The meaning of time in the theory of relativity and “Einstein's later view of the Twin Paradox”

Rodrigues, Waldyr A.; Rosa, M. A. F.

doi:10.1007/bf00731906

“…F A Comment on the Gauge Theory of Gravitation of References [19] and [44] In footnote 1 we recalled that in GRT a gravitational field is defined by an equivalence class of pentuples, where (M, g, D, τ g , ↑) and (M ′ , g ′ , D ′ , τ ′ g , ↑ ′ ) are said equivalent if there is a diffeomorphism f : M → M ′ , such that g ′ = f * g, D ′ = f * D, τ ′ g = f * τ g , ↑ ′ = f * ↑, (where f * here denotes the pullback mapping). For more details, see, e.g., [81,74,77]. Moreover, in GRT when one is studying the coupling of the matter fields, represented, say by some sections of the exterior algebra bundle 66 ψ 1,..., ψ n , which satisfy certain coupled differential equations (the equations of motion 67 ), two models {(M, g, D, τ g , ↑), ψ 1,..., ψ n } and (M ′ , g ′ , D ′ , τ ′ g , ↑ ′ ), ψ ′ 1,..., ψ ′ n } are said to be dynamically equivalent iff there is a diffeomorphism h : M → M ′ , such that (M, g, D, τ g , ↑) and (M ′ , g ′ , D ′ , τ ′ g , ↑ ′ ) and moreover ψ ′ i = f * ψ i , i = 1, ..., n. This kind of equivalence is not a particularity of GRT, it is an obvious mathematical requirement that any theory formulated with tensor fields living in an arbitrary manifold must satisfy 68 .…”

Section: Proof Of Eq(351)mentioning

confidence: 99%

“…We then get 74 In [44] Eq.(??) is written DµM = ∂µM + ω(gµ) × M. We must immediately conclude that ω(gµ) = ω(h † (eµ)) = Ω(eµ), i.e., ω = h ♣ Ω.…”

Section: Proof Of Eq(351)mentioning

confidence: 99%

Gravitation as a Plastic Distortion of the Lorentz Vacuum

Fernández

¹

,

Rodrigues

²

2010

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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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“…For more details, see, e.g., [81,74,77]. Moreover, in GRT when one is studying the coupling of the matter fields, represented, say by some sections of the exterior algebra bundle 66 ψ 1,..., ψ n , which satisfy certain coupled differential equations (the equations of motion .., n. This kind of equivalence is not a particularity of GRT, it is an obvious mathematical requirement that any theory formulated with tensor fields living in an arbitrary manifold must satisfy 68 .…”

Section: Proof Of Eq(351)mentioning

confidence: 99%

Gravitation as Plastic Distortion of the Lorentz Vacuum

Fernández

¹

,

Rodrigues

²

2010

Gravitation as a Plastic Distortion of the Lorentz Vacuum

Self Cite

View full text Add to dashboard Cite

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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“…Each integral line of I is called an inertial observer. The coordinate functions x µ , µ = 0, 1, 2, 3 of a chart of the maximal atlas of M are said to be a naturally adapted coordinate system to I (nacs/I) if I = ∂/∂x 0 [11,12]. Putting I = e 0 we can find e i = ∂/∂x i , i = 1, 2, 3 such that g(e µ , e ν ) = η µν and the coordinate functions x µ are the usual Einstein-Lorentz ones and have a precise operational meaning: 8 where t is measured by "ideal clocks" at rest on I and synchronized "à la Einstein", x i , i = 1, 2, 3 are determined with ideal rules [13].…”

Section: Maxwell Theory In Cℓ(m ) and The Hertz Potentialmentioning

confidence: 99%

“…In [19] the Φ < Jn are called the n-th order non-diffracting Bessel beams. 11 Bessel beams are examples of undistorted progressive waves (UPWs). They are "subluminal" waves.…”

Section: Subluminal and Superluminal Solutions Of The Hwementioning

confidence: 99%

A unified theory for construction of arbitrary speeds (0 ≤ v < ∞) solutions of the relativistic wave equations

Rodrigues¹,

Maiorino²

1996

Random Operators and Stochastic Equations

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Representing the relativistic physical fields as sections of the Clifford Bundle (or of the Spin-Clifford Bundle) of Minkowski spacetime we show that all the relativistic wave equations satisfied by these fields possess solutions traveling with arbitrary speeds 0 < V < OO. By giving rigorous mathematical definitions of reference frames and of the Principle of Relativity (PR) we prove that physical realizations of the V > 1 solutions of, e.g., the Maxwell equations imply in a breakdown of the PR, but in no contradiction at all with known physical facts.where J is the bilateral ideal in TV generated by elements of the form u υ + ν <8> u,u,v G V. The elements of f\V will be called multivectors, or multiforms if they arc elements of f \V*.

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The meaning of time in the theory of relativity and “Einstein's later view of the Twin Paradox”

Cited by 30 publications

References 12 publications

Gravitation as a Plastic Distortion of the Lorentz Vacuum

Gravitation as a Plastic Distortion of the Lorentz Vacuum

Gravitation as Plastic Distortion of the Lorentz Vacuum

A unified theory for construction of arbitrary speeds (0 ≤ v < ∞) solutions of the relativistic wave equations

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