Imagine two identical black holes in a circular orbit, and Alice is smack-dab in the middle of the system (at the barycenter). Bob is at infinity.
Let's assume that Alice and Bob are stationary relative to each other. Alice does not experience any net acceleration (from being at the barycenter of the system).
Does Alice experience any time dilation relative to Bob?
Here are some arguments for why she should not:
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Alice is stationary relative to Bob – so no special relativistic effects.
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Alice is not accelerating – so by the equivalence principle, she is not experiencing gravitational effects and therefore, no time dilation.
However, this feels like the wrong conclusion because she's still stuck in the gravitational well of this binary system and requires a non-zero escape velocity to exit the system and meet with Bob.
In literature, lot of discussion about time dilation talks about escape velocity which is straightforward when talking about single spherically symmetric masses – but I am not sure how it applies to this system.
Of course, the resolution here might just be they both don't experience any relative time dilation, but any signals they try to send each other will always be gravitationally redshifted. And there is no way for them to meet up and compare clocks that show the same passage of time.
Best Answer
Lets suppose that the orbit is very large, so we could apply the linearized theory at the center. The spacetime metric is then $$\tag{1} ds^2 \approx (1 + 2 \phi) \, dt^2 - (1 - 2 \phi) (dx^2 + dy^2 + dz^2), $$ where $\phi$ is the newtonian potential. For two black holes on the circular orbit: $$\tag{2} \phi = \sum_k \phi_k = -\, \frac{2 G M}{r}. $$ The time dilation is defined by the following formula (for stationary observers at the center and at infinity): $$\tag{3} d\tau = \sqrt{g_{00}} \, dt \approx (1 + \phi) \, dt. $$ So the time retardation of the central observer would be $$\tag{4} \Delta \tau \approx \frac{2 G M}{r} \, \Delta t. $$ Notice that if the orbit is very large so that $r \gg 2 G M$, then $\Delta \tau \approx 0$. The time dilation would be negligible.