Special relativity: One spaceship moving towards another

Question

Question

Imagine two spaceships: A and B

B moves at a significant % of the speed of light to A.

Here's the confusion when thinking about B's frame of reference:

According to special relativity, moving clocks appear slower. Assuming B's frame of reference, A is the one that is moving. B should be seeing A age slower.

But, since B is moving towards A, light should take less and less time to reach from A to B. So B should be seeing A aging faster.

So, how can B see A age slower (to satisfy time dilation) but also see A age faster (to satisfy constant speed of light needing to travel less further)?

I know that when B reaches A, A will be older since B moved into the frame of reference of A. My question isn't about when they both meet. That one I do think I understand. It's about why time dilation seemingly doesn't work in this instance.

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Research I have done to try and figure out an answer for this myself

I am not a physicist but I'm sure there's an explanation here I don't follow. I tried to look into the twin paradox as well as the relativistic doppler effect as I think the answer lies somewhere there.

A part of my question is sort of shown in the return leg of the twin paradox for example. But even there when the twin turns to return back, time appears to be going faster for the earth twin from the frame of reference of the space twin, so again seems to contradict special relativity.

From what I understand, the time dilation equation doesn't care about direction of travel. It just solves to time going slower to an object moving at your frame of reference.

Answer 1

You are correct, the answer is in the relativistic Doppler effect.

The classical Doppler effect is a change in the observed frequency due to the motion, and there is no time dilation involved. So even without time dilation you can get a change in the frequency just due to ordinary motion. This is directionally dependent and quite large. As the emitter moves towards the observer the frequency is blue shifted and as it moves away it is redshifted.

Now, on top of the large classical Doppler effect there is also the small relativistic time dilation effect. This is not directionally dependent and always produces a redshift. Regardless of what direction the emitter travels the frequency of the source itself is slowed.

The relativistic Doppler is the combination of the large classical Doppler and the small relativistic time dilation. Overall, it is directionally dependent. As the emitter approaches the receiver the frequency is blue shifted but not as much as it would without time dilation. As the emitter leaves the receiver the frequency is redshifted, and a little more than it would without time dilation.

Importantly, when the emitter is moving tangentially to the receiver then there is a redshift which is entirely due to the time dilation. Classical Doppler predicts no shift in that geometry, but relativistic Doppler does predict one. This is known as the transverse Doppler, and is a key prediction of relativity that has been experimentally confirmed.

Answer 2

The phrase "moving clocks appear slower" is both ambiguous and misleading.

If you are on Spaceship B viewing a clock on Spaceship A which is moving directly toward you, then the clock will seem to be running fast- if Spaceship A were moving away from you, then the clock would appear to be running slow. That is a consequence of the Doppler effect. It does not contradict special relativity.

Time dilation is a different effect. According to SR, the elapsed time between two events is frame dependent. Specifically, the elapsed time between two events in a frame in which they occur in the same place is less than the elapsed time between the same events in any other frame.

Answer 3

The answer is relativity of simultaneity. When $t = 0$ for B frame, A has at this very instant (for A frame) $t_A = vX_B$, where $X_B$ is negative if A is approaching with positive velocity $v$.(I use here $c=1$).

Because A is approaching, the radio signals received by B will show there a clock running faster that its own clock. That is the effect of relative velocity and is not related to time dilation.

But it is because the initial A time is negative, that is possible that the time runs slower in A frame, and even so the total elapsed time from B perspective is greater there from $t=0$ to the meeting time.

One way to measure the time dilation is to have some ships in a convoy with A, all of them with synchronized clocks. Each time one of them meets B, the rate of clocks can be compared.