# Self-similar solutions and critical behavior in Einstein-Maxwell-dilaton theory sourced by charged null fluids

## TL;DR

This paper explores self-similar solutions in Einstein-Maxwell-dilaton theory with charged null fluids, analyzing critical collapse behavior, deriving a general critical exponent, and discovering a new dyonic black hole solution.

## Contribution

It provides new self-similar solutions supported by charged null fluids, computes the critical exponent for black hole formation, and introduces a novel dyonic black hole spacetime.

## Key findings

- Critical exponent for black hole mass scaling is =a^2/(1+a^2).
- Self-similar solutions reduce to Roberts' solution in vacuum case.
- Discovered a time-dependent vacuum dyonic black hole solution.

## Abstract

We investigate continuously self-similar solutions of four-dimensional Einstein-Maxwell-dilaton theory supported by charged null fluids. We work under the assumption of spherical symmetry and the dilaton coupling parameter $a$ is allowed to be arbitrary. First, it is proved that the only such vacuum solutions with a time-independent asymptotic value of the dilaton necessarily have vanishing electric field, and thus reduce to Roberts' solution of the Einstein-dilaton system. Allowing for additional sources, we then obtain Vaidya-like families of self-similar solutions supported by charged null fluids. By continuously matching these solutions to flat spacetime along a null hypersurface one can study gravitational collapse analytically. Capitalizing on this idea, we compute the critical exponent defining the power-law behavior of the mass contained within the apparent horizon near the threshold of black hole formation. For the heterotic dilaton coupling $a=1$ the critical exponent takes the value $1/2$ typically observed in similar analytic studies, but more generally it is given by $\gamma=a^2(1+a^2)^{-1}$. The analysis is complemented by an assessment of the classical energy conditions. Finally, and on a different note, we report on a novel dyonic black hole spacetime, which is a time-dependent vacuum solution of this theory. In this case, the presence of constant electric and magnetic charges naturally breaks self-similarity.