Search

Riemann-Cartan geometries are geometries that admit non-zero curvature and torsion tensors. These geometries have been investigated as geometric frameworks for potential theories in physics including quantum gravity theories and have many important differences when compared to Riemannian geometries. One notable difference, is the number of symmetries for a Riemann-Cartan geometry is potentially smaller than the number of Killing vector fields for the metric. In this paper, we will review the investigation of symmetries in Riemann-Cartan geometries and the mathematical tools used to determine geometries that admit a given group of symmetries. As an illustration, we present new results by determining all static spherically symmetric and all stationary spherically symmetric Riemann-Cartan geometries. Furthermore, we have determined the subclasses of spherically symmetric Riemann-Cartan geometries that admit a seven-dimensional group of symmetries.

Theories of gravity based on teleparallel geometries are characterized by the torsion, which is a function of the coframe, derivatives of the coframe, and a zero curvature and metric compatible spin connection. The appropriate notion of a symmetry in a teleparallel geometry is that of an affine symmetry. Due to the importance of the de Sitter geometry and Einstein spaces within general relativity, we shall describe teleparallel de Sitter geometries and discuss their possible generalizations. In particular, we shall analyse a class of Einstein teleparallel geometries which have a 4-dimensional Lie algebra of affine symmetries, and display two one-parameter families of explicit exact solutions., Comment: 24 pages, no figure, Will appear soon in EPJC

Two recently published papers (J.D. Barrow and A. Paliathanasis, Eur. Phys. J. C. (2018, 2019)) claim to have found exact solutions of Einstein's field equations belonging to the class of non-trivial Szekeres models, whose source is a mixture of dust and a homogeneous time-dependent scalar field, where the energy-momentum tensors of both mixture components are independently conserved. We prove that the independent conservation of these two mixture components necessarily leads to solutions belonging to the set of spatially homogeneous subcases of the Szekeres family: Friedmann-Lema\^itre-Robertson-Walker for class I, and Kantowski-Sachs, Bianchi-Behr I or Bianchi-Behr $\mbox{VI}_{\tiny{\mbox{-1}}}$ for class II., Comment: 2 pages. Comment on "Szekeres universes with homogeneous scalar fields" and "Cyclic Szekeres universes" by Barrow and Paliathanasis, Eur. Phys. J. C. (2018, 2019); arXiv:1808.00173 and arXiv:1901.09173

We discuss black hole spacetimes with a geometrically defined quasi-local horizon on which the curvature tensor is algebraically special relative to the alignment classification. Based on many examples and analytical results, we conjecture that a spacetime horizon is always more algebraically special (in all of the orders of specialization) than other regions of spacetime. Using recent results in invariant theory, such geometric black hole horizons can be identified by the alignment type $\textbf{II}$ or $\textbf{D}$ discriminant conditions in terms of scalar curvature invariants, which are not dependent on spacetime foliations. The above conjecture is, in fact, a suite of conjectures (isolated vs dynamical horizon; four vs higher dimensions; zeroth order invariants vs higher order differential invariants). However, we are particularly interested in applications in four dimensions and especially the location of a black hole in numerical computations., Comment: 10 pages

We introduce three approaches to generate curvature invariants that transform covariantly under a conformal transformation of a four dimensional spacetime. For any black hole conformally related to a stationary black hole, we show how a set of conformally covariant invariants can be combined to produce a conformally covariant invariant that detects the event horizon of the conformally related black hole. As an application we consider the rotating dynamical black holes conformally related to the Kerr-NUT-(Anti)-de Sitter spacetimes and construct an invariant that detects the conformal Killing horizon along with a second invariant that detects the conformal stationary limit surface. In addition, we present necessary conditions for a dynamical black hole to be conformally related to a stationary black hole and apply these conditions to the ingoing Kerr-Vaidya and Vaidya black hole solutions to determine if they are conformally related to stationary black holes for particular choices of the mass function. While two of the three approaches cannot be generalized to higher dimensions, we discuss the existence of a conformally covariant invariant that will detect the event horizon for any higher dimensional black hole conformally related to a stationary black hole which admits at least two conformally covariant invariants, including all vacuum spacetimes., Comment: 9 pages

We construct a scalar polynomial curvature invariant that transforms covariantly under a conformal transformation from any spherically symmetric metric. This invariant has the additional property that it vanishes on the event horizon of any black hole that is conformal to a static spherical metric., Comment: 5 pages, minor corrections to two equations and references