On their 25th birthday, twins Alice and Bob part ways: Alice the astronaut board

Question

On their 25th birthday, twins Alice and Bob part ways: Alice the astronaut boards a spaceship traveling at beta = 0.85 to an extrasolar planet zeta chi. Bob the builder remains on earth. Immediately after reaching zeta chi, Alice reverses her spaceship engines and returns to earth at the same speed. She arrives on her 43rd birthday as measured by her own wristwatch. (a) How old is Bob? (b) How far away is the planet zeta chi? Call the outbound spaceship frame S, the inbound one S^-, and the earth frame S. All three frames have their space-time coordinates set such that x = x^- = x^~ = 0 and t = t^- = t^~ = 0 at the moment of Alice's departure from Earth. (c) What are the coordinates (x, t) of the spaceship engine reversal event in S? (d) What are the coordinates (x^-, t^-) of the same event in S^-? (e) What are the coordinates (x^~, t^~) of the same event in S^~? (f) If Alice is asked "How old is your brother right now?", what is the correct reply (i) just before she reverses her engines, (ii) just after she reverses her engines? (g) How many earth years does the return trip take?

Explanation / Answer

(a) So Alice leaves at the age of 25

=0.85=(1-v2/c2)

v= 9124 m/s

It takes 18 years for Alice to reach and come back.

So by the time Alice reaches home bob is 47 years old

(b) T=2d/v =18 years

2d = 18 years x 9124 m/s

d = 2.59 x 109 km

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