Beyond Schwarzschild-de Sitter spacetimes: II. An exact non-Schwarzschild metric in pure $R^2$ gravity and new anomalous properties of $R^2$ spacetimes

Abstract

In our recent publication (Phys. Rev. D 106, 104004 (2022)), we advanced a program that Buchdahl originated but prematurely abandoned circa 1962 (Nuovo Cimento 23, 141 (1962)). Therein we obtained an exhaustive class of metrics that constitute the branch of non-trivial solutions to the pure $R^2$ field equation in vacuo. The Buchdahl-inspired metrics in general possess non-constant scalar curvature, thereby defeating the generalized Lichnerowicz theorem previously advocated for quadratic gravity. We found that the said theorem makes an overly strong assumption on the asymptotic falloff in the spatial derivatives of the Ricci scalar, rendering it violable against the Buchdahl-inspired metrics. In this paper, we shall further extend our work mentioned above by showing that, within the class of Buchdahl-inspired metrics, the asymptotically flat member takes on an exact closed analytical expression. The new metric is characterized by a horizon radius $r_{\text{s}}$ and the Buchdahl parameter $k$, the latter of which arises via the higher-derivative nature of $R^2$ gravity. For $k=0$, the new metric recovers the classic Schwarzschild metric. Equipped with the exact expression of the new metric, we shall analytically construct the Kruskal-Szekeres diagram for pure $R^2$ spacetime. We find that the Buchdahl parameter $k$ fundamentally modifies the properties of $R^2$ spacetime structures in a variety of ways.

23 pages, 13 figures

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• Observational tests of asymptotically flat ${\mathcal{R}}^2$ spacetimes
• Observational test of ${\mathcal{R}}^2$ spacetimes with the S2 star in the Milky Way galactic center
• 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
• Emerging Newtonian potential in pure $R^2$ gravity on a de Sitter background
• 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