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Appendix C Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Raising and Lowering Indices

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B-2. Contraction of the Kronecker Delta

Metric Tensor (C.3)

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B-3. Metric and Inverse Metric of the Sphere

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B-4. Asymptotic Minkowski Property of the Schwarzschild Metric

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

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B-5. Christoffel Symbols Vanish in Flat Space

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

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Medium

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

Problem

Consider a particle with rest mass \(m\) moving at velocity \(v\) in the \(x\)-direction in \((1+1)\)-dimensional spacetime.

(1) Write the four-momentum \(p^\mu = (p^0, p^1)\) using \(m\), \(v\), and the speed of light \(c\).

(2) Show that the following mass-shell condition holds:

\[ \eta_{\mu\nu} p^\mu p^\nu = -m^2 c^2 \]

Here, the metric signature is \(\eta_{\mu\nu} = \mathrm{diag}(-1, +1)\).

(3) In string theory, massless states (\(m = 0\)) and tachyon states (\(m^2 < 0\)) appear. Explain the physical meaning of the mass-shell condition for each.

Solution

(1) Four-momentum

The Lorentz factor is:

\[ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \]

The four-momentum is defined as \(p^\mu = m \gamma (c, v)\), so:

\[ p^0 = \frac{mc}{\sqrt{1 - v^2/c^2}}, \quad p^1 = \frac{mv}{\sqrt{1 - v^2/c^2}} \]

Here, \(p^0 = E/c\) (where \(E\) is energy), and \(p^1\) is the spatial momentum.

(2) Derivation of the mass-shell condition

\[ \eta_{\mu\nu} p^\mu p^\nu = -(p^0)^2 + (p^1)^2 \]

Substituting:

\[ = -\frac{m^2 c^2}{1 - v^2/c^2} + \frac{m^2 v^2}{1 - v^2/c^2} \]
\[ = \frac{m^2(-c^2 + v^2)}{1 - v^2/c^2} \]
\[ = \frac{-m^2 c^2(1 - v^2/c^2)}{1 - v^2/c^2} \]
\[ = -m^2 c^2 \]

Thus, \(\eta_{\mu\nu} p^\mu p^\nu = -m^2 c^2\) is proven. \(\blacksquare\)

(3) Physical meaning

For massless states (\(m = 0\)):

\[ \eta_{\mu\nu} p^\mu p^\nu = 0 \quad \Longleftrightarrow \quad E^2 = (pc)^2 \]

The momentum lies on the light cone. This corresponds to states propagating at the speed of light, such as gravitons and photons.

For tachyon states (\(m^2 < 0\)):

Writing \(m^2 = -\mu^2\) (\(\mu^2 > 0\)):

\[ \eta_{\mu\nu} p^\mu p^\nu = \mu^2 c^2 > 0 \]

The momentum becomes spacelike. This is physically interpreted as a signal of vacuum instability (the potential has an unstable extremum). In bosonic string theory, the ground state of the open string becomes a tachyon with \(m^2 = -1/\alpha'\), indicating that the vacuum of this theory is unstable.

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

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

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Advanced

A-1. Bianchi Identity and Energy Conservation

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