# 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.

• See physics.stackexchange.com/questions/4546/…, where a similar question is discussed. Since photons can carry entropy, the same would be true for gravitons/gravitational waves. – Peter Diehr May 21 '16 at 22:28
• I saw that but it did not say how photons have entropy either. I understand that black body photon emission has entropy, a lot of random freqs and phases, controlled by the blackbody radiatio probability density function. But for grav radiation we know about semi-isolated emitters, like the merged black holes. Can that be considered a blackbody emission at some temperature? I am not talking about Hawking radiation, but about the macroscopic radiation we detected. – Bob Bee May 22 '16 at 0:34
• And physics.stackexchange.com/questions/119872/…. The implication is that the change in entropy depends upon what happens at the emitter - the changes in the number of states available for a microscopic analysis, or change in temperature for a macroscopic study. In either case, the emitted radiation can carry away entropy. – Peter Diehr May 22 '16 at 1:05
• I had not seen that. Thanks. So for a gravitational wave from the merged black holes, which is basically a chirp (till they merge and you get the pseudo normal modes), it almost looks like the entropy would be zero or pretty low, as the instantaneous freq is well defined (Maybe there is some distribution around the instantaneous freq, if so it'd have some entropy even if small). Is that more or less correct, in principle? While for gravitational cosmological radiation it probably has more as multiple and more random processes created them? – Bob Bee May 22 '16 at 1:37

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:

1. 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;

2. 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;

3. 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.

• Rod, Lawrence and Peter: thanks much for your explanations. I saw yesterday in a paper in arXiv a similar statement that gravitational radiation will mostly have less entropy than EM radiation because it interacts very weakly with matter-energy. Makes sense. Curious that two high entropy black holes could create low entropy gravitational waves. It certainly does not seem it was blackbody or anywhere near. I do know that black hole thermodynamics held up ok. – Bob Bee May 23 '16 at 0:18
• @BobBee Note that my answer obviously does not take account of quantum effects; I'm talking about Einstein field equation solutions for no-hair black holes here, which is why it seems a little odd: such classical, Einsteinian objects have no entropy, unlike the adjusted model conceived by Hawking since the 1970s. It might therefore be worth a follow up question as to whether the answer is thought to change in the light of any modern thoughts on quantum gravity, in particular in the light of Susskind's conception of the holographic horizon. – WetSavannaAnimal May 23 '16 at 14:34
• I agree and will post it if nobody does before – Bob Bee May 23 '16 at 18:17

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.

• So if you aim a large number of highly powered gravitational waves so that they all cross in one spot, (think of waves 20 times more powerful than came from GW150914 and run it in reverse) you make a black hole, with entropy defined by the energy left in the black hole region by the departing waves. Thus we might be able to connect this quantum entropy to the gravitational waves entropy. – Tom Andersen May 28 '16 at 18:07
• Imploding gravity waves can form a geon where the curvature of spacetime assumes a self-attracting form. If strong enough this can indeed lead to a black hole. I doubt in reality this set of circumstances actually materializes in the world, with the possible exception of the very earliest moments in the universe. – Lawrence B. Crowell May 29 '16 at 21:09