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Ch. 3 Problems

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Basic

Medium

Basic

B-1. Derivation of the Hyperbolic Function Identity

Derive the hyperbolic function identity \(\cosh^2\varphi - \sinh^2\varphi = 1\) directly from the definitions

\[ \cosh\varphi = \frac{e^\varphi + e^{-\varphi}}{2}, \qquad \sinh\varphi = \frac{e^\varphi - e^{-\varphi}}{2} \]
Hint

Square each expression, take their difference, and simplify the products of exponential functions.

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B-2. Relationship between Rapidity and the Lorentz Factor

Using the relationship between rapidity \(\varphi\) and velocity \(v\), given by \(\tanh\varphi = v/c\), derive \(\cosh\varphi = \gamma\) and \(\sinh\varphi = \gamma v/c\). Here, \(\gamma = 1/\sqrt{1 - v^2/c^2}\).

Hint

Combine \(\tanh\varphi = \sinh\varphi / \cosh\varphi\) with the identity \(\cosh^2\varphi - \sinh^2\varphi = 1\) and solve simultaneously.

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B-3. Contraction Calculation of the Lorentz Transformation Matrix

Using the Lorentz transformation matrix to an inertial frame moving with velocity \(v\) in the \(x\) direction (in units where \(c = 1\)):

\[ \Lambda^{\mu'}{}_{\nu} = \begin{pmatrix} \gamma & -\gamma v & 0 & 0 \\ -\gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

compute \(dx^{1'} = \Lambda^{1'}{}_{\nu}\,dx^\nu\) for \(dx^\mu = (dt,\, dx,\, 0,\, 0)\) following the contraction convention, and verify that \(dx' = \gamma(dx - v\,dt)\).

Hint

Write out each term for \(\nu = 0, 1, 2, 3\), and substitute \(\Lambda^{1'}{}_{0} = -\gamma v\), \(\Lambda^{1'}{}_{1} = \gamma\).

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Medium

M-1. Detailed Calculation for Determining the Lorentz Transformation Coefficients

In Section 3.5 of the main text, we obtained the system of simultaneous equations that determine the coefficients of the Lorentz transformation:

\[ \begin{aligned} -c^2\,a_1^2 + a_6^2\,v^2 &= -c^2 \quad &\text{(coefficient of }dt^2\text{)} \\ -c^2\,a_2^2 + a_6^2 &= 1 \quad &\text{(coefficient of }dx^2\text{)} \\ -2\,c^2\,a_1\,a_2 - 2\,a_6^2\,v &= 0 \quad &\text{(coefficient of the cross term }dt\,dx\text{)} \end{aligned} \]

Solve this system to derive

\[ a_1 = a_6 = \frac{1}{\sqrt{1 - v^2/c^2}}, \qquad a_2 = -\frac{v/c^2}{\sqrt{1 - v^2/c^2}} \]

Furthermore, show that the continuity condition requiring the transformation to reduce to the identity transformation (\(a_1 = a_6 = 1\), \(a_2 = 0\)) in the limit \(v \to 0\) fixes the sign of \(a_6\) to be \(+\).

Hint

(a) From the cross term equation, obtain \(a_2 = -a_6^2\,v / (c^2\,a_1)\). (b) Substitute this into the \(dx^2\) equation to obtain a relation between \(a_1^2\) and \(a_6^2\). (c) Combine with the \(dt^2\) equation to obtain \(a_6^2 = 1/(1 - v^2/c^2)\). (d) Choose the sign such that \(a_6 \to +1\) as \(v \to 0\).

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M-2. Metric Preservation Condition under Lorentz Transformation

Verify the condition for the Lorentz transformation matrix \(\Lambda^{\mu'}{}_{\nu}\) to preserve the Minkowski metric

\[ \eta_{\mu'\nu'} = \Lambda^{\alpha}{}_{\mu'}\,\Lambda^{\beta}{}_{\nu'}\,\eta_{\alpha\beta} \]

by substituting the explicit \(\Lambda\) for a boost in the \(x\)-direction and checking the \((\mu', \nu') = (0, 0)\) and \((\mu', \nu') = (0, 1)\) components.

Note: The \(\Lambda^{\alpha}{}_{\mu'}\) in the problem statement represents the "transformation matrix from primed indices (\(S'\) frame components) to unprimed indices (\(S\) frame components)." In other words, it corresponds to the inverse of the forward transformation \(x^{\mu'} = \Lambda^{\mu'}{}_{\nu}x^\nu\). However, since the matrix for an \(x\)-direction boost is symmetric, in this problem the calculation works by "reading the 0th column of the forward transformation matrix \((\gamma, -\gamma v, 0, 0)\) as \(\Lambda^{\alpha}{}_{0'}\)."

Hint

For \((\mu', \nu') = (0, 0)\), summing over \(\alpha\) and \(\beta\) yields the expression \(-\gamma^2 + \gamma^2 v^2\).

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M-3. Quantitative Consequences of the Relativity of Simultaneity

Using the Lorentz transformation equations, find the time difference \(\Delta t'\) in the \(S'\) frame for two events that are simultaneous in the \(S\) frame (\(\Delta t = 0\)) and separated by a distance \(\Delta x = L\). Explain in words what consequence of the "relativity of simultaneity" this result demonstrates.

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M-4. Relationship Between Proper Time and Coordinate Time

Using the proper time \(d\tau^2 = -ds^2 = dt^2 - dx^2 - dy^2 - dz^2\) (with \(c = 1\)), derive the relationship between coordinate time \(t\) and proper time \(\tau\):

\[ \frac{d\tau}{dt} = \frac{1}{\gamma} \]

Furthermore, explain from this result that "a moving clock runs slow."

Hint

Factor out \(dt^2\) from \(d\tau^2\) and substitute \(dx^i/dt = v^i\).

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M-5. Derivation of the Velocity Addition Formula

A problem involving two successive applications of the Lorentz transformation. Consider a frame \(S'\) moving with velocity \(v_1\) in the \(x\) direction relative to frame \(S\), and a frame \(S''\) moving with velocity \(v_2\) in the same \(x\) direction relative to frame \(S'\). Using the additivity of rapidity \(\varphi_{12} = \varphi_1 + \varphi_2\), derive the relativistic velocity addition formula

\[ v_{12} = \frac{v_1 + v_2}{1 + v_1 v_2} \]

(with \(c = 1\)).

Hint

Use the addition formula for \(\tanh(\varphi_1 + \varphi_2)\) and substitute \(\tanh\varphi = v\).

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M-6. Relativity of Simultaneity (Concrete Example)

Consider two events \(A\) and \(B\) that occur simultaneously at \(t = 0\) in the \(S\) frame, at positions \(x_A = 0\) and \(x_B = L\). In the \(S'\) frame moving with velocity \(v\) in the \(x\) direction, (a) find the time difference \(\Delta t' = t'_B - t'_A\) between the two events, and (b) determine which event occurs first, distinguishing cases by the sign of \(v\).

Hint

Substitute \(t_A = t_B = 0\) into the Lorentz transformation equation for \(t'\).

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