Conserved Currents and Superpotentials in Teleparallel Equivalent of General Relativity

Aleksey Toporensky; Alexander Petrov; Elena Emtsova

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Conserved Currents and Superpotentials in Teleparallel Equivalent of General Relativity

Elena Emtsova, Alexander Petrov, Aleksey Toporensky

General Relativity and Quantum Cosmology · Submitted 20 Oct 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim They are covariant both under coordinate transformations and local Lorentz rotations, and allow us to construct well defined conserved charges, unlike earlier approaches.

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arXiv ID1910.08960 (version 2)

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CitationAleksey Toporensky, Alexander Petrov, Elena Emtsova. "Conserved Currents and Superpotentials in Teleparallel Equivalent of General Relativity." arXiv:1910.08960 (2019). https://pith.science/paper/DJEZE66H

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What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.90 · mentions code or data availability 0.02.

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We study the Teleparallel Equivalent of General Relativity (TEGR) with Lagrangian that includes the flat (inertial) spin connection and that is evidently invariant with respect to local Lorentz rotations. Applying directly the Noether theorem, we construct new expressions for conserved currents and related superpotentials. They are covariant both under coordinate transformations and local Lorentz rotations, and allow us to construct well defined conserved charges, unlike earlier approaches. The advantage is achieved by an explicit presence of a displacement vector in the new expressions that can be interpreted as a Killing vector, as a proper vector of an observer, etc. The new expressions are used to introduce a principle for definition of an inertial spin connection that is undetermined one in the TEGR from the start. Theoretical results are applied to calculate mass for the Schwarzschild black hole and densities of conserved quantities for freely falling observers both in Friedmann-Lemaitre-Robertson-Walker world of all the three signs of curvature and in (anti-)de Sitter space.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the Teleparallel Equivalent of General Relativity (TEGR) with Lagrangian that includes the flat (inertial) spin connection and that is evidently invariant with respect to local Lorentz rotations.	contribution 0.68	0.64	0.05	asserted 0.99
s1	Applying directly the Noether theorem, we construct new expressions for conserved currents and related superpotentials.	method 0.62	0.96	0.12	asserted 1.00
s2	They are covariant both under coordinate transformations and local Lorentz rotations, and allow us to construct well defined conserved charges, unlike earlier approaches.	result 0.78	0.89	0.53	asserted 1.00
s3	The advantage is achieved by an explicit presence of a displacement vector in the new expressions that can be interpreted as a Killing vector, as a proper vector of an observer, etc.	interpretation 0.87	0.86	0.15	asserted 0.81
s4	The new expressions are used to introduce a principle for definition of an inertial spin connection that is undetermined one in the TEGR from the start.	contribution 0.85	0.91	0.17	asserted 1.00
s5	Theoretical results are applied to calculate mass for the Schwarzschild black hole and densities of conserved quantities for freely falling observers both in Friedmann-Lemaitre-Robertson-Walker world of all the three signs of curvature and in (anti-)de Sitter space.	result 0.70	0.93	0.16	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

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Original abstract

We study the Teleparallel Equivalent of General Relativity (TEGR) with Lagrangian that includes the flat (inertial) spin connection and that is evidently invariant with respect to local Lorentz rotations. Applying directly the Noether theorem, we construct new expressions for conserved currents and related superpotentials. They are covariant both under coordinate transformations and local Lorentz rotations, and allow us to construct well defined conserved charges, unlike earlier approaches. The advantage is achieved by an explicit presence of a displacement vector in the new expressions that can be interpreted as a Killing vector, as a proper vector of an observer, etc. The new expressions are used to introduce a principle for definition of an inertial spin connection that is undetermined one in the TEGR from the start. Theoretical results are applied to calculate mass for the Schwarzschild black hole and densities of conserved quantities for freely falling observers both in Friedmann-Lemaitre-Robertson-Walker world of all the three signs of curvature and in (anti-)de Sitter space.

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11 arXiv papers cite this in 39 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 23 sentences without a confident label

Uses its method or tool 2 papers

Let us outline the covariant formalism [ref] .

A moving black hole in TEGR as a moving matter ball · Introduction · 0.91
With the given symmetry, the conserved charges are obtained from the $tr$ components of the superpotential [ref] [ref] .

Kerr-Schild metrics in teleparallel gravity · Asymptotic form · 0.84

Compares with it 1 paper

We compute [torsion_nonrotating_lorentz] T̃^0 = - M/r √( r (r - 2 M) ) t ∧r , T̃^1 = 0 , T̃^2 = [ 1 - ( 1 - 2 M/r )^-1/2 ] r ∧θ, T̃^3 = sinθ[ 1 - ( 1 - 2 M/r )^-1/2 ] r ∧ϕ, which then agrees with the Schwarzschild solution of [ref] after expressing their spacetime components in terms of tetrad ones.

Kerr-Schild metrics in teleparallel gravity · Non-rotating limit · 0.99

Cites it as background 5 papers

Here we will only briefly outline the main idea, while all the details can be find in [ref] .

On Conserved Quantities for the Schwarzschild Black Hole in Teleparallel Gravity · Total energy-momentum and Noether conserved charges · 0.81
There are no universal methods applicable to any physical system yet, though certain proposals based on underlying symmetries [ref] and "switching off" gravity [ref] are currently under discussions.

Spin connection and cosmological perturbations in $f(T)$ gravity · Covariant formulation of teleparallel gravity. · 0.88
Extentions to non-Riemannian geometries were studied in [ref] , and recently in [ref] .

Off-Shell ADT Conserved Quantities in Palatini Gravity · Introduction · 0.80

What it cites · 28 sentences without a confident label

Contests 1 paper

At last, let us note that the formalism developed in [ref] does not permit to define a proper tetrad (and inertial spin connection related to diagonal tetrad) for the (A)dS space, whereas we define them here, they are ([ref]) and ([ref]).

Modified teleparallel theories of gravity in symmetric spacetimes · Discussion · contest 0.86

Compares with 4 papers

The difference from our approach is, first, that we take into account $^i_jμ$ from the start as a necessary structure of the formalism in whole, whereas in [ref] it is introduced as a compensating structure in constructing conserved quantities.

Regularizing role of teleparallelism · Discussion · 1.00
Comparing our results with those in [ref] , we remark that, first, in the case of the Schwarzschild black hole they are identical, second, in the case of the FLRW universe they add one other, third, in the case of the (A)dS space we define a proper tetrad, whereas the authors [ref] do not.

Modified teleparallel theories of gravity in symmetric spacetimes · Concluding remarks · 1.00
4) Discussing a philosophy of constructing conserving quantities in the covariant TEGR we compare it with constructing the covariantized both Einstein's pseudotensor and Freud's superpotential developed in [ref] .

Relativistic conservation laws and integral constraints for large cosmological perturbations · Concluding remarks · 1.00

Cites as support 1 paper

It turns out that all modified teleparallel theories with second order field equations belong the same property, see [ref] .

Teleparallel theories of gravity as analogue of non-linear electrodynamics · Concluding remarks · 0.96

Cites as background 13 papers

The system ([ref]) - ([ref]) was engineered by Klein, see general and detail discussion in [ref] .

Covariant Differential Identities and Conservation Laws in Metric-Torsion Theories of Gravitation. I. General Consideration · The Noether theorem in a covariant field theory · 0.90
The system ([ref]) - ([ref]) was engineered by Klein, see general and detail discussion in [ref] .

Covariant Differential Identities and Conservation Laws in Metric-Torsion Theories of Gravitation. II. Manifestly Generally Covariant Theories · The Noether theorem in a covariant field theory · 0.90
Last years the interest to Teleparallel Equivalent of General Relativity (TEGR) and its modifications, like $f(T)$-, $f(T_ax, T_ten,T_vec)$-theories and others, arises significantly; see, for example, book [ref] , reviews [ref] and numerous references therein.

The teleparallel equivalent of general relativity · Introduction · 1.00

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