# A Gravitational Entropy proposal from Ellis, Tavakol, and Clifton

1. Mar 25, 2013

### marcus

It's interesting that the entropy of the gravitational field (i.e. the entropy of geometry) has never been satisfactorily defined. Since geometry is such an important part of the picture near the start of expansion, this means that total entropy around then has been undefinable, leaving thermodynamic issues up in the air and subject to speculation.

Another interesting fact is that as long as gravity is attractive we expect the geometric entropy to increase with structure formation. That is, entropy increases with clumping, clustering, star-formation etc.
But when gravity is zero or repellent (as happens in LQG at extreme density, causing the cosmological bounce) we expect entropy to increase with structure dissolution.

A missing piece to the puzzle---namely a more precise definition for the gravitational entropy---has recently been offered by George Ellis and friends. Happily enough it behaves as we would like in the sense that it increases with the formation of structure:

http://arxiv.org/abs/1303.5612
A Gravitational Entropy Proposal
Timothy Clifton, George F R Ellis, Reza Tavakol
(Submitted on 22 Mar 2013)
We propose a thermodynamically motivated measure of gravitational entropy based on the Bel-Robinson tensor, which has a natural interpretation as the effective super-energy-momentum tensor of free gravitational fields. The specific form of this measure differs depending on whether the gravitational field is Coulomb-like or wave-like, and reduces to the Bekenstein-Hawking value when integrated over the interior of a Schwarzschild black hole. For scalar perturbations of a Robertson-Walker geometry we find that the entropy goes like the Hubble weighted anisotropy of the gravitational field, and therefore increases as structure formation occurs. This is in keeping with our expectations for the behaviour of gravitational entropy in cosmology, and provides a thermodynamically motivated arrow of time for cosmological solutions of Einstein's field equations. It is also in keeping with Penrose's Weyl curvature hypothesis.
17 pages

2. Mar 26, 2013

### marcus

I think it's a fascinating topic and would like to get some other people's comments. Here is the introduction, which clarifies the issues:

==quote Clifton Ellis Tavakol==
A key question in cosmology is how to define the entropy in gravitational fields. A suitable definition already exists for the important case of stationary black holes [1], but in the cosmological setting a well-motivated and universally agreeable analogue has yet to be found. Addressing this deficit is an important problem, as in the presence of gravitational interactions the usual statements about matter becoming more and more uniform are incorrect. Instead, structure develops spontaneously when gravitational attraction dominates the dynamics [2, 3]. This behaviour is crucial to the existence of complex structures, and indeed life, in the Universe. The question then arises, how can evolution under the gravitational interaction be made compatible with the second law of thermodynamics? If the second law is valid in the presence of gravity, such that entropy increases monotonically into the future, then the current state of the universe must be considered more probable than the initial state, even though it is more structured. For this to be true, the gravitational field itself must be carrying entropy.
For a candidate definition of gravitational entropy to be compatible with cosmological processes, such as structure formation in the Universe, it needs to be valid in non-stationary and non-vacuum spacetimes. We will argue that an appropriate definition of gravitational entropy should only involve the free gravitational field, as specified by the Weyl part of the curvature tensor, Cabcd [4], and that a particular promising candidate...
==endquote==