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

Abstract

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.

arXiv comment38 pages, submitted to JMP

References in corpus (30)

• f(T) teleparallel gravity and cosmology
• Modified teleparallel gravity: inflation without inflaton
• Einstein's Other Gravity and the Acceleration of the Universe
• Teleparallel Gravity: From Theory to Cosmology
• On Born-Infeld Gravity in Weitzenbock spacetime
• Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach
• Modified teleparallel theories of gravity in symmetric spacetimes
• Regularizing role of teleparallelism
• Spin Connection and Renormalization of Teleparallel Action
• Photon sphere and perihelion shift in weak $f(T)$ gravity
• Solar System Tests in Modified Teleparallel Gravity
• Perturbations in Non-Flat Cosmology for $f(T)$ gravity
• Stationary Axisymmetric Solutions and their Energy Contents in Teleparallel Equivalent of Einstein Theory
• The Spherically Symmetric Vacuum in Covariant $F(T)=T+\frac{\alpha }{2}{T}^2+\mathcal{O}(T^{\gamma })$ Gravity Theory
• Cosmic acceleration in non-flat $f(T)$ cosmology
• Static spherically symmetric black holes in weak f(T)-gravity
• Symmetry and Equivalence in Teleparallel Gravity
• Exploring Axial Symmetry in Modified Teleparallel Gravity
• Complete classification of cosmological teleparallel geometries
• General cosmological perturbations in teleparallel gravity
• Teleparallel axions and cosmology
• Symmetric Teleparallel Geometries
• Riemann-Cartan Space-times of Gödel Type
• Bounce Models within Teleparallel modified gravity
• Teleparallel Killing Vectors of Spherically Symmetric Spacetimes
• Locally Homogeneous Spaces, Induced Killing Vector Fields and Applications to Bianchi Prototypes
• $f(T,B)$ gravity in a Friedmann-Lema\^ıtre-Robertson-Walker universe with nonzero spatial curvature
• $\mathcal{I}$-degenerate pseudo-Riemannian metrics
• Teleparallel Geometry with a Single Affine Symmetry
• A quick guide to spacetime symmetry and symmetric solutions in teleparallel gravity

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• Symmetric Teleparallel Geometries
• Spherically symmetric teleparallel geometries
• Generalized Teleparallel de Sitter geometries
• Static spherically symmetric perfect fluid solutions in teleparallel F(T) gravity
• Kantowski-Sachs spherically symmetric solutions in teleparallel $F(T)$ gravity
• Bianchi Type Cosmological Models in $f(T)$ Tele-parallel Gravity
• Spatially Homogeneous Teleparallel Gravity: Bianchi I
• Scalar Field Kantowski--Sachs Solutions in Teleparallel $F(T)$ Gravity
• Scalar field source Teleparallel Robertson-Walker F(T)-gravity solutions
• Teleparallel Geometry with Spherical Symmetry: The diagonal and proper frames
• Teleparallel Minkowski Spacetime with Perturbative Approach for Teleparallel Gravity on a Proper Frame
• Locally-homogeneous Riemann-Cartan geometries with the largest symmetry group
• Symmetries in Riemann-Cartan Geometries
• On black holes in teleparallel torsion theories of gravity
• Spatially homogeneous teleparallel spacetimes with four-dimensional groups of motions
• Electromagnetic Kantowski--Sachs Solutions in Teleparallel $F(T)$ Gravity
• On black holes in new general relativity
• Scalar Field Static Spherically Symmetric Solutions in Teleparallel $F(T)$ Gravity
• Electromagnetic Sources Teleparallel Robertson--Walker $F(T)$-Gravity Solutions
• Chaplygin and Polytropic gases Teleparallel Robertson-Walker $F(T)$ gravity solutions