Problems Week 6
1. The timelike unit vectors ˆu, ˆv and ˆw lie in a 2-plane in spacetime.
Assume ˆu · ˆv and ˆv · ˆw are known.
a) Calculate ˆu · ˆw.
b) Write −ˆu·ˆv = cosh ζ, −ˆv· ˆw = cosh η and −ˆu· ˆw = cosh ξ. Express
ξ in terms of ζ, η.
2. Two galaxies have four-velocities ˆu and ˆv respectively. A light signal
is emitted from one of them (event R1) which is absorbed by the other
(event R2). Calculate the Doppler shift.
3. Two unaccelerated spaceships are about to meet. A light signal is sent
from ship A to ship B and the Doppler shift is given by ωB/ωA = d. Ship
A measures proper time τA from emission to meeting and B measures
time τB from receiving the signal to meeting. Calculate τA/τB.
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