B Gravitational Time Dilation

pervect

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Ok, 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.
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 is

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

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.

wiki said:
The superposition principle,[1] also known as superposition 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.
This principle only applies to linear systems, and strong field GR isn't linear.

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

Janus

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Would it be possible for the high man to read a newspaper that, from the low man's standpoint, hasn't been printed yet?
Let's examine that using a different question. Would the high man see a sunrise that hasn't occurred yet for the low man? (for this we will ignore the fact that the horizon would be further away for the higher person).

If this is true, and time runs faster for the high man, each successive sunrise should occur even earlier and earlier for the high man. The upshot would be that he would rack up sunrises faster than the low man over a long enough period.

That simply does not make any sense. What really occurs is the each man would see the same number of sunrises, but that by the higher man's local measure of time, the sunrises are spaced further apart than they are according to the low man by his measure of time. So for example if the low man measures exactly 24 hr between sunrises by his local clock, the high man might measure 24 hr and 1 nanosecond between sunrises by his local clock.
 

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