# Overlapping gravitational field of non-black holes formally creating a black hole situation?

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.

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:

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?

• 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. Commented Sep 17, 2023 at 5:57
• 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. Commented Sep 17, 2023 at 6:51
• 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. Commented Sep 17, 2023 at 6:52
• Note: the way I chose the radii, there is no black hole, but the comments apply anyway, of course. Commented Sep 17, 2023 at 15:01