(3+1)-Formulation for gravity with torsion and non-metricity: The stress-energy-momentum equation

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 $$3+1$$ 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 $$3+1$$ 3 + 1 formulation for general relativity is recovered as well as the tetrad $$3+1$$ 3 + 1 formulation of the teleparallel equivalent of general relativity and the metric $$3+1$$ 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.

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“…On the other end of the spectrum, Refs. [40,41] explored important properties in the formulation of the 3 + 1 decomposition for STEGR.…”

Section: Introductionmentioning

confidence: 99%

The 3+1 formalism in teleparallel and symmetric teleparallel gravity

et al. 2021

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“…Under the perspective described above, MAG theories result to be rather general and powerful [11,12,[15][16][17][18][19][20][21][22][23], and can possibly lead to diverse scenarios beyond GR, which may have relevant consequences, for instance, in the cosmological context. For some relevant results already obtained in MAG cosmology we refer the reader to, e.g., [24][25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

“…Besides, let us also mention that Einstein-Cartan-Weyl spaces in three dimensions, involving torsion and non-metricity, have recently proved to describe the near-horizion geometry of some supergravity BPS black holes solutions[10]. Furthermore, the (3 + 1)-formulation for gravity with torsion and non-metricity has been recently developed in[11,12].…”

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confidence: 99%

Cosmology of Quadratic Metric-Affine Gravity

Iosifidis,

Ravera

2021

Preprint

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We investigate the cosmological aspects of the most general parity preserving Metric-Affine Gravity theory quadratic in torsion and non-metricity in the presence of a cosmological hyperfluid. The equations of motion are obtained by varying the action with respect to the metric and the independent affine connection. Subsequently, considering a Friedmann-Lemaître-Robertson-Walker background, we derive the most general form of the modified Friedmann equations for the full quadratic theory. We then focus on a characteristic sub-case involving only two quadratic contributions given in terms of torsion and non-metricity vectors. In this setup, studying the modified Friedmann equations along with the conservation laws of the perfect cosmological hyperfluid, we provide exact solutions both for purely dilation and for purely spin hypermomentum sources. We then discuss the physical consequences of our model and the prominent role of torsion and non-metricity in this cosmological setup.

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“…On the other end of the spectrum, Refs. [36,37] explored important properties in the formulation of the 3 + 1 decomposition for STEGR.…”

Section: Introductionmentioning

confidence: 99%

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

Capozziello,

Finch,

Said

et al. 2021

Preprint

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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.

show abstract

(3+1)-Formulation for gravity with torsion and non-metricity: The stress-energy-momentum equation

Cited by 12 publications

References 53 publications

The 3+1 formalism in teleparallel and symmetric teleparallel gravity

The 3+1 formalism in teleparallel and symmetric teleparallel gravity

Cosmology of Quadratic Metric-Affine Gravity

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

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