Spacelike Hypersurfaces Research Articles

Uniqueness theorem for generalized Maxwell electric and magnetic black holes in higher dimensions

Based on the conformal energy theorem we prove the uniqueness theorem for static higher dimensional electrically and magnetically charged black holes being the solution of Einstein (n-2)-gauge forms equations of motion. Black hole spacetime contains an asymptotically flat spacelike hypersurface with compact interior and non-degenerate components of the event horizon.

Some estimates for the curvatures of complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes

In this paper we obtain some estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in a generalized Robertson–Walker spacetime, under certain assumptions on the warped function of the ambient space. Our results will be an application of a generalized maximum principle due to Omori.

Killing initial data revisited

Considering the Killing vector fields on a Lorentzian manifold in terms of their lapse and shift along a space-like hypersurface, we give a new definition of the Killing Initial Data (KID) of Beig and Chruściel [Class. Quantum Grav. 14-1A, A83 (1996)]. We define, on these new KID’s, the bracket operation induced by the usual Lie bracket of Killing vector fields on the Lorentzian manifold, and study the conditions for our new KID’s to form a Lie algebra. The interesting fact is that these conditions only depend on data along the hypersurface.

EXPONENTIAL STRETCH–ROTATION FORMULATION OF EINSTEIN'S EQUATIONS

We study an exponential stretch–rotation (ESR) transformation, γij=e∊ikmθmeϕke-∊jknθn, of a three-dimensional metric γij of space-like hypersurfaces embedded in a four-dimensional space–time, where ϕk are logarithms of the eigenvalues of γij, θk are rotation angles, and ∊ijk is a fully anti-symmetric symbol. This tensorial exponential transformation generalizes particular exponential transformations used previously in cases of spatial symmetry. General formulae are derived that relate γij and its differentials to ϕk, θk and their differentials in a compact form. The evolution part of Einstein's equations formulated in terms of ESR variables describes time evolution of the metric at every point of a hyper-surface as a continuous stretch and rotation of a triad associated with the main axes of the metric tensor in a tangential space. An ESR 3+1 formulation of Einstein's equations may have certain advantages for long-term stable integration of these equations.

A note on Hamilton–Jacobi formalism and D-brane effective actions

We first review the canonical formalism with general space-like hypersurfaces developed by Dirac by rederiving the Hamilton–Jacobi equations which are satisfied by on-shell actions defined on such hypersurfaces. We compare the case of gravitational systems with that of the flat space. Next, we remark as a supplement to our previous results that the effective actions of D-brane and M-brane given by arbitrary embedding functions are on-shell actions of supergravities.

A characterization of totally umbilical hypersurfaces in de Sitter space

It is shown that a compact spacelike hypersurface contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if there exist nonnegative constants C1,C2,…,Cl−1, at least one Ci is positive, such that Hl=∑i=1l−1CiHi.This extends the previous result in [J. Geom. Phys. 39 (2001), Theorem 2].

Finding apparent horizons and other 2-surfaces of constant expansion

Apparent horizons are structures of spacelike hypersurfaces that can be determined locally in time. Closed surfaces of constant expansion (CE surfaces) are a generalization of apparent horizons. I present an efficient method for locating CE surfaces. This method uses an explicit representation of the surface, allowing for arbitrary resolutions and, in principle, shapes. The CE surface equation is then solved as a nonlinear elliptic equation. It is reasonable to assume that CE surfaces foliate a spacelike hypersurface outside an interior region, thus defining an invariant (but still slicing-dependent) radial coordinate. This can be used to determine gauge modes and compare time evolutions with different gauge conditions. CE surfaces also provide an efficient way of finding new apparent horizons as they appear, for example, in binary black hole simulations.

Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces

We use the half-space model for the open set of a de Sitter space associated to the steady state space to obtain some sharp a priori estimates for the height and the slope of certain constant mean curvature spacelike graphs. These estimates allow us to prove some existence and uniqueness theorems about complete non-compact constant mean curvature spacelike hypersurfaces in de Sitter spaces with prescribed asymptotic future boundary. Their geometric properties are studied.

Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in $\mathbb R^{n,1}$

We study the Dirichlet problem for the fully nonlinear elliptic partial differential equation of second order expressing the prescription of the mth symmetric function of the principal curvatures of a spacelike hypersurface in the Minkowski space \(\mathbb R^{n,1}.\) We completely solve the prescribed lorentzian scalar curvature equation (m=2) in ambiant dimension 4, if the datas are strictly convex.

On Level Sets of Lorentzian Distance Function

Level sets of Lorentzian distance functions with respect to a point and with respect to an achronal spacelike hypersurface, are analyzed. Some bounds for the Laplacian of such Lorentzian distance functions are obtained and, in relation to them, some spacetime singularity theorems are given.

Complete spacelike hypersurfaces with constant mean curvature in the de Sitter space: a gap theorem

Let $M^n$ be a complete spacelike hypersurface with constant mean curvature $H$ in the de Sitter space $S_1^{n+1}$. We use the operator $\phi =A-HI$, where $A$ is the second fundamental form of $M$, and the roots $B_H^- \le B_H^+$ of a certain second order polynomial, to prove that either $\vert\phi\vert^2\equiv 0$ and $M$ is totally umbilical, or $B_H^-\le\sqrt{\sup \vert\phi\vert^2}\le B_H^+$. For the case $H\geq 2\sqrt{n-1}/n$ we prove the following results: for every number $B$ in the interval $[\max\{0,B_H^-\},B_H^+]$ there is an example of a complete spacelike hypersurface such that $\sqrt{\sup \vert\phi\vert^2}=B$; if $\sqrt{\sup \vert\phi\vert^2}=B_H^-$ is attained at some point, then the corresponding $M$ is a hyperbolic cylinder. We characterize the hyperbolic cylinders as the only complete spacelike hypersurfaces in $S_1^{n+1}$ with constant mean curvature, non-negative Ricci curvature and having at least two ends. We also characterize all complete spacelike hypersurfaces of constant mean curvature with two distinct principal curvatures as rotation hypersurfaces or generalized hyperbolic cylinders.

Complete space-like hypersurfaces in locally symmetric Lorentz spaces

The purpose of this paper is to study complete space-like hypersurfaces with constant mean curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.

Positivity of quasilocal mass.

Motivated by the important work of Brown and York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with a positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface.

A Bernstein theorem for complete spacelike hypersurfaces of constant mean curvature in Minkowski space

In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).

INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

AbstractIn this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their $r$-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their $r$-mean curvatures are linearly related or one of them is constant.AMS 2000 Mathematics subject classification: Primary 53C42. Secondary 53B30; 53C50; 53Z05; 83C99

Uniqueness theorem of static degenerate and nondegenerate charged black holes in higher dimensions

We prove the uniqueness theorem for static higher dimensional charged black holes spacetime containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non-degenerate components of the event horizon.

On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes

In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime \(\overline M \) equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when \(\overline M \) is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when \(\overline M \) is a generalized Robertson-Walker spacetime.

Mean Curvature Flow of Spacelike Hypersurfaces near Null Initial Data

We prove an interior estimate for the gradient function of spacelike hypersurfaces which move by mean curvature in a Lorentzian manifold. This estimate depends only on a time function which measures how far the hypersurfaces are from being null. As a consequence, we show that under mean curvature flow a weakly spacelike initial hypersurface instantaneously becomes smooth and strictly spacelike except along null geodesics which extend to its boundary.

On the Differentiability of Compact Cauchy Horizons

We construct an example of a compact Cauchy horizon that is not a differentiable manifold. This answers in the negative the question of whether a compact Cauchy horizon that arises from a space-like hypersurface is necessarily smooth.

Dirac Observables and the Phase Space of General Relativity

In the canonical approach to general relativity it is customary to parametrize the phase space by initial data on spacelike hypersurfaces. However, if one seeks a theory dealing with observations that can be made by a single localized observer, it is natural to use a different description of the phase space. This results in a different set of Dirac observables from that appearing in the conventional formulation. It also suggests a possible solution to the problem of time, which has been one of the obstacles to the development of a satisfactory quantum theory of gravity.