Robinson-Trautman solutions with scalar hair and Ricci flow

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.

Figures (2)

• Figure 1: Trapping horizons for $\varepsilon=1$, $\omega=g=1$ (left) and for $\varepsilon=-1$, $\omega=g=1$ (right). We have set $k=1$ and $r_0=0$. The dotted lines for the left figure represent singularities $u=u_{\rm s}^{(\pm )}$. The curve $u=u_{\rm TH}^{(+)}$ is immaterial in the present discussion, since it lies outside the domain of Lorentz signature. For $\omega<1/2$, the allowed region $R^2>0$ is entirely untrapped. The black dot at $u=r=0$ for $\varepsilon=-1$ denotes the instantaneous singularity.
• Figure 2: Global causal structures of the Petrov D Robinson-Trautman solution (\ref{['petrovDsym']}) for (I) $\varepsilon=+1$ and (II) $\varepsilon=-1$ with $k=1$ and $\omega>0$. The case for a normal scalar field splits further into (I-i) $\omega>1/2$, (I-ii) $\omega =1/2$ and (I-iii) $\omega<1/2$. Zigzag lines in (I) and the black dot in (II) denote curvature singularities.