arXiv subjects

R. J. van den Hoogen

Publications and source records attributed to R. J. van den Hoogen.

At least 19 recordsLinked to original sources

arXiv2026

Horizon Singularities in the Schwarzschild Geometry of the Teleparallel Equivalent of General Relativity

Certain torsion scalar invariants are known to diverge at the horizon of the Schwarzschild solution in the Teleparallel Equivalent of General Relativity (TEGR), obstructing its interpretation as a black hole spacetime. We show that Schwarzschild TEGR geometries split into two distinct subclasses determined by the Lorentz sector of the geometry -- the elements of the tetrad and spin connection not encoded in the metric but appearing in the torsion. In the regular subclass, the divergences are absent and the horizon belongs to the manifold, supporting a consistent black hole interpretation. In the singular subclass, the divergences are genuine and the horizon is excluded from the manifold. As part of this analysis, we clarify the role of inertial contributions in teleparallel gravity, showing that the proper frame does not, by itself, eliminate inertial effects and that the horizon singularities are independent of the inertial structure of the frame. Using three independent approaches -- the inertial frame condition, horizon-penetrating coordinates, and the horizon regularity criterion -- we determine the complete class of Lorentz sector functions compatible with a regular horizon in Schwarzschild TEGR geometries. For this class, all torsion scalar invariants remain finite at the horizon and analytic extensions across it are admitted.

gr-qc↗

arXiv2026

Using Gauge Covariant Lie Derivatives in Poincaré Gauge and Metric Teleparallel Theories of Gravity

A procedure to determine the initial ansatz for the co-frame and spin connection characterizing a Riemann-Cartan geometry respecting a given group of continuous symmetries is illustrated. Given a particular group of symmetries and assuming an orthonormal gauge we can determine the co-frame and corresponding spin connection having this symmetry group by employing an gauge covariant Lie derivative. This gauge covariant Lie derivative when applied to the metric and co-frame determines the values of an antisymmetric compensating matrix. The derivative of this matrix then yields the corresponding spin connection. The procedure is straightforward and can be employed for any Riemann-Cartan geometry having symmetries including those with a non-trivial isotropy subgroup. Here we illustrate the procedure with numerous examples, including, spherically symmetric, plane symmetric, locally rotationally symmetric Bianchi type III, Gödel, de Sitter and anti-de Sitter geometries. Further, we have also solved the zero curvature constraint to obtain the resulting spin connection for the corresponding metric teleparallel geometry having this same symmetry group. We complete this investigation by including the Lorentz transformation that yields the proper frame for some of these metric teleparallel geometries.

gr-qc↗

arXiv2025

On black holes in new general relativity

New General Relativity (NGR) is a class of teleparallel theories defined by three free parameters, effectively reduced to two after appropriate normalization, which are subject to experimental constraints. In this framework, matter couples minimally to the metric, ensuring that test particles follow geodesics and that null congruence expansions can be employed to detect local horizons. Assuming such horizons exist, we demonstrate that all physically viable NGR models--including the Teleparallel Equivalent of General Relativity (TEGR) and the one-parameter Hayashi and Shirafuji model (1P-H&S)--inevitably exhibit divergences in torsion scalars at the local horizon. This singular behavior obstructs the interpretation of these models and their associated teleparallel geometries as black hole configurations.

gr-qc↗

arXiv2024

Locally-homogeneous Riemann-Cartan geometries with the largest symmetry group

The symmetry frame formalism is an effective tool for computing the symmetries of a Riemann-Cartan geometry and, in particular, in metric teleparallel geometries. In the case of non-vanishing torsion in a four dimensional Riemann-Cartan geometry, the Minkowski geometry is the only geometry admitting ten affine frame symmetries. Excluding this geometry, the maximal number of affine frame symmetries is seven. A natural question is to ask what four dimensional geometries admit a seven-dimensional group of affine frame symmetries. Such geometries are locally homogeneous and admit the largest isotropy group permitted, and hence are called maximally isotropic. Using the symmetry frame formalism to compute affine frame symmetries along with the additional structure of the torsion tensor, we employ the Cartan-Karlhede algorithm to determine all possible seven-dimensional symmetry groups for Riemann-Cartan geometries.

gr-qc↗

arXiv2024

Spherically symmetric teleparallel geometries

We are interested in the development of spherically symmetric geometries in $F(T)$ teleparallel gravity which are of physical importance. We first express the general forms for the spherically symmetric frame and the zero curvature, metric compatible, spin connection. We then analyse the antisymmetric field equations (the solutions of which split into two cases, which we subsequently consider separately), and derive and analyse the resulting symmetric field equations. In order to further study the applications of spherically symmetric teleparallel models, we study $3$ subcases in which there is an additional affine symmetry so that the resulting field equations reduce to a system of ordinary differential equations. First, we study static spherical symmetric geometries and solve the antisymmetric field equations and subsequently derive the full set of symmetric field equations. In particular, we investigate vacuum spacetimes and obtain a number of new solutions. Second, we consider an additional affine frame symmetry in order to expand the affine frame symmetry group to that of a spatially homogeneous Kantowski-Sachs geometry. Third, we study the special case of spherical symmetry with an additional fourth similarity affine vector.

gr-qc↗

arXiv2023

Spatially Homogeneous Teleparallel Gravity: Bianchi I

Using a recently developed algorithm that chooses preferred coordinates and a preferred co-frame, we will determine the completely general Bianchi type I teleparallel geometry. In using this algorithm, any remaining gauge freedom is allocated to the choice of spin connection. We then solve the symmetry constraints placed on the spin connection to derive a general class of Bianchi type I teleparallel geometries. We find that this general class of Bianchi type I teleparallel geometries breaks naturally into two sub-classes. We then illustrate some simple power-law solutions in $F(T)$ teleparallel gravity for each case to illustrate the differences.

gr-qc↗

arXiv2023

Bianchi Type Cosmological Models in $f(T)$ Tele-parallel Gravity

Symmetry assumptions on the geometrical framework have provided successful mechanisms to develop physically meaningful solutions to many problems. In tele-parallel gravity, invariance of the frame and spin-connection under a group of motions defines an affine symmetry group. Here, we assume there exists a three-dimensional group of affine symmetries acting simply transitively on a spatial hypersurface and that this group of symmetry actions defines our affine frame symmetry group. We determine the general form of the co-frame and spin connection for each spatially homogeneous Bianchi type. We then construct the corresponding field equations for $f(T)$ tele-parallel gravity. We show that if the symmetry group is of Bianchi type A ($I$, $II$, $VI_0$, $VII_0$, $VIII$ or $IX$) then there exists a co-frame/spin connection pair that is consistent with the antisymmetric part of the field equations of $f(T)$ tele-parallel gravity. For those geometries having a Bianchi type B symmetry group ($IV$, $V$, $VI_h$, $VII_h$), we find that in general these geometries are inconsistent with the antisymmetric part of the $f(T)$ tele-parallel gravity field equations unless the theory reduces to an analog of General Relativity with a cosmological constant.

gr-qc↗

arXiv2023

A frame based approach to computing symmetries with non-trivial isotropy groups

A frame approach to determining the most general solution admitting a desired symmetry group has been examined previously in Riemannian and teleparallel geometries with some success. In teleparallel geometries, one must determine the general form of the frame and spin connection to generate a general solution admitting the desired symmetry group. Current approaches often rely on the use of the proper frame, where the spin connection is zero. However this leads to particular theoretical and practical problems. In this paper we introduce an entirely general approach to determining the most general Riemann-Cartan geometries which admit a given symmetry group and apply these results to teleparallel geometries. To illustrate the approach we determine the most general geometries, with the minimal number of arbitrary functions, for particular choices of symmetry groups with dimension one, three, six and seven. In addition, we rigorously show how the teleparallel analogues of the Robertson-Walker, de Sitter and Einstein static spacetimes can be determined.

gr-qc↗

arXiv2022

Symmetric Teleparallel Geometries

In teleparallel gravity and, in particular, in $F(T)$ teleparallel gravity, there is a challenge in determining an appropriate (co-)frame and its corresponding spin connection to describe the geometry. Very often, the "proper" frame, the frame in which all inertial effects are absent, is not the simplest (e.g, diagonal) (co-)frame. The determination of the frame and its corresponding spin connection for $F(T)$ teleparallel gravity theories when there exist affine symmetries is of much interest. In this paper we present the general form of the coframe and its corresponding spin connection for teleparallel geometries which are invariant under a $G_6$ group of affine symmetries. The proper coframe and the corresponding $F(T)$ field equations are also shown for these Teleparallel Robertson Walker (TRW) geometries. Further, with the addition of an additional affine symmetry, it is possible to define a Teleparallel de Sitter (TdS) geometry.

gr-qc↗

arXiv2021

Teleparallel geometries not characterized by their scalar polynomial torsion invariants

A teleparallel geometry is an n-dimensional manifold equipped with a frame basis and an independent spin connection. For such a geometry, the curvature tensor vanishes and the torsion tensor is non-zero. A straightforward approach to characterizing teleparallel geometries is to compute scalar polynomial invariants constructed from the torsion tensor and its covariant derivatives. An open question has been whether the set of all scalar polynomial torsion invariants, $\mathcal{I}_T$ uniquely characterize a given teleparallel geometry. In this paper we show that the answer is no and construct the most general class of teleparallel geometries in four dimensions which cannot be characterized by $\mathcal{I}_T$. As a corollary we determine all teleparallel geometries which have vanishing scalar polynomial torsion invariants.

gr-qc↗

arXiv2020

Kantowski-Sachs Einstein-Aether Scalar Field Cosmological Models: The Sequel

Utilizing the autonomous system of ordinary differential equations derived in arXiv:1809.01458 to define the evolution, we further investigate a class of cosmological models within an Einstein-aether gravitational framework by introducing a non-trivial coupling between the shear of the aether field to the scalar field on the future asymptotic solution. We subsequently conduct qualitative and numerical stability analysis on the new set of equilibrium points and paramountly determine that the expansionary power-law inflationary attractor becomes anisotropic rather than isotropic in the presence of such a coupling. It is further shown that the stability of this solution is dependent on the value of the shear coupling parameter $a3$. We also discover a family of asymptotically stable periodic orbits which exist for a particular range of parameter values within the Bianchi I invariant set and vanish in the absence of coupling between the aether field and the scalar field.

gr-qc↗

arXiv2020

Symmetry and Equivalence in Teleparallel Gravity

In theories such as teleparallel gravity and its extensions, the frame basis replaces the metric tensor as the primary object of study. A choice of coordinate system, frame basis and spin-connection must be made to obtain a solution from the field equations of a given teleparallel gravity theory. It is worthwhile to express solutions in an invariant manner in terms of torsion invariants to distinguish between different solutions. In this paper we discuss the symmetries of teleparallel gravity theories, describe the classification of the torsion tensor and its covariant derivative and define scalar invariants in terms of the torsion. In particular, we propose a modification of the Cartan-Karlhede algorithm for geometries with torsion (and no curvature or nonmetricity). The algorithm determines the dimension of the symmetry group for a solution and suggests an alternative frame-based approach to calculating symmetries. We prove that the only maximally symmetric solution to any theory of gravitation admitting a non-zero torsion tensor is Minkowski space. As an illustration we apply the algorithm to six particular exact teleparallel geometries. From these examples we notice that the symmetry group of the solutions of a teleparallel gravity theory is potentially smaller than their metric-based analogues in General Relativity.

gr-qc↗

arXiv2019

Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach

Teleparallel gravity and its popular generalization $f(T)$ gravity can be formulated as fully invariant (under both coordinate transformations and local Lorentz transformations) theories of gravity. Several misconceptions about teleparallel gravity and its generalizations can be found in the literature, especially regarding their local Lorentz invariance. We describe how these misunderstandings may have arisen and attempt to clarify the situation. In particular, the central point of confusion in the literature appears to be related to the inertial spin connection in teleparallel gravity models. While inertial spin connections are commonplace in special relativity, and not something inherent to teleparallel gravity, the role of the inertial spin connection in removing the spurious inertial effects within a given frame of reference is emphasized here. The careful consideration of the inertial spin connection leads to the construction of a fully invariant theory of teleparallel gravity and its generalizations. Indeed, it is the nature of the spin connection that differentiates the relationship between what have been called good tetrads and bad tetrads and clearly shows that, in principle, any tetrad can be utilized. The field equations for the fully invariant formulation of teleparallel gravity and its generalizations are presented and a number of examples using different assumptions on the frame and spin connection are displayed to illustrate the covariant procedure. Various modified teleparallel gravity models are also briefly reviewed.

gr-qc↗

arXiv2018

Kantowski-Sachs Einstein-Aether Scalar Field Cosmological Models

A class of positive curvature spatially homogeneous but anisotropic cosmological models within an Einstein-aether gravitational framework are investigated. The matter source is assumed to be a scalar field which is coupled to the expansion of the aether field through a generalized exponential potential. The evolution equations are expressed in terms of expansion-normalized variables to produce an autonomous system of ordinary differential equations suitable for a numerical and qualitative analysis. An analysis of the local stability of the equilibrium points indicates that there exists a range of values of the parameters in which there exists an accelerating expansionary future attractor. In general relativity, scalar field models with an exponential potential $V=V_0e^{-2kϕ}$ have a late-time inflationary attractor for $k^2<\frac{1}{2}$; however, it is found that the existence of the coupling between the aether and scalar fields allows for arbitrarily large values of the parameter $k$.

gr-qc↗

arXiv2017

Spatially Homogeneous Einstein-Aether Cosmological Models: Scalar Fields with a Generalized Harmonic Potential

Inflationary spatially homogeneous cosmological models within an Einstein-Aether gravitational framework are investigated. The matter source is assumed to be a scalar field which is coupled to the aether field expansion and shear scalars through the generalized harmonic scalar field potential. The evolution equations are expressed in terms of expansion-normalized variables to produce an autonomous system of ordinary differential equations suitable for numerical and local stability analysis. An analysis of the local stability of the equilibrium points indicate that there exists a range of values of the parameters in which there exists an accelerating expansionary future attractor.

gr-qc↗

arXiv2010

Averaging Spacetime: Where do we go from here?

The construction of an averaged theory of gravity based on Einstein's General Relativity is very difficult due to the non-linear nature of the gravitational field equations. This problem is further exacerbated by the difficulty in defining a mathematically precise covariant averaging procedure for tensor fields over differentiable manifolds. Together, these two ideas have been called the averaging problem for General Relativity. In the first part of the talk, an attempt to review some the various approaches to this problem will be given, highlighting strengths, weaknesses, and commonalities between them. In the second part of the talk, an argument will be made, that if one wishes to develop a well-defined averaging procedure, one may choose to parallel transport along geodesics with respect to the Levi-Cevita connection or, use the Weitzenböck connection and ensure the transportation is independent of path. The talk concludes with some open questions to generate further discussion.

gr-qc↗

arXiv2009

A Complete Cosmological Solution to the Averaged Einstein Field Equations as found in Macroscopic Gravity

A formalism for analyzing the complete set of field equations describing Macroscopic Gravity is presented. Using this formalism, a cosmological solution to the Macroscopic Gravity equations is determined. It is found that if a particular segment of the connection correlation tensor is zero and if the macroscopic geometry is described by a flat Robertson-Walker metric, then the effective correction to the averaged Einstein Field equations of General Relativity i.e., the backreaction, is equivalent to a positive spatial curvature term. This investigation completes the analysis of [Phys. Rev. Lett., vol. 95, 151102, (2005)] and the formalism developed provides a possible basis for future studies.

gr-qc↗

arXiv2008

Spherically Symmetric Solutions in Macroscopic Gravity

Schwarzschild's solution to the Einstein Field Equations was one of the first and most important solutions that lead to the understanding and important experimental tests of Einstein's theory of General Relativity. However, Schwarzschild's solution is essentially based on an ideal theory of gravitation, where all inhomogeneities are ignored. Therefore, any generalization of the Schwarzschild solution should take into account the effects of small perturbations that may be present in the gravitational field. The theory of Macroscopic Gravity characterizes the effects of the inhomogeneities through a non-perturbative and covariant averaging procedure. With similar assumptions on the geometry and matter content, a solution to the averaged field equations as dictated by Macroscopic Gravity are derived. The resulting solution provides a possible explanation for the flattening of galactic rotation curves, illustrating that Dark Matter is not real but may only be the result of averaging inhomogeneities in a spherically symmetric background.

gr-qc↗