# Constructing Lifshitz spaces using the Ricci flow

## TL;DR

This paper demonstrates how Ricci flow equations can be used to construct Lifshitz spaces with specific critical exponents, showing they are exact solutions and evolve towards flat spacetime.

## Contribution

It introduces a method to generate Lifshitz spaces with discrete or continuous critical exponents using Ricci flow, establishing these as exact solutions.

## Key findings

- Lifshitz spaces with critical exponent equal to spatial dimension are constructed.
- Lifshitz spaces with arbitrary continuous critical exponent are constructed.
- Ricci flow evolves Lifshitz spaces towards flat spacetime as a fixed point.

## Abstract

In this work we make use of the Ricci flow equations to show that, by starting from a general ansatz for the metric, we can construct two kinds of Lifshitz spaces in which: (a) the critical exponent coincides with the spatial dimension of the spacetime and therefore adopts discrete values, and (b) the critical exponent is continuous and arbitrary. These results show that Lifshitz spaces are exact solutions to the Ricci flow equations. Moreover, we found that the Ricci flow evolves towards a single fixed point for both cases which coincides with the flat spacetime.