Hamiltonian expression of curvature tensors in the York canonical basis: (I) Riemann tensor and Ricci scalars

Lusanna, Luca; Villani, Mattia

doi:10.1142/s0219887814500522

Cited by 8 publications

(18 citation statements)

References 23 publications

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“…In the family of globally-hyperbolic, asymptotically-Minkowskian space-times without super-translations discussed in [76][77][78][79][80][81][82][83], there is the possibility of defining the asymptotic ADM Poincaré generators at spatial infinity, where the instantaneous three-spaces Σ τ become Euclidean, for every kind of matter. The absence of super-translations implies that these three-spaces are relativistic non-inertial rest frames.…”

Section: Discussionmentioning

confidence: 99%

“…The four-coordinates σ A = (τ; σ r ) are named radar four-coordinates: they were introduced by Bondi [75] in general relativity and have been used for the definition of global non-inertial frames by means of 3 + 1 splittings of a certain family of Einstein space-times (see [76][77][78][79][80][81] and the reviews [82,83]). …”

Section: Relativistic Mechanics Of N Interacting Particles In Inertiamentioning

confidence: 99%

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From Relativistic Mechanics towards Relativistic Statistical Mechanics

Lusanna¹

2017

Entropy

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Till now, kinetic theory and statistical mechanics of either free or interacting point particles were well defined only in non-relativistic inertial frames in the absence of the long-range inertial forces present in accelerated frames. As shown in the introductory review at the relativistic level, only a relativistic kinetic theory of "world-lines" in inertial frames was known till recently due to the problem of the elimination of the relative times. The recent Wigner-covariant formulation of relativistic classical and quantum mechanics of point particles required by the theory of relativistic bound states, with the elimination of the problem of relative times and with a clarification of the notion of the relativistic center of mass, allows one to give a definition of the distribution function of the relativistic micro-canonical ensemble in terms of the generators of the Poincaré algebra of a system of interacting particles both in inertial and in non-inertial rest frames. The non-relativistic limit allows one to get the ensemble in non-relativistic non-inertial frames. Assuming the existence of a relativistic Gibbs ensemble, also a "Lorentz-scalar micro-canonical temperature" can be defined. If the forces between the particles are short range in inertial frames, the notion of equilibrium can be extended from them to the non-inertial rest frames, and it is possible to go to the thermodynamic limit and to define a relativistic canonical temperature and a relativistic canonical ensemble. Finally, assuming that a Lorentz-scalar one-particle distribution function can be defined with a statistical average, an indication is given of which are the difficulties in solving the open problem of deriving the relativistic Boltzmann equation with the same methodology used in the non-relativistic case instead of postulating it as is usually done. There are also some comments on how it would be possible to have a hydrodynamical description of the relativistic kinetic theory of an isolated fluid in local equilibrium by means of an effective relativistic dissipative fluid described in the Wigner-covariant framework.

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Section: Discussionmentioning

confidence: 99%

Section: Relativistic Mechanics Of N Interacting Particles In Inertiamentioning

confidence: 99%

From Relativistic Mechanics towards Relativistic Statistical Mechanics

Lusanna¹

2017

Entropy

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“…[51]) of the gravitational field are not known 15 ; they would be the two pairs of 4-scalar tidal variables in a Shanmugadhasan canonical basis adapted to all the 14 first class constraints.…”

Section: Each Solution Of These Equations Defines a Different York mentioning

confidence: 99%

Relativistic Celestial Metrology: Dark Matter as an Inertial Gauge Effect

Lusanna¹,

Stanga²

2017

Trends in Modern Cosmology

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In canonical tetrad gravity, it is possible to identify the gauge variables, describing relativistic inertial effects, in Einstein general relativity. One of these is the York time, the trace of the extrinsic curvature of the instantaneous non-Euclidean 3-spaces (global Euclidean 3-spaces are forbidden by the equivalence principle). The extrinsic curvature depends both on gauge variables and on dynamical ones like the gravitational waves after linearization. The fixation of these gauge variables is done by relativistic metrology with its identification of time and space. Till now, the International Celestial Reference Frame ICRF uses Euclidean 3-spaces outside the Solar System. It is shown that York time and non-Euclidean 3-spaces may explain the main signatures of dark matter in ordinary space-time before using cosmology. Also dark energy may be connected to these inertial gauge effects, because both red-shift and luminosity distance depend on them.

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“…(B5), we get thatW (1)τ rτ s andW (1)rsuv depend only on Rā. InsteadW (1)τ ruv depends on Rā, Πā and 3 K (1) , because we haven (1)…”

Section: The Linearized Weyl Scalars In Arbitrary Gauges Near the mentioning

confidence: 99%

“…To have quantities of order O(ζ)to be used in Eqs. (4.5) we must consider functions of the Weyl eigenvalues like either |w (1)…”

Section: B the Weyl Eigenvaluesmentioning

confidence: 99%

Hamiltonian expression of curvature tensors in the York canonical basis: (II) Weyl tensor, Weyl scalars, Weyl eigenvalues and the problem of the observables of the gravitational field

Lusanna

Villani

2014

Int. J. Geom. Methods Mod. Phys.

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We find the Hamiltonian expression in the York basis of canonical ADM tetrad gravity of the 4-Weyl tensor of the asymptotically Minkowskian space-time. Like for the 4-Riemann tensor we find a radar tensor (whose components are 4-scalars due to the use of radar 4-coordinates), which coincides with the 4-Weyl tensor on-shell on the solutions of Einstein's equations. Then, by using the Hamiltonian null tetrads, we find the Hamiltonian expression of the Weyl scalars of the NewmanPerose approach and of the four eigenvalues of the 4-Weyl tensor.After having introduced the Dirac observables of canonical gravity, whose determination requires the solution of the super-Hamiltonian and super-momentum constraints, we discuss the connection of the Dirac observables with the notion of 4-scalar Bergmann observables. Due to the use of radar 4-coordinates these two types of observables coincide in our formulation of canonical ADM tetrad gravity. However, contrary to Bergmann proposal, the Weyl eigenvalues are shown not to be Bergmann observables, so that their relevance is only in their use (first suggested by Bergmann and Komar) for giving a physical identification as point-events of the mathematical points of the space-time 4-manifold.Finally we give the expression of the Weyl scalars in the Hamiltonian Post-Minkowskian linearization of canonical ADM tetrad gravity in the family of (non-harmonic) 3-orthogonal Schwinger time gauges.

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Hamiltonian expression of curvature tensors in the York canonical basis: (I) Riemann tensor and Ricci scalars

Cited by 8 publications

References 23 publications

From Relativistic Mechanics towards Relativistic Statistical Mechanics

From Relativistic Mechanics towards Relativistic Statistical Mechanics

Relativistic Celestial Metrology: Dark Matter as an Inertial Gauge Effect

Hamiltonian expression of curvature tensors in the York canonical basis: (II) Weyl tensor, Weyl scalars, Weyl eigenvalues and the problem of the observables of the gravitational field

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