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. 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.

中文翻译：

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静态和动态布兰斯-迪克球对称模型的不变描述

摘要

我们使用不变量研究球对称静态和动态布兰斯-迪克理论精确解，特别是利用嘉当标量的纽曼彭罗斯形式主义。在静态球对称布兰斯-迪克解族中，存在一个三参数解族，它对广义相对论有相应的限制。该限制通过嘉当-卡尔赫德算法使用嘉当不变量进行检查，并且还通过标量多项式不变量分析得到支持。可以确定，这些时空中视界的出现主要取决于解族的参数之一n 。特别是，出现了无膨胀表面，对于参数值的子集，其定义了与先前工作中识别的标准表面（例如，表观地平线）不同的附加表面。当两个理论之间存在适当的极限时，静态球对称布兰斯-迪克解中的“ \(r=2M\) ”表面先前被证明对应于广义相对论中的史瓦西视界。我们还表明，在这些情况下存在其他几何定义的视界，并确定相应的广义相对论极限不是史瓦西极限但仍包含视界的所有解。这些其他表面中的一些的识别在以前的工作中已经注意到，并且在本工作中保持不变。在动态布兰斯-迪克解决方案族的情况下，我们识别了与静态情况类似的不变定义表面，并提出了其几何形状的不变特征。通过对嘉当不变量的分析，我们使用嘉当-卡尔赫德算法确定这些解决方案族中的哪些成员是局部等价的。此外，我们还利用嘉当不变量识别黑洞表面、裸奇点和虫洞。这项工作的目的是证明嘉当不变量在描述精确解的性质（例如明显不同的解之间的局部等价）以及识别特殊表面（例如黑洞视界）方面的有用性。