In teleparallel geometries, symmetries are represented by affine frame symmetries that constrain both the (co)frame basis and the spin-connection (which are the primary geometric objects). In this paper, we shall study teleparallel geometries with a single affine symmetry, utilizing the locally Lorentz covariant approach and adopting a complex null gauge. We first introduce an algorithm to study geometries with an affine frame symmetry, which consists of choosing coordinates adapted to the symmetry, constructing a canonical frame, and solving the equations describing the symmetry. All of the constraints on the geometry are determined in the case of a single affine symmetry, but there are additional constraints arising from the field equations for a given theory of teleparallel gravity. In particular, we find that in f(T) teleparallel gravity there will be severe constraints on the geometry arising from the antisymmetric part of the field equations.

1.
S. I.
Nojiri
and
S. D.
Odintsov
,
Int. J. Geom. Methods Mod. Phys.
04
,
115
(
2007
); arXiv:0601213 [hep-th].
2.
S.
Capozziello
and
M.
De Laurentis
,
Phys. Rep.
509
,
167
(
2011
); arXiv:1108.6266 [gr-qc].
4.
S.
Bahamonde
,
C. G.
Böhmer
, and
M.
Wright
,
Phys. Rev. D
92
,
104042
(
2015
); arXiv:1508.05120 [gr-qc].
5.
R.
Ferraro
and
F.
Fiorini
,
Phys. Rev. D
75
,
084031
(
2007
); arXiv:gr-qc/0610067 [gr-qc].
6.
R.
Ferraro
and
F.
Fiorini
,
Phys. Lett. B
702
,
75
(
2011
); arXiv:1103.0824 [gr-qc].
7.
Y.-F.
Cai
,
S.
Capozziello
,
M.
De Laurentis
, and
E. N.
Saridakis
,
Rep. Prog. Phys.
79
,
106901
(
2016
); arXiv:1511.07586 [gr-qc].
8.
S.
Bahamonde
,
K. F.
Dialektopoulos
,
C.
Escamilla-Rivera
,
G.
Farrugia
,
V.
Gakis
,
M.
Hendry
,
M.
Hohmann
,
J. L.
Said
,
J.
Mifsud
, and
E.
Di Valentino
, “
Teleparallel gravity: From theory to cosmology
,” (published online
2022
); arXiv:2106.13793 [gr-qc].
9.
Y. N.
Obukhov
and
G. F.
Rubilar
,
Phys. Rev. D
73
,
124017
(
2006
); arXiv:gr-qc/0605045 [gr-qc].
10.
T. G.
Lucas
,
Y. N.
Obukhov
, and
J. G.
Pereira
,
Phys. Rev. D
80
,
064043
(
2009
); arXiv:0909.2418 [gr-qc].
11.
R.
Aldrovandi
and
J. G.
Pereira
,
Teleparallel Gravity
, Fundamental Theories of Physics Vol. 173 (
Springer
,
Dordrecht
,
2013
).
12.
M.
Krššák
,
R. J.
van den Hoogen
,
J. G.
Pereira
,
C. G.
Böhmer
, and
A. A.
Coley
,
Classical Quantum Gravity
36
,
183001
(
2019
); arXiv:1810.12932 [gr-qc].
13.
M.
Krššák
and
E. N.
Saridakis
,
Classical Quantum Gravity
33
,
115009
(
2016
); arXiv:1510.08432 [gr-qc].
14.
M.
Krššák
and
J. G.
Pereira
,
Eur. Phys. J. C
75
,
519
(
2015
); arXiv:1504.07683 [gr-qc].
15.
H.
Stephani
,
D.
Kramer
,
M.
MacCallum
,
C.
Hoenselaers
, and
E.
Herlt
,
Exact Solutions of Einstein’s Field Equations
(
Cambridge University Press
,
2009
).
16.
A. A.
Coley
,
R. J.
van den Hoogen
, and
D. D.
McNutt
,
J. Math. Phys.
61
,
072503
(
2020
); arXiv:1911.03893 [gr-qc].
17.
R.
Hecht
,
F. W.
Hehl
,
J. D.
McCrea
,
E. W.
Mielke
, and
Y.
Ne’eman
,
Phys. Lett. A
172
,
13
(
1992
); arXiv:gr-qc/9210009.
18.
P. J.
Olver
,
Equivalence, Invariants and Symmetry
(
Cambridge University Press
,
1995
).
19.
J. B.
Fonseca-Neto
,
M. J.
Rebouças
, and
A. F. F.
Teixeira
,
J. Math. Phys.
33
,
2574
(
1992
).
20.
D. D.
McNutt
,
A. A.
Coley
, and
R. J.
van den Hoogen
, “
A frame based approach to computing symmetries with non-trivial isotropy groups
,”
J. Math. Phys.
(submitted) (
2023
).
21.
M.
Hohmann
,
L.
Järv
,
M.
Krssak
, and
C.
Pfeifer
,
Phys. Rev. D
100
,
084002
(
2019
); arXiv:1901.05472 [gr-qc].
22.
C.
Pfeifer
, “
A quick guide to spacetime symmetry and symmetric solutions in teleparallel gravity
,” arXiv:2201.04691 [gr-qc].
23.
M.
Hohmann
,
J. Math. Phys.
57
,
082502
(
2016
); arXiv:1505.07809 [math-ph].
24.
R. J.
van den Hoogen
,
D. D.
McNutt
, and
A. A.
Coley
, “
Bianchi type cosmological models in f(T) teleparallel gravity
” (unpublished).
25.
M.
Hohmann
,
Int. J. Geom. Methods Mod. Phys.
18
,
2140005
(
2021
); arXiv:2008.12186 [gr-qc].
26.
A. A.
Coley
,
R. J.
van den Hoogen
, and
D. D.
McNutt
,
Classical Quantum Gravity
39
,
22LT01
(
2022
); arXiv:2205.10719 [gr-qc].
27.
A.
Golovnev
,
Classical Quantum Gravity
38
,
197001
(
2021
); arXiv:2105.08586 [gr-qc].
28.
A.
Coley
,
Classical Quantum Gravity
25
,
033001
(
2008
); arXiv:0710.1598 [gr-qc].
29.
M.
Hohmann
,
L.
Järv
,
M.
Krššák
, and
C.
Pfeifer
,
Phys. Rev. D
97
,
104042
(
2018
); arXiv:1711.09930 [gr-qc].
30.
M.
Hohmann
,
L.
Järv
, and
U.
Ualikhanova
,
Phys. Rev. D
97
,
104011
(
2018
); arXiv:1801.05786 [gr-qc].
31.
A.
Golovnev
,
T.
Koivisto
, and
M.
Sandstad
,
Classical Quantum Gravity
34
,
145013
(
2017
); arXiv:1701.06271 [gr-qc].
32.
T. P.
Sotiriou
and
V.
Faraoni
,
Rev. Mod. Phys.
82
,
451
497
(
2010
); arXiv:0805.1726 [gr-qc].
33.
A.
Golovnev
and
M. J.
Guzman
,
Phys. Rev. D
103
,
044009
(
2021
); arXiv:2012.00696 [gr-qc].
You do not currently have access to this content.