Spacelike Hypersurfaces Research Articles

Spacelike Hypersurfaces Immersed in Weighted Standard Static Spacetimes: Uniqueness, Nonexistence and Stability

A spacetime endowed with a globally defined timelike Killing vector field admits a certain model of warped product, called the standard static spacetime, and, when the volume element is modified by a factor that depends on a smooth function (which is called density function), we say that this ambient is a weighted standard static spacetime. In such spacetimes, we study some aspects of the geometry of spacelike hypersurfaces through of drift Laplacian of two functions support naturally related to them. For such hypersurfaces, with some restrictions on density function and the geometry of the ambient spacetime, we begin by stating and showing some results of uniqueness and nonexistence, several of them not assuming that the hypersurface to be of constant weighted mean curvature. Versions of these results are given for entire Killing graphs, that is, graphs constructed over an integral leaf of the distribution of smooth vector fields orthogonal to timelike Killing vector field. Finally, for closed spacelike hypersurface immersed in a weighted standard static spacetime with constant weighted mean curvature, we study a notion of stability via the first eigenvalue of the drift Laplacian.

On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance [Formula: see text] from a spacelike hypersurface [Formula: see text] is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure, compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.

Complete spacelike hypersurfaces immersed in pp-wave spacetimes

We study some aspects of the geometry of complete spacelike hypersurfaces immersed into a pp-wave spacetime, namely, into a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying suitable versions of the classical Hopf and Stokes theorems and a criterion of parabolicity for complete Riemannian manifolds, we obtain sufficient conditions which guarantee that a complete spacelike hypersurface is either maximal, 1-maximal or totally geodesic. As a consequence of these results, we also establish some results of nonexistence concerning such spacelike hypersurfaces. Finally, considering constant mean curvature closed spacelike hypersurfaces immersed in a pp-wave spacetime, we study a notion of stability via the first nonzero eigenvalue of the Laplacian.

New characterizations of spacelike hyperplanes in the steady state space

In this article, we deal with complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb {S}^{n+1}_{1}$ which is known as the steady state space $\mathcal {H}^{n+1}$. Under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we are able to prove that they must be spacelike hyperplanes of $\mathcal {H}^{n+1}$. Furthermore, through the analysis of the hyperbolic cylinders of $\mathcal {H}^{n+1}$, we discuss the importance of the main hypothesis in our results. Our approach is based on a generalized maximum principle at infinity for complete Riemannian manifolds.

Rotationally symmetric spacelike translating solitons for the mean curvature flow in Minkowski space

The rotationally symmetric spacelike translating solitons for the MCF in the Minkowski space are considered in this paper. To be specific, the complete classification for such translating solitons according to their rotation axes, which are timelike, spacelike and lightlike, is provided. In particular, the explicit parametrization of the rotationally symmetric spacelike hypersurface of the lightlike rotation axis with the prescribed mean curvature is obtained.

New Calabi–Bernstein type results in Lorentzian product spaces with density

We investigate the rigidity of complete spacelike hypersurfaces immersed in a weighted Lorentzian product space of the type R1×Pfn, endowed with a weight function f which does not depend on the parameter t∈R and whose Riemannian fiber Pn has nonnegative Bakry–Émery–Ricci tensor. In this direction, supposing that the f-mean curvature is constant and assuming appropriate constraints on the norm of the gradient of f, we prove that such a spacelike hypersurface must be a slice of the ambient space. As application of our investigation, we obtain new Calabi–Bernstein type results concerning entire spacelike graphs constructed over Pn. Our approach is based on an extension for the drift Laplacian of a generalized maximum principle of Akutagawa (1987) and a Liouville-type result due to Pigola et al. (2005).

Complete spacelike hypersurfaces on symmetric spacetimes

A Lorentz manifold is said to be a conformally stationary spacetime if it is endowed with a globally defined conformal timelike vector field K, whereas it is a pp-wave when there is a globally defined parallel lightlike vector field K on M. The study of rigidity and non-existence results for spacelike hypersurfaces with constant mean curvature in these spaces has been considered in the last years by several authors. In this note we unify some known techniques and results related to both problems by considering spacelike hypersurfaces in a spacetime endowed with a globally defined conformal causal vector field. This wide family of Lorentz manifolds not only includes previous cases, but it also includes new families of spacetimes.

New class of naked singularities and their observational signatures

By imposing suitable junction conditions on a spacelike hypersurface, we obtain a two-parameter family of possible static configurations from gravitational collapse. These exemplify a new class of naked singularities. We show that these admit a consistent description via a two-fluid model, one of which might be dust. We then study lensing and accretion disk properties of our solution and point out possible differences with black hole scenarios. The distinctive features of our solution compared to the existing naked singularity solutions in the literature are discussed.

On Geometric Analysis of the Dynamics of Volumetric Expansion and its Applications to General Relativity

In this paper, we discuss the global aspect of the geometric dynamics of volumetric expansion and its applications to the problem of the existence in the space-time of compact and complete spacelike hypersurfaces and to the global geometry of generalized Robertson–Walker space-times.

Lorentzian manifolds with causal Killing vector field: causality and geodesic connectedness

We prove that a compact Lorentzian manifold $$({\overline{M}},{\overline{g}})$$ admitting a causal Killing vector field is totally vicious or it contains a compact achronal Killing horizon. In particular a compact spacetime which satisfies the null generic condition and admits a causal Killing vector field is totally vicious. If in addition, its universal Lorentzian covering is globally hyperbolic then it is geodesically connected. In the non-compact case, we prove that a chronological spacetime admitting a complete causal Killing vector field, a smooth spacelike partial Cauchy hypersurface S and satisfying the null generic condition is stably causal. If additionally S is compact then the spacetime is globally hyperbolic. We also determine the geodesic connectedness in this case.

Transversely trapping surfaces: Dynamical version

Abstract We propose new concepts, a dynamically transversely trapping surface (DTTS) and a marginally DTTS, as indicators for a strong gravity region. A DTTS is defined as a two-dimensional closed surface on a spacelike hypersurface such that photons emitted from arbitrary points on it in transverse directions are acceleratedly contracted in time, and a marginally DTTS is reduced to the photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close analogy with (marginally) trapped surfaces in many aspects. After preparing the method of solving for a marginally DTTS in the time-symmetric initial data and the momentarily stationary axisymmetric initial data, some examples of marginally DTTSs are numerically constructed for systems of two black holes in the Brill–Lindquist initial data and in the Majumdar–Papapetrou spacetimes. Furthermore, the area of a DTTS is proved to satisfy the Penrose-like inequality $A_0\le 4\pi (3GM)^2$, under some assumptions. Differences and connections between a DTTS and the other two concepts proposed by us previously, a loosely trapped surface [Prog. Theor. Exp. Phys. 2017, 033E01 (2017)] and a static/stationary transversely trapping surface [Prog. Theor. Exp. Phys. 2017, 063E01 (2017)], are also discussed. A (marginally) DTTS provides us with a theoretical tool to significantly advance our understanding of strong gravity fields. Also, since DTTSs are located outside the event horizon, they could possibly be related with future observations of strong gravity regions in dynamical evolutions.

Foliations by Spacelike Hypersurfaces on Lorentz Manifolds

In this work, we study the geometric properties of spacelike foliations by hypersurfaces on a Lorentz manifold. We find an equation that relates the foliation with the ambient manifold and applies it to investigate conditions for the leaves being totally umbilical or geodesic. Using the Maximum principle with the mentioned equation we obtain an obstruction for the existence of totally geodesic spacelike foliations in a spacetime with positive Ricci curvature on the direction N.

Generalized covariant prescriptions for averaging cosmological observables

We present two new covariant and general prescriptions for averaging scalar observables on spatial regions typical of the observed sources and intersecting the past light-cone of a given observer. One of these prescriptions is adapted to sources exactly located on a given space-like hypersurface, the other applies instead to situations where the physical location of the sources is characterized by the experimental “spread” of a given observational variable. The geometrical and physical differences between the two procedures are illustrated by computing the averaged energy flux received by distant sources located on (or between) constant redshift surfaces, and by working in the context of a perturbed ΛCDM geometry. We find significant numerical differences (of about ten percent or more, in a large range of redshift) even limiting our model to scalar metric perturbations, and stopping our computations to the leading non-trivial perturbative order.

Curvature estimate of steepest descents of 2-dimensional maximal space-like hypersurfaces on space forms

For the maximal space-like hypersurface defined on 2-dimensional space forms, based on the regularity and the strict convexity of the level sets, the steepest descents are well defined. In this paper, we come to estimate the curvature of its steepest descents by deriving a differential equality.

Conformally flat Willmore spacelike hypersurfaces inRn+1

In this paper, we give the equation satisfied by umbilics-free Willmore spacelike hypersurfaces using the conformal invariants in Lorentzian space forms. At the same time, we give the equation satisfied by hyperelastic spacelike curves in $2$-dimensional Lorentzian space forms and classify the closed hyperelastic spacelike curves. Finally conformally flat Willmore spacelike hypersurfaces are classified in terms of the hyperelastic spacelike curves in $2$-dimensional Lorentzian space forms.

Nonstationary energy in general relativity

Using the time evolution equations of (cosmological) General Relativity in the first order Fischer-Marsden form, we construct an integral that measures the amount of non-stationary energy on a given spacelike hypersurface in $D$ dimensions. The integral vanishes for stationary spacetimes; and with a further assumption, reduces to Dain's invariant on the boundary of the hypersurface which is defined with the Einstein constraints and a fourth order equation defining approximate Killing symmetries.

Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space

Abstract Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space, Pacific J. Math. 292 (2018), 1–19].

Calabi–Bernstein type results for complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes

In this article we study complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes. Under appropriate geometric assumptions we prove new parametric uniqueness results as well as obtain very general Calabi–Bernstein type results for the constant mean curvature equation for spacelike graphs in these ambient spacetimes.

Horizon molecules in causal set theory

We propose a new definition of "horizon molecules" in Causal Set Theory following pioneering work by Dou and Sorkin. The new concept applies for any causal horizon and its intersection with any spacelike hypersurface. In the continuum limit, as the discreteness scale tends to zero, the leading behaviour of the expected number of horizon molecules is shown to be the area of the horizon in discreteness units, up to a dimension dependent factor of order one. We also determine the first order corrections to the continuum value, and show how such corrections can be exploited to obtain further geometrical information about the horizon and the spacelike hypersurface from the causal set.

Editorial note to: On the Newtonian limit of Einstein’s theory of gravitation (by Jürgen Ehlers)

We give an overview of literature related to J\"urgen Ehlers' pioneering 1981 paper on Frame theory--a theoretical framework for the unification of General Relativity and the equations of classical Newtonian gravitation. This unification encompasses the convergence of one-parametric families of four-dimensional solutions of Einstein's equations of General Relativity to a solution of equations of a Newtonian theory if the inverse of a causality constant goes to zero. As such the corresponding light cones open up and become space-like hypersurfaces of constant absolute time on which Newtonian solutions are found as a limit of the Einsteinian ones. It is explained what it means to not consider the `standard-textbook' Newtonian theory of gravitation as a complete theory unlike Einstein's theory of gravitation. In fact, Ehlers' Frame theory brings to light a modern viewpoint in which the `standard' equations of a self-gravitating Newtonian fluid are Maxwell-type equations. The consequences of Frame theory are presented for Newtonian cosmological dust matter expressed via the spatially projected electric part of the Weyl tensor, and for the formulation of characteristic quasi-Newtonian initial data on the light cone of a Bondi-Sachs metric.