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Appendix G Problems

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Medium

Medium

M-1. Derivation of \(\delta\sqrt{-g}\)

Take the variation of both sides of \(g^{\mu\alpha}g_{\alpha\nu} = \delta^\mu_\nu\) and derive \(g_{\mu\nu}\delta g^{\mu\nu} = -g^{\mu\nu}\delta g_{\mu\nu}\). Furthermore, combining this with \(\delta g = g \cdot g^{\mu\nu}\delta g_{\mu\nu}\), show that \(\delta\sqrt{-g} = -\frac{1}{2}\sqrt{-g} \, g_{\mu\nu}\delta g^{\mu\nu}\).

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M-2. Newtonian Limit of Einstein-Hilbert

In the weak gravitational field (\(g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\), \(|h_{\mu\nu}| \ll 1\)), low velocity (\(v \ll c\)), and static (\(\partial_t = 0\)) limit of the Einstein field equations, accept that \(R_{00} \approx -\frac{1}{2}\nabla^2 h_{00}\), and by setting \(h_{00} = -2\Phi/c^2\), derive Newton's Poisson equation \(\nabla^2\Phi = 4\pi G\rho\).

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M-3. Variation of the Cosmological Constant Term

Consider the action obtained by adding the cosmological constant term \(-2\Lambda\) to the Einstein-Hilbert action: \(S = \frac{c^4}{16\pi G}\int (R - 2\Lambda)\sqrt{-g} \, d^4x\). Perform the variation with respect to \(g^{\mu\nu}\) and derive the Einstein field equations with cosmological constant: \(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}\).

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