General relativity prevents light from escaping a black hole, but does it also apply to gravitational waves?
Any gravitational waves emitted inside the event horizon fail to make out of the event horizon because they travel at lightspeed. And a lightspeed signal from inside stays inside.
Therefore, any waves emitted inside the event horizon are never observed on the outside of the event horizon.
Even if all the mass behind the horizon would magically disappear in one moment, you would not notice that from outside the horizon.
One could assume that in this case there was no mass left that could bend spacetime anymore and the gravitational field would disappear with c, but on the other side you have to take time dilation into account:
From the perspective of the outside observer everything that makes up the black hole is stacked up at the horizon and asymptotically approaches it as time goes to infinity, simply because the factor for the time dilation approaches 0 as an object approaches the horizon.
Therefore, what happens inside a black hole at a given proper time of an infalling observer does not even have a corresponding coordinate time on the outside of the black hole (mathematically an imaginary one, but technically after infinity), because from that perspective it takes an infinite amount of time to even get near the horizon, not to mention behind it.
If from our perspective there is nothing behind the horizon there is nothing that could create gravitational waves behind the horizon. In other words: whatever happens behind the horizon, outside the horizon it has not happened yet.
So the answear would be no, you can not send gravitational waves from the inside of a black hole to the outside, simply because you haven't even yet been falling though the horizon in the system of an outside observer. From his perspective you are always outside the black hole and never go through the horizon until infinity.
If a gravitational wave can be created inside event horizon, I do not see any reason why it would not escape from it. It may not be possible to create a GW inside event horizon, but that is a different question.
When we say nothing can escape event horizon, the reason for that is enormous gravity beyond event horizon. This argument likely will not apply to gravity which is cause for the argument itself.
Remember, gravity causes black hole, not the other way. Because, gravity was there before black hole had formed, not that gravity appeared after black hole formed. Gravity controls black hole, not that black hole controls gravity.
GR itself predicts GW, not sure if it specifies location of origin of it.
Also, if a GW could not escape EH, and suppose, it can enter EH, then that would imply a GW can be absorbed by a BH. If it can not even enter, then that would imply GW can be reflected off EH. These would be follow up questions.