Robinson-Trautman solutions with scalar hair and Ricci flow

TL;DR

This paper classifies Robinson-Trautman solutions with scalar hair in Einstein gravity, revealing new solution classes linked to Ricci flow equations, and explores their implications in supergravity and wormhole physics.

Contribution

It introduces three new classes of solutions with scalar hair, connecting Ricci flow to gravitational solutions, and provides explicit solutions and their physical interpretations.

Findings

Class I-a extends Robinson-Trautman with scalar hair satisfying Ricci flow.

Class I-b features solutions governed by Perelman's Ricci flow equations.

A new spherical symmetric solution derived from AdS-Roberts solution.

Abstract

The vacuum Robinson-Trautman solution admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. We perform a comprehensive classification of solutions exhibiting this property in Einstein's gravity with a massless scalar field, assuming that the solution belongs at least to Petrov-type II and some of the components of Ricci tensor identically vanish. We find that these solutions can be grouped into three distinct classes: (I-a) a natural extension of the Robinson-Trautman family incorporating a scalar hair satisfying the time derivative of the Ricci flow equation, (I-b) a novel non-asymptotically flat solution characterized by two functions satisfying Perelman's pair of the Ricci flow equations, and (II) a dynamical solution possessing ${\rm SO}(3)$, ${\rm ISO}(2)$ or ${\rm SO}(1,2)$ symmetry. We provide a complete list of all explicit solutions falling into Petrov type D for classes (I-a) and (I-b). Moreover, leveraging the massless solution in class (I-a), we derive the neutral Robinson-Trautman solution to the ${\cal N}=2$ gauged supergravity with the prepotential $F(X) =-iX^0X^1$. By flipping the sign of the kinetic term of the scalar field, the Petrov-D class (I-a) solution leads to a time-dependent wormhole with an instantaneous spacetime singularity. Although the general solution is unavailable for class (II), we find a new dynamical solution with spherical symmetry from the AdS-Roberts solution via AdS/Ricci-flat correspondence.