Spacelike Hypersurfaces Research Articles

Entire Scalar Curvature Flow and Hypersurfaces of Constant Scalar Curvature in Minkowski Space

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.

Entire spacelike hypersurfaces of constant Gauß curvature in Minkowski space

We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauß curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates.

On the gravitational angular momentum of rotating sources

The gravitational energy-momentum and angular momentum satisfy the algebra of the Poincare group in the full phase space of the teleparallel equivalent of general relativity. The expression for the gravitational energy-momentum may be written as a surface integral in the three-dimensional spacelike hypersurface, whereas the definition for the angular momentum is given by a volume integral. It turns out that in practical calculations of the angular momentum of the gravitational field generated by localized sources like rotating neutron stars, the volume integral reduces to a surface integral, and the calculations can be easily carried out. Similar to previous investigations in the literature, we show that the total angular momentum is finite provided a certain asymptotic behaviour is verified. We discuss the dependence of the gravitational angular momentum on the frame, and argue that it is a measure of the dragging of inertial frames.

Optimal Eigenvalue Estimate for the Dirac–Witten Operator on Bounded Domains with Smooth Boundary

Eigenvalue estimate for the Dirac–Witten operator is given on bounded domains (with smooth boundary) of spacelike spin hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS and chiral conditions). Roughly speaking, any eigenvalue of the Dirac–Witten operator satisfies $$\left|\lambda\right|^{2} \,\geq\, \frac{n}{n-1} \frak R_{0} ,$$ where \({\frak R_{0}}\) is the infimum of (the opposite of) the Lorentzian norm of the constraints vector. Equality cases are also investigated and lead to interesting geometric situations.

Covariance and time regained in canonical general relativity

Canonical vacuum gravity is expressed in generally covariant form in order that spacetime diffeomorphisms be represented within its equal-time phase space. In accordance with the principle of general covariance and ideas developed within history phase-space formalisms in [I. Kouletsis and K. V. Kuchar, Phys. Rev. D 65, 125026 (2002)], [K. Savvidou, Classical Quantum Gravity 18, 3611 (2001)], [K. Savvidou, Classical Quantum Gravity 21, 615 (2004)], [K. Savvidou, Classical Quantum Gravity 21, 631 (2004)], the time mapping T: M{yields}R and the space mapping X: M{yields}{sigma} that define the Dirac-Arnowitt-Deser-Misner (ADM) foliation are incorporated into the framework of the Hilbert variational principle. The resulting canonical action encompasses all individual Dirac-ADM actions, corresponding to different choices of foliating vacuum spacetimes by spacelike hypersurfaces. The equal-time phase space P=(g{sub ij},p{sup ij},Y{sup {alpha}},P{sub {alpha}}) includes the embeddings Y{sup {alpha}}: {sigma}{yields}M and their conjugate momenta P{sub {alpha}}. It is constrained by eight first-class constraints. The constraint surface C is determined by the super-Hamiltonian and supermomentum constraints of vacuum gravity and the vanishing of the embedding momenta. Deformations of the time and space mappings, {delta}T and {delta}X, and spacetime diffeomorphisms, V(set-membership sign)LDiffM, induce symplectic diffeomorphisms of P. While the generator D{sub ({delta}T,{delta}X)} of deformations depends on allmore » eight constraints, the generator D{sub V} of spacetime diffeomorphisms depends only on the embedding momentum constraints. As a result, spacetime observables, namely, dynamical variables F on P that are invariant under spacetime diffeomorphisms, (F,D{sub V})|{sub C}=0, are not necessarily invariant under the deformations of the mappings, (F,D{sub ({delta}T,{delta}X)})|{sub C}{ne}0, nor are they constants of the motion, (F,{integral}d{sup 3}xH)|{sub C}{ne}0. Dirac observables form only a subset of spacetime observables that are invariant under the transformations of T and X and do not evolve in time. In this generally covariant framework, the conventional interpretation of the canonical theory, due to Bergmann and Dirac, amounts to postulating that the transformations of the reference system (T,X) have no measurable consequences; i.e., that all first-class constraints generate gauge transformations. If this postulate is not deemed necessary, canonical gravity admits no classical problem of time.« less

Spacelike hypersurfaces with constant mean curvature in the steady state space

We consider complete spacelike hypersurfaces with constant mean curvature in the open region of de Sitter space known as the steady state space. We prove that if the hypersurface is bounded away from the infinity of the ambient space, then the mean curvature must be H = 1. Moreover, in the 2-dimensional case we obtain that the only complete spacelike surfaces with constant mean curvature which are bounded away from the infinity are the totally umbilical flat surfaces. We also derive some other consequences for hypersurfaces which are bounded away from the future infinity. Finally, using an isometrically equivalent model for the steady state space, we extend our results to a wider family of spacetimes.

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space S 1 n + 1 ( c ) , n ⩾ 3 , with constant normalized scalar curvature R satisfying n − 2 n c ⩽ R ⩽ c totally umbilical?

Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds

We derive, for the square operator of Yau, an analogue of the Omori–Yau maximum principle for the Laplacian. We then apply it to obtain nonexistence results concerning complete noncompact spacelike hypersurfaces immersed with constant higher order mean curvature in a conformally stationary Lorentz manifold.

SPACE-LIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACE

Let M be an n-dimensional space-like hypersurface in a locally symmetric Lorentz space, with n(n−1)R=κH(κ>0) and satisfying certain additional conditions on the sectional curvature. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M, respectively. We show that if the mean curvature is nonnegative and attains its maximum on M, then: (1) if H2<4(n−1)c/n2, M is totally umbilical; (2) if H2=4(n−1)c/n2, M is totally umbilical or is an isoparametric hypersurface; (3) if H2>4(n−1)c/n2 and S satisfies some pinching conditions, M is totally umbilical or is an isoparametric hypersurface. doi:10.1017/S1446181108000254

Axisymmetric evolution of Einstein equations and mass conservation

For axisymmetric evolution of isolated systems, we prove that there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution. The conserved mass integral controls the square of the extrinsic curvature and the square of first derivatives of the intrinsic metric. We also discuss applications of this result for the global existence problem in axial symmetry.

Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature

We prove the existence and uniqueness in Rn,1 of entire spacelike hypersurfaces contained in the future of the origin O and asymptotic to the light-cone, with scalar curvature prescribed at their generic point M as a negative function of the unit vector pointing in the direction of OM→, divided by the square of the norm of OM→ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from O (the hyperbolic space); radial upper and lower solutions are constructed which, relying on a previous result in the Cartesian setting, imply their existence.

Maximal Hypersurfaces of (m + 2)-Dimensional Lorentzian Space Forms

We determine maximal space-like hypersurfaces which are the images of subbundles of the normal bundle of m-dimensional totally geodesic space-like submanifolds of an (m + 2)-dimensional Lorentzian space form (c) under the normal exponential map. Then we construct examples of maximal space-like hypersurfaces of (c).

The hypersurfaces with conformal normal Gauss map in Hn+1 and S1n+1

In this paper, we introduce the fourth fundamental forms for hypersurfaces in Hn+1 and space-like hypersurfaces in S1n+1, and discuss the conformality of the normal Gauss map of the hypersurfaces in Hn+1 and S1n+1. Particularly, we discuss the surfaces with conformal normal Gauss map in H³ and S³1, and prove a duality property. We give a Weierstrass representation formula for space-like surfaces in S³1 with conformal normal Gauss map. We also state the similar results for time-like surfaces in S³1. Some examples of surfaces in S³1 with conformal normal Gauss map are given and a fully nonlinear equation of Monge-Ampère type for the graphs in S³1 with conformal normal Gauss map is derived.

Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σn immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space \({-\mathbb{R} \times M^n}\) , and whose boundary is contained into a slice {t} × Mn. By considering the hyperbolic caps of the Lorentz–Minkowski space \({\mathbb{L}^{n+1}}\) , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.

On marginally outer trapped surfaces in stationary and static spacetimes

In this paper, we prove that for any spacelike hypersurface containing an untrapped barrier in a stationary spacetime satisfying the null energy condition, no bounding marginally outer trapped surface lying in the exterior region where the stationary Killing vector is causal and penetrating into the timelike region can exist. In the static case, we prove that any bounding marginally outer trapped surface cannot penetrate into the exterior region where the static Killing vector is timelike. In fact, we prove this result at an initial data level, without even assuming the existence of a spacetime. The proof relies on a powerful theorem by Andersson and Metzger on the existence of an outermost marginally outer trapped surface.

Positive mass theorems for higher dimensional Lorentzian manifolds

We extend Witten’s proof [“A new proof of the positive energy theorem,” Commun. Math. Phys. 80, 381–402 (1981)] of the positive mass theorem to a rigorous proof for Lorentzian manifolds of any dimension. This includes the original higher-dimensional positive mass theorem proved by spinors [Parker, T. and Taubes, C., “On Witten’s proof of the positive energy theorem,” Commun. Math. Phys. 84, 223–238 (1982)] for dimension 4 and [Zhang, X., “Positive mass conjecture for five-dimensional Lorentzian manifolds,” J. Math. Phys. 40, 3540–3552 (1999)] for dimension 5. Stimulated by the article of Zhang [“Positive mass theorem for hypersurface in 5-dimensional Lorentzian manifolds,” Commun. Anal. Geom. 8, 635–652 (2000)], we weaken the spin condition on the spacelike hypersurface to require only a spinc structure and give a modified positive mass theorem for Lorentzian manifolds for dimensions 4, 5, and 6.

Self-similar cosmological solutions with dark energy. II. Black holes, naked singularities, and wormholes

We use a combination of numerical and analytical methods, exploiting the equations derived in a preceding paper, to classify all spherically symmetric self-similar solutions which are asymptotically Friedmann at large distances and contain a perfect fluid with equation of state $p=(\ensuremath{\gamma}\ensuremath{-}1)\ensuremath{\mu}$ with $0&lt;\ensuremath{\gamma}&lt;2/3$. The expansion of the Friedmann universe is accelerated in this case. We find a one-parameter family of self-similar solutions representing a black hole embedded in a Friedmann background. This suggests that, in contrast to the positive pressure case, black holes in a universe with dark energy can grow as fast as the Hubble horizon if they are not too large. There are also self-similar solutions which contain a central naked singularity with negative mass and solutions which represent a Friedmann universe connected to either another Friedmann universe or some other cosmological model. The latter are interpreted as self-similar cosmological white hole or wormhole solutions. The throats of these wormholes are defined as two-dimensional spheres with minimal area on a spacelike hypersurface and they are all nontraversable because of the absence of a past null infinity.

General boundary quantum field theory: Foundations and probability interpretation

We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary spacetime regions. State spaces are associated to general (not necessarily spacelike) hypersurfaces. We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms. We explain how standard quantum mechanics arises as a special case. We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework. We formulate vacuum axioms and incorporate spacetime symmetries into the framework. We show how the Schroedinger-Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form. We discuss the role of operators.

Remarks on the notion of global hyperbolicity

Global hyperbolicity is a central concept in Mathematical Relativity. Here, we review the different approaches to this concept explaining both, classical approaches and recent results. The former includes Cauchy hypersurfaces, naked singularities, and the space of the causal curves connecting two events. The latter includes structural results on globally hyperbolic spacetimes, their embeddability in Lorentz-Minkowski, and the recently revised notions of both, causal and conformal boundaries. Moreover, two criteria for checking global hyperbolicity are reviewed. The first one applies to general splitting spacetimes. The second one characterizes accurately global hyperbolicity and spacelike Cauchy hypersurfaces for standard stationary spacetimes, in terms of a naturally associated Finsler metric.

Foliations by (η−1)-umbilical spacelike hypersurfaces

Foliations by (η−1)-umbilical spacelike hypersurfaces