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Appendix C Solutions

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Raising and Lowering Indices

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\(A^\mu = (1, 2, 3, 4)\), \(\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)\).

\(A_\mu = \eta_{\mu\nu}A^\nu\):

\(A_0 = \eta_{00}A^0 = (-1)(1) = -1\)

\(A_1 = \eta_{11}A^1 = (1)(2) = 2\)

\(A_2 = \eta_{22}A^2 = (1)(3) = 3\)

\(A_3 = \eta_{33}A^3 = (1)(4) = 4\)

\(A_\mu = (-1, 2, 3, 4)\)

Only the time component has its sign flipped.

B-2. Contraction of the Kronecker Delta

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\(\delta^\mu_\nu A^\nu = \sum_{\nu=0}^{3} \delta^\mu_\nu A^\nu\)

Since \(\delta^\mu_\nu\) is 1 when \(\mu = \nu\) and 0 otherwise, only the term with \(\nu = \mu\) survives in the sum:

\(\delta^\mu_\nu A^\nu = A^\mu\)

The Kronecker delta is an operator that "passes the index through unchanged."

B-3. Metric and Inverse Metric of the Sphere

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From \(ds^2 = R^2(d\theta^2 + \sin^2\theta \, d\phi^2)\):

\(g_{\theta\theta} = R^2, \quad g_{\phi\phi} = R^2\sin^2\theta, \quad g_{\theta\phi} = g_{\phi\theta} = 0\)

The inverse metric is the reciprocal of the diagonal components:

\(g^{\theta\theta} = \frac{1}{R^2}, \quad g^{\phi\phi} = \frac{1}{R^2\sin^2\theta}\)

B-4. Asymptotic Minkowski Property of the Schwarzschild Metric

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

As \(r \to \infty\), \(\frac{2GM}{c^2 r} \to 0\), so:

\(ds^2 \to -c^2 dt^2 + dr^2 + r^2 d\Omega^2\)

This is the Minkowski metric written in spherical coordinates. At large distances, spacetime returns to flat.

B-5. Christoffel Symbols Vanish in Flat Space

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When \(g_{\mu\nu} = \eta_{\mu\nu} = \text{const}\), we have \(\partial_\alpha g_{\mu\nu} = 0\) (all partial derivatives are zero).

\(\Gamma^\nu_{\mu\alpha} = \frac{1}{2}g^{\nu\beta}(\partial_\mu g_{\alpha\beta} + \partial_\alpha g_{\mu\beta} - \partial_\beta g_{\mu\alpha}) = \frac{1}{2}g^{\nu\beta}(0 + 0 - 0) = 0\)

All Christoffel symbols are zero.

B-6. Geodesics in Flat Space are Straight Lines with Constant Velocity

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\(\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0\)

When \(\Gamma = 0\):

\(\ddot{x}^\mu = 0\)

This is uniform rectilinear motion (zero acceleration), corresponding to Newton's first law. In curved spacetime (\(\Gamma \neq 0\)), the \(\Gamma\) terms represent "gravitational acceleration."

Medium

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

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\(p^\mu = (E/c,\; p_x,\; p_y,\; p_z)\)

\(p^\mu p_\mu = \eta_{\mu\nu}p^\mu p^\nu = -(E/c)^2 + p_x^2 + p_y^2 + p_z^2 = -E^2/c^2 + |\mathbf{p}|^2\)

Setting this equal to \(-m^2c^2\):

\(-E^2/c^2 + |\mathbf{p}|^2 = -m^2c^2\)

\(E^2/c^2 = |\mathbf{p}|^2 + m^2c^2\)

\(E^2 = |\mathbf{p}|^2 c^2 + m^2 c^4 = (pc)^2 + (mc^2)^2\)

This is precisely the relativistic energy-momentum relation. When \(\mathbf{p} = 0\) (at rest), \(E = mc^2\).

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

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From \(ds^2 = dr^2 + r^2 d\theta^2\):

\(g_{rr} = 1, \quad g_{\theta\theta} = r^2, \quad g_{r\theta} = 0\)

At \(r = 0\), \(g_{\theta\theta} = 0\), and the metric becomes degenerate (\(\det g = 0\)). This is a coordinate singularity, not a physical singularity. At the origin, the \(\theta\) direction cannot be defined (no matter which direction you face, it is the same point).

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

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Formula: \(\Gamma^\nu_{\mu\alpha} = \frac{1}{2}g^{\nu\beta}(\partial_\mu g_{\alpha\beta} + \partial_\alpha g_{\mu\beta} - \partial_\beta g_{\mu\alpha})\)

\(\Gamma^\theta_{\phi\phi} = \frac{1}{2}g^{\theta\theta}(\partial_\phi g_{\phi\theta} + \partial_\phi g_{\phi\theta} - \partial_\theta g_{\phi\phi})\)

Since \(g_{\phi\theta} = 0\), the first two terms are zero:

\(= \frac{1}{2} \cdot \frac{1}{R^2} \cdot (0 + 0 - \partial_\theta(R^2\sin^2\theta))\)

\(= \frac{1}{2R^2}(-2R^2\sin\theta\cos\theta) = -\sin\theta\cos\theta\)

Advanced

A-1. Bianchi Identity and Energy Conservation

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The Einstein tensor \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\) automatically satisfies:

\(\nabla^\mu G_{\mu\nu} = 0\)

as a consequence of the Bianchi identity for the Riemann tensor (this is an identity of differential geometry, not an equation of motion).

Applying \(\nabla^\mu\) to both sides of the Einstein equation \(G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}\):

\(0 = \nabla^\mu G_{\mu\nu} = \frac{8\pi G}{c^4}\nabla^\mu T_{\mu\nu}\)

Therefore \(\nabla^\mu T_{\mu\nu} = 0\) (conservation of energy-momentum).

In other words, the mathematical structure of the Einstein equation automatically guarantees energy-momentum conservation. There is no need to separately postulate the conservation law.