How to Solve the Twin Paradox Problem Using Lorentz Transformations

[GammaLyra]

Homework Statement

This is a typical twin paradox problem as laid out in Griffith's Introduction to Electrodynamics, problem 12.16. The problem states that, on their 21st birthday, one of two twins - we'll call her Alice - departs Earth for star X at (4/5)c. Upon arriving at star X, she immediately begins the return journey also at (4/5)c. She returns home at the age of 39 (according to her watch). Her twin Bob remains at Earth during the entirety of the trip.

There are two reference frames associated with Alice: the "outbound" frame in which she travels toward star X away from Earth, and the "inbound" frame in which she returns to Earth. There is also the Earth reference frame.

Part (d) of the problem asks: what are the coordinates (x, t) of the jump (from the outbound frame to the inbound frame) in the outbound frame?

Part (g) of the problem asks: how old does Alice say her brother is right now just before she makes the jump? I.e., how old does Alice think Bob is in the outbound frame right before she jumps to the inbound frame?

Homework Equations

All that's needed are the usual Lorentz transformations:

t_alice = γ(t_bob - (v/c²)x_bob)
t_bob = γ(t_alice + (v/c²)x_alice)

The Attempt at a Solution

I correctly solved part (d) by finding the coordinates of the jump in the Earth frame and transforming them to the outbound frame. This meant transforming the coordinates (12 ly, 15 yrs) to (0, 9 yrs).

For part (g), I want to use the first Lorentz transformation above and solve for t_bob:

t_bob = t_alice/γ + (v/c²)x_bob

The problem is that I think x_bob should be nonzero, when the correct solutions use 0 for this variable. That doesn't make sense to me. As stated above, I think what part (g) is asking is what Bob's coordinates (x, t) are in the outbound frame right before the jump. In the outbound frame, Alice was stationary and her x coordinate in this frame should be 0. Consequently, Bob's x coordinate in the outbound frame should be nonzero, yet it is not. What is the error in my understanding?

Any insight is highly appreciated. Thank you.

 

[MarcusAgrippa]

Clue: what is the physical interpretation of proper time?