1. 4-vectors: Four components that Lorentz transform like ct, x, y, z.

2. The line is a worldline for a particle that is (in this frame) A] stationary B] in constant acceleration

3. The magnitude of the 4-velocity of this particle (in this frame) is A] zero B] positive C] negative

4. The 4-velocity in this frame is (c, 0, 0, 0) The magnitude of this (its “length”) is c.

5. In a different frame, this particle is moving. In the primed frame shown, is the particle moving in the [A] +x’ or [B]-x’ direction?

6. In this primed frame, what is the magnitude (i.e. length) of the particle’s 4-velocity? A] larger than c, because now it has an x-component B] smaller than c, because now it has an x-component C] c

7. The magnitude of a particle’s 4-velocity is invariant. Moving to a different frame “mixes up the components”, just as a cartesian rotation mixes up components.

8. In the primed frame, the size of the x-component of the 4-velocity increased. What happens to the size of the ct component (compared to the particle’s rest frame)? A] it decreases to “compensate” B] it increases to “compensate” C] it is unchanged

9. In the primed frame, the size of the x-component of the 4-velocity increased. What happens to the size of the ct component (compared to the particle’s rest frame)? B] it increases to “compensate” Because of the minus sign in our length formula, d(ct’)/d  must be larger than c. In other words, t’ is larger than  … no surprise, really, since the proper time is the slowest clock.

10. Just as we sometimes draw velocity vectors on a plot of object positions, we can draw 4-velocities on a spacetime diagram. What is the direction of the 4-velocity vector for the particle we have been discussing, on the diagram shown? A] straight down B] straight up C] parallel to the ct axis for the unprimed frame, but parallel to ct’ for the primed frame

11. “A worldline is a worldline is a worldline” - Gertrude Stein (maybe?) It’s the same worldline, so a small piece of it, divided by an invariant number, is the same 4-vector for all observers: straight up in this diagram.

12. Since the length of a 4-vector is invariant, the length of your 4-velocity vector is the same regardless of how fast you move! (That makes sense… there is always a frame where you are stationary!) If you are moving, both the time and the space components increase.

13. Which 4-vector is longest? Or choose E] they are the same length

14. 4-momentum