Is it possible to utilize gravitational waves as a delivery system for information between two observers straddling the event horizon of a black hole? And why ?
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2 Answers
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Yeah, great question.
Firstly, one of your underlying assumptions is spot on. You can send messages with gravity. If you have a brick in your hand and wave it back and forth a pendulum on the moon will have its period slightly changed because the direction and strength of the gravitational field around the pendulum will change slightly as the brick moves.
You can't do this with a black hole because the nature of a black hole (and specifically the nature of its gravitational field) is determined solely by its total mass, electric charge and angular momentum. This is a consequence of the equations of GR. As waving a brick around inside a black hole cannot change any of these qualities, it cannot be detected outside the black hole.
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No, you can't send information through an event horizon with gravitational waves. When a compact body contracts to form a black hole it leaves the vacuum spacetime outside the body in a curved manner. That's because certain kinds of curvature can persist on their own, even static ones,they are called vacuum solutions, and they are curved spacetimes that are curved in such as way as to cause their own future curvature. And that's the kind of curvature a black hole leaves outside as it collapses, and that's what makes things orbit it or fall into it, it's not information from inside the black hole that comes out that tells things about how massive it is.
So, certain kinds off curvature naturally beget themselves (just like a travelling electromagnetic wave causes itself to appear in a slightly different position at a slightly later time). What gravitational sources do is to allow different kinds of vacuum solutions to be sewn together.
For instance you could take the vacuum solution for the region outside a mass $M$ star, and it looks like a funnel shape. Then you could take a vacuum solution for the region outside a mass $m$ star, and it looks like another funnel shape. You can cut out the inside part of the mass $M$ solution and the outside of the mass $m$ solution in the region where the circumference was the same and sew them together. That corresponds to a solution where there is a shell of mass at that circumference.
It's natural for spacetime to be curved when it's left with a curved shape that persists. And all you have to do is leave it in one of those funnel shaped vacuum solutions as you contract and it will stay that way. All gravitational sources do is allow novel curvatures different than those natural kinds, and the sources only do that exactly where the source is located, so what gravitational sources do is piece together different vacuum solutions.
A gravitational wave is a kind of vacuum solution, but one that is not static. Instead it is one that travels like an electromagnetic vacuum solution travels. It does not cross an event horizon.
