Suppose then that we want to "bend" or otherwise change the metric of the black hole. This might happen in a black hole merger.
We assume that a "causal" change in the metric can only propagate at the local speed of light. A merger is a causal process, since we can either let the black hole to stand alone, or we can merge it with another black hole.
If the local speed of light, as seen by a faraway observer, stays extremely slow close to the horizon, then the merger process cannot change the metric very close to the horizon.
But an infinitely rigid object involves several paradoxes. One could send a superluminal signal by nudging such an object. How to resolve the paradoxes?
How to move a neutron star?
A milder version of the problem occurs with a neutron star. If we start to move a neutron star, it takes some time for the surface and the interior to "know" that it is moving.
We wrote about this problem also on March 17, 2023.
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/ \ <-- ● mass M
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neutron star
Let us assume that we suddenly move a mass M close to the neutron star. It starts pulling on the star.
It takes some time for the interior of the neutron star to know that the mass M came nearby. The metric (in the comoving frame) inside the neutron star must remain constant during that period. But the neutron star, obviously, starts moving toward M immediately.
This means that the metric in the comoving frame of the interior really describes a "perfectly rigid object" during that short time.
The matter in a neutron star is not perfectly rigid. It is possible that the outer parts of the star start moving before the central part.
Let us then assume that the local speed of light (in global Schwarzschild coordinates) inside the star is extremely slow. We may be able to move the neutron star several diameters before its center knows that anything is happening. In this case, the central part must start moving before light has time to reach it.
The solution in this case might be frame dragging. The central part moves along with the outer parts "automatically". One cannot use frame dragging to communicate a signal to an observer at the center of the star, because the observer has no means of detecting frame dragging.
If we use comoving coordinates inside the neutron star, then we have to make the metric to change outside the star, to make the star to move.
If we use static coordinates inside the neutron star, then we have to let the mass start moving inside the star, and this process propagates faster than the local speed of light inside the star. If we have banned "causal" processes which happen faster than light, then we cannot use static coordinates.
Frame dragging in our Minkowski & newtonian model of gravity
In our own gravity model, a slow speed of light comes from the extra inertia which surrounding masses give to the photon.
If we move the surrounding masses, then the photon must move along. Frame dragging is a very natural process.
Also, the mutual inertia couples masses together. If we move the outer part of a neutron star, then the inner part must move along. The speed of this process exceeds the local speed of light. The speed is probably the global speed of light in the underlying Minkowski space.
Conclusions
We want to understand how black holes move, to figure out how a merger of two black holes technically happens.
Our analysis above suggests that we have to move the black hole with a frame dragging process - and the black hole really is a perfectly rigid object.
We do not know if numerical relativity uses frame dragging to move a black hole. And if not, what implications that has. Do the calculations allow superluminal causal effects?
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