- #1
Isaac Pepper
- 31
- 1
Homework Statement
The highest energy protons have gamma factors around ##1.0*10^{12}##.
(a) Our galaxy has a disk diameter of 30 kpc, which is ##9.3*10^{20}m##. If a photon and one of these high energy protons start traversing the galaxy at the same time, by how long will the arrival of the photon on the other side precede the arrival of the proton, in the rest frame of the galaxy? Give your answer in seconds.
(b) To an observer in the reference frame of the traveling proton, what is the measured diameter of the galaxy in kpc?
(c) To an observer in the reference frame of the traveling proton, how long does the journey across the galaxy take? Give your answer in seconds.
Homework Equations
##t'=\gamma (t-\frac{xv}{c^2})##
##x'=\gamma (x-vt)##
The Attempt at a Solution
(a) I have managed to get an answer of ##1.55*10^{-12} s## for this question using a binomial expansion.
(b) I'm not sure as to how to obtain the diameter of the galaxy in the reference frame of the traveling proton.
I'm assuming it should be shorter in the reference frame of the proton due to length contraction
Do I have to use the formula ##x'=\gamma(x-vt)## where x is ##9.3*10^{20}m## and ##t=\frac{d}{v} =\frac {9.3*10^{20}}{(3*10^8 - 1.55*10^{-12})}## ? When I do I get an answer which isn't contracted, but is dilated.
(c) ... Once I've understood how to do (b) I'm sure I can work out (c)!
Thanks in advance for any help !