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 C μ ν ρ σ 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 | ψ μ | 0 〉 ≠ 0 for a vector field ψ μ 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.
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