Beyond Schwarzschild-de Sitter spacetimes: I. A new exhaustive class of metrics inspired by Buchdahl for pure $R^2$ gravity in a compact form

TL;DR

This paper solves a long-standing problem by deriving a new class of exact static spherically symmetric solutions in pure $R^2$ gravity, expanding the understanding of vacuum solutions beyond Schwarzschild-de Sitter spacetimes.

Contribution

The authors reformulate Buchdahl's intractable ODE, obtaining a comprehensive class of metrics that satisfy pure $R^2$ vacuum equations and evade previous no-go theorems.

Findings

Derived a new class of exact solutions in pure $R^2$ gravity.

Verified solutions satisfy the $R^2$ vacuum field equations.

Analyzed properties of solutions near the origin and for small parameters.

Abstract

Some sixty years ago Buchdahl pioneered a program in search of static spherically symmetric vacua for pure $R^{2}$ gravity (Nuovo Cimento, Vol 23, No 1, pp 141-157 (1962); [https://link.springer.com/article/10.1007/BF02733549]). Surpassing several obstacles, his work culminated in a non-linear second-order ordinary differential equation (ODE) which required being solved. However Buchdahl deemed the ODE intractable and prematurely abandoned his pursuit for an analytical solution. We have finally managed to break this six-decades-old impasse and accomplish his goal. Reformulating the Buchdahl ODE, we obtain a novel exhaustive class of metrics (which we shall call the Buchdahl-inspired metrics hereafter) in a compact form. We are able to verify ex post, via direct inspection, that the metric obtained herein satisfies the $R^{2}$ vacuo field equation, hence establishing its validity. We show why the Buchdahl-inspired metrics, which exhibit non-constant scalar curvature, defeat a Lichnerowicz-type no-go theorem, previously proved for quadratic gravity, by evading the overly strong restriction on the asymptotic falloff of the metric at spatial infinity. Our novel solution thereby completes Buchdahl's six-decades-old program. We also explore the mathematical properties of the Buchdahl-inspired metrics in the limit of small k and in the region around the coordinate origin.