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

TL;DR

This paper uses Cartan invariants and the Newman-Penrose formalism to analyze spherically symmetric Brans-Dicke solutions, identifying horizons, singularities, and geometrical features invariantly, and clarifying their relation to general relativity limits.

Contribution

It introduces an invariant geometric characterization of static and dynamical Brans-Dicke solutions, including horizons and singularities, using Cartan invariants and the Cartan-Karlhede algorithm.

Findings

Identified invariantly defined horizons in static and dynamical solutions.

Determined conditions for horizons and singularities using Cartan invariants.

Classified solutions based on local equivalence and geometric properties.

Abstract

We investigate spherically symmetric static and dynamical Brans-Dicke theory exact solutions using invariants and, in particular, the Newman Penrose formalism utilizing Cartan scalars. The GR limit of these solutions 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. These surfaces 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.