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 analogs in general relativity.
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July 2020
Research Article|
July 28 2020
Symmetry and equivalence in teleparallel gravity
A. A. Coley
;
A. A. Coley
a)
1
Department of Mathematics and Statistics, Dalhousie University
, Halifax, Nova Scotia B3H 3J5, Canada
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R. J. van den Hoogen
;
R. J. van den Hoogen
b)
2
Department of Mathematics and Statistics, St. Francis Xavier University
, Antigonish, Nova Scotia B2G 2W5, Canada
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D. D. McNutt
D. D. McNutt
c)
3
Department of Mathematics and Physics, University of Stavanger
, Stavanger, Norway
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c)
Author to whom correspondence should be addressed: [email protected]
J. Math. Phys. 61, 072503 (2020)
Article history
Received:
January 31 2020
Accepted:
June 29 2020
Citation
A. A. Coley, R. J. van den Hoogen, D. D. McNutt; Symmetry and equivalence in teleparallel gravity. J. Math. Phys. 1 July 2020; 61 (7): 072503. https://doi.org/10.1063/5.0003252
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