Let $(M,g)$ be a riemannian manifold with riemannian curvature $R$, and $R_2=R(e_i,e_j)R(e_i,e_j)$. I define also $R'_2=R_2/(\int_M tr(R_2)d\mu)$. The entropy $S$ of $M$ is then:
$$S= - \int_M tr(R'_2 ln(R'_2)) d \mu$$
As $R'_2$ is positiv, the integral is well defined.
Is the entropy an increasing function when it is put over a space-like hypersurface of the Einstein space-time manifold?