Time dilation caused by gravity and special relativistic velocity [closed]

I am working on an exercise for my Relativity class. The exercise is:

Two atomic clocks are transported in two aeroplanes once around the Earth in either eastern or western direction. For simplicity, assume the aeroplanes fly directly above the equator, where the rotation speed of the Earth is about $$v_E = 1674 \, \text{km}\, \text{h}^{-1}$$. Furthermore, use an average cruising speed of $$v_p = 800 \, \text{km} \, \text{h}^{-1}$$ and a mean flying altitude of $$10 \text{km}$$.

Calculate the respective time dilations the clocks exhibit compared to a clock which stayed on the ground, after the aeroplanes have landed. For this purpose, take into account the separate contributions due to the gravitational and the special relativistic velocity effect.

My thoughts so far:

• the contribution of the special relativistic velocity to the time dilation is $$t_\text{SR} = \gamma \tau = \frac{\tau}{\sqrt{1-\frac{v^2}{c^2}}}$$
• the contribution of gravity to the time dilation is $$t_\text{GR} = ( 1 + \frac{gh}{c^2}) \tau$$ for $$gh \ll c^2$$
• since the rotational velocities are small compared to $$c$$, I can just add them up:
1. plane flies parallel to earth's rotation: plane's velocity in the reference frame of the observer on earth is $$v_{p,||} = - 874 \, \text{km} \, \text{h}^{-1}$$
2. plane flies anti-parallel to earth's rotation: plane's velocity in the reference frame of the observer on earth is $$v_{p,-||} = - 2474 \, \text{km} \, \text{h}^{-1}$$

Now comes the confusing part.

When I read "seperate contributions", I thought I could just calculate both contributions and add them up like $$t_\text{total} = t_\text{SR} + t_\text{GR}$$, I just get $$t_\text{total} \approx 2$$ because $$gh \ll c^2$$ and $$v^2 \ll c^2$$. This looks weird. I think you can't just add up contributions.

I did some research and found out that in such a case both contributions should counteract each other because the faster you are, the slower time passes for you and the weaker the gravitational field, the faster time passes.

If time passes faster for weaker gravitational fields (larger $$h$$ or smaller $$g$$), my equation for $$t_\text{GR}$$ seems to be wrong but the Wikipedia article for Gravitational time dilation derived the same time dilation.

All in all, I am really confused about time dilation in the presence of gravity. Some enlightening words would be really appreciated.

I should have said that I don't want a solution to the exercise. I want to have clarification about the the problems and the confusion which arose from the exercise.

closed as off-topic by John Rennie, Jon Custer, Kyle Kanos, ZeroTheHero, user191954 Nov 22 '18 at 11:48

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• Please note that we don't answer homework or worked example type questions. Please see this page in the site help for more on what topics you can ask about here. – John Rennie Nov 17 '18 at 17:59
• See: Is gravitational time dilation different from other forms of time dilation? for some background info on this. – John Rennie Nov 17 '18 at 18:06
• @JohnRennie I should have maybe said that I don't want the solution to this exercise. Rather I wanted to clear up the confusion which arose from this exercise. The easiest way for me to communicate this confusion was by showing you the exercise. – Limechime Nov 17 '18 at 18:58