Lorentzian Geometry Research Articles

On the geometrical properties of Heisenberg groups

In [20] the existence of major differences about totally geodesic two-dimensional foliations between Riemannian and Lorentzian geometry of the Heisenberg group $H_{3}$ is proved. Our aim in this paper is to obtain a comparison on some other geometrical properties of these spaces. Interesting behaviours are found. Also the non-existence of left-invariant Ricci and Yamabe solitons and the existence of algebraic Ricci soliton in both Riemannian and Lorentzian cases are proved. Moreover, all of the descriptions of their homogeneous Riemannian and Lorentzian structures and their types are obtained. Besides, all the left-invariant generalized Ricci solitons and unit time-like vector fields which are spatially harmonic are completely determined.

De Sitter space and entanglement

We argue that the notion of entanglement in de Sitter space arises naturally from the non-trivial Lorentzian geometry of the spacetime manifold, which consists of two disconnected boundaries and a causally disconnected interior. In four bulk dimensions, we propose an holographic description of an inertial observer in terms of a thermofield double state in the tensor product of the two boundaries Hilbert spaces, whereby the Gibbons–Hawking formula arises as the holographic entanglement entropy between the past and future conformal infinities. When considering the bulk entanglement between the two causally disconnected Rindler wedges, we show that the corresponding entanglement entropy is given by one quarter of the area of the pair of codimension two minimal surfaces that define the set of fixed points of the dS orbifold.

Matrix Extension of the Manakov-Santini System and an Integrable Chiral Model on an Einstein-Weyl Background

It was demonstrated recently [Dunajski, Ferapontov and Kruglikov (2014)] that the Manakov-Santini system describes a local form of general Lorentzian Einstein-Weyl geometry. We introduce integrable matrix extension of the Manakov-Santini system and show that it describes (2+1)-dimensional integrable chiral model in Einstein-Weyl space. We develop a dressing scheme for the extended MS system and define an extended hierarchy. Matrix extension of Toda type system connected with another local form of Einstein-Weyl geometry is also considered.

Canonical Kähler metrics on classes of Lorentzian 4-manifolds

Conditions for the existence of Kahler–Einstein metrics and central Kahler metrics (Maschler in Trans Am Math Soc 355:2161–2182, 2003) along with examples, both old and new, are given on classes of Lorentzian 4-manifolds with two distinguished vector fields. The results utilize the general construction (Aazami and Maschler in Kahler metrics via Lorentzian geometry in dimension four, Complex Manifolds 7:36–61 (2020) of Kahler metrics on such manifolds. The examples include both complete and incomplete metrics, and some reside on Lie groups associated with four types of Lie algebras. An appendix includes a similar construction for scalar-flat Kahler metrics.

The future is not always open

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open and may differ from the corresponding sets defined via piecewise C^1-curves. By refining the notion of a causal bubble from Chruściel and Grant (Class Quantum Gravity 29(14):145001, 2012), we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low-regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

Higher spin fluctuations on spinless 4D BTZ black hole

We construct linearized solutions to Vasiliev’s four-dimensional higher spin gravity on warped AdS3 ×ξS1 which is an Sp(2) × U(1) invariant non-rotating BTZ-like black hole with ℝ2 × T2 topology. The background can be obtained from AdS4 by means of identifications along a Killing boost K in the region where ξ2 ≡ K2 ≥ 0, or, equivalently, by gluing together two Bañados-Gomberoff-Martinez eternal black holes along their past and future space-like singularities (where ξ vanishes) as to create a periodic (non-Killing) time. The fluctuations are constructed from gauge functions and initial data obtained by quantizing inverted harmonic oscillators providing an oscillator realization of K and of a commuting Killing boost tilde{K} . The resulting solution space has two main branches in which K star commutes and anti-commutes, respectively, to Vasiliev’s twisted-central closed two-form J. Each branch decomposes further into two subsectors generated from ground states with zero momentum on S1. We examine the subsector in which K anti-commutes to J and the ground state is mathrm{U}{(1)}_Ktimes mathrm{U}{(1)}_{tilde{K}} -invariant of which U(1)K is broken by momenta on S1 and mathrm{U}{(1)}_{tilde{K}} by quasi-normal modes. We show that a set of mathrm{U}{(1)}_{tilde{K}} -invariant modes (with n units of S1 momenta) are singularity-free as master fields living on a total bundle space, although the individual Fronsdal fields have membrane-like singularities at {tilde{K}}^2=1 . We interpret our findings as an example where Vasiliev’s theory completes singular classical Lorentzian geometries into smooth higher spin geometries.

The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces

In this work, we are interested in the differential geometry of surfaces in simply isotropic $\mathbb{I}^3$ and pseudo-isotropic $\mathbb{I}_{\mathrm{p}}^3$ spaces, which consists of the study of $\mathbb{R}^3$ equipped with a degenerate metric such as $\mathrm{d}s^2=\mathrm{d}x^2\pm\mathrm{d}y^2$. The investigation is based on previous results in the simply isotropic space [B. Pavkovi\'c, Glas. Mat. Ser. III $\mathbf{15}$, 149 (1980); Rad JAZU $\mathbf{450}$, 129 (1990)], which point to the possibility of introducing an isotropic Gauss map taking values on a unit sphere of parabolic type and of defining a shape operator from it, whose determinant and trace give the known relative Gaussian and mean curvatures, respectively. Based on the isotropic Gauss map, a new notion of connection is also introduced, the \emph{relative connection} (\emph{r-connection}, for short). We show that the new curvature tensor in both $\mathbb{I}^3$ and $\mathbb{I}_{\mathrm{p}}^3$ does not vanish identically and is directly related to the relative Gaussian curvature. We also compute the Gauss and Codazzi-Mainardi equations for the $r$-connection and show that $r$-geodesics on spheres of parabolic type are obtained via intersections with planes passing through their center (focus). Finally, we show that admissible pseudo-isotropic surfaces are timelike and that their shape operator may fail to be diagonalizable, in analogy to Lorentzian geometry. We also prove that the only totally umbilical surfaces in $\mathbb{I}_{\mathrm{p}}^3$ are planes and spheres of parabolic type and that, in contrast to the $r$-connection, the curvature tensor associated with the isotropic Levi-Civita connection vanishes identically for $any$ pseudo-isotropic surface, as also happens in simply isotropic space.

Interval topology in contact geometry

A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.

The inverse kinematics of rolling contact of timelike curves lying on timelike surfaces

Rolling contact between two surfaces plays an important role in robotics and engineering such as spherical robots, single wheel robots, and multi-fingered robotic hands to drive a moving surface on a fixed surface. The rolling contact pairs have one, two, or three degrees of freedom (DOFs) consisting of angular velocity components. Rolling contact motion can be divided into two categories: spin-rolling motion and pure-rolling motion. Spin-rolling motion has three (DOFs), and pure-rolling motion has two (DOFs). Further, it is well known that the contact kinematics can be divided into two categories: forward kinematics and inverse kinematics. In this paper, we investigate the inverse kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface in the direction of timelike unit tangent vectors of their timelike trajectory curves by determining the desired motion and the coordinates of the contact point on each surface. We get three nonlinear algebraic equations as inputs by using curvature theory in Lorentzian geometry. These equations can be reduced as a univariate polynomial of degree six by applying the Darboux frame method. This polynomial enables us to obtain rapid and accurate numerical root approximations and to analyze the rolling rate as an output. Moreover, we obtain another outputs: the rolling direction and the compensatory spin rate.

Asymptotic analysis of spin foam amplitude with timelike triangles

The large $j$ asymptotic behavior of $4$-dimensional spin foam amplitude is investigated for the extended spin foam model (Conrady-Hnybida extension) on a simplicial complex. We study the most general situation in which timelike tetrahedra with timelike triangles are taken into account. The large $j$ asymptotic behavior is determined by critical configurations of the amplitude. We identify the critical configurations that correspond to the Lorentzian simplicial geometries with timelike tetrahedra and triangles. Their contributions to the amplitude are phases asymptotically, whose exponents equal to Regge action of gravity. The amplitude may also contains critical configurations corresponding to non-degenerate split signature $4$-simplices and degenerate vector geometries. But for vertex amplitudes containing at least one timelike tetrahedron and one spacelike tetrahedron, critical configurations only give Lorentzian $4$-simplices, while the split signature and degenerate $4$-simplices do not appear.

Causal structure of evaporating black holes

We offer a mathematically rigorous basis for the widely held suspicion that full black hole evaporation is in tension with predictability. Based on conditions expressing the global causal structure of evaporating black hole spacetimes, we prove two theorems in Lorentzian geometry showing that such spacetimes either fail to be causally simple or fail to be causally continuous. These theorems, when combined with recent results (Aké et al 2018 (arXiv:1808.04412 [gr-qc])) on the causal structure of spacetimes with timelike boundary, bear significantly on the question of whether these spacetimes permit for a predictable evolution.

Airplane boarding meets express line queues

Airlines have been implementing several policies which partition passengers into groups of fast and slow passengers and placing them in different queue/row combinations. For example, the policy which provides boarding precedence to passengers needing assistance and families traveling with small children, or a policy which provides precedence to passengers with no overhead bin luggage. The idea of separating customers into groups with different service times also appears in the very different setting of express line queues, for example, in the supermarket or in server farms. Such queues have been extensively explored in the last 20 years in the queueing theoretic literature.We show that the two systems, airplane boarding with slow and fast passenger groups and express line queues are intimately related in the asymptotic regime where the ratio between the slowest and fastest customer becomes large. We produce good algorithms for placing different groups of passengers in the airplane boarding setting with competitive guarantees compared to optimal placements. We then show how we can analyze the airplane boarding setting using results from the express line setting. This relation provides a novel bridge between the theory of project management and the critical path method to which airplane boarding belongs and queueing theory to which express line queues belong.The analysis uses very basic notions and arguments from geometry, but in the setting of space-time (Lorentzian) geometry, a geometry which was invented to study relativity theory.

A universal linear algebraic model for conformal geometries

This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown analytically, through a framework comparing the measurement of distances and angles in Cayley–Klein geometries, including Lorentzian geometries, as done by F. Bachmann and later R. Struve. On the other hand, such a relationship may also be expressed in a purely linear algebraic manner, as explained by D. Hestens, H. Li and A. Rockwood. The model described in this article unifies these approaches via a generalization of Lie sphere geometry. Like the work of N. Wildberger, it is a purely algebraic construction, and as such it works over any field of odd characteristic. It is shown that measurement of distances and angles is an inherent property of the model that is easy to identify, and the possible models are classified over the real, complex and finite fields, and partially in characteristic 2, revealing a striking analogy between the real and finite geometries. This is an abbreviated version of a previous manuscript, with certain expository parts removed for the sake brevity. The original manuscript is available on the website arXiv, with more motivation, examples, properties. Several definitions and theorems can be extended to include the characteristic 2 case, which are also omitted here.

Nonrelativistic string theory and T-duality

Nonrelativistic string theory in flat spacetime is described by a two-dimensional quantum field theory with a nonrelativistic global symmetry acting on the worldsheet fields. Nonrelativistic string theory is unitary, ultraviolet complete and has a string spectrum and spacetime S-matrix enjoying nonrelativistic symmetry. The worldsheet theory of nonrelativistic string theory is coupled to a curved spacetime background and to a Kalb-Ramond two-form and dilaton field. The appropriate spacetime geometry for nonrelativistic string theory is dubbed string Newton-Cartan geometry, which is distinct from Riemannian geometry. This defines the sigma model of nonrelativistic string theory describing strings propagating and interacting in curved background fields. We also implement T-duality transformations in the path integral of this sigma model and uncover the spacetime interpretation of T-duality. We show that T-duality along the longitudinal direction of the string Newton-Cartan geometry describes relativistic string theory on a Lorentzian geometry with a compact lightlike isometry, which is otherwise only defined by a subtle infinite boost limit. This relation provides a first principles definition of string theory in the discrete light cone quantization (DLCQ) in an arbitrary background, a quantization that appears in nonperturbative approaches to quantum field theory and string/M-theory, such as in Matrix theory. T-duality along a transverse direction of the string Newton-Cartan geometry equates nonrelativistic string theory in two distinct, T-dual backgrounds.

Regular instantons in the Eddington-inspired-Born-Infeld gravity: Lorentzian wormholes from bubble nucleations

The O(4)-symmetric regular instanton solutions are studied within the framework of the Eddington-inspired-Born-Infeld gravity (EiBI). We find that the behavior of the solution would deviate from that in Einstein gravity when the kinetic energy of the scalar field is sufficiently large. The tunneling probability is calculated numerically in different parameter space. We find that the tunneling probability would increase with the Born-Infeld coupling constant, which is assumed to be positive in this paper. Furthermore, we discover a neck feature in the physical instanton solutions when the kinetic energy of the scalar field is sufficiently large. This feature can be interpreted as a Lorentzian time-like wormhole geometry formed during the bubble materialization.

Lorentzian length spaces

We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way, we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

Lorentzian geometry for detecting qubit entanglement

We describe a new approach based on Lorentzian geometry to detect qubit entanglement. The treatment is based physically, on the causal structure of Minkowski spacetime, and mathematically, on a Lorentzian Singular Value Decomposition. A surprising feature is the natural emergence of “Energy conditions” used in Relativity. All states satisfy a “Dominant Energy Condition” (DEC) and separable states satisfy the Strong Energy Condition(SEC), while entangled states violate the SEC. We thus propose a test for two qubit entanglement which is an alternative to the positive partial transpose (PPT) test. This test is based on the partial Lorentz transformation (PLT) on individual qubits. Apart from testing for entanglement, our approach also enables us to construct a separable form for the density matrix in those cases where it exists. Our approach leads to a simple graphical three dimensional representation of the state space which shows the entangled states within the set of all states.

Antipodal correlation on the meron wormhole and a bang-crunch universe

We present a covariant euclidean wormhole solution to Einstein Yang-Mills system and study scalar perturbations analytically. The fluctuation operator has a positive definite spectrum. We compute the Euclidean Green's function, which displays maximal antipodal correlation on the smallest three sphere. Upon analytic continuation, it corresponds to the Feynman propagator on a compact bang-crunch universe. We present the connection matrix that relates past and future modes. We thoroughly discuss the physical implications of the antipodal map in both the Euclidean and Lorentzian geometries.

Curvature-dimension bounds for Lorentzian splitting theorems

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-\'Emery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when $N=1$. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry-\'Emery-Ricci tensor for all $N\in (-\infty, 2]\cup (n,\infty)$ and for the $N=\infty$ case as well, with reduced rigidity if $N=2$. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-\'Emery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension.

Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics

An analogy with real Clifford algebras on even-dimensional vector spaces suggests assigning an ordered pair (s, t) of space and time dimensions (or equivalently an ordered pair (m, n) of metric and KO dimensions) modulo 8 to any algebraic structure (that we call CPT corepresentation) represented over a Hilbert space by two self-adjoint involutions and an anti-unitary operator having specific commutation relations. It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product of two CPT corepresentations are the sums of the space and time dimensions of its factors, and the same holds for the metric and KO dimensions. This could provide an interpretation of the presence of such algebras in PT-symmetric Hamiltonians or the description of topological matter. This construction is used to build an indefinite (i.e., pseudo-Riemannian) version of the spectral triple of noncommutative geometry, defined over a Krein space and classified by the pair (m, n) instead of the KO dimension only. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.