Skip to content

Ch. 7 Problems

← Back to chapter　|　View solutions

Table of Contents

Basic

Medium

Advanced

Basic

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

Using the Minkowski metric \(\eta_{\alpha\beta} = \mathrm{diag}(-1,1,1,1)\), calculate the line element \(ds^2 = \eta_{\alpha\beta}\,dx^\alpha\,dx^\beta\) for the coordinate difference between two events \(dx^\alpha = (dt,\,dx,\,dy,\,dz) = (3,\,1,\,2,\,0)\), and determine whether this interval is timelike, spacelike, or lightlike.

Hint

Expand \(ds^2 = -dt^2 + dx^2 + dy^2 + dz^2\) and classify based on the sign. If \(ds^2 < 0\), the interval is timelike.

→ View solution

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

Given a 2-dimensional metric \(ds^2 = a^2\,d\theta^2 + a^2\sin^2\theta\,d\varphi^2\) (where \(a\) is a constant), read off the components of the metric tensor \(g_{\theta\theta}\), \(g_{\varphi\varphi}\), and \(g_{\theta\varphi}\). Also, find the components of the inverse metric \(g^{\theta\theta}\) and \(g^{\varphi\varphi}\).

Hint

The inverse of a diagonal metric is simply the reciprocal of each diagonal component. \(g^{\theta\theta} = 1/g_{\theta\theta}\).

→ View solution

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

In the metric (6.12) for flat spacetime in polar coordinates, find the proper length \(dL\) when \(\varphi\) is varied by \(d\varphi\) at a point fixed at \(r = R\), \(\theta = \pi/4\).

Hint

Set \(dt = dr = d\theta = 0\) and compute \(dL^2 = g_{\varphi\varphi}\,d\varphi^2\). Use \(\sin(\pi/4) = \frac{\sqrt{2}}{2}\).

→ View solution

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

In the Schwarzschild metric (6.14), express the proper time \(d\tau\) of an observer at rest at \(r = 10M\) in terms of coordinate time \(dt\).

Hint

For a static observer (\(dr = d\theta = d\varphi = 0\)), substitute \(r = 10M\) into \(d\tau = \sqrt{1 - 2M/r}\,dt\).

→ View solution

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

From the metric tensor components (6.15) of the Schwarzschild metric (6.14), find the values of \(g_{00}\), \(g_{11}\), \(g_{22}\), and \(g_{33}\) (at \(\theta = \pi/2\)) at \(r = 4M\).

Hint

Substitute \(r = 4M\) into \(g_{00} = -(1-2M/r)\). For \(g_{11}\), pay attention to the sign of its reciprocal.

→ View solution

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

Given the metric \(ds^2 = -dt^2 + e^{2Ht}(dx^2 + dy^2 + dz^2)\) (where \(H\) is a constant), write down all independent non-zero components of the metric tensor, and find all non-zero components of the inverse metric \(g^{\alpha\beta}\).

Hint

Since this is a diagonal metric, \(g^{\alpha\alpha} = 1/g_{\alpha\alpha}\). The reciprocal of \(e^{2Ht}\) is \(e^{-2Ht}\).

→ View solution

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

In the Schwarzschild metric (6.14), find the proper length \(dL\) when moving by \(d\varphi\) in the \(\varphi\) direction from the position \(r = 6M\), \(\theta = \pi/2\) at a given instant. Furthermore, compare this with the proper length for the same \(r = 6M\), \(\theta = \pi/2\), \(d\varphi\) in flat spacetime polar coordinates (6.11), and state whether the two agree or not.

Hint

Note that the metric component \(g_{33}\) in the \(\varphi\) direction is \(r^2\sin^2\theta\) in both the Schwarzschild and flat spacetime cases.

→ View solution

B-8. Proper Time in the Rindler Metric

In the 2-dimensional Rindler metric \(ds^2 = -\alpha^2 x^2\,dt^2 + dx^2\) (where \(\alpha\) is a constant and \(x > 0\)), express the proper time \(d\tau\) of an observer at rest at \(x = x_0\) in terms of the coordinate time \(dt\).

Hint

Set \(dx = 0\) and use \(d\tau^2 = -ds^2 = \alpha^2 x_0^2\,dt^2\).

→ View solution

Medium

M-1. Calculating the Area of a Sphere

In the polar coordinates (6.11) of flat spacetime, fix \(t\) and compute the area of the sphere at \(r = R\) (constant) from the metric. Specifically, find the area element \(dA = \sqrt{\det(g_{ij})}\,d\theta\,d\varphi\) (\(i, j = \theta, \varphi\)) for infinitesimal changes in \(\theta\) and \(\varphi\), and show that integrating over the entire sphere yields \(4\pi R^2\).

Hint

The induced metric on \(r = R\) is \(ds^2_{(2)} = R^2\,d\theta^2 + R^2\sin^2\theta\,d\varphi^2\). Compute the determinant of the 2-dimensional submetric and integrate over \(\theta: 0 \to \pi\), \(\varphi: 0 \to 2\pi\).

→ View solution

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

In the polar coordinates (6.11) of flat spacetime, fix \(t\), and calculate the circumference of a circle on the equatorial plane where \(r = R\) (constant) and \(\theta = \pi/2\).

→ View solution

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

In the Schwarzschild metric (6.14), an observer at rest at \(r = r_0 > 2M\) emits light (frequency \(\nu_\text{em}\)), which is received by an observer at \(r = \infty\). Express the received frequency \(\nu_\text{obs}\) in terms of \(\nu_\text{em}\) and \(r_0\), and derive the gravitational redshift formula

\[ \frac{\nu_\text{obs}}{\nu_\text{em}} = \sqrt{1 - \frac{2M}{r_0}} \]

Here, use the fact that the frequency of light is proportional to the inverse of proper time (\(\nu \propto 1/d\tau\)).

Hint

The proper time of a stationary observer is \(d\tau = \sqrt{-g_{00}}\,dt\). Due to the static nature of the spacetime, the coordinate time \(dt\) is common to both the emitter and receiver. At \(r = \infty\), \(g_{00} = -1\).

→ View solution

M-4. Radial Proper Length in the Schwarzschild Metric

In the Schwarzschild metric (6.14), write down the proper length measured radially from \(r = r_1\) to \(r = r_2\) (\(r_2 > r_1 \gg 2M\)) as

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

and perform the integration using the approximation \((1 - 2M/r)^{-1/2} \approx 1 + M/r\). Find the difference \(\delta L = \Delta L - (r_2 - r_1)\) between the proper length \(\Delta L\) and the coordinate difference \(r_2 - r_1\), and express it in terms of \(M\), \(r_1\), and \(r_2\).

Hint

Use \(\int_{r_1}^{r_2} \frac{M}{r}\,dr = M\ln(r_2/r_1)\).

→ View solution

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

For a general diagonal metric

\[ ds^2 = g_{00}\,dt^2 + g_{11}\,dr^2 + g_{22}\,d\theta^2 + g_{33}\,d\varphi^2 \]

confirm that the number of independent components is 4, using the fact that \(g_{\alpha\beta}\) is a symmetric tensor and is diagonal. Next, show that the number of independent components of a general 4-dimensional metric tensor (including off-diagonal components) is \(\frac{4 \times 5}{2} = 10\), based on the symmetry \(g_{\alpha\beta} = g_{\beta\alpha}\).

Hint

The number of independent components of an \(n \times n\) symmetric matrix is \(n(n+1)/2\). For a diagonal matrix, all off-diagonal components are additionally zero.

→ View solution

Advanced

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

Consider the following 2-dimensional metric:

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

where \(k\) is a constant, \(r \geq 0\), and \(0 \leq \varphi < 2\pi\).

(a) Calculate the proper circumference \(C(r_0)\) of a "circle" of coordinate radius \(r = r_0\) centered at the origin.

(b) Write down the radial proper length ("radius") \(\mathcal{R}(r_0)\) from the origin to coordinate radius \(r_0\) in integral form, and for the case \(k > 0\), compare \(C(r_0)\) with \(2\pi \mathcal{R}(r_0)\). Determine whether \(C < 2\pi\mathcal{R}\) or \(C > 2\pi\mathcal{R}\), and explain the geometric meaning of this result.

(c) It is known that this metric represents a space of constant curvature. Using the result from (b) and intuitive arguments, explain why \(k > 0\), \(k = 0\), and \(k < 0\) correspond to a sphere, a plane, and a hyperbolic surface, respectively.

Hint

(a) On \(r = r_0\), set \(dr = 0\) so that \(dL = \frac{r_0}{1+kr_0^2/4}\,d\varphi\), and integrate from \(0\) to \(2\pi\). (b) \(\mathcal{R} = \int_0^{r_0}\frac{dr}{1+kr^2/4}\) can be expressed in terms of the arctangent function when \(k > 0\). On a sphere, "circumference < \(2\pi\) × radius."

→ View solution

A-2. GPS and Gravitational Redshift

Using the Schwarzschild metric (6.13) (in the form with \(c\) and \(G\) explicit), compare the proper times of a clock at rest on the Earth's surface (\(r = R_\oplus\)) and a clock at rest at the GPS satellite orbit (\(r = R_\text{GPS}\)).

(a) For a distant coordinate time \(\Delta t = 1\) day, find the proper times \(\Delta\tau_\oplus\) and \(\Delta\tau_\text{GPS}\) ticked by each clock under the approximation \(2GM_\oplus/(c^2 r) \ll 1\).

(b) Estimate the proper time difference \(\Delta\tau_\text{GPS} - \Delta\tau_\oplus\) numerically. Use the following values:

\[ \frac{2GM_\oplus}{c^2} \approx 8.87 \times 10^{-3}\;\mathrm{m},\quad R_\oplus \approx 6.37 \times 10^6\;\mathrm{m},\quad R_\text{GPS} \approx 2.66 \times 10^7\;\mathrm{m} \]

(c) Estimate the GPS positioning error per day if this proper time difference is not corrected, using \(c \approx 3 \times 10^8\;\mathrm{m/s}\). Discuss why this result cannot be neglected in practice.

Hint

(a) Use \(\sqrt{1-\epsilon} \approx 1 - \epsilon/2\). (b) \(\Delta\tau_\text{GPS} - \Delta\tau_\oplus \approx \frac{1}{2}\left(\frac{r_s}{R_\oplus} - \frac{r_s}{R_\text{GPS}}\right)\Delta t\) where \(r_s = 2GM_\oplus/c^2\). (c) Distance error \(\sim c \times (\text{time difference})\). Note that in actual GPS systems, in addition to the gravitational redshift effect calculated here, the special relativistic time dilation (due to satellite motion) must also be corrected.

→ View solution