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In the weak field case, you can justify adding the Newtonian potentials together to get another Newtonian potential, and you can probably get away with saying that the time dilation factor isOk, so in the example above say there is a spot between the two black holes where GRAVITATIONAL POTENTIAL is equal zero. I would say this would mean that any time dilation effects caused by one body (black hole A) would be cancelled out by the other body (black hole B) due to gravitational potential being equal to zero at that point. Thanks.

$$g_{00} = 1 - 2U + 2U^2$$

Here U is the Newtonian potential, which is for your two body cases the sum of the Newtonian potential due to each body.

That's based on the PPN formulation, and it's not really complete, I've ignored a lot of terms. See https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism for the full expression of ##g_{00}## in the PPN formalism.

But let's take the best case and assume all the stuff I'm ignoring doesn't matter. Even in that case, the potential at the midpoint still won't be zero. At least not if you normalize the potential to be 0 at inifintiy, which is the standard way of normalizing things in GR. With this sort of normalization, the clocks at infinity which are far away from any other mass, can be assumed to be not time dilated. The point is that clocks closer to the masses, including a clock at your midpoint, will be running slower than the clocks at infinity due to gravitational time dilation.

So, basically, you're wrong, even in the best possible case. And this formula is just approximatee - it's definitely not something you can apply to general, strong-field situations, as the Einstein field equations aren't linear. Because it's not linear, the superposition doesn't really apply.

This principle only applies to linear systems, and strong field GR isn't linear.wiki said:Thesuperposition principle,[1] also known assuperposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.

There are other issues, but this should do for starters.