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) → O(3) × R, where O(3) is the rotational symmetry of the Friedmann–Lemaître–Robertson–Walker spacetime. The Weyl curvature tensor Cμνρσ 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 〈0|ψμ|0〉 6= 0 for a vector field ψμ acts as an order parameter and at the critical temperature Tc a phase transition occurs breaking the Lorentz symmetry spontaneously. During the ordered O(3) symmetry phase the entropy is vanishingly small and for T < Tc 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.