Could information be transmitted from inside a black hole using gravitational waves? Could you use gravitational waves to communicate from inside the event horizon of a black hole to someone who was outside?
A black hole is defined as a region of spacetime where the gravitational effects are so strong that the escape velocity within the event horizon is greater than the speed of light.  According to relativity, nothing can propagate faster than the speed of light, so at most gravitational waves would propagate at this speed.  According to particle physics, mediation of gravity would be controlled by massless particles called gravitons, which, by virtue of being massless, would travel at the same speed as photons, i.e. the speed of light.  Suffice to say gravitational waves (analogous to electromagnetic waves) would not be able to escape from inside the event horizon.

The question of information, in regards to black holes is open however. The No-Hair Theorem would seem to imply the destruction of information as the black hole consumes matter.  However, the holographic principle, if correct would imply that information regarding the matter that formed the black hole would be encoded on the 2-dimensional boundary of the event horizon.  This information could very well be exhausted by Hawking radiation or gravitational waves.  Since this information was outside of the event horizon this would be fully consistent with the formulation of black holes.