12.E: Spacetime Diagrams (Exercises)

12.1 In the $(c t, x)$ coordinates of an inertial frame $S$ three events occur at $G_1=(1,2), G_2=(5,4)$ and $G_3=(3,6)$. Which of these events could be causally linked?

12.2 Two astronauts (we'll call them A and B) leave Earth on January 1st in opposite directions. Astronaut A initially travels as a speed $4 c / 5$, and B with $3 c / 5$. The astronauts keep traveling until they measure their distance to Earth at 5 light years, at which point they turn around and travel back. On the way home, astronaut A travels at $3 c / 5$, while B travels at $4 c / 5$.
(a) Determine how far each astronaut has traveled in Earth's frame of reference before (s)he returns.
(b) Draw a spacetime diagram showing the trips of the two astronauts. NB: You may want to take your space for this, it will help see details clearly.
(c) Which astronaut turns around first?
(d) Find the length of the spacetime interval between the events 'astronaut A reaches his turning point' and 'astronaut B reaches her turning point', as measured by each of the astronauts, and by an observer back on Earth, and verify that the spacetime interval is indeed invariant.
(e) Which astronaut returns to Earth first?
(f) As measured by astronaut A , what is the total distance that astronaut B travels?
(g) Before they left Earth, the astronauts promised to send each other a message by radio every new year. How many such messages does each astronaut send?
(h) Draw the lines representing the radio signals of each astronaut in the spacetime diagram.
(i) For each astronaut, determine how many signals they sent on the way out, way back, and when already back on Earth.
(j) After both astronauts have returned to Earth, which of them has aged more?

12.3 You observe a spaceship with astronauts A and B (fill in your own favorite SciFi characters if you like) in it, moving at constant velocity $u>0$ along the $x$-axis of your rest frame $S$. At time $t=t_0$, when the spaceship is at $\left(x_0, y_0, z_0\right)$, astronaut B gets on board a small shuttle and leaves with constant velocity $v>u$ in the $x$-direction to a distant star, which is stationary in your rest frame $S$. At $t=t_0$, you measure the distance between the spaceship and distant star to be $d$. Astronaut B reaches the star at $t=t_1$ on your clock, does whatever (s)he wanted to do quickly, then returns to the mother ship with the same speed $v$, reaching it at $t=t_2$ on your clock. The mother ship meanwhile kept the same speed $u$ throughout.
(a) Draw a spacetime diagram with the worldlines of astronauts A and B . Indicate all relevant quantities, and specify the slopes of any lines you draw.
(b) Express $t_2$ and $x_2$ (the place where the shuttle and mother ship meet) in terms of the quantities $x_0$, $t_0, u, v$, and $d$.
(c) Both astronauts carry high-precision watches which they synchronize when B leaves the mother ship. Express the times given by both watches upon return of $B$ in terms of the given quantities.
(d) Which astronaut has aged more between the departure and return of $B$ ?

12.4 Having negotiated a cease-fire, an imperial and a rebel spaceship leave their meeting place with synchronized clocks (we'll set the meeting place at $x=0$, and the time of leaving at $t=0$ ), departing with relative velocity (i.e, the velocity at which they each measure the other ship to be moving) of $(3 / 5) c$ in opposite directions. Once he has thought about it for a day, the imperial captain comes to the conclusion that the emperor will reward him higher for eliminating the rebels than making peace with them, so he fires a photon torpedo at the rebel ship. Likewise, having pondered the option of peace for a day, the rebel captain decides that offense is the best defense, and fires a photon torpedo at the imperial ship. Both ships are equipped with launchers that can fire such torpedos with speed $(4 / 5) c$ with respect to the ship.
(a) Draw a spacetime diagram with the two spaceships and the two photon torpedos as seen from the imperial ship. Use units of $c$. (half a day) on the tim(b) When, as measured on the clock of the imperial ship, does the torpedo fired by them hit the rebel ship?
(c) When, as measured on the clock of the imperial ship, does the torpedo fired by the rebels hit them?
(d) Fortunately for the rebel ship, their shield is quite strong, being able to withstand an impact with a momentum up to $3.0 \cdot 10^{11} \mathrm{~kg} \cdot \mathrm{~m} / \mathrm{s}$. The torpedo fired by the imperial ship has a mass of 1000 kg . Does the rebel ship survive the impact?
(e) Both ships have a home base 1 light week away from the meeting point. When the rebels at the home base see the imperial ship fire on their friends, they decide to mount a counterattack. They launch another ship at $v=(3 / 5) c$, which immediately after launch also fires a torpedo with (4/5)c relative to the ship. Does that torpedo hit the imperial ship before it returns home? To answer this question, you may assume that the imperial and rebel ship left the meeting point with equal speeds, as measured from the rest frame of the bases. You may also find drawing another spacetime diagram useful.

12.5 A train with proper length $L$ moves at speed $4 c / 5$ with respect to the ground. A passenger lets a toy train run from the back of the (moving) train to the front, with speed $c / 3$ as measured on the large train.
(a) Draw a spacetime diagram with the worldlines of the front of the large train, the back of the large train, and the small train, from the point of view of a (stationary) observer on the ground.

How much time does it take the toy train to cross the train, and what distance does the toy train cover, in:
(b) The frame of the large train?
(c) The frame of the toy train?
(d) The frame of a stationary observer on the ground?
(e) Verify that the length of the invariant interval (equation 12.4) is the same in all three frames.

e axis, and half a light-day on the space axis. Provide the calculation for each value that you have to compute. NB: make your drawing large enough for details to be properly visible and use a ruler; don't worry about the paper, we promise to recycle it in due time.