Type II hidden symmetries for the homogeneous heat equation in some general classes of Riemannian spaces

TL;DR

This paper investigates the reduction of the heat equation in specific Riemannian and spacetime geometries, identifying sources of Type II hidden symmetries through symmetry analysis and applying results to cosmologically relevant models.

Contribution

It systematically classifies the origins of Type II hidden symmetries in heat equations reduced by various geometric symmetries in Riemannian and Petrov spacetimes.

Findings

Gradient KV reductions lead to linear heat equations in nondecomposable spaces.

Gradient HV reductions produce Laplace equations with symmetries from proper CKVs.

No Type II hidden symmetries in Petrov spacetimes with nongradient HV reductions.

Abstract

We study the reduction of the heat equation in Riemannian spaces which admit a gradient Killing vector, a gradient homothetic vector and in Petrov Type D,N,II and Type III space-times. In each reduction we identify the source of the Type II hidden symmetries. More specifically we find that a) If we reduce the heat equation by the symmetries generated by the gradient KV the reduced equation is a linear heat equation in the nondecomposable space. b) If we reduce the heat equation via the symmetries generated by the gradient HV the reduced equation is a Laplace equation for an appropriate metric. In this case the Type II hidden symmetries are generated from the proper CKVs. c) In the Petrov spacetimes the reduction of the heat equation by the symmetry generated from the nongradient HV gives PDEs which inherit the Lie symmetries hence no Type II hidden symmetries appear. We apply the general results to cases in which the initial metric is specified. We consider the case that the irreducible part of the decomposed space is a space of constant nonvanishing curvature and the case of the spatially flat Friedmann-Robertson-Walker space time used in Cosmology. In each case we give explicitly the Type II hidden symmetries provided they exist.