https://doi.org/10.1140/epjc/s10052-021-09944-6
Regular Article - Theoretical Physics
The 3+1 formalism in teleparallel and symmetric teleparallel gravity
1
Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II”, Via Cinthia, 80126, Naples, Italy
2
Istituto Nazionale di Fisica Nucleare (INFN), sez. di Napoli, Via Cinthia 9, 80126, Naples, Italy
3
Scuola Superiore Meridionale, Largo San Marcellino 10, 80138, Naples, Italy
4
Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050, Tomsk, Russia
5
Institute of Space Sciences and Astronomy, University of Malta, Msida, Malta
6
Department of Physics, University of Malta, Msida, Malta
Received:
17
November
2021
Accepted:
10
December
2021
Published online:
28
December
2021
Teleparallel and symmetric teleparallel gravity offer platforms in which gravity can be formulated in interesting geometric approaches, respectively given by torsion and nonmetricity. In this vein, general relativity can be expressed in three dynamically equivalent ways which may offer insights into the different properties of these decompositions such as their Hamiltonian structure, the efficiency of numerical analyses, as well as the classification of gravitational field degrees of freedom. In this work, we take a 
decomposition of the teleparallel equivalent of general relativity and the symmetric teleparallel equivalent of general relativity which are both dynamically equivalent to curvature based general relativity. By splitting the spacetime metric and corresponding tetrad into their spatial and temporal parts as well as through finding the Gauss-like equations, it is possible to set up a general foundation for the different formulations of gravity. Based on these results, general 3-tetrad and 3-metric evolution equations are derived. Finally through the choice of the two respective connections, the metric formulation for general relativity is recovered as well as the tetrad formulation of the teleparallel equivalent of general relativity and the metric formulation of symmetric teleparallel equivalent of general relativity. The approach is capable, in principle, of resolving common features of the various formulations of general relativity at a fundamental level and pointing out characteristics that extensions and alternatives to the various formulations can present.
© The Author(s) 2021
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