1,721,017 research outputs found
The distance formula in algebraic spacetime theories
No full textThe Lorentzian distance formula, conjectured several years ago by Parfionov and Zapatrin, has been recently proved by the second author. In this work we focus on the derivation of an equivalent expression in terms of the geometry of 2-spinors by using a partly original approach due to the first author. Our calculations clearly show the independence of the algebraic distance formula of the observer
Hints at a relationship between friction and relativistic physics
Get PDFIn applied mechanics Reye's law (1860) establishes, via energy arguments,
that the mass of the debris produced by dry friction in the contact of rigid
bodies is proportional to the work done by friction forces. This result has
long been used for the determination of the distribution of pressure in the
contact of rigid bodies, and hence for the design of brakes. In this work I
show that, when bodies losing mass due to friction are treated, as they should,
as variable mass systems, a relationship analogous to the relativistic mass
formula is recovered. This result suggests that mathematical structures typical
of relativistic physics could have been discovered prior to 1905, without
making any reference to electromagnetism, group theory or the speed of light.
Also this result could point to the existence of a physical theory depending on
two constants, the speed of light (Reye's constant) and a universal frictional
deceleration with respect to an absolute frame. The limit of the theory for
vanishing friction would give Special Relativity, as the absolute frame would
become unobservable, while the limit for the speed of light going to infinity
would lead to Aristotelian mechanics, i.e. a classical mechanics type theory
presenting universal friction. Finally, I present a reference frame
transformation that displays these features and I apply the theory to some open
cosmological problems.Comment: Latex, 18 pages, 9 figures. Contribution to the conference DICE2022
`Quantum riddles and spacetime oddities', Castiglioncello, Sep. 19-23, 2022.
arXiv admin note: text overlap with arXiv:1412.001
Indicatrix Geometry Clarifies that Finsler Length can be Larger than Relative Length
No full textI show that Matsumoto conjectured inequality between relative length and Finsler length is false. The incorrectness of the claim is easily inferred from the geometry of the indicatrix
A gravitational collapse singularity theorem consistent with black hole evaporation
Get PDFThe global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work, I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in the absence of predictability and hence compatibly with quantum field theory and black hole evaporation
Lorentzian causality theory
No full textI review Lorentzian causality theory paying particular attention to the optimality and generality of the presented results. I include complete proofs of some foundational results that are otherwise difficult to find in the literature (e.g. equivalence of some Lorentzian length definitions, upper semi-continuity of the length functional, corner regularization, etc.). The paper is almost self-contained thanks to a systematic logical exposition of the many different topics that compose the theory. It contains new results on classical concepts such as maximizing curves, achronal sets, edges, horismos, domains of dependence, Lorentzian distance. The treatment of causally pathological spacetimes requires the development of some new versatile causality notions, among which I found particularly convenient to introduce: biviability, chronal equivalence, araying sets, and causal versions of horismos and trapped sets. Their usefulness becomes apparent in the treatment of the classical singularity theorems, which is here considerably expanded in the exploration of some variations and alternatives
Inclusion of a perfect fluid term into the Einstein-Hilbert action
No full textI show that it is possible to obtain the stress-energy tensor of the perfect fluid by adding a suitable term to the Einstein-Hilbert action. Variation should be understood with respect to the metric
Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime
No full textI characterize the Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime (i.e., the Lorentzian submanifolds of Minkowski spacetime that are also closed subsets). Moreover, I prove that the Lorentzian manifolds that can be properly conformally embedded in Minkowski spacetime coincide with the globally hyperbolic spacetimes. Finally, by taking advantage of the embedding, I obtain an infinitesimal version of the distance formula
Globally hyperbolic spacetimes can be defined without the `causal' condition
No full textReasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non C1,1 metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity
On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures
No full textWe 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 from a spacelike hypersurface 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
Lorentzian metric spaces and their Gromov–Hausdorff convergence
No full textWe present an abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, and an alternative approach to Lorentzian length spaces that does not use auxiliary "positive signature" metrics or other unobserved fields. We begin by defining a notion of (abstract) bounded Lorentzian metric space which is sufficiently general to comprise compact causally convex subsets of globally hyperbolic spacetimes and causets. We define the Gromov-Hausdorff distance and show that two bounded Lorentzian metric spaces at zero GH distance are indeed both isometric and homeomorphic. Then we show how to define from the Lorentzian distance, beside topology, the causal relation and the causal curves for these spaces, obtaining useful limit curve theorems. Next, we define Lorentzian (length) prelength spaces via suitable (maximal) chronal connectedness properties. These definitions are proved to be stable under GH limits. Furthermore, we define bounds on sectional curvature for our Lorentzian length spaces and prove that they are also stable under GH limits. We conclude with a (pre)compactness theorem
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