Neglected solutions in quadratic gravity

Abstract

We report on several previously overlooked families of static spherically symmetric solutions in quadratic gravity. Our main result concerns the existence of solutions whose leading exponents depend on the ratio ${ω=α/(3β)}$ of the four-derivative couplings. We demonstrate that the space of models with ${ω>1}$ contains a dense set that admits non-Frobenius solutions ${(s_*, 2 - 3 s_*)_0}$ (in standard Schwarzschild coordinates), with certain rational numbers $s_*(ω)$. These solutions correspond to a singular core at ${\bar{r}=0}$. Another related non-Frobenius family, $(s_*, 2 - 3 s_*)_\infty$, exists for a dense set of models with ${1/4 < ω< 1}$, describing a singular boundary at ${\bar{r}\to\infty}$. Both families are uncovered by recasting the metric into special coordinates in which the solutions become Frobenius. Additionally, for models with real ratios ${ω\neq 1}$ we identify six novel families of non-Frobenius solutions around points ${\bar{r}=\bar{r}_0} \neq 0$, describing horizons and wormhole throats. Finally, we re-derive and summarize all known families of solutions in modified as well as in the standard Schwarzschild coordinates.