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## Standard derivation of gravitational field equations

Hilbert introduced the Einstein-Hilbert action in 1915.  It is useful to appreciate the action simply on space components.  It is an action on a fixed space $X$:

$S = \int_X \sqrt{-g}(R + L_M) dx$.

The Euler-Lagrange equations arise by setting variation with respect to $g^{\mu\nu}$ to zero.  You can find the derivation of the gravitational field equations here and I won’t repeat it.  The key points are that $\partial R / \partial g^{\mu\nu} = R_{\mu\nu}$ the right side being the Ricci curvature and $T_{\mu\nu} = (-2/\sqrt{-g}) \delta (\sqrt{-g} L_M )/\delta g^{\mu\nu} = -2 \delta L_M/\delta g^{\mu\nu} + g_{\mu\nu}L_M$.  So the variation is over metric tensor over the same topological space.

It is useful to consider a slightly different action principle where the integrand of the the Einstein-Hilbert action remains the same but we consider integrals over hypersurfaces of a fixed ambient manifold $S^4(1/h)$.  We two things change: we are not restricting the topology of the three dimensional manifolds that can occur as critical points of this action but we are restricting the domain of search for new possible metrics to those achievable as the hypersurfaces of the ambient four-manifold.  Technically the setting is familiar from the calculus of variations for minimal hypersurfaces which studies the variations of the area functional.