Let’s say I’m in a space ship, headed straight for a nearby star at 92% of the speed of light. (92% of the speed of light works out to a gamma factor of 2.55, which I know isn’t a nice round number, but still.)
I know the light from that star will be horribly blue-shifted. My calculations put the blue-shift factor at 4.9, i.e. every photon that star emits will reach my eyes with 4.9 times the frequency it had when the star emitted it (or 1/4.9 of its wavelength).
What I’m wondering is:
(1) How much BRIGHTER will the star be at those wavelengths? Besides the fact that a photon at 4.9 times its initial frequency has 4.9 times its initial energy, I’m moving toward this light source, so I should be slamming into MORE photons every second.
(2) How much does the Lorentz Contraction factor into this? From my reference frame on the space ship, the star and all the light it emits is moving toward me at gamma = 2.55, so the space between each emitted photon should be 2.55 times shorter, meaning 2.55 times as many photons should be reaching me each second, ON TOP OF the “slamming into the light source head-on” factor.
(3) Does time dilation exactly cancel out the Lorentz contraction? Yes, the spaces between the photons are squished to be 2.55 times shorter, but the all the processes in the star that CAUSE the photons to be emitted in the first place should be happening 2.55 times slower from my reference frame.