The $3+1$ Formalism in the Geometric Trinity of Gravity

Abstract

The geometric trinity of gravity offers a platform in which gravity can be formulated in three analogous approaches, namely curvature, 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 $3+1$ 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 $3+1$ formulation for general relativity is recovered as well as the tetrad $3+1$ formulation of the teleparallel equivalent of general relativity and the metric $3+1$ 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.

23 pages

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Cited by in corpus (20)

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• Comparing Equivalent Gravities: common features and differences
• Slow-roll inflation in $f(Q)$ non-metric gravity
• Static and spherically symmetric wormholes in metric-affine theories of gravity
• Exploring the stability of $f(Q)$ cosmology near general relativity limit with different connections
• Charged strange star model in Tolman-Kuchowicz spacetime in the background of 5D Einstein-Maxwell-Gauss-Bonnet gravity
• Gravitational tensor and scalar modes in $f(Q,B)$ non-metric gravity
• The origin of Weyl gauging in metric-affine theories
• Geometric Perfect Fluids from Extended Gravity
• Gravity generated by four one-dimensional unitary gauge symmetries and the Standard Model
• Hamilton's equations in the covariant teleparallel equivalent of general relativity
• Extended Geometric Trinity of Gravity
• Kinematics in Metric-Affine Geometry
• First-order hyperbolic formulation of the pure tetrad teleparallel gravity theory
• The Hamiltonian constraint in the symmetric teleparallel equivalent of general relativity
• Foliation-generating observers under Lorentz transformations
• Equivalent Gravities and Equivalence Principle: Foundations and experimental implications
• Lagrangian Formulation of the Raychaudhuri Equation in Non-Riemannian Geometry
• On the Effects of Non-metricity in an Averaged Universe
• Lost in translation: the Abelian affine connection (in the coincident gauge)