Do gravitational waves have entropy? We know, according the current understanding of black holes and General Relativity, as well as quantum fields in General Relativity, that black holes have an entropy proportional to the area of the event horizon. No problem with that. My question is simpler, but something I am not clear on. Does the gravitational radiation, eg that produced by the binary black holes that merged that carried away about 3 solar masses of energy, also carried away entropy? (There is no problem with the black hole entropies, the final black hole still had more than the initial ones added up)
I am not referring to Hawking radiation nor to any entanglement issue. This is more basic. The question is what statistical or microscopic property of the gravitational wave can be described as representing that entropy? The different ways it could have been produced? Some statistics on the (linearized theory) gravitons? Or does the question or concept of entropy of a gravitational wave not make sense? I've read up what I could find, no clear answer. 
 A: As stated in the comment by Peter Diehr, the question is in principle no different whether you ask it for electromagnetic, gravitational or any other kind of wave. The wave's entropy is simply the conditional Shannon entropy of the specification needed to define the wave's full state given knowledge of its macroscopically measured variables. A theoretical gravitational wave defined by a full solution of the Einstein Field equations has an entropy of zero just as a full solution of Maxwell's equations does; if you know at the outset that the wave has come from a lone black hole whose state is known, then measurement of the amplitude, polarization and arrival time alone will fully define the wave (the six independent, modulo gauge, components of the metric tensor at your position). 
But from these perfectly defined states, gravitational wave and light wave systems can take on "imprints" from their interactions with the World around them in many ways, so that any set of macroscopic measurements of a wave leaves much about the wave's state that is unknown:


*

*As in Lawrence Crowell's Answer, the source could have an unknown configuration. There may be a complicated system of gravitating black holes generating the waves, so our ignorance of this configuration means that we cannot infer the full state of the wave from macroscopic measurements. There could even be some advanced society of creatures modulating the waves for communication purposes; the message that they encode has Shannon entropy that helps compose the wave's total entropy;

*Waves scatter from objects; unless the scattering is very simple, the scattering will lead to changes in the full wave state that cannot be gleaned from macroscopic measurements alone. The Optical Grasp of light scattered from rough surfaces increases as properties of the surfaces become encoded into the light's full state which are inaccessible to a macroscopic observer. In theory, gravitational waves are perfectly analogous: their grasp will be increased by interactions with complicated matter systems;

*Gravitational waves, like light, can in theory thermalize, so that gravitational black body radiation is in theory possible. One could imagine gravitational waves bouncing back and forth and interacting with vast regions of space filled by black holes and hot gas.
However, I suspect in practice the entropy of gravitational waves will be much lower than  that of light. The interaction between gravitational waves and matter is vastly weaker than that between light and matter, simply by dent of (1) the weakness of the gravitational force's action on matter in comparison with that of the electromagnetic force and (2) the fact that gravitational wave sources are quadrupolar and higher order unlike light sources which can be dipolar. Therefore, the thermalization and increase of grasp theorized above are probably just that: theoretical possibilities that seldom if ever arise in our Universe, at least over timescales of the order of the Universe's present age.
A: The answer is a definite yes and no. Gravitational waves have entropy in that we can think of them travelling from their source to our detector as a channel that sends units of information in the sense of the Shannon formula. The ringing of our detector is then the reception of that information. The Shannon formula $S=-k\sum_n p_n log(p_n)$ would give a nonzero entropy.  The probabilities $p_n$ are based on a Bayesian inference on the signal strength for Fourier modes for an expected linearized gravity wave.
The reason I also say no is that gravitational radiation does not have entropy in the same way a black hole does. There is no event horizon with a certain area that define entropy.
