Problem 33
(a) By exchanging \(x_{1}\) and \(x_{2}\), write down the Lorentz transformation for a boost of velocity \(V\) along the \(x_{2}\) axis and the corresponding \(4 \times 4\) matrix \(\Lambda_{\mathrm{B} 2}\). ( \(\mathbf{b}\) ) Write down the \(4 \times 4\) matrices \(\Lambda_{\mathrm{R}+}\) and \(\Lambda_{\mathrm{R}-}\) that represent rotations of the \(x_{1} x_{2}\) plane through \(\pm \pi / 2,\) with the angle of rotation measured counterclockwise. (c) Verify that \(\Lambda_{\mathrm{B} 2}=\Lambda_{\mathrm{R}-} \Lambda_{\mathrm{B} 1} \Lambda_{\mathrm{R}+},\) where \(\Lambda_{\mathrm{B} 1}\) is the standard boost along the \(x_{1}\) axis, and interpret this result.
Problem 34
Let \(\Lambda_{\mathrm{B}}(\theta)\) denote the 4 \(\times 4\) matrix that gives a pure boost in the direction that makes an angle \(\theta\) with the \(x_{1}\) axis in the \(x_{1} x_{2}\) plane. Explain why this can be found as \(\Lambda_{\mathrm{B}}(\theta)=\Lambda_{\mathrm{R}}(-\theta) \Lambda_{\mathrm{B}}(0) \Lambda_{\mathrm{R}}(\theta)\) where \(\Lambda_{\mathrm{R}}(\theta)\) denotes the matrix that rotates the \(x_{1} x_{2}\) plane through angle \(\theta\) and \(\Lambda_{\mathrm{B}}(0)\) is the standard boost along the \(x_{1}\) axis. Use this result to find \(\Lambda_{\mathrm{B}}(\theta)\) and check your result by finding the motion of the spatial origin of the frame \(\mathcal{S}\) as observed in \(\mathcal{S}^{\prime}\).
Problem 35
Prove the following useful result, called the zero-component theorem: Let \(q\) be a four-vector, and suppose that one component of \(q\) is found to be zero in all inertial frames. (For example, \(q_{4}=0\) in all frames.) Then all four components of \(q\) are zero in all frames.
Problem 36
We have seen that the scalar product \(x \cdot x\) of any four-vector \(x\) with itself is invariant under Lorentz transformations. Use the invariance of \(x \cdot x\) to prove that the scalar product \(x \cdot y\) of any two four-vectors \(x\) and \(y\) is likewise invariant.
Problem 37
Verify directly that \(x^{\prime} \cdot y^{\prime}=x \cdot y\) for any two four- vectors \(x\) and \(y,\) where \(x^{\prime}\) and \(y^{\prime}\) are related to \(x\) and \(y\) by the standard Lorentz boost along the \(x_{1}\) axis.
Problem 38
As an observer moves through space with position \(\mathbf{x}(t),\) the four- vector \((\mathbf{x}(t), c t)\) traces a path through space-time called the observer's world line. Consider two events that occur at points \(P\) and \(Q\) in space-time. Show that if, as measured by the observer, the two events occur at the same time \(t,\) then the line joining \(P\) and \(Q\) is orthogonal to the observer's world line at the time \(t\); that is, \(\left(x_{P}-x_{Q}\right) \cdot d x=0,\) where \(d x\) joins two neighboring points on the world line at times \(t\) and \(t+d t\).
Problem 42
Prove that if \(x\) is time-like and \(x \cdot y=0,\) then \(y\) is space-like.
Problem 43
(a) Show that if a body has speed \(v < c\) in one inertial frame, then \(v < c\) in all frames. [Hint: Consider the displacement four-vector \(d x=(d \mathbf{x}, c d t),\) where \(d \mathbf{x}\) is the three-dimensional displacement in a short time \(d t .]\) (b) Show similarly that if a signal (such as a pulse of light) has speed \(c\) in one frame, its speed is \(c\) in all frames.
Problem 47
Consider the tale of the physicist who is ticketed for running a red light and argues that because he was approaching the intersection, the red light was Doppler shifted and appeared green. How fast would he have to have been going? \(\left(\lambda_{\text {red }} \approx 650 \mathrm{nm} \text { and } \lambda_{\text {green }} \approx 530 \mathrm{nm} .\right)\)
Problem 49
Show that the four-velocity of any object has invariant length squared \(u \cdot u=-c^{2}\)
