How to compute the composition of 2 gravitational time dilations?

Physics Asked by foolishmuse on September 12, 2020

This question could apply to stars orbiting a black hole, or planets orbiting a star, or moons orbiting a planet. The example numbers don’t matter, just the concept.

Lets say that a star has a gravitational time dilation factor of 2 of the level of an observer on Earth (i.e. time runs twice as slow as an observer on earth.) At a certain altitude above the star the time dilation factor is 1.5. This should mean that anything at that altitude should also be running at that same time rate of 1.5. But a planet, which would have its own time dilation factor of 1.5 if it was out in distant space, is orbiting the star at that altitude.

So what is the total time dilation on the planet to the observer on earth? Is it 1.5 x 1.5 = 2.25? Or does the planet just retain its own factor of 1.5? Or something else?

As an approximation you can add the gravitational potentials and therefore multiply the time dilation factors. This will be a good approximation in weak fields, but factors of $$1.5$$ or $$2$$ are not weak fields! So for such large amounts of time dilation, multiplying them together is a very rough approximation; really it is only good for some sort of qualitative guide. Indeed I think that the factor 2 is about as large as it can get for a star (any larger and the star must collapse to a black hole).

For an exact treatment one would need to solve the field equation, taking into account its non-linearity. This is non-trivial and would require numerical methods.

Correct answer by Andrew Steane on September 12, 2020

Yes, I think you are right, and the factors should be multiplied by each other. This must work correctly for, at least, the first order of approximation. The time dilation depends on the gravitational potentials, which, in your example, the gravitational potentials of the star and the planet should be added up together. This means, to some extent, that the time dilation factors are multiplied by each other.

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