Time dilation between perihelion and aphelion of earth with the combined effects of Special Relativity and General Relativity

Question

I am a high school student and a total newbie in the field so I apologize for any misconceptions or mistakes I may do in advance.

I am trying to calculate the time dilation observed on earth at it's aphelion state by an observer on earth at it's perihelion state.

I thought because earth's movements must be parallel to each other at those states their velocities can be added to find their relative velocity to calculate time dilation predicted by special relativity with the formula: $ t'=\frac{t}{\sqrt{1-\frac{v ^2}{c^2}}}$

And that I can use $ t_0=t_f\sqrt{1- \frac{2GM}{rc^2}}$ for both situations to form a ratio between the time dilations predicted by general relativity for these situations.

And finally add them up to find the overall time dilation observed in earth's aphelion state by an observer on earth on its perihelion state.

However, I have also seen that the Schwarzschild metric is used in an answer to a very similar question by Mr. John Rennie: Does Earth experience any significant, measurable time dilation at perihelion?

Is my method correct or should I use the Schwarzschild metric to answer the question? If the latter, does this include the effects of time dilation related with Special Relativity? If not, can these effects be combined together by simply adding them, or is there another method to do this?

Answer 1

Your question is ill-posed. General relativity doesn't have time dilation as a generic concept. Time dilation exists in GR as a description of a special case in which you compare two static observers in a static spacetime. The earth is in motion relative to the sun, so you're not dealing with a static observer here.

A less fancy way of saying this is that what we would actually observe experimentally would be a Doppler shift of a signal. So we could imagine that we put a satellite in orbit around the sun so that it's 180 degrees away from the earth, but shares the earth's orbit. Then we could ask what the Doppler shift would be when we send and receive light signals to and from the satellite. The problem here is that such a signal will have more than one path by which it can travel, because the light rays get bent by the sun's gravity. I haven't estimated it, but I would generically expect this bending effect to be on the same order of magnitude as the Doppler shift, which means that you wouldn't get a single well-defined Doppler shift. If you transmit a pulse, it will be received more than once.

Having said all that, if I had to guess some kind of average value for the effect, averaged over possible propagation paths, I would guess that it would be well approximated as a product of a gravitational time dilation factor and a kinematic Doppler shift. The kinematic Doppler shift would be approximately a transverse Doppler shift, which would be approximately equal to the kinematic time dilation factor you wrote down. So yes, on the average I expect that the product of the two factors you wrote down would come out about right.