn. See Belinfante-Rosenfeld tensor $-\backslash frac\{1\}\{2\}\; T\_\{mn\}\; :=\; \backslash frac\{1\}\{\backslash sqrt\{|\backslash det(g)|\}\}\; \backslash frac\{\backslash delta\}\{\backslash delta\; g^\{mn\}\}\; \backslash sqrt\{|\backslash det(g)|\}\; L\_\backslash mathrm\{M\}$ 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 $\backslash delta\; R^r\{\}\_\{mln\}\; =\; \backslash nabla\_l\; (\backslash delta\; \backslash Gamma^r\_\{nm\})\; -\; \backslash nabla\_n\; (\backslash delta\; \backslash Gamma^r\_\{lm\})$ 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 $\backslash delta\; R\; =\; R\_\{mn\}\; \backslash delta\; g^\{mn\}\; +\; \backslash nabla\_s\; (\; g^\{mn\}\; \backslash delta\backslash Gamma^s\_\{nm\}\; -\; g^\{ms\}\backslash delta\backslash Gamma^r\_\{rm\}\; )$ 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 $\backslash ,\backslash !\; \backslash det(g)\; =\; \backslash exp\; \backslash mathrm\{tr\}\; \backslash ln\; g$ to determine the variation $\backslash ,\backslash !\; \backslash delta\; \backslash det(g)=\backslash det(g)\; g^\{mn\}\; \backslash delta\; g\_\{mn\}$ $\backslash delta\backslash sqrt\{|\backslash det(g)$ =\frac{1}{2} \sqrt{|\det(g) (-g_{mn} \delta g^{mn}) where δ(gmngmn)=0 has been used. Equation of motion From $\backslash delta\; S=\backslash int\; d^nx\; \backslash sqrt\{|\backslash det(g)$

\left( R_{mn} - \frac{1}{2} g_{mn} R) - \frac{1}{2} T_{mn} \right \delta g^{mn}

$R\_\{mn\}\; -\; \backslash frac\{1\}\{2\}\; g\_\{mn\}\; R\; =\; \backslash frac\{8\; \backslash pi\; G\}\{c^4\}\; T\_\{mn\}$

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

$T\_\{mn\}\; =\; g\_\{mn\}\; L\_\backslash mathrm\{M\}\; -\; 2\; \backslash frac\{\backslash delta\; L\_\backslash mathrm\{M\}\}\{\backslash delta\; g^\{mn\}\}$

See also

• Palatini action
• Einstein-Cartan theory
• teleparallelism

References

• Carroll, Sean M. (Dec, 1997). Lecture Notes on General Relativity, NSF-ITP-97-147, 231pp,

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