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This is a follow-up to the answer about time dilation in the middle between two gravitational forces: John Rennie's diagram describes the situation where the question was about the time dilation between two masses.

enter image description here

Important is not the net force, this can be zero, but the total gravitational potential. Time dilation is described in that answer as $$\frac{dt}{dt_0} = \sqrt{1 - \frac{4GM}{c^2r}}$$ where $t_0$ is measured by a clock far away from the masses.

With the Schwarzschild Radius $r_s = 2GM/c^2$, this can be written as $$\frac{dt}{dt_0} = \sqrt{1 - \frac{2r_s}{r}} $$

which gets 0 and then complex valued as $r$ passes the value of $2r_s$ from above. So we can construct a situation where the radius of the masses $M$ is $R$ such that $r_s<R<r<2r_s$:

overlapping potentials generating black hole area?

Can someone explain what happens within the overlapping area with regard to time dilation, the speed of light, curvature? Is this an "empty" black hole? Or is the computation wrong, because the fields are too strong there? What would be the correct way to compute time dilation then?

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  • $\begingroup$ Note to self: r<2r_s so I start out already with a problematic situation. Coming back here I wonder if I asked the wrong question back then. What if r is a bit larger than r_s on both sides? Likely the same answer applies, since solving then equations for two separated masses is surely non-triviall. $\endgroup$
    – Harald
    Commented Sep 17, 2023 at 5:57
  • $\begingroup$ The gravitational time dilation equation $\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}$ comes from the Schwarzschild metric. John Rennie's equation uses the approximation $$(1+a)(1+b)\approx(1+a+b)$$, which is ok when $|a|$ and $|b|$ are both small. So it's fine for calculating dilation in weak fields, but not between two nearby black holes, which is quite different to the simple Schwarzschild geometry. $\endgroup$
    – PM 2Ring
    Commented Sep 17, 2023 at 6:51
  • $\begingroup$ General Relativity is non-linear, so the fields of 2 BHs don't simply add together. And 2 neighbouring BHs don't just sit there, their relative velocity is relativistic, and such a system radiates considerable energy in the form of gravitational waves. As John says below, calculating (approximately) what happens in that system requires extreme number crunching. $\endgroup$
    – PM 2Ring
    Commented Sep 17, 2023 at 6:52
  • $\begingroup$ Note: the way I chose the radii, there is no black hole, but the comments apply anyway, of course. $\endgroup$
    – Harald
    Commented Sep 17, 2023 at 15:01

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The equation I gave in the previous answer you link to applies only to the weak field case i.e. when the spacetime curvature is small. You cannot use it to locate event horizons.

In the example you give of the two masses if the midpoint is behind a horizon that means the two masses cannot remain stationary with respect to each other. Instead they must merge. So we have a situation like two black holes merging i.e. the geometry changes with time so the position and shape of the horizon changes with time. This turns out to be a horrendously complicated problem and it's only recently that computers have become powerful enough to calculate how the horizon evolves as the two black holes merge.

So there is no simple answer to your question. You cannot simply calculate what happens at the midpoint of the masses in situations like merging black holes where the curvature is high.

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