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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.

1 Answers1

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The first thing to say is that this experiment has actually been done using commercial airliners. Reading up on that might be enough to clear up any confusion.

Although in reality the two time dilations should probably be multiplied, they are both tiny enough that adding will get you the same answer (ignoring higher order terms). For example, the gravitational time dilation of leaving the Earth, exiting the Solar System, and traveling far away from the Milky Way into intergalactic space, amounts to about 0.2 seconds per year. This corresponds to a time dilation factor of about 1.000000006 or about 6 parts per billion. The change from just increasing altitude on the Earth a little is much smaller than that.

The factor for special-relativistic time dilation is always less than 1, so the two effects are in opposite directions and will partially cancel each other. For example, suppose you got 0.99999995 for that (-50 ppb) and the gravitational effect was 1.000000001 (+1 ppb). Then the completely correct calculation would be 0.99999995 * 1.000000001 = 0.99999995099999995, but the simple addition would get you 1 + 1 ppb - 50 ppb = 0.999999951. Your actual numbers will be different, but the general size and behavior will be the same. If you round even a little bit, either method will give the same answer. This is because (1+a)(1+b) = 1 + a + b + ab, and if both a and b are small then you can ignore the ab term.

Notice that in the adding method, you do not add two copies of the 1. This may be where you went wrong.

Howard A. Landman
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