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

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.

This recently accepted paper is the first installment of a three-paper "Beyond SdS spacetimes" series, being under review at Phys. Rev. D. Link to PRD: https://journals.aps.org/prd/accepted/e907dQ06D1310133219288223264bd8b133a125f1

References in corpus (4)

• Black Holes in Higher-Derivative Gravity
• Static Solutions for 4th order gravity
• New Exact Solutions of Quadratic Curvature Gravity
• Black Holes in Higher Dimensional Gravity Theory with Quadratic in Curvature Corrections

Cited by in corpus (16)

• Traversable Morris-Thorne-Buchdahl wormholes in quadratic gravity
• Beyond Schwarzschild-de Sitter spacetimes: II. An exact non-Schwarzschild metric in pure $R^2$ gravity and new anomalous properties of $R^2$ spacetimes
• Gravitational Lensing and Image Distortion by Buchdahl Inspired Metric in ${\mathcal{R}}^2$ Gravity
• Closed Timelike Curves Induced by a Buchdahl-inspired Vacuum Spacetime in $R^2$ Gravity
• Observational test of ${\mathcal{R}}^2$ spacetimes with the S2 star in the Milky Way galactic center
• Observational tests of asymptotically flat ${\mathcal{R}}^2$ spacetimes
• Beyond Schwarzschild-de Sitter spacetimes: III. A perturbative vacuum with non-constant scalar curvature in $R+R^2$ gravity
• A stationary axisymmetric vacuum solution for pure $R^2$ gravity
• Buchdahl-inspired spacetimes and wormholes: Unearthing Hans Buchdahl's other 'hidden' treasure trove
• Revisiting Weak Energy Condition and wormholes in Brans-Dicke gravity
• Emerging Newtonian potential in pure $R^2$ gravity on a de Sitter background
• Epicyclic oscillations and accretion disk around a special Buchdahl-inspired spacetime
• Time-reversed information flow through a wormhole in scalar-tensor gravity
• Asymptotically-Flat Black Hole Solutions in Symmergent Gravity
• Nonstatic Reissner-Nordström metric in the perturbative $f(R)$ theory: Embedding in the background of the FLRW cosmology, uniqueness of solutions, the TOV equation
• Non-triviality of asymptotically flat Buchdahl-inspired metrics in pure $R^2$ gravity