Affine Sphere Relativity

Minguzzi, E.

doi:10.1007/s00220-016-2802-9

Cited by 18 publications

(22 citation statements)

References 93 publications

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“…Since we can always find another C 2 (subsidiary) Finsler Lagrangian L with the same light cones, the unparametrized lightlike geodesics are indeed well defined provided we can show that they are independent ofL . This is precisely what we proved in [1].…”

Section: Geodesic Flow On the Lightlike Cotangent Bundlesupporting

confidence: 86%

“…In a series of recent works we have stressed the importance of the Lorentz-Finsler spaces for which I α = 0, see [1,7,8]. They are precisely the affine sphere spacetime which we met before.…”

Section: Lorentz-finsler Geometrymentioning

confidence: 88%

“…A remarkable fact concerning the concept of affine sphere spacetime is that it embodies in a nice way the non-relativistic limit [1]. In fact, non-relativistic spacetimes admit the same description of (relativistic) affine sphere spacetimes: the cotangent indicatrix is determined by a distribution of affine spheres which, in the non-relativistic case, have center at infinity.…”

Section: Lorentz-finsler Geometrymentioning

confidence: 99%

“…In this work we are going to present some recent results on anisotropic gravity theories obtained in [1,2]. Although, ultimately the theory is of Lorentz-Finsler type we wish to introduce it from a different angle, emphasizing the role of affine differential geometry in its development.…”

Section: Introductionmentioning

confidence: 99%

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Gravity theory through affine spheres

Minguzzi

2017

J. Phys.: Conf. Ser.

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Abstract. In this work it is argued that in order to improve our understanding of gravity and spacetime our most successful theory, general relativity, must be destructured. That is, some geometrical assumptions must be dropped and recovered just under suitable limits. Along this line of thought we pursue the idea that the roundness of the light cone, and hence the isotropy of the speed of light, must be relaxed and that, in fact, the shape of light cones must be regarded as a dynamical variable. Mathematically, we apply some important results from affine differential geometry to this problem, the idea being that in the transition we should preserve the identification of the spacetime continuum with a manifold endowed with a cone structure and a spacetime volume form. To that end it is suggested that the cotangent indicatrix (dispersion relation) must be described by an equation of Monge-Ampère type determining a hyperbolic affine sphere, at least whenever the matter content is negligible. Non-relativistic spacetimes fall into this description as they are recovered whenever the center of the affine sphere is at infinity. In the more general context of Lorentz-Finsler theories it is shown that the lightlike unparametrized geodesic flow is completely determined by the distribution of light cones. Moreover, the transport of lightlike momenta is well defined though there could be no notion of affine parameter. Finally, we show how the perturbed indicatrix can be obtained from the perturbed light cone.

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Section: Geodesic Flow On the Lightlike Cotangent Bundlesupporting

confidence: 86%

Section: Lorentz-finsler Geometrymentioning

confidence: 88%

Section: Lorentz-finsler Geometrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gravity theory through affine spheres

Minguzzi

2017

J. Phys.: Conf. Ser.

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“…Let us consider a positive homogeneous function F : C → [0, +∞) such that F −1 (0) = ∂C, L := −F 2 /2 is such that the vertical Hessian on C, d 2 y L , is smooth and Lorentzian, and dL = ∅ on ∂C. This function exists by [19,Prop. 13], see also [12,Cor.…”

Section: The Signed Distance Function and Its Regularitymentioning

confidence: 99%

On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

Minguzzi

2020

Rev. Math. Phys.

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We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance dS from a spacelike hypersurface S is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity of the hypersurface. We also show that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result. Lemma 2.3. Let (M, C) be a closed cone structure. Let S be a locally stably acausal Cauchy hypersurface then J + (S) and J − (S) are closed, J + (S) ∩ J − (S) = S, J + (S) ∪ J − (S) = M . The set M \S is disconnected, for it is the union of the disjoint non-empty open sets J + (S)\S = M \J − (S) and J − (S)\S = M \J + (S). Proof. Through every point passes an inextendible continuous causal curve intersecting S, thus J + (S) ∪ J − (S) = M . The acausality of S implies J + (S) ∩ J − (S) = S.Suppose that there is a point p ∈ J + (S)\J + (S) then necessarily p ∈ J − (S)\S. Since p ∈ J + (S)\J + (S) by a standard limit curve argument [20, Thm. 2.14]

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Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

Caponio,

Corona,

Giambò

et al. 2024

Annali di Matematica

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We consider an autonomous, indefinite Lagrangian L admitting an infinitesimal symmetry K whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point p to a flow line $$\gamma =\gamma (t)$$ γ = γ ( t ) of K that does not cross p. By utilizing the invariance of L under the flow of K, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the “arrival time” t, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When L is positively homogeneous of degree 2 in the velocities, the resulting equation establishes a variational principle that extends the Fermat’s principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.

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Affine Sphere Relativity

Cited by 18 publications

References 93 publications

Gravity theory through affine spheres

Gravity theory through affine spheres

On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

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