Timelike Surfaces Research Articles

Constant Angle Surfaces in Lorentzian Berger Spheres

In this work, we study helix spacelike and timelike surfaces in the Lorentzian Berger sphere $${\mathbb S}_{\varepsilon }^3$$ , that is the three-dimensional sphere endowed with a 1-parameter family of Lorentzian metrics, obtained by deforming the round metric on $${\mathbb S}^3$$ along the fibers of the Hopf fibration $${\mathbb S}^3\rightarrow {\mathbb S}^2({1}/{2})$$ by $$-\varepsilon ^2$$ . Our main result provides a characterization of the helix surfaces in $${\mathbb S}_{\varepsilon }^3$$ using the symmetries of the ambient space and a general helix in $${\mathbb S}_{\varepsilon }^3$$ , with axis the infinitesimal generator of the Hopf fibers. Also, we construct some explicit examples of helix surfaces in $${\mathbb S}_{\varepsilon }^3$$ .

Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space

Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular points on spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces do not admit fold singular points. Moreover, we show that the singular point set of any conjugate CMC surface of a spacelike Delaunay surface with conelike singular points consists of $(2,5)$-cuspidal edges.

Perturbations of spiky strings in AdS3

Perturbations of a class of semiclassical spiky strings in three dimensional Anti-de Sitter (AdS) spacetime, are investigated using the well-known Jacobi equations for small, normal deformations of an embedded timelike surface. We show that the equation for the perturbation scalar which governs the behaviour of such small deformations, is a special case of the well-known Darboux-Treibich-Verdier (DTV) equation. The eigenvalues and eigensolutions of the DTV equation for our case are obtained by solving certain continued fractions numerically. These solutions are thereafter utilised to further demonstrate that there do exist finite perturbations of the AdS spiky strings. Our results therefore establish that the spiky string configurations in AdS3 are indeed stable against small fluctuations. Comments on future possibilities of work are included in conclusion.

Timelike surfaces in Minkowski space with a canonical null direction

Given a constant vector field Z in Minkowski space, a timelike surface is said to have a canonical null direction with respect to Z if the projection of Z on the tangent space of the surface gives a lightlike vector field. For example in the three-dimensional Minkowski space: a surface has a canonical null direction if and only if it is minimal and flat. When the ambient has arbitrary dimension, if a surface has a canonical null direction and has parallel mean curvature vector then it is minimal. We give different ways for building these surfaces in the four-dimensional Minkowski space. On the other hand, we describe several properties in the non ruled general case in four-dimensional Minkowski space. One property is that the tangent part of Z is an asymptotic direction of the surface. We describe these surfaces in the ruled case in arbitrary dimension. Finally use the Gauss map for describe another properties of these surfaces in dimension four.

A refined description of evolving interfaces in certain nonlinear wave equations

We improve on recent results that establish the existence of solutions of certain semilinear wave equations possessing an interface that roughly sweeps out a timelike surface of vanishing mean curvature in Minkowski space. Compared to earlier work, we present sharper estimates, in stronger norms, of the solutions in question.

Black hole mass formula in the membrane paradigm

The membrane paradigm approach adopts a timelike surface, stretched out off the null event horizon, to study several important black hole properties. We use this powerful tool to give a direct derivation of the black hole mass formula in the static and stationary cases without and with electric field. Since here the membrane is a self-gravitating material system we go beyond the usual applicability on test particles and test fields of the paradigm.

Computation of Smarandache curves according to Darboux frame in Minkowski 3-space

Abstract In this paper, we study Smarandache curves according to Darboux frame in the three-dimensional Minkowski space E 1 3 . Using the usual transformation between Frenet and Darboux frames, we investigate some special Smarandache curves for a given timelike curve lying fully on a timelike surface. Finally, we defray a computational example to confirm our main results.

On extremal surfaces and de Sitter entropy

We study extremal surfaces in the static patch coordinatization of de Sitter space, focusing on the future and past universes. We find connected timelike codim-2 surfaces on a boundary Euclidean time slice stretching from the future boundary I+ to the past boundary I−. In a limit, these surfaces pass through the bifurcation region and have minimal area with a divergent piece alone, whose coefficient is de Sitter entropy in 4-dimensions. These are reminiscent of rotated versions of certain surfaces in the AdS black hole. We close with some speculations on a possible dS/CFT interpretation of 4-dim de Sitter space as dual to two copies of ghost-CFTs in an entangled state. For a simple toy model of two copies of ghost-spin chains, we argue that similar entangled states always have positive norm and positive entanglement.

Rindler fluid with weak momentum relaxation

We realize the weak momentum relaxation in Rindler fluid, which lives on the time-like cutoff surface in an accelerating frame of flat spacetime. The translational invariance is broken by massless scalar fields with weak strength. Both of the Ward identity and the momentum relaxation rate of Rindler fluid are obtained, with higher order correction in terms of the strength of momentum relaxation. The Rindler fluid with momentum relaxation could also be approached through the near horizon limit of cutoff AdS fluid with momentum relaxation, which lives on a finite time-like cutoff surface in Anti-de Sitter(AdS) spacetime, and further could be connected with the holographic conformal fluid living on AdS boundary at infinity. Thus, in the holographic Wilson renormalization group flow of the fluid/gravity correspondence with momentum relaxation, the Rindler fluid can be considered as the Infrared Radiation(IR) fixed point, and the holographic conformal fluid plays the role of the ultraviolet(UV) fixed point.

Curvature Dependent Energy of Surface Curves in Minkowski Space

In this paper, we firstly introduce kinematics properties of the moving particle lying on a surface S. We assume that the particle corresponds to a different type of surface curves such that they are characterized by using the Darboux vector field W in Minkowski spacetime. Based on this result, we present geometrical understanding of the energy of the particle in each Darboux vector fields whether they lie on a spacelike surface or a timelike surface. Then, we also determine the bending elastic energy functional for the same particle on a surface S by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy of the particle in each Darboux vector field W.

Minimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula which is the unification of representation formulas for minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gaus map of minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discussed.

Global existence for the minimal surface equation on ℝ^{1,1}

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on R 1 , 1 \mathbb {R}^{1,1} describing graphical timelike minimal surfaces embedded in R 1 , 2 \mathbb {R}^{1,2} enjoy small data global existence for compactly supported initial data, using Christodoulou’s conformal method. Here we give a different, geometric proof of the same fact, which exposes more clearly the inherent null structure of the equations, and which allows us to also close the argument using relatively few derivatives and mild decay assumptions at infinity.

A Time–Space Symmetry Based Cylindrical Model for Quantum Mechanical Interpretations

Following a bi-cylindrical model of geometrical dynamics, our study shows that a 6D-gravitational equation leads to geodesic description in an extended symmetrical time–space, which fits Hubble-like expansion on a microscopic scale. As a duality, the geodesic solution is mathematically equivalent to the basic Klein–Gordon–Fock equations of free massive elementary particles, in particular, the squared Dirac equations of leptons. The quantum indeterminism is proved to have originated from space–time curvatures. Interpretation of some important issues of quantum mechanical reality is carried out in comparison with the 5D space–time–matter theory. A solution of lepton mass hierarchy is proposed by extending to higher dimensional curvatures of time-like hyper-spherical surfaces than one of the cylindrical dynamical geometry. In a result, the reasonable charged lepton mass ratios have been calculated, which would be tested experimentally.

Constant angle surfaces in the Lorentzian Heisenberg group

In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.

Minimal surfaces in Lorentzian Heisenberg group and Damek–Ricci spaces via the Weierstrass representation

Abstract In this paper we discuss a Weierstrass type representation for minimal surfaces in the 3 -dimensional Heisenberg group and in the 4 -dimensional Damek–Ricci spaces, endowed with left invariant Riemannian or Lorentzian metrics. For the case of spacelike surfaces we employ the complex analysis, and for timelike surfaces our approach makes use of the paracomplex analysis. Then, we exhibit various examples of spacelike and timelike minimal surfaces in these spaces.

Quadrics and Scherk towers

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net.

Computation of Smarandache curves according to Darboux frame in Minkowski 3-space

Abstract In this paper, we study Smarandache curves according to Darboux frame in the three-dimensional Minkowski space E 1 3 . Using the usual transformation between Frenet and Darboux frames, we investigate some special Smarandache curves for a given timelike curve lying fully on a timelike surface. Finally, we defray a computational example to confirm our main results.

Lifespan of smooth solutions for timelike extremal surface equation in de Sitter spacetime

In this paper, we study the generalized timelike extremal surface equation in the de Sitter spacetime, which plays an important role in both mathematics and physics. Under the assumption of small initial data with compact support, we investigate the lower bound of lifespan of smooth solutions by weighted energy estimates.

Extension of photon surfaces and their area: Static and stationary spacetimes

We propose a new concept, the transversely trapping surface (TTS), as an extension of the static photon surface characterizing the strong gravity region of a static/stationary spacetime in terms of photon behavior. The TTS is defined as a static/stationary timelike surface $S$ whose spatial section is a closed two-surface, such that arbitrary photons emitted tangentially to $S$ from arbitrary points on $S$ propagate on or toward the inside of $S$. We study the properties of TTSs for static spacetimes and axisymmetric stationary spacetimes. In particular, the area $A_0$ of a TTS is proved to be bounded as $A_0\le 4\pi(3GM)^2$ under certain conditions, where $G$ is the Newton constant and $M$ is the total mass. The connection between the TTS and the loosely trapped surface proposed by us [arXiv:1701.00564] is also examined.

Enneper representation of minimal surfaces in the three-dimensional Lorentz–Minkowski space

In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz–Minkowski space $${\mathbb {L}}^{3}$$ , using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples of minimal surfaces in $${\mathbb {L}}^{3}$$ constructed via the Enneper representation formula that it is equivalent to the Weierstrass representation obtained by Kobayashi (for spacelike immersions) and by Konderak (for the timelike ones).