Extension of Buchdahl's Theorem on Reciprocal Solutions

TL;DR

This paper extends Buchdahl's method to include the cosmological constant and scalar-tensor theories, deriving new solutions such as Schwarzschild-de Sitter and Nariai in these extended frameworks.

Contribution

It introduces a generalized approach to Buchdahl's solutions for scalar-tensor theories with a cosmological constant, including quadratic potentials.

Findings

Derived solutions for scalar-tensor gravity with a cosmological constant.

Obtained specific metrics like Schwarzschild-de Sitter and Nariai.

Extended Buchdahl's method to new gravitational contexts.

Abstract

Since the development of Brans-Dicke gravity, it has become well-known that a conformal transformation of the metric can reformulate this theory, transferring the coupling of the scalar field from the Ricci scalar to the matter sector. Specifically, in this new frame, known as the Einstein frame, Brans-Dicke gravity is reformulated as General Relativity supplemented by an additional scalar field. In 1959, Hans Adolf Buchdahl utilized an elegant technique to derive a set of solutions for the vacuum field equations within this gravitational framework. In this paper, we extend Buchdahl's method to incorporate the cosmological constant and to the scalar-tensor cases beyond the Brans-Dicke archetypal theory, thereby, with a conformal transformation of the metric, obtaining solutions for a version of Brans-Dicke theory that includes a quadratic potential. More specifically, we obtain synchronous solutions in the following contexts: in scalar-tensor gravity with massless scalar fields, Brans-Dicke theory with a quadratic potential, where we obtain specific synchronous metrics to the Schwarzschild-de Sitter metric, the Nariai solution, and a hyperbolically foliated solution.