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Ch. 6 Problems

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Basic

Medium

Advanced

Basic

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

Regarding the Einstein equation \(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}\):

(a) State in one sentence each the physical meaning of the left-hand side (Einstein tensor \(G_{\mu\nu}\)) and the right-hand side (energy-momentum tensor \(T_{\mu\nu}\)).

(b) Show that in vacuum \(T_{\mu\nu} = 0\), the Einstein equation reduces to \(R_{\mu\nu} = 0\) (the key point is to take the trace of both sides).

(c) The Schwarzschild metric is a solution of \(R_{\mu\nu} = 0\). What does this mean physically?

Hint

(b) Contracting both sides with \(g^{\mu\nu}\) gives \(R - 2R = \frac{8\pi G}{c^4}T\) (where \(T = g^{\mu\nu}T_{\mu\nu}\)). If \(T_{\mu\nu} = 0\) then \(R = 0\), and substituting back into the original equation yields \(R_{\mu\nu} = 0\). (c) It describes the spacetime structure outside a star (in vacuum). The energy-momentum of the star itself exists only in the region \(r < R_{\text{star}}\).

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Medium

M-1. The Convenient Action and Constraints

The "convenient action" for a particle of mass \(m\):

\[ S_{\text{useful}} = \frac{1}{2}\int d\tau\,g_{\mu\nu}(x)\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} \]

Show that the geodesic equation

\[ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0 \]

is obtained from this action under the constraint \(g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu = -c^2\).

(a) Write down the Euler-Lagrange equation with respect to \(x^\sigma\), and simplify using the symmetry of \(g_{\mu\nu}\) and relabeling of dummy indices.

(b) Multiply both sides by the inverse metric \(g^{\sigma\mu}\) to solve for \(\ddot{x}^\mu\), and obtain the defining expression for the Christoffel symbols \(\Gamma^\mu_{\alpha\beta}\).

(c) Argue that the constraint \(g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu = -c^2\) is automatically preserved during time evolution (i.e., it holds once \(\tau\) is chosen to be proper time).

Hint

(b) The definition of the Christoffel symbols is \(\Gamma^\mu_{\alpha\beta} = \frac{1}{2}g^{\mu\lambda}(\partial_\alpha g_{\beta\lambda} + \partial_\beta g_{\alpha\lambda} - \partial_\lambda g_{\alpha\beta})\). Refer to the derivation in the main text and General Relativity Ch. 8. (c) From the Euler-Lagrange equations, one can show that \(g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu\) is a conserved quantity with respect to \(\tau\).

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

The convenient action for a particle \(S_{\text{useful}} = \frac{1}{2}\int d\tau\,g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu\) naturally generalizes to the Polyakov action in string theory:

\[ S_{\text{P}} = -\frac{T}{2}\int d^2\sigma\sqrt{-h}\,h^{ab}\,\partial_a X^\mu\,\partial_b X^\nu\,g_{\mu\nu}(X) \]

Organize the following correspondences.

(a) Establish the symbol correspondences that describe the difference between the particle's "worldline" (1-dimensional) and the string's "worldsheet" (2-dimensional) (\(\tau \leftrightarrow \sigma^a\), \(\dot{x}^\mu \leftrightarrow \partial_a X^\mu\), etc.).

(b) Discuss what the string theory "constraint" corresponding to the particle's constraint \(g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu = -c^2\) becomes, using the equation of motion (variation) of \(h_{ab}\).

(c) List three structural features common to both the particle action and the string action (the role of the metric \(g_{\mu\nu}(X)\), reparametrization invariance, and the advantage of eliminating the square root).

Hint

(a) Particle: \(\tau\) is a single parameter, and the worldline is \(x^\mu(\tau)\). String: \(\sigma^a = (\tau, \sigma)\) are two parameters, and the worldsheet is \(X^\mu(\tau, \sigma)\). (b) Varying \(h_{ab}\) yields the energy-momentum tensor \(T_{ab} = 0\). This is the constraint (the classical version of the Virasoro constraints). Details are covered in Ch. 13.

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M-3. Clock Delay in a Weak Gravitational Field

In a spacetime with a weak gravitational field \(g_{00} \approx -(1 + 2\Phi/c^2)\), the relationship between the proper time \(d\tau\) measured by a clock at rest and the coordinate time \(dt\) is

\[ d\tau = dt\sqrt{-g_{00}} = dt\sqrt{1 + 2\Phi/c^2} \]

(a) Show that \(d\tau/dt \approx 1 + \Phi/c^2\) in the approximation \(|\Phi|/c^2 \ll 1\).

(b) Calculate the ratio of proper times between a GPS satellite (altitude \(h \approx 20000\) km above the surface, with Earth's gravitational potential \(\Phi(r) = -GM_\oplus/r\)) and a clock on the ground, and estimate the drift per day (in microseconds) (\(M_\oplus = 5.97 \times 10^{24}\) kg, \(R_\oplus = 6.37 \times 10^6\) m).

(c) Estimate the effect of a 1-microsecond time drift per day on position measurement, in units of \(c \cdot 10^{-6}\) s. Explain why corrections are necessary for GPS position accuracy.

Hint

(a) Use \(\sqrt{1+x} \approx 1 + x/2\). (b) Calculate the difference between \(\Phi\) at the surface and \(\Phi\) at the satellite. \(\Phi_{\text{satellite}} - \Phi_{\text{surface}} > 0\) (the satellite has a higher gravitational potential, i.e., a smaller absolute value). (c) The distance light travels in 1 microsecond is approximately 300 m. This means without corrections, the drift would be several hundred meters per day, and several kilometers over a few days.

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Advanced

A-1. Singularities and the Need for Quantum Gravity

Consider \(r \to 0\) in the Schwarzschild metric.

(a) Using the curvature invariant \(K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} = 48 G^2 M^2/(c^4 r^6)\), estimate the value of \(r\) at which \(K\) becomes comparable to the Planck scale \(1/\ell_P^4\) (\(\ell_P = \sqrt{\hbar G/c^3} \approx 1.6 \times 10^{-35}\) m, \(M\) is the solar mass).

(b) From the value of \(r\) obtained in (a), evaluate the size of the region where the Planck scale is reached for a solar-mass black hole, and confirm that the "region where quantum gravity becomes necessary" lies far inside the Schwarzschild radius \(r_s \approx 3\) km.

(c) Briefly discuss how the existence of singularities relates to the falsifiability of general relativity (from the standpoint that "models are hypotheses").

Hint

(a) Solve \(48 G^2 M^2/(c^4 r^6) \sim 1/\ell_P^4\) for \(r\). (b) For a solar-mass black hole, the region where quantum gravity is needed satisfies \(r \ll r_s\). (c) General relativity predicts its own breakdown—this is evidence that "models are provisional."

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