How do you calculate time dilation if there's two gravitational pulls acting at once?

Question

I was wondering how you'd calculate time dilation if there's two masses acting on you at once. For example lets say I wanted to calculate the time dilation between Earth and PlanetB (I just made that up idk) but PlanetB was really close to a black hole who's gravity was acting on it. How would I combine the gravity of the black hole and the gravity of PlanetB itself to find the time dilation. (Ignoring time dilation from velocity)

Answer 1

It is, as it turns out, very difficult to do this.

The reason is because the key equations of general relativity - the Einstein field equations - are highly nonlinear. This means that if I have two solutions to it (e.g. the solution for PlanetB and the solution for the related black hole), adding them together does not produce another solution. In the weak-field case, as for anything less-intense than a neutron star or such, you can approximate the fields in a linearized way so that you can get a good estimate of what things should be; for a black hole, though, that's not possible, because the linearized approximation fails in such cases.

For two planets, gravity is roughly linear, so time dilation would just be

$$\sqrt{1-\frac{2G}{c^2}\left(\frac{M_1}{r_1}+\frac{M_2}{r_2}\right)}$$

for the mass of and distance to Planet $i$ being $M_i$ and $r_i$, respectively. If you're talking about a black hole and a planet orbiting close enough that that matters, the gravitational time dilation will be very different; determining it will require identifying a two-body vacuum solution of such a form, which to my knowledge has not yet been accomplished.