Black holes in $f(\mathbb Q)$ Gravity

Abstract

We systematically study the field equations of $f(\mathbb Q)$ gravity for spherically symmetric and stationary metric-affine spacetimes. Such spacetimes are described by a metric as well as a flat and torsionless affine connection. In the Symmetric Teleparallel Equivalent of GR (STEGR), the connection is pure gauge and hence unphysical. However, in the non-linear extension $f(\Q)$, it is promoted to a dynamical field which changes the physics. Starting from a general metric-affine geometry, we construct the most general static and spherically symmetric forms of the metric and the affine connection. We then use these symmetry reduced geometric objects to prove that the field equations of $f(\Q)$ gravity admit GR solutions as well as beyond-GR solutions, contrary to what has been claimed in the literature. We formulate precise criteria, under which conditions it is possible to obtain GR solutions and under which conditions it is possible to obtain beyond-GR solutions. We subsequently construct several perturbative corrections to the Schwarzschild solution for different choices of $f(\Q)$, which in particular include a hair stemming from the now dynamical affine connection. We also present an exact beyond-GR vacuum solution. Lastly, we apply this method of constructing spherically symmetric and stationary solutions to $f(\T)$ gravity, which reproduces similar solutions but without a dynamical connection.