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Ch. 7 Solutions

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Calculation of Line Element and Spacetime Classification Using the Minkowski Metric

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Solution strategy: Substitute the coordinate differences into \(ds^2 = \eta_{\alpha\beta}\,dx^\alpha\,dx^\beta = -dt^2 + dx^2 + dy^2 + dz^2\).

Calculation:

\[ ds^2 = -(3)^2 + (1)^2 + (2)^2 + (0)^2 = -9 + 1 + 4 + 0 = -4 \]

Classification: Since \(ds^2 = -4 < 0\), this interval is timelike.

Verification: For the time component \(dt = 3\), the magnitude of the spatial components is \(\sqrt{1^2 + 2^2 + 0^2} = \sqrt{5} \approx 2.24 < 3\), which corresponds to motion at less than the speed of light, consistent with the interval being timelike.

B-2. Metric Tensor and Inverse Metric of the 2-Dimensional Sphere

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Solution strategy: Read off each component from \(ds^2 = a^2\,d\theta^2 + a^2\sin^2\theta\,d\varphi^2\). Since the metric is diagonal, the inverse metric components are simply the reciprocals of each component.

Metric tensor components:

\[ g_{\theta\theta} = a^2, \qquad g_{\varphi\varphi} = a^2\sin^2\theta, \qquad g_{\theta\varphi} = g_{\varphi\theta} = 0 \]

Inverse metric components:

\[ g^{\theta\theta} = \frac{1}{g_{\theta\theta}} = \frac{1}{a^2}, \qquad g^{\varphi\varphi} = \frac{1}{g_{\varphi\varphi}} = \frac{1}{a^2\sin^2\theta} \]

Verification: \(g_{\theta\theta}\,g^{\theta\theta} = a^2 \cdot \frac{1}{a^2} = 1\), \(g_{\varphi\varphi}\,g^{\varphi\varphi} = a^2\sin^2\theta \cdot \frac{1}{a^2\sin^2\theta} = 1\). The inverse matrix condition \(g_{\alpha\gamma}\,g^{\gamma\beta} = \delta_\alpha^{\ \beta}\) is satisfied. ✓

B-3. Proper Length in the \(\varphi\) Direction in Polar Coordinates

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Solution strategy: The metric for flat spacetime in polar coordinates (6.12) is \(g_{\alpha\beta} = \mathrm{diag}(-1, 1, r^2, r^2\sin^2\theta)\). Set \(dt = dr = d\theta = 0\) and find \(dL\).

Calculation:

Fixing \(r = R\), \(\theta = \pi/4\), and varying only \(\varphi\):

\[ dL^2 = g_{\varphi\varphi}\,d\varphi^2 = r^2\sin^2\theta\,d\varphi^2 = R^2\sin^2\!\left(\frac{\pi}{4}\right)d\varphi^2 = R^2\left(\frac{\sqrt{2}}{2}\right)^2 d\varphi^2 = \frac{R^2}{2}\,d\varphi^2 \]
\[ \boxed{dL = \frac{R}{\sqrt{2}}\,d\varphi} \]

Sanity check: For \(\theta = \pi/2\) (the equator), \(dL = R\,d\varphi\), which is the usual arc length. Since \(\theta = \pi/4\) is closer to the pole than the equator, the circumference should be shorter, and \(R/\sqrt{2} < R\) is consistent with this. ✓

B-4. Proper Time of a Static Observer in Schwarzschild Metric

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Solution strategy: Assume stationary (\(dr = d\theta = d\varphi = 0\)) and compute \(d\tau^2 = -ds^2 = -g_{00}\,dt^2\).

Calculation:

\[ d\tau^2 = \left(1 - \frac{2M}{r}\right)dt^2 \]

Substituting \(r = 10M\):

\[ d\tau^2 = \left(1 - \frac{2M}{10M}\right)dt^2 = \left(1 - \frac{1}{5}\right)dt^2 = \frac{4}{5}\,dt^2 \]
\[ \boxed{d\tau = \sqrt{\frac{4}{5}}\,dt = \frac{2}{\sqrt{5}}\,dt = \frac{2\sqrt{5}}{5}\,dt} \]

Numerically, \(d\tau \approx 0.894\,dt\).

Verification: Since \(r = 10M \gg 2M\), \(d\tau\) should be slightly smaller than \(dt\). Indeed \(0.894 < 1\), which is consistent. The limits \(d\tau \to dt\) as \(r \to \infty\) and \(d\tau \to 0\) as \(r \to 2M\) are also correct. ✓

B-5. Metric Components of the Schwarzschild Metric at \(r = 4M\)

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Solution strategy: Substitute \(r = 4M\), \(\theta = \pi/2\) into the metric tensor (6.15).

Calculation:

\[ 1 - \frac{2M}{r} = 1 - \frac{2M}{4M} = 1 - \frac{1}{2} = \frac{1}{2} \]

Each component:

\[ g_{00} = -\left(1 - \frac{2M}{r}\right) = -\frac{1}{2} \]
\[ g_{11} = \left(1 - \frac{2M}{r}\right)^{-1} = \left(\frac{1}{2}\right)^{-1} = 2 \]
\[ g_{22} = r^2 = (4M)^2 = 16M^2 \]
\[ g_{33} = r^2\sin^2\theta = (4M)^2\sin^2\!\left(\frac{\pi}{2}\right) = 16M^2 \]
\[ \boxed{g_{00} = -\frac{1}{2},\quad g_{11} = 2,\quad g_{22} = 16M^2,\quad g_{33} = 16M^2} \]

Verification: From \(g_{00} \cdot g^{00} = 1\) we get \(g^{00} = -2\). From \(g_{11} \cdot g^{11} = 1\) we get \(g^{11} = 1/2\). We have \(g_{00} \cdot g_{11} = -1/2 \times 2 = -1\), which satisfies the relation \(g_{00} \cdot g_{11} = -1\) between \(g_{00} = -(1-2M/r)\) and \(g_{11} = (1-2M/r)^{-1}\). ✓

B-6. Components and Inverse Metric of the de Sitter Type Metric

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Solution strategy: Read off the diagonal components from \(ds^2 = -dt^2 + e^{2Ht}(dx^2 + dy^2 + dz^2)\) and take their reciprocals.

Non-zero components of the metric tensor:

\[ g_{00} = -1, \qquad g_{11} = e^{2Ht}, \qquad g_{22} = e^{2Ht}, \qquad g_{33} = e^{2Ht} \]

(All off-diagonal components are zero)

Non-zero components of the inverse metric:

\[ g^{00} = \frac{1}{g_{00}} = -1, \qquad g^{11} = \frac{1}{g_{11}} = e^{-2Ht}, \qquad g^{22} = e^{-2Ht}, \qquad g^{33} = e^{-2Ht} \]

Verification: \(g_{11}\,g^{11} = e^{2Ht} \cdot e^{-2Ht} = 1\). ✓ Also, when \(H = 0\), we recover \(g_{\alpha\beta} = \eta_{\alpha\beta}\). ✓

B-7. Proper Length in the \(\varphi\) Direction in the Schwarzschild Metric

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Solution strategy: Set \(dt = dr = d\theta = 0\) to calculate the proper length in the \(\varphi\) direction, and compare with the flat spacetime case.

Calculation in the Schwarzschild metric:

At \(r = 6M\), \(\theta = \pi/2\), moving by \(d\varphi\) in the \(\varphi\) direction:

\[ dL^2 = g_{33}\,d\varphi^2 = r^2\sin^2\theta\,d\varphi^2 = (6M)^2 \cdot 1 \cdot d\varphi^2 = 36M^2\,d\varphi^2 \]
\[ dL = 6M\,d\varphi \]

Calculation in flat spacetime polar coordinates:

In flat spacetime (6.12), \(g_{33} = r^2\sin^2\theta\) as well, so for the same \(r = 6M\), \(\theta = \pi/2\):

\[ dL_{\text{flat}} = 6M\,d\varphi \]

Comparison: The two are identical.

\[ \boxed{dL = dL_{\text{flat}} = 6M\,d\varphi} \]

This is because \(g_{33} = r^2\sin^2\theta\) in the Schwarzschild metric has the same form as in flat spacetime. The only components of the Schwarzschild metric that differ from flat spacetime are \(g_{00}\) and \(g_{11}\); the angular components in the \(\theta\) and \(\varphi\) directions remain unchanged. The coordinate \(r\) in Schwarzschild coordinates is defined such that "the area of a sphere at coordinate radius \(r\) is \(4\pi r^2\)" (the areal radius), so the arc length in the \(\varphi\) direction is the same as in flat spacetime.

Verification: As \(r \to \infty\), the Schwarzschild metric approaches flatness, so it is natural for the angular directions to agree. Moreover, since \(g_{22}\) and \(g_{33}\) have the same form as in flat spacetime even at finite \(r\), the agreement holds for any \(r\). ✓

B-8. Proper Time in the Rindler Metric

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Solution strategy: Calculate \(d\tau^2 = -ds^2\) with \(dx = 0\) (at rest).

Calculation:

\[ ds^2 = -\alpha^2 x_0^2\,dt^2 + 0 = -\alpha^2 x_0^2\,dt^2 \]
\[ d\tau^2 = -ds^2 = \alpha^2 x_0^2\,dt^2 \]
\[ \boxed{d\tau = \alpha x_0\,dt} \]

(Since \(x_0 > 0\) and \(\alpha > 0\), we have \(d\tau > 0\).)

Verification: The larger \(x_0\) is, the larger \(d\tau\) becomes (time runs faster). The Rindler metric describes spacetime as seen by an observer with uniform acceleration \(\alpha\), and larger \(x_0\) (farther in the direction of acceleration) corresponds to a position with higher gravitational potential, so time running faster is physically correct. This has a structure analogous to the Schwarzschild metric \(d\tau = \sqrt{1-2M/r}\,dt\) (the square root of \(g_{00}\) gives the ratio of proper time to coordinate time). ✓

Medium

M-1. Calculating the Area of a Sphere

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Solution strategy: From the flat spacetime polar coordinates (6.11), set \(t\) constant and \(r = R\) constant to obtain the induced metric, find the area element, and integrate over the entire sphere.

Derivation of the induced metric:

Setting \(dt = 0\), \(dr = 0\), the line element on the sphere at \(r = R\) is:

\[ ds^2_{(2)} = R^2\,d\theta^2 + R^2\sin^2\theta\,d\varphi^2 \]

The metric matrix of the 2-dimensional subspace is:

\[ g_{ij} = \begin{pmatrix} R^2 & 0 \\ 0 & R^2\sin^2\theta \end{pmatrix} \]

Calculation of the determinant:

\[ \det(g_{ij}) = R^2 \cdot R^2\sin^2\theta = R^4\sin^2\theta \]

Area element:

\[ dA = \sqrt{\det(g_{ij})}\,d\theta\,d\varphi = R^2\sin\theta\,d\theta\,d\varphi \]

Integration over the entire sphere:

\[ A = \int_0^{2\pi}\int_0^{\pi} R^2\sin\theta\,d\theta\,d\varphi \]

\(\varphi\) integration:

\[ \int_0^{2\pi} d\varphi = 2\pi \]

\(\theta\) integration:

\[ \int_0^{\pi} \sin\theta\,d\theta = [-\cos\theta]_0^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2 \]

Therefore:

\[ \boxed{A = R^2 \cdot 2\pi \cdot 2 = 4\pi R^2} \]

Verification: This is precisely the formula for the surface area of a sphere of radius \(R\), correctly derived from the metric. Dimensionally, it has units of \([R^2]\), which is the dimension of area. ✓

M-2. Circumference of a Circle on the Equatorial Plane

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Solution strategy: Fix \(r = R\), \(\theta = \pi/2\), and integrate the line element in the \(\varphi\) direction.

Calculation:

At constant \(t\), \(r = R\), \(\theta = \pi/2\), we have \(dt = dr = d\theta = 0\), so the line element is

\[ dL^2 = g_{\varphi\varphi}\,d\varphi^2 = R^2\sin^2(\pi/2)\,d\varphi^2 = R^2\,d\varphi^2 \]

Therefore \(dL = R\,d\varphi\). The circumference is

\[ C = \int_0^{2\pi} R\,d\varphi = 2\pi R \]
\[ \boxed{C = 2\pi R} \]

Verification: This is simply the standard formula for a circumference in flat space. In polar coordinates of flat spacetime, space is not curved, so agreement with the Euclidean geometry result is expected. In the Schwarzschild metric, \(g_{\varphi\varphi} = r^2\sin^2\theta\) has the same form, but the interpretation differs because \(r\) is not the "distance from the center" but rather the "areal radius" (see Ch. 8). ✓

M-3. Derivation of Gravitational Redshift (Complete Derivation from the Schwarzschild Metric)

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Solution Strategy: Use the fact that the frequency of light is inversely proportional to proper time (\(\nu \propto 1/d\tau\)) and that in a static spacetime, the coordinate time interval \(dt\) is common to both the emitter and receiver.

Derivation:

Step 1: Proper time of a stationary observer

The proper time of an observer at rest at \(r = r_0\) (\(dr = d\theta = d\varphi = 0\)) is, from (6.16):

\[ d\tau_{\text{em}} = \sqrt{1 - \frac{2M}{r_0}}\,dt \]

The proper time of an observer at \(r = \infty\) (where \(g_{00} \to -1\) as \(r \to \infty\)) is:

\[ d\tau_{\text{obs}} = \sqrt{1 - \frac{2M}{\infty}}\,dt = 1 \cdot dt = dt \]

Step 2: Universality of the coordinate time \(dt\)

Since the Schwarzschild metric is static (\(g_{\alpha\beta}\) does not depend on \(t\)), the coordinate time required for light to propagate from \(r_0\) to \(\infty\) is constant. Therefore, if one period of light is emitted during a coordinate time interval \(dt\) at the emitter, one period of light is also received during the same coordinate time interval \(dt\) at the receiver.

Step 3: Ratio of frequencies

Since frequency is inversely proportional to one period of proper time:

\[ \nu_{\text{em}} \propto \frac{1}{d\tau_{\text{em}}} = \frac{1}{\sqrt{1 - 2M/r_0}\,dt} \]
\[ \nu_{\text{obs}} \propto \frac{1}{d\tau_{\text{obs}}} = \frac{1}{dt} \]

Taking the ratio:

\[ \frac{\nu_{\text{obs}}}{\nu_{\text{em}}} = \frac{d\tau_{\text{em}}}{d\tau_{\text{obs}}} = \frac{\sqrt{1 - 2M/r_0}\,dt}{dt} \]
\[ \boxed{\frac{\nu_{\text{obs}}}{\nu_{\text{em}}} = \sqrt{1 - \frac{2M}{r_0}}} \]

Physical Interpretation: When \(r_0 > 2M\), we have \(\sqrt{1 - 2M/r_0} < 1\), so \(\nu_{\text{obs}} < \nu_{\text{em}}\). That is, light escaping from a gravitational field decreases in frequency (its wavelength increases). This is the gravitational redshift.

Verification:

  • \(r_0 \to \infty\): \(\nu_{\text{obs}}/\nu_{\text{em}} \to 1\) (no redshift when gravity is weak). ✓
  • \(r_0 \to 2M\): \(\nu_{\text{obs}}/\nu_{\text{em}} \to 0\) (light from the event horizon is infinitely redshifted). ✓
  • Weak-field approximation: \(\sqrt{1 - 2M/r_0} \approx 1 - M/r_0 = 1 - GM/(c^2 r_0)\) (in units where \(c = 1\)). Using the Newtonian gravitational potential \(\Phi = -GM/r_0\), this corresponds to \(1 + \Phi/c^2\), which agrees with the prediction from the equivalence principle. ✓

M-4. Radial Proper Length in the Schwarzschild Metric

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Solution strategy: Set \(dt = d\theta = d\varphi = 0\), integrate the radial proper length, and expand in the weak-gravity approximation.

Proper length expression:

At a given instant (\(dt = 0\)), varying only the \(r\) direction:

\[ dL^2 = g_{11}\,dr^2 = \frac{dr^2}{1 - 2M/r} \]
\[ \Delta L = \int_{r_1}^{r_2} \frac{dr}{\sqrt{1 - 2M/r}} \]

Weak-gravity approximation:

When \(r \gg 2M\):

\[ \frac{1}{\sqrt{1 - 2M/r}} = \left(1 - \frac{2M}{r}\right)^{-1/2} \approx 1 + \frac{1}{2}\cdot\frac{2M}{r} = 1 + \frac{M}{r} \]

Evaluating the integral:

\[ \Delta L \approx \int_{r_1}^{r_2} \left(1 + \frac{M}{r}\right)dr = \int_{r_1}^{r_2} dr + M\int_{r_1}^{r_2} \frac{dr}{r} \]
\[ = (r_2 - r_1) + M\left[\ln r\right]_{r_1}^{r_2} \]
\[ = (r_2 - r_1) + M\ln\frac{r_2}{r_1} \]

Difference from the coordinate separation:

\[ \boxed{\delta L = \Delta L - (r_2 - r_1) = M\ln\frac{r_2}{r_1}} \]

Physical interpretation: The proper length \(\Delta L\) is longer than the coordinate separation \(r_2 - r_1\) by \(M\ln(r_2/r_1)\). This quantitatively demonstrates that in Schwarzschild spacetime, the radial direction of space is "stretched." The effect becomes larger for greater mass \(M\) and for smaller \(r_1\) (closer to the star).

Verification:

  • Dimensional analysis: In units where \(G = c = 1\), \(M\) has dimensions of length (\(M \to GM/c^2\)). Since \(\ln(r_2/r_1)\) is dimensionless, \(\delta L = M\ln(r_2/r_1)\) has dimensions of length. ✓
  • \(M \to 0\) limit: \(\delta L \to 0\), so in flat spacetime the proper length equals the coordinate separation. ✓
  • \(r_1 = r_2\) limit: \(\delta L = M\ln 1 = 0\). ✓
  • Sign: Since \(r_2 > r_1\), we have \(\ln(r_2/r_1) > 0\), so \(\delta L > 0\). This is consistent with the proper length being longer than the coordinate separation. ✓

M-5. Number of Independent Components of the Metric Tensor

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Solution strategy: Use the formula for the number of independent components of a symmetric matrix.

Diagonal metric case

For a diagonal metric, \(g_{\alpha\beta} = 0\) (\(\alpha \neq \beta\)), so the only non-zero components are the diagonal components \(g_{00}\), \(g_{11}\), \(g_{22}\), \(g_{33}\)—4 in total. These are all independent.

\[ \boxed{\text{Number of independent components of a diagonal metric} = 4} \]

General 4-dimensional metric tensor case

The \(4 \times 4\) matrix \(g_{\alpha\beta}\) generally has \(4 \times 4 = 16\) components. However, due to the symmetry \(g_{\alpha\beta} = g_{\beta\alpha}\), components with \(\alpha \neq \beta\) are pairwise equal.

Counting independent components:

  • Diagonal components (\(\alpha = \beta\)): \(g_{00}\), \(g_{11}\), \(g_{22}\), \(g_{33}\) — 4
  • Off-diagonal components (\(\alpha < \beta\)): \((0,1)\), \((0,2)\), \((0,3)\), \((1,2)\), \((1,3)\), \((2,3)\) — 6

Total: \(4 + 6 = 10\).

In general, the number of independent components of an \(n \times n\) symmetric matrix is:

\[ \frac{n(n+1)}{2} \]

For \(n = 4\):

\[ \frac{4 \times 5}{2} = 10 \]
\[ \boxed{\text{Number of independent components of a general 4-dimensional metric tensor} = 10} \]

Verification: The same result is obtained by subtracting the number of constraints due to symmetry from the total number of components. There are \(\binom{4}{2} = 6\) pairs with \(\alpha \neq \beta\), and for each pair there is one constraint \(g_{\alpha\beta} = g_{\beta\alpha}\). Therefore, the number of independent components is \(16 - 6 = 10\). ✓

Advanced

A-1. Geometry of Constant Curvature 2-Dimensional Spaces

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Given metric:

\[ ds^2 = \frac{dr^2 + r^2\,d\varphi^2}{\left(1 + \frac{k}{4}r^2\right)^2} \]

(a) Proper length of the circumference of a circle with coordinate radius \(r_0\)

Strategy: Set \(r = r_0\) (constant) so that \(dr = 0\). Integrate \(\varphi\) from \(0\) to \(2\pi\).

Calculation:

The line element on \(r = r_0\) is:

\[ dL = \frac{r_0}{\left(1 + \frac{k}{4}r_0^2\right)}\,d\varphi \]

The proper circumference:

\[ C(r_0) = \int_0^{2\pi} \frac{r_0}{1 + \frac{k}{4}r_0^2}\,d\varphi = \frac{2\pi r_0}{1 + \frac{k}{4}r_0^2} \]
\[ \boxed{C(r_0) = \frac{2\pi r_0}{1 + \frac{k}{4}r_0^2}} \]

(b) Comparison of the radial proper length and the circumference

Integration of the proper radius:

At constant \(\varphi\), from \(r = 0\) to \(r = r_0\):

\[ \mathcal{R}(r_0) = \int_0^{r_0} \frac{dr}{1 + \frac{k}{4}r^2} \]

Integration for \(k > 0\):

Setting \(\frac{k}{4} = a^2\) (i.e., \(a = \frac{\sqrt{k}}{2}\)):

\[ \mathcal{R}(r_0) = \int_0^{r_0} \frac{dr}{1 + a^2 r^2} = \frac{1}{a}\arctan(a r_0) = \frac{2}{\sqrt{k}}\arctan\!\left(\frac{\sqrt{k}}{2}r_0\right) \]
\[ \boxed{\mathcal{R}(r_0) = \frac{2}{\sqrt{k}}\arctan\!\left(\frac{\sqrt{k}}{2}r_0\right)} \]

Comparison of \(C(r_0)\) and \(2\pi\mathcal{R}(r_0)\):

Setting \(u = \frac{\sqrt{k}}{2}r_0 > 0\):

\[ C = \frac{2\pi r_0}{1 + u^2}, \qquad 2\pi\mathcal{R} = 2\pi \cdot \frac{2}{\sqrt{k}}\arctan u = \frac{2\pi r_0}{u}\arctan u \]

(where we used \(r_0 = \frac{2u}{\sqrt{k}}\).)

Taking the ratio:

\[ \frac{C}{2\pi\mathcal{R}} = \frac{\frac{2\pi r_0}{1+u^2}}{\frac{2\pi r_0}{u}\arctan u} = \frac{u}{(1+u^2)\arctan u} \]

For \(u > 0\), we have \(\arctan u < u\) (a property of the arctangent function), but since the denominator contains \((1+u^2)\arctan u\), we need to examine this more carefully.

We investigate the sign of \(f(u) = u - (1+u^2)\arctan u\). We have \(f(0) = 0\), and:

\[ f'(u) = 1 - 2u\arctan u - (1+u^2)\cdot\frac{1}{1+u^2} = 1 - 2u\arctan u - 1 = -2u\arctan u \]

For \(u > 0\), \(f'(u) = -2u\arctan u < 0\), so \(f(u) < f(0) = 0\).

That is, \(u < (1+u^2)\arctan u\), and therefore:

\[ \frac{C}{2\pi\mathcal{R}} = \frac{u}{(1+u^2)\arctan u} < 1 \]
\[ \boxed{k > 0 \text{: } C(r_0) < 2\pi\mathcal{R}(r_0)} \]

Geometric interpretation: In Euclidean flat space, circumference \(= 2\pi \times\) radius. The fact that \(C < 2\pi\mathcal{R}\) means that "the circumference is shorter than expected relative to the actual distance from the center (proper radius)." This is a characteristic of spaces with positive curvature.

Intuitively, on a sphere, if one measures the great-circle distance from the north pole to a line of latitude (proper radius), the circumference of that latitude circle is shorter than \(2\pi \times\) (great-circle distance). For example, on a sphere the great-circle distance from the north pole to the equator is \(\pi R/2\), but the circumference of the equator is \(2\pi R\), and indeed \(2\pi R < 2\pi \cdot \pi R/2 = \pi^2 R\).

(c) Correspondence between the sign of \(k\) and geometry

\(k > 0\) (spherical):

As shown in (b), \(C < 2\pi\mathcal{R}\). This is a characteristic of spherical geometry. On a sphere, as one moves away from the center, the rate of increase of the circumference is slower than a factor of \(2\pi\). The sum of interior angles of a triangle is greater than \(\pi\). The space is "closed" and has finite area.

\(k = 0\) (flat):

The metric becomes \(ds^2 = dr^2 + r^2\,d\varphi^2\), which is simply the standard Euclidean plane in polar coordinates. \(C = 2\pi r_0 = 2\pi\mathcal{R}\) (since \(\mathcal{R} = r_0\)), giving flat geometry.

\(k < 0\) (hyperbolic):

For \(k < 0\), setting \(|k|/4 = b^2\):

\[ \mathcal{R}(r_0) = \int_0^{r_0} \frac{dr}{1 - b^2 r^2} = \frac{1}{b}\mathrm{arctanh}(b r_0) = \frac{2}{\sqrt{|k|}}\mathrm{arctanh}\!\left(\frac{\sqrt{|k|}}{2}r_0\right) \]

(valid for \(r_0 < 2/\sqrt{|k|}\).)

Since \(\mathrm{arctanh}(v) > v\) (for \(0 < v < 1\)), we have \(\mathcal{R} > r_0\), while:

\[ C = \frac{2\pi r_0}{1 - b^2 r_0^2} > 2\pi r_0 \]

A similar analysis shows that \(C > 2\pi\mathcal{R}\). This is a characteristic of spaces with negative curvature (hyperbolic surfaces, saddle-shaped surfaces), where as one moves away from the center, the rate of increase of the circumference is faster than a factor of \(2\pi\). The sum of interior angles of a triangle is less than \(\pi\). The space is "open" and has infinite area.

Sign of \(k\) Geometry Circumference-radius relation Sum of triangle angles
\(k > 0\) Spherical \(C < 2\pi\mathcal{R}\) \(> \pi\)
\(k = 0\) Flat \(C = 2\pi\mathcal{R}\) \(= \pi\)
\(k < 0\) Hyperbolic \(C > 2\pi\mathcal{R}\) \(< \pi\)

Verification: For \(k > 0\) as \(r_0 \to 0\), we have \(C \to 2\pi r_0\) and \(\mathcal{R} \to r_0\), so \(C \to 2\pi\mathcal{R}\) (locally the space appears flat). ✓ Furthermore, this metric appears as the spatial part of the FLRW metric in cosmology, where \(k > 0\), \(k = 0\), and \(k < 0\) are known to correspond to a closed universe, a flat universe, and an open universe, respectively. ✓

A-2. GPS and Gravitational Redshift

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(a) Approximate Calculation of Proper Time

Solution strategy: In the Schwarzschild metric (6.13), the proper time of a static observer is \(d\tau = \sqrt{1 - \frac{2GM}{c^2 r}}\,dt\). Since \(\frac{2GM}{c^2 r} \ll 1\), we use the approximation \(\sqrt{1-\epsilon} \approx 1 - \epsilon/2\).

Definition of the Schwarzschild radius:

\[ r_s = \frac{2GM_\oplus}{c^2} \approx 8.87 \times 10^{-3}\;\mathrm{m} \]

Observer on the Earth's surface (\(r = R_\oplus\)):

\[ \Delta\tau_\oplus = \sqrt{1 - \frac{r_s}{R_\oplus}}\,\Delta t \approx \left(1 - \frac{r_s}{2R_\oplus}\right)\Delta t \]

Observer in GPS satellite orbit (\(r = R_\text{GPS}\)):

\[ \Delta\tau_\text{GPS} = \sqrt{1 - \frac{r_s}{R_\text{GPS}}}\,\Delta t \approx \left(1 - \frac{r_s}{2R_\text{GPS}}\right)\Delta t \]
\[ \boxed{\Delta\tau_\oplus \approx \left(1 - \frac{r_s}{2R_\oplus}\right)\Delta t, \qquad \Delta\tau_\text{GPS} \approx \left(1 - \frac{r_s}{2R_\text{GPS}}\right)\Delta t} \]

(b) Numerical Estimate of the Proper Time Difference

Calculation:

\[ \Delta\tau_\text{GPS} - \Delta\tau_\oplus \approx \frac{r_s}{2}\left(\frac{1}{R_\oplus} - \frac{1}{R_\text{GPS}}\right)\Delta t \]

Substituting numerical values:

\[ \frac{r_s}{2} = \frac{8.87 \times 10^{-3}}{2} = 4.435 \times 10^{-3}\;\mathrm{m} \]
\[ \frac{1}{R_\oplus} = \frac{1}{6.37 \times 10^6} = 1.570 \times 10^{-7}\;\mathrm{m^{-1}} \]
\[ \frac{1}{R_\text{GPS}} = \frac{1}{2.66 \times 10^7} = 3.759 \times 10^{-8}\;\mathrm{m^{-1}} \]
\[ \frac{1}{R_\oplus} - \frac{1}{R_\text{GPS}} = 1.570 \times 10^{-7} - 3.759 \times 10^{-8} = 1.194 \times 10^{-7}\;\mathrm{m^{-1}} \]
\[ \frac{r_s}{2}\left(\frac{1}{R_\oplus} - \frac{1}{R_\text{GPS}}\right) = 4.435 \times 10^{-3} \times 1.194 \times 10^{-7} = 5.295 \times 10^{-10} \]

For \(\Delta t = 1\) day \(= 86400\;\mathrm{s}\):

\[ \Delta\tau_\text{GPS} - \Delta\tau_\oplus \approx 5.295 \times 10^{-10} \times 86400\;\mathrm{s} \]
\[ \boxed{\Delta\tau_\text{GPS} - \Delta\tau_\oplus \approx 4.57 \times 10^{-5}\;\mathrm{s} \approx 45.7\;\mu\mathrm{s/day}} \]

Physical interpretation: Since GPS satellites are located at a higher gravitational potential than the Earth's surface, the satellite clocks run faster than ground clocks by approximately \(45.7\;\mu\mathrm{s}\) per day.

(c) Estimate of GPS Positioning Error

Calculation:

If the time difference is not corrected, the distance error for signals propagating at the speed of light is:

\[ \delta x \approx c \times (\Delta\tau_\text{GPS} - \Delta\tau_\oplus) = 3 \times 10^8\;\mathrm{m/s} \times 4.57 \times 10^{-5}\;\mathrm{s} \]
\[ \boxed{\delta x \approx 1.37 \times 10^4\;\mathrm{m} \approx 13.7\;\mathrm{km/day}} \]

Practical discussion:

While the civilian GPS positioning accuracy is on the order of a few meters, without correcting for gravitational redshift, an error of approximately 14 km per day would accumulate. This is an enormous error that is completely unacceptable in practice, demonstrating that general relativistic time corrections are essential for the proper operation of GPS.

In the actual GPS system, in addition to the gravitational redshift effect calculated here (satellite clocks running fast: \(+45.7\;\mu\mathrm{s/day}\)), the special relativistic time dilation (due to the satellite's orbital motion, causing satellite clocks to run slow: approximately \(-7.2\;\mu\mathrm{s/day}\)) must also be corrected. The net effect combining both is approximately \(+38.5\;\mu\mathrm{s/day}\), with gravitational redshift being the dominant contribution.

Verification:

  • Dimensional analysis: \([c] \times [t] = \mathrm{m/s} \times \mathrm{s} = \mathrm{m}\). ✓
  • Order-of-magnitude estimate: \(r_s/R_\oplus \sim 10^{-3}/10^{7} \sim 10^{-10}\). One day \(\sim 10^5\;\mathrm{s}\), so the time difference \(\sim 10^{-10} \times 10^5 = 10^{-5}\;\mathrm{s}\). Distance error \(\sim 3 \times 10^8 \times 10^{-5} = 3 \times 10^3\;\mathrm{m}\). As an order of magnitude, \(\sim 10\;\mathrm{km}\), which is consistent. ✓
  • Comparison with literature values: The gravitational redshift effect of approximately \(45\;\mu\mathrm{s/day}\) and a distance error of approximately \(10\;\mathrm{km/day}\) are consistent with standard textbook values. ✓