Lorentzian Surfaces Research Articles

Relation between the minimal Lorentz surfaces in ℝ24 and ℝ13

In this paper, we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space [Formula: see text] using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in [Formula: see text] and the pairs of null curves in [Formula: see text]. Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in [Formula: see text] and the pairs of minimal Lorentz surfaces in [Formula: see text]. As an application, we give examples showing how we can obtain a minimal Lorentz surface in [Formula: see text] using a pair of minimal Lorentz surfaces in [Formula: see text].

Carroll black holes

Despite the absence of a lightcone structure, some solutions of Carroll gravity show black hole-like behaviour. We define Carroll black holes as solutions of Carroll gravity that exhibit Carroll thermal properties and have a Carroll extremal surface, notions introduced in our work. The latter is a Carroll analogue of a Lorentzian extremal surface. As examples, we discuss the Carroll versions of Schwarzschild, Reissner-Nordström, and BTZ black holes and black hole solutions of generic 1+1 dimensional Carroll dilaton gravity, including Carroll JT and Carroll Witten black holes.

Combinatorial Quantum Gravity and Emergent 3D Quantum Behaviour

We review combinatorial quantum gravity, an approach that combines Einstein’s idea of dynamical geometry with Wheeler’s “it from bit” hypothesis in a model of dynamical graphs governed by the coarse Ollivier–Ricci curvature. This drives a continuous phase transition from a random to a geometric phase due to a condensation of loops on the graph. In the 2D case, the geometric phase describes negative-curvature surfaces with two inversely related scales: an ultraviolet (UV) Planck length and an infrared (IR) radius of curvature. Below the Planck scale, the random bit character survives; chunks of random bits of the Planck size describe matter particles of excitation energy given by their excess curvature. Between the Planck length and the curvature radius, the surface is smooth, with spectral and Hausdorff dimension 2. At scales larger than the curvature radius, particles see the surface as an effective Lorentzian de Sitter surface, the spectral dimension becomes 3, and the effective slow dynamics of particles, as seen by co-moving observers, emerges as quantum mechanics in Euclidean 3D space. Since the 3D distances are inherited from the underlying 2D de Sitter surface, we obtain curved trajectories around massive particles also in 3D, representing the large-scale gravity interactions. We thus propose that this 2D model describes a generic holographic screen relevant for real quantum gravity.

Gauss–Bonnet Theorems for Lorentzian and Spacelike Surfaces Associated to Canonical Connections in the Lorentzian Heisenberg Group

Gauss–Bonnet Theorems for Lorentzian and Spacelike Surfaces Associated to Canonical Connections in the Lorentzian Heisenberg Group

Timelike loxodromes on Lorentzian helicoidal surfaces in Minkowski n-space

In this paper, we examine timelike loxodromes on three kinds of Lorentzian helicoidal surfaces in Minkowski nn-space. First, we obtain the first order ordinary differential equations which determine timelike loxodromes on the Lorentzian helicoidal surfaces in En1E1n according to the causal characters of their meridian curves. Then, by finding general solutions, we get the explicit parametrizations of such timelike loxodromes. In particular, we investigate the timelike loxodromes on the three kinds of Lorentzian right helicoidal surfaces in En1E1n. Finally, we give an example to visualize the results.

Spinorial Representation of Surfaces in Lorentzian Homogeneous Spaces of Dimension $$\varvec{3}$$

We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension 3. We in particular obtain a representation theorem for surfaces in \(\mathbb {L}(\kappa ,\tau )\) spaces. We then recover the Calabi correspondence between minimal surfaces in \(\mathbb {R}^3\) and maximal surfaces in \(\mathbb {R}_1^3\), and obtain a new Lawson type correspondence between CMC surfaces in \(\mathbb {R}_1^3\) and in the 3-dimensional pseudo-hyperbolic space \(\mathbb {H}_1^{3}.\)

Maximal simply connected Lorentzian surfaces with a Killing field and their completeness

In the first part of this paper, we give a global description of simply connected maximal Lorentzian surfaces whose group of isometries is of dimension 1 (i.e. with a complete Killing field), in terms of a 1-dimensional generally non-Hausdorff manifold (the space of Killing orbits) and a smooth function defined there. In the second part, we study the completeness of such surfaces and prove in particular that under the hypothesis of bounded curvature, completeness is equivalent to null completeness. We also give completeness criterions involving the topological structure of the space of Killing orbits, or both the topology of this space and the geometry of the surface.

Explicit solving of the system of natural PDEs of minimal Lorentz surfaces in [formula omitted

A minimal Lorentz surface in R24 is said to be of general type if its corresponding null curves are non-degenerate. These surfaces admit canonical isothermal and canonical isotropic coordinates. It is known that the Gauss curvature K and the normal curvature ϰ of such a surface considered as functions of the canonical coordinates satisfy a system of two natural PDEs. Using the Weierstrass type representations of the corresponding null curves, we solve explicitly the system of natural PDEs, expressing any solution by means of four real functions of one variable. We obtain the transformation formulas for the functions in the Weierstrass representation of a null curve under a proper motion in R24. Using this, we find the relation between two quadruples of real functions generating one and the same solution to the system of natural PDEs.

Lorentzian Approximations and Gauss–Bonnet Theorem for E(1,1) with the Second Lorentzian Metric

In this paper, we consider the Lorentzian approximations of rigid motions of the Minkowski plane . By using the method of Lorentzian approximations, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface in E(1,1) with the second Lorentzian metric away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in E(1,1) with the second left‐invariant Lorentzian metric g2.

Canonical Coordinates and Natural Equation for Lorentz Surfaces in R13

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

Weierstrass Representations of Lorentzian Minimal Surfaces in $$\mathbb R^4_2$$

The minimal Lorentzian surfaces in $$\mathbb {R}^4_2$$ whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature $$\varkappa $$ satisfy $$K^2-\varkappa ^2 >0$$ are called minimal Lorentzian surfaces of general type. These surfaces admit canonical parameters and with respect to such parameters are determined uniquely up to a motion in $$\mathbb {R}^4_2$$ by the curvatures K and $$\varkappa $$ satisfying a system of two natural PDEs. In the present paper we study minimal Lorentzian surfaces in $$\mathbb {R}^4_2$$ and find a Weierstrass representation with respect to isothermal parameters of any minimal surface with two-dimensional first normal space. We also obtain a Weierstrass representation with respect to canonical parameters of any minimal Lorentzian surface of general type and solve explicitly the system of natural PDEs expressing any solution to this system by means of four real functions of one variable.

Pseudo-Isotropic Centro-Affine Lorentzian Surfaces

In this paper, we study centro-affine Lorentzian surfaces M2 in ℝ3 which have pseudo-isotropic or lightlike pseudo-isotropic difference tensor. We first show that M2 is pseudo-isotropic if and only if the Tchebychev form T=0. In that case, M2 is a an equi-affine sphere. Next, we will get a complete classification of centro-affine Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic.

Global properties of warped solutions in general relativity with an electromagnetic field and a cosmological constant. II.

We consider general relativity with a cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional spacetime manifold is a warped product of two surfaces with Lorentzian and Euclidean signature metrics. Field equations imply that at least one of the surfaces must be of constant curvature leading to the symmetry of the metric (``spontaneous symmetry emergence''). We classify all global solutions in the case when the Lorentzian surface is of constant curvature (case C). These solutions are invariant with respect to the Lorentz $\mathbb{S}\mathbb{O}(1,2)$ or Poincare $\mathbb{I}\mathbb{O}(1,1)$ groups acting on the Lorentzian surface.

Extensions maximales et classification des tores lorentziens munis d’un champ de Killing

We study the simply connected inextendable Lorentzian surfaces admitting a Killing vector field. We construct a natural family of such surfaces, that we call extensions. They are characterized by a condition of symmetry, the reflexivity, and a by a rather weak completeness assumption, the absence of saddles at infinity. Considering these surfaces as model spaces, we study their minimal quotients, divisible open sets and conjugate points. We show uniformisation results (by an open subset of one of these universal extensions, which is uniquely determined) in the following cases: compact surfaces and analytical surfaces. It allows us to give a classification of Lorentzian tori and Klein bottles with a Killing vector field.

Pseudo‐hyperbolic Gauss maps of Lorentzian surfaces in anti‐de Sitter space

AbstractIn this paper, we determine the type numbers of the pseudo‐hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non‐diagonalizable shape operator in the 3‐dimensional anti‐de Sitter space. Also, we investigate the behavior of type numbers of the pseudo‐hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the 3‐dimensional anti‐de Sitter space. Furthermore, we investigate the type number of the pseudo‐hyperbolic Gauss map of one of Lorentzian hypersurfaces of B‐scroll type in a general dimensional anti‐de Sitter space.

Projective properties of Lorentzian surfaces

We investigate projective properties of Lorentzian surfaces. In particular, we prove that if T is a non-flat torus, then the index of its isometry group in its projective group is at most two. We also prove that any topologically finite non-compact surface can be endowed with a metric having a non-isometric projective transformation of infinite order.

Null scrolls asB-scrolls in Lorentz–Minkowski 3-space

Null scrolls, i.e. ruled surfaces whose base curve and rulings are both lightlike (null), are Lorentzian surfaces having no Euclidean counterparts. In this work we present reparametrization of nondegenerate null scroll as a B-scroll, i.e. as a ruled surface whose rulings correspond to the binormal vectors of a base curve. We prove that the curvature of a base curve, which determines the Gaussian and mean curvature of a null scroll, is invariant under such a reparametrization. We also determine a one-parameter family of null curves on null scroll which serve as base curves for this kind of reparametrization.

Minimal Lorentz surfaces in pseudo-Euclidean 4-space with neutral metric

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature ϰ satisfy the inequality K2−ϰ2>0. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that any minimal Lorentz surface of general type is determined (up to a rigid motion) by two invariant functions satisfying a system of two natural partial differential equations. Using a concrete solution to this system we construct an example of a minimal Lorentz surface of general type.

Lorentzian surfaces and the curvature of the Schmidt metric

The b-boundary is a mathematical tool used to attach a topological boundary to incomplete Lorentzian manifolds using a Riemaniann metric called the Schmidt metric on the frame bundle. In this paper we give the general form of the Schmidt metric in the case of Lorentzian surfaces. Furthermore, we write the Ricci scalar of the Schmidt metric in terms of the Ricci scalar of the Lorentzian manifold and give some examples. Finally, we discuss some applications to general relativity.

On the Gauss map of minimal Lorentzian surfaces in 4-dimensional semi-Riemannian space forms with index 2

On the Gauss map of minimal Lorentzian surfaces in 4-dimensional semi-Riemannian space forms with index 2