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Ch. 6 Solutions

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Left-Hand Side and Right-Hand Side of the Einstein Equation

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(a) Physical Meaning

  • Left-hand side \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\): A geometric quantity representing the curvature of spacetime. It automatically satisfies \(\nabla^\mu G_{\mu\nu} = 0\) due to the Bianchi identity.
  • Right-hand side \(T_{\mu\nu}\): A tensor that encapsulates the energy density, momentum density, and stress (pressure and shear stress) at that location. The conservation law \(\nabla^\mu T_{\mu\nu} = 0\) represents the conservation of energy and momentum of matter.

In a nutshell: The left-hand side is "the geometry of spacetime," and the right-hand side is "matter and energy." In Wheeler's words, "Spacetime tells matter how to move; matter tells spacetime how to curve."

(b) \(R_{\mu\nu} = 0\) in Vacuum

Calculation: Contract both sides of the Einstein equation with \(g^{\mu\nu}\):

\[ g^{\mu\nu}R_{\mu\nu} - \frac{1}{2}g^{\mu\nu}g_{\mu\nu}R = \frac{8\pi G}{c^4}g^{\mu\nu}T_{\mu\nu} \]

Using \(g^{\mu\nu}R_{\mu\nu} = R\), \(g^{\mu\nu}g_{\mu\nu} = 4\) (trace in 4 dimensions), and \(g^{\mu\nu}T_{\mu\nu} = T\):

\[ R - 2R = \frac{8\pi G}{c^4}T \quad \Longrightarrow \quad R = -\frac{8\pi G}{c^4}T \]

In vacuum \(T_{\mu\nu} = 0\), so \(T = 0\) and \(R = 0\). Substituting back into the original Einstein equation:

\[ R_{\mu\nu} - 0 = 0 \quad \Longrightarrow \quad \boxed{R_{\mu\nu} = 0} \]

(c) Physical Meaning of the Schwarzschild Metric

The Schwarzschild metric is a solution of \(R_{\mu\nu} = 0\) — it describes the spacetime structure in the vacuum region outside a star. Inside the star (\(r < R_{\text{star}}\)), \(T_{\mu\nu} \neq 0\), and a separate interior solution is required (for the spherically symmetric case, uniqueness is guaranteed by Birkhoff's theorem).

Applications: Planetary motion in the solar system occurs outside the Sun, so it can be described by the Schwarzschild solution. The perihelion precession of Mercury, the deflection of light, and GPS time corrections all emerge from this vacuum solution.

Medium

M-1. The Convenient Action and Constraints

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(a) Euler-Lagrange Equation

Strategy: Vary the Lagrangian \(\mathcal{L} = g_{\mu\nu}(x)\dot{x}^\mu\dot{x}^\nu / 2\) with respect to \(x^\sigma\).

Calculation:

From \(\partial\mathcal{L}/\partial\dot{x}^\sigma\):

\[ \frac{\partial\mathcal{L}}{\partial\dot{x}^\sigma} = \frac{1}{2}\left(g_{\sigma\nu}\dot{x}^\nu + g_{\mu\sigma}\dot{x}^\mu\right) = g_{\sigma\nu}\dot{x}^\nu \]

(We used \(g_{\mu\nu} = g_{\nu\mu}\) and relabeling of dummy indices.)

Time derivative:

\[ \frac{d}{d\tau}\left(g_{\sigma\nu}\dot{x}^\nu\right) = \partial_\alpha g_{\sigma\nu}\,\dot{x}^\alpha\dot{x}^\nu + g_{\sigma\nu}\ddot{x}^\nu \]

From \(\partial\mathcal{L}/\partial x^\sigma\):

\[ \frac{\partial\mathcal{L}}{\partial x^\sigma} = \frac{1}{2}\partial_\sigma g_{\mu\nu}\,\dot{x}^\mu\dot{x}^\nu \]

Substituting into the Euler-Lagrange equation \(\frac{d}{d\tau}\partial\mathcal{L}/\partial\dot{x}^\sigma - \partial\mathcal{L}/\partial x^\sigma = 0\):

\[ g_{\sigma\nu}\ddot{x}^\nu + \partial_\alpha g_{\sigma\nu}\,\dot{x}^\alpha\dot{x}^\nu - \frac{1}{2}\partial_\sigma g_{\mu\nu}\,\dot{x}^\mu\dot{x}^\nu = 0 \]

Symmetrizing the middle term using the symmetry of \(\dot{x}^\alpha\dot{x}^\nu\) under \(\alpha \leftrightarrow \nu\):

\[ \partial_\alpha g_{\sigma\nu}\dot{x}^\alpha\dot{x}^\nu = \frac{1}{2}\left(\partial_\alpha g_{\sigma\nu} + \partial_\nu g_{\sigma\alpha}\right)\dot{x}^\alpha\dot{x}^\nu \]

Substituting this (and relabeling \(\mu, \nu\) → \(\alpha, \beta\)):

\[ g_{\sigma\beta}\ddot{x}^\beta + \frac{1}{2}\left(\partial_\alpha g_{\sigma\beta} + \partial_\beta g_{\sigma\alpha} - \partial_\sigma g_{\alpha\beta}\right)\dot{x}^\alpha\dot{x}^\beta = 0 \]

(b) Reduction to the Geodesic Equation

Calculation: Multiply both sides by the inverse metric \(g^{\mu\sigma}\), using \(g^{\mu\sigma}g_{\sigma\beta} = \delta^\mu{}_\beta\):

\[ \ddot{x}^\mu + \frac{1}{2}g^{\mu\sigma}\left(\partial_\alpha g_{\sigma\beta} + \partial_\beta g_{\sigma\alpha} - \partial_\sigma g_{\alpha\beta}\right)\dot{x}^\alpha\dot{x}^\beta = 0 \]

Defining the Christoffel symbols as

\[ \boxed{\Gamma^\mu_{\alpha\beta} = \frac{1}{2}g^{\mu\sigma}\left(\partial_\alpha g_{\sigma\beta} + \partial_\beta g_{\sigma\alpha} - \partial_\sigma g_{\alpha\beta}\right)} \]

we obtain the desired geodesic equation:

\[ \boxed{\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0} \]

(c) Conservation of the Constraint

Argument: Differentiate \(Q(\tau) \equiv g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu\) with respect to \(\tau\):

\[ \frac{dQ}{d\tau} = \partial_\alpha g_{\mu\nu}\dot{x}^\alpha\dot{x}^\mu\dot{x}^\nu + 2g_{\mu\nu}\ddot{x}^\mu\dot{x}^\nu \]

Substituting the geodesic equation \(\ddot{x}^\mu = -\Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta\) and expanding the Christoffel symbols using their definition, one can show upon simplification that \(dQ/d\tau = 0\) (this is equivalent to the vanishing of the covariant derivative of \(g_{\mu\nu}\)).

Therefore, if we impose the initial condition \(Q(0) = -c^2\), then \(Q(\tau) = -c^2\) is maintained for all values of \(\tau\)—the constraint is automatically preserved. This is the mathematical content of the claim that "\(\tau\) can be chosen as proper time."

Check: In flat spacetime \(g_{\mu\nu} = \eta_{\mu\nu}\), we have \(\Gamma^\mu_{\alpha\beta} = 0\) so \(\ddot{x}^\mu = 0\): uniform rectilinear motion. The condition \(\eta_{\mu\nu}\dot{x}^\mu\dot{x}^\nu = -c^2\) is precisely the normalization condition for the four-velocity. ✓

M-2. From the Convenient Action to the String Action

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(a) Correspondence of Notation

Particle (worldline, 1-dimensional) String (worldsheet, 2-dimensional)
Parameter \(\tau\) Parameters \(\sigma^a = (\tau, \sigma)\) (\(a = 0, 1\))
Worldline \(x^\mu(\tau)\) Worldsheet \(X^\mu(\tau, \sigma)\)
Derivative \(\dot{x}^\mu = dx^\mu/d\tau\) Partial derivatives \(\partial_a X^\mu\)
Action = length of worldline \(\int d\tau\sqrt{-g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu}\) Action = area of worldsheet \(\int d^2\sigma\sqrt{-\det(g_{\mu\nu}\partial_a X^\mu\partial_b X^\nu)}\)
Convenient action \(\frac{1}{2}\int d\tau\,g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu\) Polyakov action \(-\frac{T}{2}\int d^2\sigma\sqrt{-h}\,h^{ab}g_{\mu\nu}\partial_a X^\mu\partial_b X^\nu\)
Constraint: \(g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu = -c^2\) Constraints: \(T_{ab} = 0\) (Virasoro constraints, Ch. 14)

(b) Constraint Conditions in String Theory

Solution strategy: Vary the Polyakov action \(S_{\text{P}}\) with respect to the auxiliary metric \(h^{ab}\).

Calculation: \(h^{ab}\) is an auxiliary field (playing the role of a Lagrange multiplier) that has no equations of motion within the Polyakov action. Varying \(S_{\text{P}}\) with respect to \(h^{ab}\) yields the condition

\[ T_{ab} \equiv \partial_a X^\mu\,\partial_b X_\mu - \frac{1}{2}h_{ab}\,h^{cd}\partial_c X^\mu\,\partial_d X_\mu = 0 \]

This is the classical string constraint condition, which upon quantization becomes the Virasoro constraints.

In the particle case, \(g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu = -c^2\) is a single constraint, while in the string case, \(T_{ab} = 0\) imposes constraints with 3 independent components at each point on the worldsheet (among \(T_{00}, T_{01}, T_{11}\), symmetry and trace conditions reduce the independent ones to 2).

Consistency check: For a particle (1-dimensional), the constraint is 1 condition; for a string (2-dimensional), the constraints increase — it is natural that constraints increase as the dimension goes up. ✓

(c) Three Common Structures

  1. Role of the metric \(g_{\mu\nu}(X)\): In both the point particle and the string, \(g_{\mu\nu}\) plays the same role as the "externally given background spacetime" (the target space metric in which the string moves).
  2. Reparametrization invariance: Both the worldline parameter \(\tau\) and the worldsheet parameters \(\sigma^a\) leave the action invariant under reparametrizations (worldline reparametrization / worldsheet diffeomorphisms). This enables the separation of physical degrees of freedom from gauge degrees of freedom.
  3. Elimination of the square root: In both cases, introducing an auxiliary field (constraint condition / \(h_{ab}\)) removes the square root, making calculation and quantization tractable.

M-3. Clock Delay in a Weak Gravitational Field

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(a) Derivation of the Approximation

For \(|\Phi|/c^2 \ll 1\), we have \(\sqrt{1 + 2\Phi/c^2} \approx 1 + \Phi/c^2\) (using \(\sqrt{1+x} \approx 1 + x/2\)). Therefore:

\[ \boxed{\frac{d\tau}{dt} \approx 1 + \frac{\Phi}{c^2}} \]

At locations with lower gravitational potential (large \(|\Phi|\), \(\Phi < 0\)), \(d\tau/dt < 1\)—clocks run slower. At higher locations (larger \(\Phi\), i.e., smaller \(|\Phi|\)), clocks run relatively faster.

(b) GPS Satellite and Ground Clocks

Ground surface: \(r_{\text{ground}} = R_\oplus = 6.37 \times 10^6\) m

\(\Phi_{\text{ground}} = -\frac{GM_\oplus}{R_\oplus} = -\frac{(6.674 \times 10^{-11})(5.97 \times 10^{24})}{6.37 \times 10^6} \approx -6.25 \times 10^7\ \mathrm{m^2/s^2}\)

Satellite (altitude \(h = 20000\) km \(= 2.0 \times 10^7\) m): \(r_{\text{sat}} = R_\oplus + h \approx 2.637 \times 10^7\) m

\(\Phi_{\text{sat}} = -\frac{GM_\oplus}{r_{\text{sat}}} \approx -1.51 \times 10^7\ \mathrm{m^2/s^2}\)

Potential difference:

\(\Delta\Phi = \Phi_{\text{sat}} - \Phi_{\text{ground}} \approx -1.51 \times 10^7 - (-6.25 \times 10^7) = +4.74 \times 10^7\ \mathrm{m^2/s^2}\)

Ratio of proper times:

\[ \frac{d\tau_{\text{sat}}}{d\tau_{\text{ground}}} - 1 \approx \frac{\Delta\Phi}{c^2} = \frac{4.74 \times 10^7}{8.99 \times 10^{16}} \approx 5.28 \times 10^{-10} \]

The satellite clock runs approximately 5.3 × 10⁻¹⁰ seconds faster per second than the ground clock.

Drift per day:

\(5.28 \times 10^{-10} \times 86400 \approx 4.6 \times 10^{-5}\ \mathrm{s} = 46\ \mu\mathrm{s}\)

\[ \boxed{\text{Approximately 46 microseconds of drift per day}} \]

Note: This accounts for general relativistic effects only. Since GPS satellites move at high speed, special relativistic effects (time dilation due to \(v \approx 3.87\) km/s, approximately \(-7\) μs per day) also contribute, giving an overall drift of +38 μs/day.

(c) Impact on Position Accuracy

The distance light travels in 1 microsecond:

\(c \cdot 10^{-6}\ \mathrm{s} = 3 \times 10^8 \times 10^{-6} = 300\ \mathrm{m}\)

Since GPS positioning is based on precise measurement of signal travel times from satellites, clock drift translates directly into distance errors. The 46 μs/day from (b) would produce an error of ≈ 14 km/day—without corrections, the drift would exceed 10 km in a single day.

Physical significance: General relativity is essential for everyday technology (smartphone GPS, car navigation, surveying, air traffic control). The correction algorithm directly incorporates \(d\tau/dt = 1 + \Phi/c^2\).

Consistency check: For Earth, \(GM/R \sim c^2 \cdot 10^{-9}\), so the gravitational potential is 9 orders of magnitude smaller than the square of the speed of light. The weak-field approximation is well justified. ✓

Advanced

A-1. Singularities and the Need for Quantum Gravity

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(a) The Scale Where Quantum Gravity Becomes Relevant

Calculation: Solve \(K = 48 G^2 M^2/(c^4 r^6) \sim 1/\ell_P^4\) for \(r\):

\[ r^6 \sim 48 G^2 M^2 \ell_P^4/c^4 \]

Take the sixth root of \(r\). Substituting numerical values for solar mass \(M = M_\odot \approx 2.0 \times 10^{30}\) kg:

\(G M_\odot / c^2 \approx 1477\) m (corresponds to half the Schwarzschild radius)

\(48 \cdot (G M_\odot)^2 / c^4 \approx 48 \cdot 1477^2 \approx 1.05 \times 10^8\ \mathrm{m^2}\)

\(\ell_P^4 \approx (1.6 \times 10^{-35})^4 \approx 6.6 \times 10^{-140}\ \mathrm{m^4}\)

\(r^6 \sim 1.05 \times 10^8 \times 6.6 \times 10^{-140} \approx 6.9 \times 10^{-132}\ \mathrm{m^6}\)

\[ r \sim (6.9 \times 10^{-132})^{1/6} \approx 10^{-22}\ \mathrm{m} \]

(b) Comparison with the Schwarzschild Radius

The Schwarzschild radius of a solar-mass black hole is \(r_s = 2GM_\odot/c^2 \approx 3\) km \(= 3 \times 10^3\) m.

The value \(r \sim 10^{-22}\) m found in (a) has a ratio to \(r_s\) of \(10^{-22}/(3 \times 10^3) \sim 3 \times 10^{-26}\)—quantum gravity becomes relevant only in an extremely deep region well inside the Schwarzschild radius. From the event horizon to the center, classical general relativity describes most of the region with sufficient accuracy.

However: The size of the region where quantum gravity is needed is itself \(10^{-22}\) m, which is much larger than the Planck length \(\ell_P \sim 10^{-35}\) m. This represents "the boundary where quantum gravitational effects begin to become non-negligible"; going further down to the true Planck scale, the full picture of quantum gravity is even more critically demanded.

(c) Relation to Falsifiability

General relativity predicts its own limits of applicability (singularities) from within. This means:

  1. Models are not perfect: The appearance of infinities signifies the breakdown of the current model. This is clear evidence that it is a "hypothesis," not a "law."
  2. A better model is needed: This is precisely why the pursuit of quantum gravity theories (string theory, loop quantum gravity, etc.) is a frontier challenge in physics.
  3. The true power of falsifiability: By explicitly showing "where it breaks down," general relativity concretizes what the next model must satisfy (e.g., singularities are resolved at the Planck scale, the black hole information paradox is resolved, etc.).

Here we find a concrete example of the stance emphasized in the prologue: "models are hypotheses."