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Appendix D Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Schwarzschild \(\Gamma^r_{\ tt}\)

$$\Gamma^\mu{}{\nu\sigma} = \frac{1}{2}g^{\mu\alpha}\left(\partial\nu g_{\alpha\sigma} + \partial_\sigma g_{\alpha\nu} - \partial_\alpha g_{\nu\sigma}\right) $$ Using this expression, derive \(\Gamma^r{}_{tt}\). In the process, explicitly write out \(g^{rr}\) and \(\partial_r g_{tt}\).

Hint

For a diagonal metric, \(g^{rr} = 1/g_{rr}\), and in the calculation of \(\Gamma^r{}_{tt}\), only \(\alpha = r\) contributes in the summation over the index \(\alpha\). Compute \(\partial_r g_{tt}\) first.

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B-2. Schwarzschild \(\Gamma^r_{\ rr}\)

Hint

Differentiating \(g_{rr} = (1-2M/r)^{-1}\) with respect to \(r\) requires the chain rule. The result is \(\Gamma^r{}_{rr} = \frac{1}{2}g^{rr}\,\partial_r g_{rr}\).

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B-3. \(\Gamma^\theta_{\ \varphi\varphi}\) in Minkowski Spherical Coordinates

From $$ds^2 = -dt^2 + dr^2 + r^2 d\theta^2 + r^2\sin^2\theta\, d\varphi^2 $$ compute \(\Gamma^\theta{}_{\varphi\varphi}\) according to its definition.

Hint

The key is the \(\theta\) derivative of \(g_{\varphi\varphi} = r^2\sin^2\theta\). Use \(g^{\theta\theta} = 1/r^2\).

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B-4. FRW \(\Gamma^r_{\ tr}\)

$$ds^2 = -dt^2 + a^2(t)\left[dr^2 + r^2 d\theta^2 + r^2\sin^2\theta\, d\varphi^2\right] $$ For this metric, compute \(\Gamma^r{}_{tr}\) from its definition and verify that it equals \(\dot{a}/a\).

Hint

Since \(g_{rr} = a^2(t)\), we have \(\partial_t g_{rr} = 2a\dot{a}\). Compute \(\Gamma^r{}_{tr} = \frac{1}{2}g^{rr}\,\partial_t g_{rr}\).

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B-5. FRW \(\Gamma^t_{\ \theta\theta}\)

Hint

Find the time derivative of \(g_{\theta\theta} = a^2(t)r^2\), and noting that \(g^{tt} = -1\), compute \(\Gamma^t{}_{\theta\theta} = -\frac{1}{2}g^{tt}\,\partial_t g_{\theta\theta}\).

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B-6. Verification of Schwarzschild Orthonormal Basis

Hint

Compute \(g(\mathbf{e}_{\hat{r}},\, \mathbf{e}_{\hat{r}}) = g_{\alpha\beta}\,(\mathbf{e}_{\hat{r}})^\alpha\,(\mathbf{e}_{\hat{r}})^\beta\). The only non-zero component is \(\alpha = \beta = r\).

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B-7. Christoffel Symbol Verification for General Spherical Symmetry

Hint

Set \(\nu = \ln(1-2M/r)\), find \(\nu' = d\nu/dr\), and substitute using \(e^{\nu-\lambda} = (1-2M/r)^2\).

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B-8. Application of Riemann Tensor Symmetries

$$R_{\alpha\beta\gamma\delta} = -R_{\beta\alpha\gamma\delta}, \quad R_{\alpha\beta\gamma\delta} = -R_{\alpha\beta\delta\gamma}, \quad R_{\alpha\beta\gamma\delta} = R_{\gamma\delta\alpha\beta} $$ Using these symmetries, find the orthonormal basis component \(R_{\hat{t}\hat{r}\hat{t}\hat{r}}\) of the Schwarzschild spacetime from \(R_{\hat{r}\hat{t}\hat{r}\hat{t}} = -2M/r^3\).

Hint

You can obtain the result by applying the antisymmetry in the first and second index pairs twice, or by using the pair exchange symmetry in a single step.

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B-9. Schwarzschild \(R_{tt} = 0\)

Hint

Expand \(R_{\hat{t}\hat{t}} = R^{\hat{\rho}}{}_{\hat{t}\hat{\rho}\hat{t}} = R^{\hat{t}}{}_{\hat{t}\hat{t}\hat{t}} + R^{\hat{r}}{}_{\hat{t}\hat{r}\hat{t}} + R^{\hat{\theta}}{}_{\hat{t}\hat{\theta}\hat{t}} + R^{\hat{\varphi}}{}_{\hat{t}\hat{\varphi}\hat{t}}\). In the orthonormal basis, use \(\eta_{\hat{\alpha}\hat{\beta}}\) for raising and lowering indices.

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B-10. Scalar Curvature of FRW

Hint

Compute \(G^{\hat{\alpha}}{}_{\hat{\alpha}} = \eta^{\hat{\alpha}\hat{\beta}}G_{\hat{\alpha}\hat{\beta}}\). The trace of \(G_{\hat{\mu}\hat{\nu}} = R_{\hat{\mu}\hat{\nu}} - \frac{1}{2}\eta_{\hat{\mu}\hat{\nu}}R\) gives \(G^{\hat{\alpha}}{}_{\hat{\alpha}} = R - 2R = -R\).

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Medium

M-1. Newtonian Limit of Schwarzschild Geodesics

Substitute the Christoffel symbols from the formula collection into $$\frac{d^2 r}{d\tau^2} + \Gamma^r{}{tt}\left(\frac{dt}{d\tau}\right)^2 + \Gamma^r{}}\left(\frac{dr}{d\tau}\right)^2 + \Gamma^r{{\theta\theta}\left(\frac{d\theta}{d\tau}\right)^2 + \Gamma^r{}\right)^2 = 0. $$ Furthermore, take the limit of low velocity (}\left(\frac{d\varphi}{d\tau\(dr/d\tau \approx 0\), \(d\theta/d\tau \approx 0\), \(d\varphi/d\tau \approx 0\)) and weak gravitational field (\(r \gg 2M\), \(dt/d\tau \approx 1\)), and derive Newton's equation of motion \(d^2r/dt^2 \approx -M/r^2\).

Hint

In the low-velocity, weak-field limit, \(\Gamma^r{}_{tt} \approx M/r^2\), and terms containing other Christoffel symbols can be neglected. Also use \(d\tau \approx dt\).

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M-2. FRW and Friedmann Equations - Conservation Law

$$\dot{\rho} + 3\frac{\dot{a}}{a}(\rho + p) = 0 $$ Derive this equation.

Hint

Differentiate the first Friedmann equation with respect to time, and eliminate \(\ddot{a}\) using the second Friedmann equation.

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M-3. Kretschmann Scalar for Schwarzschild

Compute $$K = R_{\alpha\beta\gamma\delta}\,R^{\alpha\beta\gamma\delta} $$ Note that in an orthonormal basis \(R^{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}} = \eta^{\hat{\alpha}\hat{\mu}}\eta^{\hat{\beta}\hat{\nu}}\eta^{\hat{\gamma}\hat{\rho}}\eta^{\hat{\delta}\hat{\sigma}}R_{\hat{\mu}\hat{\nu}\hat{\rho}\hat{\sigma}}\), and use the symmetries of the Riemann tensor to enumerate the contributions from independent components. Confirm that the result is \(K = 48M^2/r^6\).

Hint

The independent nonzero components are the 6 components \(R_{\hat{r}\hat{t}\hat{r}\hat{t}}\), \(R_{\hat{\theta}\hat{t}\hat{\theta}\hat{t}}\), \(R_{\hat{\varphi}\hat{t}\hat{\varphi}\hat{t}}\), \(R_{\hat{r}\hat{\theta}\hat{r}\hat{\theta}}\), \(R_{\hat{r}\hat{\varphi}\hat{r}\hat{\varphi}}\), \(R_{\hat{\theta}\hat{\varphi}\hat{\theta}\hat{\varphi}}\). Pay attention to the multiplicity factors arising from symmetries when each component contributes to \(K\).

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M-4. Derivation of the Mass Function

Define

\[e^{-\lambda} = 1 - \frac{2m(r)}{r} \]

Rewrite \(G_{\hat{t}\hat{t}} = 8\pi\rho\) as a relation between \(m(r)\) and \(\rho\), and derive

\[\frac{dm}{dr} = 4\pi r^2 \rho \]

State the physical meaning of this equation.

Hint

It is more concise to directly differentiate \(e^{-\lambda} = 1 - 2m/r\) with respect to \(r\) to obtain \(\lambda' e^{-\lambda}\). Substitute into \(G_{\hat{t}\hat{t}}\) and simplify.

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Advanced

A-1. Schwarzschild Circular Orbits and Tidal Forces

(a) Using the geodesic equation and the Christoffel symbols from the formula collection, show that the angular velocity \(\Omega = d\varphi/dt\) of a circular orbit satisfies

$$\Omega^2 = \frac{M}{r^3} $$ (the general relativistic version of Kepler's third law).

(b) Evaluate the tidal force experienced by an observer on a circular orbit using the Riemann tensor components in an orthonormal basis. Specifically, express the magnitude of the relative acceleration experienced by two free particles separated by a distance \(\delta r\) in the radial direction in terms of \(M\), \(r\), and \(\delta r\).

(c) Find the ratio of the tidal force at the innermost stable circular orbit (ISCO) \(r = 6M\) to the tidal force at the event horizon \(r = 2M\).

Hint

(a) For a circular orbit, set \(dr/d\tau = 0\), \(d^2r/d\tau^2 = 0\), \(\theta = \pi/2\), and substitute \(\Gamma^r{}_{tt}\) and \(\Gamma^r{}_{\varphi\varphi}\) into the \(r\)-component of the geodesic equation. (b) Use the geodesic deviation equation \(D^2\xi^{\hat{\alpha}}/d\tau^2 = -R^{\hat{\alpha}}{}_{\hat{\beta}\hat{\gamma}\hat{\delta}}u^{\hat{\beta}}u^{\hat{\gamma}}\xi^{\hat{\delta}}\). (c) Substitute each value of \(r\) into the Riemann components and take the ratio.

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A-2. de Sitter and the Cosmological Constant

\[G_{\hat{\mu}\hat{\nu}} + \Lambda\,\eta_{\hat{\mu}\hat{\nu}} = 8\pi T_{\hat{\mu}\hat{\nu}} \]

can be written as above.

(a) Using the FRW Einstein tensor components from the formula sheet, derive the modified Friedmann equations including \(\Lambda\):

\[H^2 = \frac{8\pi\rho}{3} + \frac{\Lambda}{3} - \frac{k}{a^2} $$ $$\frac{\ddot{a}}{a} = -\frac{4\pi}{3}(\rho + 3p) + \frac{\Lambda}{3} \]

(b) In the case of no matter and no spatial curvature (\(\rho = p = 0\), \(k = 0\)), show that \(a(t) \propto e^{Ht}\) (de Sitter spacetime) and express \(H\) in terms of \(\Lambda\).

(c) Confirm that the Riemann tensor of de Sitter spacetime takes the maximally symmetric space form \(R_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}} = \frac{\Lambda}{3}(\eta_{\hat{\alpha}\hat{\gamma}}\eta_{\hat{\beta}\hat{\delta}} - \eta_{\hat{\alpha}\hat{\delta}}\eta_{\hat{\beta}\hat{\gamma}})\), by obtaining the Ricci tensor and scalar curvature from the FRW Einstein tensor components and comparing with the general form of the Riemann tensor for a 4-dimensional maximally symmetric space.

Hint

(a) Note that \(\Lambda\,\eta_{\hat{t}\hat{t}} = -\Lambda\). (b) Verify the constant solution \(H^2 = \Lambda/3\). (c) For a maximally symmetric space, \(R_{\alpha\beta\gamma\delta} = \frac{R}{n(n-1)}(g_{\alpha\gamma}g_{\beta\delta} - g_{\alpha\delta}g_{\beta\gamma})\) (where \(n\) is the number of dimensions), and use the scalar curvature of de Sitter spacetime \(R = 4\Lambda\).

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