Lorentzian Distance Research Articles

Lorentzian distance on the Lobachevsky plane *

Left-invariant Lorentzian structures on the 2D solvable non-Abelian Lie group are studied. Sectional curvature, attainable sets, Lorentzian length maximizers, distance, spheres, and infinitesimal isometries are described.

Lorentzian metric spaces and their Gromov–Hausdorff convergence

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.

Minkowski Space from Quantum Mechanics

Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.

Further observations on the definition of global hyperbolicity under low regularity

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–Sämann’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.

Distribution Disagreement via Lorentzian Focal Representation.

Error disagreement-based active learning (AL) selects the data that maximally update the error of a classification hypothesis. However, poor human supervision (e.g., few labels, improper classifier parameters) may weaken or clutter this update; moreover, the computational cost of performing a greedy search to estimate the errors using a deep neural network is intolerable. In this paper, a novel disagreement coefficient based on distribution, not error, provides a tighter bound on label complexity, which further guarantees its generalization in hyperbolic space. The focal points derived from the squared Lorentzian distance, present more effective hyperbolic representations on aspherical distribution from geometry, replacing the typical euclidean, kernelized, and Poincaré centroids. Experiments on different deep AL tasks show that, the focal representation adopted in a tree-likeliness splitting, significantly performs better than typical baselines of centroid representations and error disagreement, and state-of-the-art neural network architectures-based AL, dramatically accelerating the learning process.

Lorentzian distance functions in contact geometry

An important tool to analyse the causal structure of a Lorentzian manifold is given by the Lorentzian distance function. We define a class of Lorentzian distance functions on the group of contactomorphisms of a closed contact manifold depending on the choice of a contact form. These distance functions are continuous with respect to the Hofer norm for contactomorphisms defined by Shelukhin [The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if and only if the group of contactomorphisms is orderable. To prove this, we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined in [D. Rosen and J. Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165–1194] is nondegenerate. In this case, similar results hold for a Lorentzian distance functions on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.

Computing Expectiles Using k-Nearest Neighbours Approach

Expectiles have gained considerable attention in recent years due to wide applications in many areas. In this study, the k-nearest neighbours approach, together with the asymmetric least squares loss function, called ex-kNN, is proposed for computing expectiles. Firstly, the effect of various distance measures on ex-kNN in terms of test error and computational time is evaluated. It is found that Canberra, Lorentzian, and Soergel distance measures lead to minimum test error, whereas Euclidean, Canberra, and Average of (L1,L∞) lead to a low computational cost. Secondly, the performance of ex-kNN is compared with existing packages er-boost and ex-svm for computing expectiles that are based on nine real life examples. Depending on the nature of data, the ex-kNN showed two to 10 times better performance than er-boost and comparable performance with ex-svm regarding test error. Computationally, the ex-kNN is found two to five times faster than ex-svm and much faster than er-boost, particularly, in the case of high dimensional data.

Spectral distance on Lorentzian Moyal plane

We present here a completely operatorial approach, using Hilbert–Schmidt operators, to compute spectral distances between time-like separated “events”, associated with the pure states of the algebra describing the Lorentzian Moyal plane, using the axiomatic framework given by [N. Franco, The Lorentzian distance formula in noncommutative geometry, J. Phys. Conf. Ser. 968(1) (2018) 012005; N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26(8) (2014) 1430007]. The result shows no deformations of non-commutative origin, as in the Euclidean case, if the pure states are constructed out of Glauber–Sudarshan coherent states.

Splitting theorems for hypersurfaces in Lorentzian manifolds

This paper looks at the splitting problem for globally hyperbolic spacetimes with timelike Ricci curvature bounded below containing a (spacelike, acausal, future causally complete) hypersurface with mean curvature bounded from above. For such spacetimes we show a splitting theorem under the assumption of either the existence of a ray of maximal length or a maximality condition on the volume of Lorentzian distance balls over the hypersurface. The proof of the first case follows work by Andersson, Galloway and Howard and uses their geometric maximum principle for level sets of the (Lorentzian) Busemann function. For the second case we give a more elementary proof.

The distance formula in algebraic spacetime theories

The 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.

The Lorentzian distance formula in noncommutative geometry

For almost twenty years, a search for a Lorentzian version of the well-known Connes’ distance formula has been undertaken. Several authors have contributed to this search, providing important milestones, and the time has now come to put those elements together in order to get a valid and functional formula. This paper presents a historical review of the construction and the proof of a Lorentzian distance formula suitable for noncommutative geometry.

Rodrigues parameters on dual hyperbolic unit sphere

Rodrigues parameters depend on the tangent of the half rotation angle in Euclidean space but in Dual space, dual Rodrigues parameters contain both rotation angle and distance corresponding the shortest distance between the straight lines in ℝ3. In this paper, we give Cayley's formula for the dual hyperbolic spherical motion and explain 3×3 type L-Dual skew symmetric matrices by using properties of this formula. Then, we obtain Rodrigues pa­rameters of dual Hyperbolic unit sphere and show that Rodrigues parameters contain the hyperbolic rotation angle which is being between timelike lines and distance which is the minimal Lorentzian distance between the timelike lines of .

Active contours driven by novel LGIF energies for image segmentation

An active contour model driven by novel local and global intensity fitting (LGIF) energies for image segmentation is presented. First, the GIF energy is defined by the squared-chord distance, which is more robust to noise. Second, the LIF energy is defined by the Lorentzian distance to calculate the local intensity information, which improves the generality of the presented model. Experiments are carried out on synthetic and real images and the results illustrate that the presented model can obtain higher segmentation accuracy and better segmentation efficiency; moreover, it is not sensitive to initial contour.

Corrigendum to “Generalised time functions and finiteness of the Lorentzian distance” [J. Geom. Phys. 106 (2016) 108–121

Corrigendum to “Generalised time functions and finiteness of the Lorentzian distance” [J. Geom. Phys. 106 (2016) 108–121

The use of Lorentzian distance metric in classification problems

In this paper, we introduce Lorentzian distance metric into classification problem. Here we benefit from the interesting properties of the distance metric of the Lorentzian space. A preprocessing step composed of basic mathematical operations such as compression and shifting is necessary to prepare for using the Euclidean data in the Lorentzian space. For defining validity and usability of this method, 6 public datasets (CLIMATE, GESTURE, PARKINSON, RELAX, VERTEBRAL, WINE) are used in our experiments. The experimental results are compared with k Nearest Neighbor (kNN) and other well-known classification methods. Test results show that our method produces better classification rates in most cases. Furthermore, for increasing the classification rate, we improve our base method with some extensions. We investigate the influence of parameters in compression matrix and obtain optimum values. On the other hand, we add a rotation operation into the preprocessing step. These improvements increase significantly the classification rate.

Geometric analysis of the Lorentzian distance function on trapped submanifolds

We use geometric analysis of the Lorentzian distance on an n-dimensional spacetime to derive some non-existence results and some sharp mean curvature estimates for trapped submanifolds contained in the chronological future of either a point or of a space-like achronal hypersurface . Our results are given as an application of the corresponding Hessian comparison results for the Lorentzian distance function and the so-called weak maximum principle for the Laplacian operator.

Generalised time functions and finiteness of the Lorentzian distance

We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and manipulate achronal sets. As a consequence of these techniques we obtain a functional description of the Lorentzian distance extending the work of Franco (2010) and Moretti (2003).

Raychaudhuri equation and singularity theorems in Finsler spacetimes

The Raychaudhuri equation and its consequences for chronality are studied in the context of Finsler spacetimes. It is proved that the notable singularity theorems of Lorentzian geometry extend to the Finslerian domain. Indeed, so do the theorems by Hawking, Penrose, Hawking and Penrose, Geroch, Gannon, Tipler and Kriele, and also the Topological Censorship theorem and so on. It is argued that the notable results in causality theory connected to achronal sets, future sets, domains of dependence, limit curve theorems, length functional, Lorentzian distance and geodesic connectedness, extend to the Finslerian domain. Results concerning the spacetime asymptotic structure, horizons differentiability and conformal transformations are also included.

Comportamiento de causalidad en el espacio-tiempo de Carter

En esta investigación nos enfocamos en el carácter causal del espaciotiempo de Carter (ver [2], [10]). Este espaciotiempo es importante por la siguiente razón: para los espaciotiempos con un comportamiento causal óptimo, es decir, los globalmente hiperbólicos, existen varias caracterizaciones o definiciones alternativas. En algunos casos se ha demostrado que ciertas condiciones de causalidad requeridas en tales caracterizaciones pueden relajarse.Pero el espaciotiempo de Carter nos da un contraejemplo que hace imposible la relajación en una de ellas. Basándonos en estos resultados previos, estudiamos la posibilidad de que el espaciotiempo de Carter sea también un contraejemplo para otra caracterización.En particular, demostraremos que la separación temporal o distancia Lorentziana entre dos puntos del espaciotiempo de Carter es infinita. Si bien este espaciotiempo resultó no ser el contraejemplo buscado, la conclusión es de por sí interesante y aporta ideas alternativas para estudiar la posibilidad o no de rebajar la condición en la caracterización mencionada.

Temporal Lorentzian spectral triples

We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3 + 1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal–Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.