General relativity limit of Brans-Dicke gravity

In Brans-Dicke gravity a scalar field non-minimally coupled with the curvature acts as a new gravitational degree of freedom in addition to the usual metric field of General Relativity. Usually, when the coupling ω of the scalar field diverges, the latter approaches a constant and the solutions of the theory reduce to those generated by the Einstein equations with the same energy-momentum tensor as source. However, in the limit ω → ∞, the static and spherically symmetric vacuum solutions, called Brans solutions, do not reduce to the Schwarzschild metric, the only solution of General Relativity with the same type of symmetry. In this work we test their limit with a new method. Unlike the usual way, we use the Brans solutions to take the limit at the level of the equations. Since these are non-linear field equations depending on ω, in principle our procedure could lead to a different outcome. In particular, we take advantage of the Einstein frame formulation of the Brans-Dicke theory and we check if the solutions are able to satisfy the equations of the so called Minimal Geometric Deformation approach. Namely, we study if they can be seen as a geometric deformation - due to the Brans-Dicke scalar - of the Schwarzschild solution. Indeed, the peculiar feature of the geometric deformation is that it is built in such a way to guarantee a clear limit to the vacuum equations of General Relativity. However, we find that the Brans solutions do not meet all the requirements imposed by the structure of the Minimal Geometric Deformation approach. This result supports the conclusion that there is no simple limiting procedure in which the complete Brans-Dicke theory just reduces to General Relativity.