Following the standard conventions for drawing Minkowski diagrams, draw two diagrams, with all the named points labeled on your diagram.

1. the first from Maria’s point of view. (Her worldline should be vertical.)
2. the second from Lucille’s point of view. (Her worldline should be vertical.)

Maria is floating inertially. Lucille is approaching her (at a constant velocity) from the right (relative to the viewer of the page), and Kevin is approaching her (at a constant velocity) from the left. All three worldlines intersect at point p.
The trajectories of the three people are unaffected by their encounter. A few minutes after the encounter, Maria sneezes. This occurs at spacetime point ps. With respect to Maria’s rest frame, her sneeze is (unbeknownst to her at the time) simultaneous with Kevin’s cough, which occurs at spacetime point pc, and with Lucille’s falling asleep, which occurs at pf. Light pulses are emitted at pc and at pf and they reach Maria at pm. She uses this information, together with certain other facts, and is eventually able to calculate that, with respect to her rest frame, Lucille fell asleep just when Kevin coughed.

Questions:

1. Spacetime point p is ________ from spacetime point pm.

a. timelike separated

b. spacelike separated

c. light like separated

d. timelike separated with respect to some inertial frames, lightlike separate with respect to other inertial frames, and spacelike separated with respect to still other frames

2. Spacetime point Pc is _______ from spacetime point pf.

a. timelike separated

b. spacetime separated

c. lightlike separated

d. timelike separated with respect to some inertial frames, lightlike separate with respect to other inertial frames, and spacelike separated with respect to still other frames

3. Spacetime point pc is _____ from spacetime point pm.

a. timelike separated

b. spacetime separated

c. lightlike separated

d. timelike separated with respect to some inertial frames, lightlike separate with respect to other inertial frames, and spacelike separated with respect to still other frames

4. With respect to Lucille’s rest frameL

a. Lucille falls asleep before Kevin coughs.

b. Lucille falls asleep at the same time as Kevin coughs.

c. Lucille falls asleep after Kevin coughs.

d. Not enough information

5. With respect to Kevin’s rest frame:

a. Maria sneezes before Lucille falls asleep.

b. Maria sneezes at the same time as Lucille falls asleep.

c. Maria sneezes after Lucille falls asleep.

d. Not enough information.

Kevin, Maria, and Lucille had their stopwatches set to 0 and started them at p. Their stopwatches display numbers, representing the number of minutes that have elapsed since the stopwatch was started, and the stopwatches are accurate and arbitrarily precise (‘as many decimal places as you like’). At pc, Kevin’s stopwatch reads ‘2’. They all inhabit Minkowski spacetime. The set-up from question #1 remains in effect.

6. We can conclude that

a. At ps, Maria’s stopwatch displays a number greater than two, and at pf, Lucille’s stopwatch displays a number less than two.

b. At ps, Maria’s stopwatch displays a number less than two, and at pf, Lucille’s stopwatch displays a number even less than the number displayed by Maria’s stopwatch at ps.

c. At ps, Maria’s stopwatch displays the number two, and at pf, Lucille’s stopwatch displays the number two.

d. At ps, Maria’s stopwatch displays some number greater than two, and at pf, Lucille’s stopwatch displays the number two.

e. at ps, Maria’s stopwatch displays some number less than two, and at pf, Lucille’s stopwatch displays the number two.

7. With respect to Kevin’s rest frame, when he coughs, what does Lucille’s stopwatch read?

a. Some number less than 2.
b. 2.
c. Some number greater than 2.
d. Whatever number Lucille’s stopwatch reads at pf.

8. Whose speed is the closest to lightspeed?

a. Lucille’s.
b. Maria’s.
c. Kevin’s.
d. Lucille and Kevin are travelling at the same speed (in different directions), so their speeds are equally close to lightspeed, and closer to lightspeed than Maria’s is. With respect to Maria’s frame, (d) is correct. With respect to Lucille’s frame, (c) is correct. With respect to Kevin’s frame, (a) is correct. But there is no absolute, frame-invariant fact of the matter. There are no (invariant) facts about subluminal speeds in Minkowski spacetime.

9. Spacetime point ps is in the __________ of spacetime point pf.

a. causal future.

b. causal past.

c. absolute elsewhere.

d. absolute past.

10. In terms of our formula for the interval posted on the slides, if I is the interval between p and ps, then I is

a. imaginary.

b. 0.

c. positive real.

d. Not enough information.