Mass and energy in General Relativity

Bozhkov, Yuri; Rodrigues, Waldyr A.

doi:10.1007/bf02113065

Cited by 18 publications

(27 citation statements)

References 11 publications

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“…If we recall the well known definition of the integral of a differential form [3,7] we see that a coordinate free result depends fundamentally on the fact that the differential form being integrated defines a true tensor. However, as already mentioned in Remark 1, the S µ are not true indexed forms, and so their integration will be certainly coordinate dependent [2]. In Appendix C for completeness and hoping that the present paper may be of some utility for people trying to understand this issue, we find also from our formalism the so called Einstein and the Landau-Lifshitz "inertial" masses (concepts which have the same problems as the one defined in Eq.…”

Section: Freud's Identity and The Energy-momentum "Conservationmentioning

confidence: 72%

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Freud’s Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

Notte-Cuello

Rodrigues²

2008

AACA

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We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold, the true meaning of Freud's identity of differential geometry discovered in 1939 (as a generalization of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud's identity (which is a decomposition of the Einstein indexed 3-forms G a in two gauge dependent objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way that Freud's identity is usually applied in the formulation of energy-momentum "conservation laws" in GR. In order for this paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms) all calculations are done with all details (disclosing some of the "tricks of the trade" of the subject). (2000). Primary 83C40; Secondary 15A66. Mathematics Subject Classification

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Section: Freud's Identity and The Energy-momentum "Conservationmentioning

confidence: 72%

“…Some of them we briefly discuss below. 2 A sample on the kind of the misconceptions associated to the interpretation of Freud's identity (and which served as inspiration for preparing the present paper) show up when we read 3 , e.g., in [29] :…”

Section: Introductionmentioning

confidence: 99%

Freud’s Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

Notte-Cuello

Rodrigues²

2008

AACA

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“…(363) also for coordinate basis of the linear frame bundle of the Lorentzian spacetime, thus obtaining expressions different 'energy-mometum pseudo-tensors', for which the analogous integral to Eq. (371) depends on the coordinate system used for their calculations [5,64].…”

Section: Remark 64mentioning

confidence: 99%

“…The signature of a g is arbitrary. So, even in the case where (M, g, ∇) is part of a Lorentzian spacetime structure and g has signature −2 we can have another GSS (M, g ′ , ∇) where g ′ has an another signature 5 . Now the objects that characterize a GSS are the following tensor fields:…”

Section: Introductionmentioning

confidence: 99%

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Gravitation as Plastic Distortion of the Lorentz Vacuum

Fernández

Rodrigues

2010

Gravitation as a Plastic Distortion of the Lorentz Vacuum

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In this paper we present a theory of the gravitational field where this field, represented by a (1, 1)-extensor field h describing a plastic distortion of the Lorentz vacuum (a real substance that lives in a Minkowski spacetime) due to the presence of matter. The field h distorts the Minkowski metric extensor η generating what may be interpreted as an effective Lorentzian metric extensor g = h † ηh and also it permits the introduction of different kinds of parallelism rules on the world manifold, which may be interpreted as distortions of the parallelism structure of Minkowski spacetime and which may have non null curvature and/or torsion and/or non metricity tensors. We thus have different possible effective geometries which may be associated to the gravitational field and thus its description by a Lorentzian geometry is only a possibility, not an imposition from Nature. Moreover, we developed with enough details the theory of multiform functions and multiform functionals that permitted us to successfully write a Lagrangian for h and to obtain its equations of motion, that results equivalent to Einstein field equations of General Relativity (for all those solutions where the manifold M is diffeomorphic to R 4 ). However, in our theory, differently from the case of General Relativity, trustful energy-momentum and angular momentum conservation laws exist. We express also the results of our theory in terms of the gravitational potentials g µ = h † (ϑ µ ) where {ϑ µ } is an orthonormal basis of Minkowski spacetime in order to have results which may be easily expressed with the * Some (odd) misprints and typos have been corrected, some sentences have been changed for better intelligibility and Appendix F has new important remarks which result from discussions that W.A .R. had with A. Lasenby at ICCA10 (Tartu) in August 2014.

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Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes

Rodrigues

Oliveira

2016

The Many Faces of Maxwell, Dirac and Einstein Equations

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Mass and energy in General Relativity

Cited by 18 publications

References 11 publications

Freud’s Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

Freud’s Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

Gravitation as Plastic Distortion of the Lorentz Vacuum

Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes

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