Chapter 15

Consider two events that occur simultaneously at $t=0$ in frame $\mathcal{S},$ both on the $x$ axis at $x=0$ and $x=a$. (a) Find the times of the two events as measured in a frame ${\mathcal{S}}^{\prime }$ traveling in the positive direction along the $x$ axis with speed $V$. (b) Do the same for a second frame ${\mathcal{S}}^{\prime \prime }$ traveling at speed $V$ but in the negative direction along the $x$ axis. Comment on the time ordering of the two events as seen in the three different frames. This startling result is discussed further in Section 15.10 .

A traveler in a rocket of proper length 2d sets up a coordinate system ${\mathcal{S}}^{\prime }$ with its origin $O^{\prime }$ anchored at the exact middle of the rocket and the $x^{\prime }$ axis along the rocket's length. At $t^{\prime }=0$ she ignites a flashbulb at $O^{\prime }.$ (a) Write down the coordinates $x_{\mathrm{F}}^{\prime },t_{\mathrm{F}}^{\prime }$ and $x_{\mathrm{B}}^{\prime },t_{\mathrm{B}}^{\prime }$ for the arrival of the light at the front and back of the rocket. (b) Now consider the same experiment as observed from a frame $\delta$ relative to which the rocket is traveling with speed $V$ (with $\delta$ and $S$ ' in the standard configuration). Use the inverse Lorentz transformation to find the coordinates $x_{\mathrm{F}},t_{\mathrm{F}}$ and $x_{\mathrm{B}},t_{\mathrm{B}}$ for the arrival of the two signals. Explain clearly in words why the two arrivals are simultaneous in ${\mathcal{S}}^{\prime }$ but not in $\mathcal{S}.$ This phenomenon is called the relativity of simultaneity.

Newton's first law can be stated: If an object is isolated (subject to no forces), then it moves with constant velocity. We know that this is invariant under the Galilean transformation. Prove that it is also invariant under the Lorentz transformation. [Assume that it is true in an inertial frame $\mathcal{S}$, and use the relativistic velocity-addition formula to show that it is also true in any other ${\mathcal{S}}^{\prime }.$ ]

A rocket traveling at speed $\frac{1}{2}c$ relative to frame $\mathcal{S}$ shoots forward bullets traveling at speed $\frac{3}{4}c$ relative to the rocket. What is the speed of the bullets relative to $\mathcal{S}?$

A rocket is traveling at speed $0.9c$ along the $x$ axis of frame $\delta$. It shoots a bullet whose velocity ${\mathbf{v}}^{\prime }${ measured in the rocket's rest frame } ${\mathcal{S}}^{\prime })$ is $0.9c$ along the $y^{\prime }$ axis of ${\mathcal{S}}^{\prime }.$ What is the bullet's velocity (magnitude and direction) as measured in $\mathcal{S}$ ?

As seen in frame $\mathcal{S}$, two rockets are approaching one another along the $x$ axis traveling with equal and opposite velocities of $0.9c.$ What is the velocity of the rocket on the right as measured by observers in the one on the left? [This and the previous two problems illustrate the general result that in relativity the "sum" of two velocities that are less than $c\text{ is always less than }c.\text{ See Problem }15.43.]$

A robber's getaway vehicle, which can travel at an impressive 0.8c, is pursued by a cop, whose vehicle can travel at a mere 0.4c. Realizing that he cannot catch up with the robber, the cop tries to shoot him with bullets that travel at $0.5c$ (relative to the cop). Can the cop's bullets hit the robber?

A rocket is traveling at speed $V$ along the $x$ axis of frame S. It emits a signal (for example, a pulse of light) that travels with speed $c$ along the $y^{\prime }$ axis of the rocket's rest frame ${\mathcal{S}}^{\prime }.$ What is the speed of the signal as measured in $\mathcal{S}?$

Frame ${\mathcal{S}}^{\prime }$ travels at speed $V_1$ along the $x$ axis of frame $\mathcal{S}$ (in the standard configuration). Frame ${\mathcal{S}}^{\prime \prime }$ travels at speed $V_2$ along the $x^{\prime }$ axis of frame ${\mathcal{S}}^{\prime }$ (also in the standard configuration). By applying the standard Lorentz transformation twice find the coordinates $x^{\prime \prime },y^{\prime \prime },z^{\prime \prime },t^{\prime \prime }$ of any event in terms of $x,y,z,t.$ Show that this transformation is in fact the standard Lorentz transformation with velocity $V$ given by the relativistic "sum" of $V_1$ and $V_2$

(a) Find the 3 $\times 3$ matrix $\mathbf{R}(\theta )$ that rotates three- dimensional space about the $x_3$ axis, so that ${\mathbf{e}}_1$ rotates through angle $\theta$ toward ${\mathbf{e}}_2$. (b) Show that $[\mathbf{R}(\theta ){]}^2=\mathbf{R}(2\theta ),$ and interpret this result.