arXiv subjects
Publications and source records attributed to E. Minguzzi.
Recently, an order completeness property has attracted attention in some low regularity spacetime geometry literature, where it was named `chronocompleteness'. In this work we prove that, in the framework of standard Lorentzian geometry, this property is equivalent to global hyperbolicity. The equivalence of global hyperbolicity with other more traditional forms of order completeness is also established.
We provide a short introduction to ``Lorentzian metric spaces" i.e., spacetimes defined solely in terms of the two-point Lorentzian distance. As noted in previous work, this structure is essentially unique if minimal conditions are imposed, such as the continuity of the Lorentzian distance and the relative compactness of chronological diamonds. The latter condition is natural for interpreting these spaces as low-regularity versions of globally hyperbolic spacetimes. Confirming this interpretation, we prove that every Lorentzian metric space admits a Cauchy time function. The proof is constructive for this general setting and it provides a novel argument that is interesting already for smooth spacetimes.
In this work, we seek characterizations of global hyperbolicity in smooth Lorentzian manifolds that do not rely on the manifold topology and that are inspired by metric geometry. In particular, strong causality is not assumed, so part of the problem is precisely that of recovering the manifold topology so as to make sense of it also in rough frameworks. After verifying that known standard characterizations do not meet this requirement, we propose two possible formulations. The first is based solely on chronological diamonds and is interesting due to its analogies with the Hopf-Rinow theorem. The second uses only properties of the Lorentzian distance function and it is suitable for extension to abstract `Lorentzian metric' frameworks. It turns out to be equivalent to the definition of `Lorentzian metric space' proposed in our previous joint work with S. Suhr, up to slightly strengthening weak $d$-distinction to `future or past $d$-distinction'. The role of a new property which we term {\em $d$-reflectivity} is also discussed. We then investigate continuity properties of the Lorentzian distance and the property of $d$-reflectivity in non-smooth frameworks. Finally, we establish a result of broader interest: the exponential map of a smooth spray is $C^{1,1}$ (smooth outside the zero section). Additionally, we derive a Lorentz-Finsler version of the Busemann-Mayer formula and demonstrate that, in strongly causal smooth Finsler spacetimes, the Finsler fundamental function can be reconstructed from the distance. As a consequence, distance-preserving bijections are shown to be Lorentz-Finsler isometries in the conventional smooth sense.
We introduce a notion of Lorentzian metric space which drops the boundedness condition from our previous work and argue that the properties defining our spaces are minimal. In fact, they are defined by three conditions given by (a) the reverse triangle inequality for chronologically related events, (b) Lorentzian distance continuity and relative compactness of chronological diamonds, and (c) a distinguishing condition via the Lorentzian distance function. By adding a countably generating condition we confirm the validity of desirable properties for our spaces including the Polish property. The definition of (pre)length space given in our previous work on the bounded case is generalized to this setting. We also define a notion of Gromov-Hausdorff convergence for Lorentzian metric spaces and prove that (pre)length spaces are GH-stable. It is also shown that our (sequenced) Lorentzian metric spaces bring a natural quasi-uniformity (resp. quasi-metric). Finally, an explicit comparison with other recent constructions based on our previous work on bounded Lorentzian metric spaces is presented.
We point out that the geometry of connected totally geodesic compact null hypersurfaces in Lorentzian manifolds is only slightly more specialized than that of Riemannian flows over compact manifolds, the latter mathematical theory having been much studied in the context of foliation theory since the work by Reinhart (Ann Math 69:119, 1959). We are then able to import results on Riemannian flows to the horizon case, so obtaining theorems on the dynamical structure of compact horizons that do not rely on (non-)degeneracy assumptions. Furthermore, we clarify the relation between isometric/geodesible Riemannian flows and non-degeneracy conditions. This work also contains some positive results on the possibility of finding, in the degenerate case, lightlike fields tangent to the horizon that have zero surface gravity.
Available proofs of the regularity of stationary black hole event horizons rely on certain assumptions on the existence of sections that imply a $C^1$ differentiability assumption. By using a quotient bundle approach, we remedy this problem by proving directly that, indeed, under the null energy condition event horizons of stationary black holes are totally geodesic null hypersurfaces as regular as the metric. Only later, by using this result, we show that the cross-sections, whose existence was postulated in previous works, indeed exist.These results hold true under weak causality conditions. Subsequently, we prove that under the dominant energy condition stationary black hole event horizons indeed admit constant surface gravity, a result that does not require any non-degeneracy assumption, requirements on existence of cross-sections or a priori smoothness conditions. We are able to make sense of the angular velocity and of the value (not just sign) of surface gravity as quantities related to the horizon, without the need of assuming Einstein's vacuum equations and the Killing extension. Physically, this implies that under very general conditions every stationary black hole has indeed a constant temperature (the zeroth law of black hole thermodynamics).
I 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.
The definitions of global hyperbolicity for closed cone structures and topological preordered spaces are known to coincide. In this work we clarify the connection with definitions of global hyperbolicity proposed in recent literature on Lorentzian length spaces and Lorentzian optimal transport, suggesting possible corrections for the terminology adopted in these works. It is found that in Kunzinger-S\"amann's Lorentzian length spaces the definition of global hyperbolicity coincides with that valid for closed cone structures and, more generally, for topological preordered spaces: the causal relation is a closed order and the causally convex hull operation preserves compactness. In particular, it is independent of the metric, chronological relation or Lorentzian distance.
In 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.
We 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.
We study an action integral for Finsler gravity obtained by pulling back an Einstein-Cartan-like Lagrangian from the tangent bundle to the base manifold. The vacuum equations are obtained imposing stationarity with respect to any section (observer) and are well posed as they are independent of the section. They imply that in vacuum the metric is actually independent of the velocity variable so the dynamics becomes coincident with that of general relativity.
The theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator E, E^2 = 0, that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product (D-module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.
In the standard approach to Finsler geometry the metric is defined as a vertical Hessian and the Chern or Cartan connections appear as just two among many possible natural linear connections on the pullback tangent bundle. Here it is shown that the Hessian nature of the metric, the non-linear connection and the Chern or Cartan connections can be derived from a few compatibility axioms between metric and Finsler connection. This result provides a metrical formulation of Finsler geometry which is well adapted to field theory, and which has proved useful in Einstein-Cartan-like approaches to Finsler gravity.
In this paper we extend the projective symmetry of the full metric-affine Einstein-Hilbert theory to a new symmetry transformation in the space of affine connections called the amplified symmetry. We prove that the Lagrangian of the Standard Model of particle physics is invariant under this new symmetry. We also show that the gravitational Lagrangian can be modified so that the amplified symmetry extends to the gravitational sector and hence to the whole action. The new theory so constructed is shown to be dynamically equivalent to Einstein-Cartan's though genuinely metric-affine.
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 $d_S$ 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 as 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.
The 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.
Reasonable 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 $C^{1,1}$ 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.
We prove that for continuous Lorentz-Finsler spaces timelike completeness implies inextendibility. Furthermore, we prove that under suitable locally Lipschitz conditions on the Finsler fundamental function the continuous causal curves that are locally length maximizing (geodesics) have definite causal character, either lightlike almost everywhere or timelike almost everywhere. These results generalize previous theorems by Galloway, Ling and Sbierski, and by Graf and Ling.