Does inertial time dilation cause red shift? Or how does doppler shift work in light when it's always moving at C relative to anything?

Question

If we start with the proven fact light travels at the same speed in all reference frames, and add the proven fact a difference in speed causes a red or blue shift in frequency. Then it seems like doppler shifts in light do not work the same as sound.

It's my understanding, and could be wrong, that sound experiences a doppler shift because the source's differing speed relative to sound can be different. An oncoming object piles the sound up a bit compressing it's wave. Away going objects stretch it. All because their relative speed of sound through air is different. Sonic booms happen when an object outruns it's own sound.

However light is always the same speed relative to whatever emitted it, it can't pile up like sound. An object going 0.999999 the speed of C, and emitting a photon will see the photon race ahead at 1C and so will an object moving 0.000001 C. So what I'm wondering about is how doppler shifts work in light. My best guess is a blue shifted objected has a different relative clock than mine. From my perspective it's experiencing time faster, so any light it emits is going to have a higher frequency than it would other wise because any clocks it would have would tick faster, and frequency is a function of time. Red shifted objects would have a slower clock.

Is this correct or what's the correct way to conceptualize red shift and blue shift?

Answer 1

it seems like doppler shifts in light do not work the same as sound.

It does seem that way because it is typically presented as though they are very different. I like the mathpages Doppler Shift for Sound and Light page for exactly this reason. Basically, they derive the following formula, which holds exactly the same for both sound and for light: $$\frac{f_a}{f_e}=\frac{1-v_a/c_s}{1+v_e/c_s}\sqrt{\frac{1-(v_e/c)^2}{1-(v_a/c)^2}}$$ where $f$ is frequency, $v$ is velocity, $c_s$ is the speed of the signal with respect to its medium, $c$ is the speed of light, and $a$ and $e$ refer to the absorber and emitter respectively. This expression applies in any frame where $c_s$ is isotropic, and for light $c_s=c$ and $c$ is isotropic in any inertial frame.

An oncoming object piles the sound up a bit compressing it's wave. Away going objects stretch it.

This happens with light also. In a reference frame where the emitter is moving the wave is compressed in front and stretched behind the emitter.

However light is always the same speed relative to whatever emitted it, it can't pile up like sound. An object going 0.999999 the speed of C, and emitting a photon will see the photon race ahead at 1C and so will an object moving 0.000001 C.

Here you are mixing up two frames.

In the frame where the object is moving the wavefronts do pile up, as shown in the image above. Note that each wavefront expands at the same speed, $c$, but because each wavefront is emitted from a different location they do pile up in front, just like with sound.

In the object's frame it is at rest. In the object's frame the wavefronts are isotropic and each is emitted from the same location so they do not pile up. In that frame the Doppler shift is due to the absorber which is now moving. An inward-moving absorber crosses more wavefronts than they would if they were stationary.

The issue that you were running into in this analysis is that you started with a frame where an emitter is moving and then jumped to a frame where it is not moving. You need to be careful when switching the analysis to a different frame.