Space-time edge geometry

Dodson, C. T. J.

doi:10.1007/bf00670382

Cited by 41 publications

(28 citation statements)

References 41 publications

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“…The fact that, in this case, G = R greatly simplifies computations, and is not an essential limitation [5]. In the full ten dimensional case, more ones appear in the diagonal of this matrix with no effect on our general conclusions.…”

Section: Noncommutative Geometry Of the Closed Friedman Modelmentioning

confidence: 94%

A noncommutative Friedman cosmological model

Heller

2010

Ann. Phys.

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The closed Friedman cosmological model, based on noncommutative geometry, is presented. Two global effects exhibited by the model are discussed. The first effect is the "generation of matter out of geometry". Gravitational field equation in this model has the form of the eigenvalue equation for the Einstein operator. It turns out that the eigenvalues of this operator reproduce components of the energy-momentum tensor. The second effect concerns the existence of the initial and final singularities. Because of the strongly probabilistic character of the noncommutative dynamics on the fundamental level, although singularities do exist, they are probabilistically irrelevant.

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Section: Noncommutative Geometry Of the Closed Friedman Modelmentioning

confidence: 94%

A noncommutative Friedman cosmological model

Heller

2010

Ann. Phys.

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“…The central role of V in relativity was extended by Schmidt (1971) to the characterization of singularities by incorporating them in the b-boundary OM of space-time. Details of this and subsequent modifications are given in Dodson (1978). We have shown (Dodson and Radivoiovici, 1981) that the connection V in L(2)M induced by V in LM allows another view of singularities by means of the /~-boundary 0M, which contains OM.…”

Section: Space-time Boundariesmentioning

confidence: 96%

Second-order tangent structures

Dodson

Radivoiovici

1982

Int J Theor Phys

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Second-order differential processes have special significance for physics. Two reasonable generalizations of the procedure for constructing a tangent bundle over a smooth n-manifold M yield different second-order structures, each projecting onto the standard first-order structure TM. One approach, based on the work of Ehresmann generalizes the notion of a tangent vector as a derivation. The other, based on the work of Yano and Ishihara generalizes the notion of a tangent vector as the velocity of a curve. The former leads to J2M, the 2-jet vector bundle consisting of second-order derivations, the latter leads to T~2)M, the bundle of curves agreeing up to acceleration. Both project naturally onto TM because the 1-jet bundle of first-order derivations and the bundle of curves agreeing up to velocity are isomorphs of TM. Both generalizations admit extension to higher orders but the second-order case illustrates their differences and is important in applications. It is always true that J2M is a vector bundle: but T~2~M is a vector bundle if and only if M has a linear connection and then TI2~M :-TM@TM with fiber R-'", whereas J2M always has fiber R ~': +3.~/2 We compare these constructions and give some results about TC2)M and the principal bundle L~Z~M to which it is associated. In a space-time there is a distinguished linear connection induced by the Lorentz metric, so both secondorder tangent structures are available and the reduction of J~-M to T~-~M is a considerable simplification in the case n = 4. We show also that both second-order bundles have applications to the study of space-time boundaries.

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“…∂ µ → 1 R(η) ∂ µ . The choice of the structural group G = R results in great computational simplifications but is not an essential limitation (see [4] and [8,Sect. 3.1]).…”

Section: Geometry Of the Gravitational Sectormentioning

confidence: 99%

Noncommutative closed Friedman universe

et al. 2008

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Heller et al. (J Math Phys 46:122501, 2005; Int J Theor Phys 46:2494, 2007 proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra A defined on a groupoid having the frame bundle over spacetime as its base space. The generalized Einstein equation is assumed in the form of the eigenvalue equation of the Einstein operator on a module of derivations of the algebra A. No matter sources are assumed. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, generalized eigenvalues of the Einstein operator reproduce components of the perfect fluid energy-momentum tensor for the usual Friedman model together with the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space-time geometry. These results are briefly discussed.

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Space-time edge geometry

Cited by 41 publications

References 41 publications

A noncommutative Friedman cosmological model

A noncommutative Friedman cosmological model

Second-order tangent structures

Noncommutative closed Friedman universe

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