A coarse-grained entropy for self-gravitating matter
In plain words
For an ordinary gas, spreading out raises entropy, but under gravity clumping is the likely direction, so the usual formula fails. A gravitational entropy that grows as stars and galaxies form would make the second law consistent from the Big Bang onward.
Precise statement
Construct a coarse-grained entropy functional $S_{\mathrm{grav}}$ for classical self-gravitating systems or perturbed Friedmann spacetimes (for example built from the Weyl tensor, the tidal part of spacetime curvature, following the Weyl curvature hypothesis) that is nondecreasing under gravitational clustering and reduces to the Bekenstein-Hawking entropy $A/(4\ell_{P}^{2})$ for black holes. An answer is a definition with a monotonicity proof for a stated class of dynamics, or a no-go result.
What would settle it
A definition shown to increase monotonically in N-body or perturbed cosmological dynamics and to match black-hole entropy, or a proof that no local functional can.