Spacelike Surfaces Research Articles

A Note on Spacelike Submanifolds Through Light Cones in Lorentzian Space Forms

We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones $$\Lambda _{c}(p)$$ of de Sitter and anti-de Sitter spacetimes. Every light cone $$\Lambda _{c}(p)$$ contains the lightlike geodesics starting from p and, essentially, it coincides with the horisms $$E^{+}(p)\cup E^{-}(p)$$ . The analysis works by means of an explicit correspondence with the spacelike submanifolds through the light cone in the Lorentz–Minkowski spacetime. In particular, a characterization of totally umbilical compact surfaces through light cones in de Sitter and anti-de Sitter is shown and we obtain an estimation of the first eigenvalue of the Laplace operator on a compact spacelike surface in a light cone.

On Differential Analysis of Flows on Normal Congruence of Spacelike Surfaces

The present paper examines the differential analysis of fows on normal congruence of spacelike surfaces with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters which are related by three partial differential equations are discussed. Then, it is seen that the curl of tangent vector field does not include any component with principal normal direction. Thus there exists a surface which contains both s-lines and b-lines. Also, we examine a normal congruence of spacelike surfaces containing the s-lines and b-lines. By compatibility conditions, Gauss-Mainardi-Codazzi equations for this normal congruence of spacelike surface are obtained: Intrinsic geometric properties of this normal congruence of spacelike surfaces are given.

THE SHAPE OPERATOR OF THE BEZIER SURFACES IN MINKOWSKI-3 SPACE

Bezier surfaces are commonly used in Computer-Aided Geometric Design since it enables in geometric modeling of the objects. In this study, the shape operator of the timelike and spacelike surfaces has been analyzed in Minkowski-3 space. Then, the obtained results were applied to a numeric example

The complete classification of a class of new linear Weingarten surfaces in Minkowski space

We consider spacelike and timelike surfaces satisfying an interesting geometric property that the gradient of the mean curvature is a principal direction in a Minkowski 3-space. It is proved that these surfaces are linear Weingarten surfaces and contain all biconservative surfaces in a Minkowski 3-space. We call them generalized biconservative surfaces (or GB surfaces for short). We give complete explicit classifications of spacelike and timelike GB surfaces in Minkowski 3-space. Our results show that the non-constant-mean-curvature GB surfaces in Minkowski 3-space are locally either surfaces of revolution or null scrolls.

Proof of a momentum/complexity correspondence

We show that the holographic Complexity = Volume proposal satisfies a very general notion of Momentum/Complexity correspondence (PC), based on the Momentum Constraint of General Relativity. It relates the rate of complexity variation with an appropriate matter momentum flux through spacelike extremal surfaces. This formalizes the intuitive idea that `gravitational clumpling' of matter increases complexity, and the required notion of `infall momentum' is shown to have a Newtonian avatar which expresses this idea. The proposed form of the PC correspondence is found to be exact for any solution of Einstein's equations in 2+1 dimensions, and any spherically symmetric solution in arbitrary dimensions, generalizing all previous calculations using spherical thin shells. Gravitational radiation enters through a correction which does not have a straightforward interpretation as a PC correspondence. Other obstructions to an exact PC duality have a topological origin and arise in the presence of wormholes.

SPACELIKE TRANSLATION SURFACES IN MINKOWSKI 4-SPACE E_1^4

In the present paper, we consider spacelike translation surfaces in $4$-dimensio\-nal Minkowski space. We characterize such surfaces in terms of their Gaussian curvature and mean curvature functions. We classify flat and minimal spacelike translation surfaces in $\mathbb{E}_{1}^{4}$.

Schwarzschild de Sitter and extremal surfaces

We study extremal surfaces in the Schwarzschild de Sitter spacetime with real mass parameter. We find codim-2 timelike extremal surfaces stretching between the future and past boundaries that pass through the vicinity of the cosmological horizon in a certain limit. These are analogous to the surfaces in http://arxiv.org/abs/1711.01107 [hep-th]. We also find spacelike surfaces that never reach the future/past boundaries but stretch indefinitely through the extended Penrose diagram, passing through the vicinity of the cosmological and Schwarzschild horizons in a certain limit. Further, these exhibit interesting structure for de Sitter space (zero mass) as well as in the extremal, or Nariai, limit.

Conformal homogeneous spacelike surfaces in 3-dimensional Lorentz space forms

Let M13(c) be an 3-dimensional Lorentz space form and C(M13(c)) denote the conformal transformation group of M13(c). A spacelike surface x:M2→M13(c) is called a conformal homogeneous spacelike surface. If there exists a subgroup G⊂C(M13(c)) such that the orbit G(p)=x(M2),p∈x(M2). In this paper, we classify completely conformal homogeneous spacelike surfaces up to a conformal transformation of M13(c).

Aspects of marginally trapped and antitrapped surfaces in a D -dimensional evolving dust model

In this paper, we explore the formation of the marginally trapped and marginally anti-trapped surfaces that arise from the evolution of homogeneous dust in D-dimensions with and without the cosmological constant, this is seen through the analytical expressions for such surfaces. We obtain closed form expressions for the Norm of the normal to the Horizon that decides their causal nature and also look at several interesting features of these surface evolution that are significantly different from the four dimensional counterpart. We obtain the expressions for the Ashtekar-Badrikrihnan's Area-balance law for dynamical horizon (spacelike surface) tailored for the case of spherically symmetric dust evolution in D-dimensions.

A consistent model of non-singular Schwarzschild black hole in loop quantum gravity and its quasinormal modes

We investigate the interior structure, perturbations, and the associated quasinormal modes of a quantum black hole model recently proposed by Bodendorfer, Mele, and Münch (BMM). Within the framework of loop quantum gravity, the quantum parameters in the BMM model are introduced through polymerization, consequently replacing the Schwarzschild singularity with a spacelike transition surface. By treating the quantum geometry corrections as an `effective' matter contribution, we first prove the violation of energy conditions (in particular the null energy condition) near the transition surface and then investigate the required junction conditions on it. In addition, we study the quasinormal modes of massless scalar field perturbations, electromagnetic perturbations, and axial gravitational perturbations in this effective model. As expected, the quasinormal spectra deviate from their classical counterparts in the presence of quantum corrections. Interestingly, we find that the quasinormal frequencies of perturbations with different spins share the same qualitative tendency with respect to the change of the quantum parameters in this model.

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.

Constant mean curvature spacelike hypersurfaces in standard static spaces: rigidity and parabolicity

Our purpose in this paper is investigate the geometry of complete constant mean curvature spacelike hypersurfaces immersed in a standard static space, that is, a Lorentzian manifold endowed with a globally defined timelike Killing vector field. In this setting, supposing that the ambient space is a warped product of the type $M^n\times_{\rho}\mathbb{R}_1$ whose Riemannian base $M^n$ has nonnegative sectional curvature and the warping function $\rho$ is convex on $M^n$, we use the generalized maximum principle of Omori-Yau in order to establish rigidity results concerning these spacelike hypersurfaces. We also study the parabolicity of maximal spacelike surfaces in $M^2\times_{\rho}\mathbb{R}_1$ and we obtain uniqueness results for entire Killing graphs constructed over $M^n$.

Cylinder transition amplitudes in pure AdS3 gravity

A spacelike surface with cylinder topology can be described by various sets of canonical variables within pure AdS3 gravity. Each is made of one real coordinate and one real momentum. The Hamiltonian can be either H = 0 or it can be nonzero and we display the canonical transformations that map one into the other, in two relevant cases. In a choice of canonical coordinates, one of them is the cylinder aspect q, which evolves nontrivially in time. The time dependence of the aspect is an analytic function of time t and an “angular momentum” J . By analytic continuation in both t and J we obtain a Euclidean evolution that can be described geometrically as the motion of a cylinder inside the region of the 3D hyperbolic space bounded by two “domes” (i.e. half spheres), which is topologically a solid torus. We find that for a given J the Euclidean evolution cannot connect an initial aspect to an arbitrary final aspect; moreover, there are infinitely many Euclidean trajectories that connect any two allowed initial and final aspects. We compute the transition amplitude in two independent ways; first by solving exactly the time-dependent Schrödinger equation, then by summing in a sensible way all the saddle contributions, and we discuss why both approaches are mutually consistent.

Surfaces in pseudo-Riemannian space forms with zero mean curvature vector

We characterize a space-like surface in a pseudo-Riemannian space form with zero mean curvature vector, in terms of complex quadratic differentials on the surface as sections of a holomorphic line bundle. In addition, combining them, we have a holomorphic quartic differential. If the ambient space is $S^4$, then this differential is just one given in [5]. If the space is $S^4_1$, then the differential coincides with a holomorphic quartic differential in [6] on a Willmore surface in $S^3$ corresponding to the original surface through the conformal Gauss map. We define the conformal Gauss maps of surfaces in $E^3$ and $H^3$, and space-like surfaces in $S^3_1$, $E^3_1$, $H^3_1$ and the cone of future-directed light-like vectors of $E^4_1$, and have results which are analogous to those for the conformal Gauss map of a surface in $S^3$.

Construction of the spacelike constant angle surface family in Minkowski 3-space

A spacelike constant angle surface in $E_1^3$ (Minkowski 3-space) is defined by its unit normal vector field forms a constant angle with some fixed vector direction. In this paper, we use the Frenet frame to express the surface and construct a family of spacelike constant angle surfaces which possess the given curve as isoparametric curve. The normal vector of each surface forms a constant angle with a fixed vector lying in the spacelike cone and in the timelike cone respectively. Finally, we give some representative examples.

Gravitational corner conditions in holography

Contrary to popular belief, asymptotically anti-de Sitter solutions of gravitational theories cannot be obtained by taking initial data (satisfying the constraints) on a spacelike surface, and choosing an arbitrary conformal metric on the timelike boundary at infinity. There are an infinite number of corner conditions that also must be satisfied where the initial data surface hits the boundary. These are well known to mathematical relativists, but to make them more widely known we give a simple explanation of why these conditions exist and discuss some of their consequences. An example is given which illustrates their power. Some implications for holography are also mentioned.

Holographic complexity from Crofton’s formula in Lorentzian AdS3

We study Crofton’s formula in the Lorentzian AdS3 and find that the area of a generic spacelike two-dimensional surface is given by the flux of spacelike geodesics. The “complexity[Formula: see text]=[Formula: see text]volume” conjecture then implies a new holographic representation of complexity in terms of the number of geodesics. Finally, we explore the possible explanation of this result from the standpoint of information theory.

Yang–Mills Theory of Gravity

The canonical formulation of general relativity (GR) is based on decomposition space–time manifold M into R × Σ , where R represents the time, and Ksi is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general coordinates, and local Lorentz transformations. These symmetries are associated with conserved currents that are coupled to gravity. These symmetries are studied on a three dimensional space-like hypersurface Σ embedded in a four-dimensional space–time manifold. This implies continuous symmetries and conserved currents by Noether’s theorem on that surface. We construct a three-form E i ∧ D A i (D represents covariant exterior derivative) in the phase space ( E i a , A a i ) on the surface Σ , and derive an equation of continuity on that surface, and search for canonical relations and a Lagrangian that correspond to the same equation of continuity according to the canonical field theory. We find that Σ i 0 a is a conjugate momentum of A a i and Σ i a b F a b i is its energy density. We show that there is conserved spin current that couples to A i , and show that we have to include the term F μ ν i F μ ν i in GR. Lagrangian, where F i = D A i , and A i is complex S O ( 3 ) connection. The term F μ ν i F μ ν i includes one variable, A i , similar to Yang–Mills gauge theory. Finally we couple the connection A i to a left-handed spinor field ψ , and find the corresponding beta function.

Conic Lagrangian Varieties and Localized Asymptotic Solutions of Linearized Equations of Relativistic Gas Dynamics

We study asymptotic solution of the Cauchy problem for the linearized system of relativistic gas dynamics. We assume that initial condiditiopns are strongly localized near a space-like surface in the Minkowsky space. We prove that the solution can be decomposed into three modes, corresponding to different routsb of the equations of characteristics. One of these roots is twice degenerate and the there are no focal points in the corresponding miode. The other two roots are simple; in order to describe the corresponding modes, we use the modificication of the Maslov’s canonical operator which was obtained recently.

Rotating black holes in higher order gravity

We develop a new technique for finding black hole solutions in modified gravity that have "stealth" hair, i.e., hair whose only gravitational effect is to tune the cosmological constant. We consider scalar-tensor theories in which gravitational waves propagate at the speed of light, and show that Einstein metrics can be painted with stealth hair provided there exists a family of geodesics always normal to spacelike surfaces. We also present a novel scalar-dressed rotating black hole that has finite scalar field at both the black hole and cosmological event horizons.