Gravitational Entropy and the Second Law of Thermodynamics

The spontaneous violation of Lorentz and diffeomorphism invariance in a phase near the big bang lowers the entropy, allowing for an arrow of time and the second law of thermodynamics. The spontaneous symmetry breaking leads to $O(3,1)\rightarrow O(3)\times R$, where $O(3)$ is the rotational symmetry of the Friedmann-Lema\^{i}tre-Robertson-Walker spacetime. The Weyl curvature tensor $C_{\mu\nu\rho\sigma}$ vanishes in the FLRW spacetime satisfying the Penrose zero Weyl curvature conjecture. The requirement of a measure of gravitational entropy is discussed. The vacuum expectation value $\langle 0\vert\psi_\mu\vert 0\rangle\neq 0$ for a vector field $\psi_\mu$ acts as an order parameter and at the critical temperature $T_c$ a phase transition occurs breaking the Lorentz symmetry spontaneously. During the ordered $O(3)$ symmetry phase the entropy is vanishingly small and for $T < T_c$ as the universe expands the anti-restored $O(3,1)$ Lorentz symmetry leads to a disordered phase and a large increase in entropy creating the arrow of time.