Twins Separated at Birth: Do Ages Match?

[the_kid]

Homework Statement

Twins are separated at birth on earth. Twin B travels in a straight line back and forth at a velocity, V. According to twin A, who stays on earth, twin B travels away for 10 years, and then back toward Earth for ten years.

They send light signals to each other at every birthday each one celebrates. When do they receive another's signals and does each count agree with their ages upon return?

Homework Equations

The Attempt at a Solution

Event 1: A emits a pulse
Event 2: Y receives the pulse

Event 1, in earth-frame: (t_{1}, 0)
Event 1, in moving frame of B: (t^{'}_{1}, -Vt_{1})

Event 2, in earth-frame: (t_{2}, Vt_{2})
Event 2, in moving frame of B: (t^{'}_{2}, 0)

This gives me both events in each coordinate system. They can be related by the expression c=\frac{x_{2}}{t_{2}-t_{1}}. After substituting x_{2}=Vt_{2} and simplifying, we have t_{2}=\frac{ct_{1}}{c-V}. This gives the relationship in the earth-frame.

Now how do I get the relationship in the moving frame?

 

[Simon Bridge]

You can get an understanding of what is happening by constructing the space-time diagram. Try this description.