Global structure and physical interpretation of the Fonarev solution for a scalar field with exponential potential

Abstract

We discuss the physical interpretation of a dynamical and inhomogeneous spherically symmetric solution obtained by Fonarev for a scalar field with an exponential potential. There is a single parameter $w$ in the solution which can be set to $\pm1$ if it is non-zero, in addition to the steepness parameter $\lambda$ in the potential. The spacetime is conformally static and asymptotically flat Friedmann-Robertson-Walker spacetime. The solution reduces to the Friedmann-Robertson-Walker solution for $w=0$. There are two curvature singularities, of which one is a timelike central singularity and the other is a big-bang or big-crunch type singularity. Depending on the parameters, the spacetime can possess a future outer trapping horizon in the collapsing case. Then the solution represents a dynamical black hole in the sense of Hayward although there is a locally naked singularity at the center and no black-hole event horizon. This demonstrates a weak point of the local definition of a black hole in terms of a trapping horizon.