Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow

Abstract

By starting from a Perelman entropy functional and considering the Ricci-DeTurck flow equations we analyze the behaviour of Einstein-Hilbert and Einstein-Proca theories with Lifshitz geometry as functions of a flow parameter. In the former case, we found one consistent fixed point that represents flat space-time as the flow parameter tends to infinity. Massive vector fields in the latter theory enrich the system under study and have the same fixed point achieved at the same rate as in the former case. The geometric flow is parametrized by the metric coefficients and represents a change in anisotropy of the geometry towards an isotropic flat space-time as the flow parameter evolves. Indeed, the flow of the Proca fields depends on certain coefficients that vanish when the flow parameter increases, rendering these fields constant. We have been able to write down the evolving Lifshitz metric solution with positive, but otherwise arbitrary, critical exponents relevant to geometries with spatially anisotropic holographic duals. We show that both the scalar curvature and matter contributions to the Ricci-DeTurck flow vanish under the flow at a fixed point consistent with flat space-time geometry. Thus, the behaviour of the scalar curvature always increases, homogenizing the geometry along the flow. Moreover, the theory under study keeps positive-definite but decreasing the entropy functional along the Ricci-DeTurck flow.