# Is there a definition for a *geometric entropy*?

In statistical mechanics, entropy of a system is usually defined as a measure of the system's micro-state randomness, or as an averaged "surprise" of its micro-state: $$\begin{equation}\tag{1} S_{\text{stat}} = - k \sum_n p_n \, \ln{p_n}, \end{equation}$$ where $$p_n$$ is the probability of state $$n$$, such that $$\sum_n p_n = 1$$, and $$k$$ is an arbitrary positive constant.

In classical general relativity, there is no definition of spacetime entropy, even if there is a black hole in that spacetime.

Now, imagine some curved geometry (a curvy sheet of rubber, for example). It has some local curvature on every point on it: $$R^{\lambda}_{\; \kappa \mu \nu}(x)$$.

As a functional of its curvature tensor, is it possible to define a "geometric entropy" $$S_{\text{geo}}$$ for that sheet, as a kind of global scalar measure of the randomness of its curvature? In comparison, a flat space (one for which the curvature tensor is $$R^{\lambda}_{\; \kappa \mu \nu} = 0$$ at every point) would have $$S_{\text{geo}} = 0$$.

The first idea I have as an answer is the Hilbert geometric action ($$k$$ is an arbitrary constant): $$\begin{equation}\tag{2} S = k \int_{\mathcal{M}} R \, \sqrt{-g} \, d^n x. \end{equation}$$ But what else? According to (2), spacetimes with zero scalar curvature: $$R = 0$$, would have 0 action (interpreted as a "geometric entropy") even if the curvature isn't 0 ($$R = 0$$ doesn't imply $$R^{\lambda}_{\; \kappa \mu \nu} = 0$$).

• Where is the probabilistic character entering in classical general relativity? What are the different 'microstate spacetimes' that one 'macrostate spacetime' has the options to choose from? – Avantgarde May 30 at 19:34
• In the view I'm considering, there is no probability on the geometry. I'm just asking about a way to define a kind of "geometric randomness", or "complexity" if you prefer. More curvy and complex is the geometry, higher should be the "geometric entropy". – Cham May 30 at 19:46
• You might be interested in reading about the complexity equals action conjecture. – Avantgarde May 30 at 22:02

This paper by Clifton et al. proposes a definition of spacetime entropy based on the square root of Bel–Robinson tensor:

From the paper:

… we therefore make the following list of requirements on the gravitational entropy, $$S_\text{grav}$$, that we expect to guide us:

• E1: It should be non-negative.
• E2: It should vanish if and only if $$C_{abcd}=0$$.
• E3: It should measure the local anisotropy of the free gravitational field.
• E4: It should reproduce the Bekenstein–Hawking entropy of a black hole.
• E5: It should increase monotonically, as structure forms in the universe.

That paper offers several example calculations of spacetime entropy and its evolution, most notably it reproduces Bekenstein–Hawking entropy for a Schwarzschild black hole.

A more involved example for the spacetime entropy evolution during an Oppenheimer–Snyder–Datt spherically symmetric collapse into a black hole could be found in this paper.

• Wow ! Thanks, I'll check these papers. I knew (felt from the guts) that something could be defined as a measure of the geometry complexity, from curvature. – Cham May 31 at 14:04