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 $R^{\lambda}_{\; \kappa \mu \nu} = 0$).
 A: This paper by Clifton et al. proposes a definition of spacetime entropy based on the square root of Bel–Robinson tensor:

*

*Clifton, T., Ellis, G. F., & Tavakol, R. (2013). A gravitational entropy proposal. Classical and Quantum Gravity, 30(12), 125009, doi:10.1088/0264-9381/30/12/125009, arXiv:1303.5612.

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.
