Let (M,g) be a riemannian manifold with riemannian curvature R, and R2=R(ei,ej)R(ei,ej). I define also R′2=R2/(∫Mtr(R2)dμ). The entropy S of M is then:
S=−∫Mtr(R′2ln(R′2))dμ
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?