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Appendix C Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Raising and Lowering Indices

Using the Minkowski metric \(\eta_{\mu\nu}\), compute the covariant components \(A_\mu = \eta_{\mu\nu}A^\nu\) of the 4-vector \(A^\mu = (1, 2, 3, 4)\).

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B-2. Contraction of the Kronecker Delta

Using Einstein's summation convention, compute the contraction of the Kronecker delta \(\delta^\mu_\nu\) (equal to 1 when \(\mu = \nu\), and 0 otherwise) with a contravariant vector \(A^\nu\):

\[ \delta^\mu_\nu A^\nu \]

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B-3. Metric and Inverse Metric of a Sphere

From the metric of a 2-dimensional sphere (radius \(R\)), \(ds^2 = R^2(d\theta^2 + \sin^2\theta \, d\phi^2)\), read off \(g_{\theta\theta}\), \(g_{\phi\phi}\), \(g_{\theta\phi}\). Also find the inverse metric components \(g^{\theta\theta}\), \(g^{\phi\phi}\).

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B-4. Asymptotic Minkowski Property of the Schwarzschild Metric

Covariant Derivatives and Curvature (C.5-C.6)

For the Schwarzschild metric

\[ ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right)c^2 dt^2 + \frac{dr^2}{1 - 2GM/(c^2 r)} + r^2 d\Omega^2 \]

show that in the limit \(r \to \infty\), it asymptotically approaches the Minkowski metric.

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B-5. Christoffel Symbols Vanish in Flat Space

Verify that all Christoffel symbols are zero in flat space (\(g_{\mu\nu} = \eta_{\mu\nu}\)).

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B-6. Geodesics in Flat Space Reduce to Uniform Straight-Line Motion

Confirm that the geodesic equation \(\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0\) reduces to uniform straight-line motion \(\ddot{x}^\mu = 0\) when \(\Gamma = 0\).

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Medium

M-1. Four-Momentum and the Mass-Shell Condition

Compute the inner product \(p^\mu p_\mu = \eta_{\mu\nu}p^\mu p^\nu\) of the four-momentum \(p^\mu = (E/c, p_x, p_y, p_z)\), and show that \(p^\mu p_\mu = -m^2c^2\) is equivalent to the relativistic energy-momentum relation \(E^2 = (pc)^2 + (mc^2)^2\).

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M-2. Polar Coordinate Metric and Behavior at \(r=0\)

Write down the metric of two-dimensional flat space in polar coordinates \((r, \theta)\), \(ds^2 = dr^2 + r^2 d\theta^2\), and read off \(g_{rr}\) and \(g_{\theta\theta}\). Discuss what happens at \(r = 0\).

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M-3. \(\Gamma^\theta_{\phi\phi}\) on the Sphere

Calculate \(\Gamma^\theta_{\phi\phi}\) from the metric of the two-dimensional sphere using the formula for Christoffel symbols.

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Advanced

A-1. Bianchi Identity and Energy Conservation

Accept that the left-hand side of the Einstein equation \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\) satisfies \(\nabla^\mu G_{\mu\nu} = 0\) (a consequence of the Bianchi identity), and explain how this is consistent with energy-momentum conservation \(\nabla^\mu T_{\mu\nu} = 0\) on the right-hand side.

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