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.

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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 $$\frac{dt}{dt_0} = \sqrt{1 - \frac{2r_2}{r}} $$

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?


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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