Invariant description of static and dynamical Brans–Dicke spherically symmetric models

We investigate spherically symmetric static and dynamical Brans–Dicke theory exact solutions using invariants and, in particular, the Newman Penrose formalism utilizing Cartan scalars. In the family of static, spherically symmetric Brans–Dicke solutions, there exists a three-parameter family of solutions, which have a corresponding limit to general relativity. This limit is examined through the use of Cartan invariants via the Cartan–Karlhede algorithm and is additionally supported by analysis of scalar polynomial invariants. It is determined that the appearance of horizons in these spacetimes depends primarily on one of the parameters, n, of the family of solutions. In particular, expansion-free surfaces appear which, for a subset of parameter values, define additional surfaces distinct from the standard surfaces (e.g., apparent horizons) identified in previous work. The “\(r=2M\)” surface in static spherically symmetric Brans–Dicke solutions was previously shown to correspond to the Schwarzschild horizon in general relativity when an appropriate limit exists between the two theories. We show additionally that other geometrically defined horizons exist for these cases, and identify all solutions for which the corresponding general relativity limit is not a Schwarzschild one, yet still contains horizons. The identification of some of these other surfaces was noted in previous work and is characterized invariantly in this work. In the case of the family of dynamical Brans–Dicke solutions, we identify similar invariantly defined surfaces as in the static case and present an invariant characterization of their geometries. Through the analysis of the Cartan invariants, we determine which members of these families of solutions are locally equivalent, through the use of the Cartan–Karlhede algorithm. In addition, we identify black hole surfaces, naked singularities, and wormholes with the Cartan invariants. The aim of this work is to demonstrate the usefulness of Cartan invariants for describing properties of exact solutions like the local equivalence between apparently different solutions, and identifying special surfaces such as black hole horizons.