Spacelike Hypersurfaces Research Articles

Manifestly gauge invariant theory of the nonlinear cosmological perturbations in the leading order of the gradient expansion

In the full nonlinear cosmological perturbation theory in the leading order of the gradient expansion, all the types of the gauge invariant perturbation variables are defined. The metric junction conditions across the spacelike transition hypersurface are formulated in a manifestly gauge invariant manner. It is manifestly shown that all the physical laws such as the evolution equations, the constraint equations, and the junction conditions can be written using the gauge invariant variables which we defined only. Based on the existence of the universal adiabatic growing mode in the nonlinear perturbation theory and the $\ensuremath{\rho}$ philosophy where the physical evolution are described using the energy density $\ensuremath{\rho}$ as the evolution parameter, we give the definitions of the adiabatic perturbation variable and the entropic perturbation variables in the full nonlinear perturbation theory. In order to give the analytic order estimate of the nonlinear parameter ${f}_{\mathrm{NL}}$, we present the exponent evaluation method. As the models where ${f}_{\mathrm{NL}}$ changes continuously and becomes large, using the $\ensuremath{\rho}$ philosophy, we investigate the non-Gaussianity induced by the entropic perturbation of the component which does not govern the cosmic energy density, and we show that in order to obtain the significant non-Gaussianity it is necessary that the scalar field which supports the entropic perturbation is extremely small compared with the scalar field which supports the adiabatic perturbation.

Slant geometry of spacelike hypersurfaces in the lightcone

In this paper, we consider one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in the lightcone. These geometries are constructed by applying two of one-parameter families of Legendrian dualities between pseudo-spheres in Lorentz-Minkowski space. These Legendrian dualities have been recently given as a part of extensions of the mandala of Legendrian dualities in the previous research of the authors.

On the geometry of conformally stationary Lorentz spaces

We study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending a result of J. Simons on the minimality of cones in Euclidean space to these spaces, and apply it to the construction of complete, noncompact minimal Lorentz submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hypersurface to be of constant mean curvature. Finally, we study the strong r-stability of spacelike hypersurfaces of constant r-th mean curvature in a conformally stationary Lorentz manifold of constant sectional curvature, extending previous results in the current literature.

Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field

A new technique to study spacelike hypersurfaces of constant mean curvature in a spacetime which admits a timelike gradient conformal vector field is introduced. As an application, the leaves of the natural spacelike foliation of such spacetimes are characterized in some relevant cases. The global structure of this class of spacetimes is analyzed and the relation with its well-known subfamily of generalized Robertson–Walker spacetimes is exposed in detail. Moreover, some known uniqueness results for compact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes are widely extended. Finally, and as a consequence, several Calabi–Bernstein problems are solved obtaining all the entire solutions on a compact Riemannian manifold to the constant mean curvature spacelike hypersurface equation, under natural geometric assumptions.

The gravitational light shift and the Sachs-Wolfe effect

Using a 3+1 decomposition of spacetime, we derive a new formula to compute the gravitational light shifts as measured by two observers which are normal to the spacelike hypersurfaces defining the foliation. This formula is quite general and is also independent of the existence of Killing fields. Known examples are considered to illustrate the usefulness of the formula. In particular, we focus on the Sachs-Wolfe effect that arises in a perturbed Friedman-Robertson-Walker cosmology.

Odd-parity perturbations of the self-similar LTB spacetime

We consider the behaviour of odd-parity perturbations of those self-similar Lemaître–Tolman–Bondi spacetimes which admit a naked singularity. We find that a perturbation which evolves from initially regular data remains finite on the Cauchy horizon. Finiteness is demonstrated by considering the behaviour of suitable energy norms of the perturbation (and pointwise values of these quantities) on natural spacelike hypersurfaces. This result holds for a general choice of initial data and initial data surface. Finally, we examine the perturbed Weyl scalars in order to provide a physical interpretation of our results. Taken on its own, this result does not support cosmic censorship; however, a full perturbation of this spacetime would include even-parity perturbations, so we cannot conclude that this spacetime is stable to all linear perturbations.

On complete spacelike hypersurfaces with two distinct principal curvatures in a de Sitter space

We investigate complete spacelike hypersurfaces in a de Sitter space with two distinct principal curvatures and constant m-th mean curvature. By using Otsukiʼs idea, we obtain some global classification results. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in de Sitter ( n + 1 ) -spaces S 1 n + 1 ( 1 ) ( n ⩾ 3 ) of constant m-th mean curvature H m ( | H m | ⩾ 1 ) with two distinct principal curvatures λ and μ satisfying inf ( λ − μ ) 2 > 0 are the hyperbolic cylinders. We also obtain some global rigidity results for hyperbolic cylinders and obtain some non-existence results.

On Vanishing Theorems for Vector Bundle Valued p-Forms and their Applications

Let $F: [0, \infty) \to [0, \infty)$ be a strictly increasing $C^2$ function with $F(0)=0$. We unify the concepts of $F$-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce $F$-Yang-Mills fields, $F$-degree, $F$-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. When $F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,$ and $1-\sqrt{1-2t},$ the $F$-Yang-Mills field becomes an ordinary Yang-Mills field, $p$-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the $E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) and derive the first variational formula of the $E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors, linked to $F$-conservation laws yield monotonicity formulae. A "macroscopic" version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for $p-$forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for $F-$harmonic maps (e.g. $p$-harmonic maps), and $F-$Yang-Mills fields (e.g. generalized Yang-Mills-Born-Infeld fields on manifolds). We also obtain generalized Chern type results for constant mean curvature type equations for $p-$forms on $\Bbb{R}^m$ and on manifolds $M$ with the global doubling property by a different approach. The case $p=0$ and $M=\mathbb{R}^m$ is due to Chern.

Positive mass theorems for high-dimensional spacetimes with black holes

We give a rigorous proof of the positive mass theorem for high-dimensional spacetimes with black holes if the spacetime contains an asymptotically flat spacelike spin hypersurface and satisfies the dominant energy condition along the hypersurface. We also weaken the spin structure on the spacelike hypersurface to spinc structure and give a modified positive mass theorem for spacetimes with black holes in dimensions 4, 5 and 6.

Complete spacelike CMC hypersurfaces in a Lorentzian space form

Let x : Mn �¨ M n+1 1 (c) be a complete spacelike hypersurface immersed into a Lorentzian space form, where M n+1 1 (c) is a Lorentz{Minkowski space Ln+1 = Rn+1 1 , a de Sitter space Sn+1 1 �¼ Rn+2 1 or an anti-de Sitter space Hn+1 1 �¼ Rn+2 2 , according to c = 0, c = 1 or c = .1, respectively. Let . = .x, a. and �O = ..H , a., where .H is the mean curvature vector eld of Mn and a is a xed nonzero vector in the corre- sponding pseudo-Euclidean space. We prove that if Mn has constant mean curvature (CMC), and . = �E�O, for some real number �E, then Mn is a spacelike isoparametric hypersurface of M n+1 1 (c). Furthermore, it is either a totally umbilical hypersurface or a hyperbolic cylinder.

Critical Points of Wang–Yau Quasi-Local Energy

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $\Sigma$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $\Omega$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $\Sigma$ has positive Gaussian curvature and all boundary components of $\Omega$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $\Sigma$ in $\Omega$ and $H_0$ is the mean curvature of $\Sigma$ when isometrically embedded in $R^3$. If $\Omega$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $\Sigma$ in $\Omega$ is a strict local minimum of the Wang-Yau quasi-local energy of $\Sigma$, 2. on a small perturbation $\tilde{\Sigma}$ of $\Sigma$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tilde{\Sigma}$.

Deformation of codimension-2 surfaces and horizon thermodynamics

The deformation equation of a spacelike submanifold with an arbitrary codimension is given by a general construction without using local frames. In the case of codimension-1, this equation reduces to the evolution equation of the extrinsic curvature of a spacelike hypersurface. In the more interesting case of codimension-2, after selecting a local null frame, this deformation equation reduces to the well known (cross) focusing equations. We show how the thermodynamics of trapping horizons is related to these deformation equations in two different formalisms: with and without introducing quasilocal energy. In the formalism with the quasilocal energy, the Hawking mass in four dimension is generalized to higher dimension, and it is found that the deformation of this energy inside a marginal surface can be also decomposed into the contributions from matter fields and gravitational radiation as in the four dimension. In the formalism without the quasilocal energy, we generalize the definition of slowly evolving future outer trapping horizons proposed by Booth to past trapping horizons. The dynamics of the trapping horizons in FLRW universe is given as an example. Especially, the slowly evolving past trapping horizon in the FLRW universe has close relation to the scenario of slow-roll inflation. Up to the second order of the slowly evolving parameter in this generalization, the temperature (surface gravity) associated with the slowly evolving trapping horizon in the FLRW universe is essentially the same as the one defined by using the quasilocal energy.

On the Gauss map of complete spacelike hypersurfaces with bounded mean curvature in the Minkowski space

In this paper, under an appropriated restriction on the Gauss map, we obtain an extension of the Xin-Aiyama theorem concerning to complete spacelike hypersurfaces immersed with bounded mean curvature in the Minkowski space.

Relative Lorentzian volume comparison with integral Ricci and scalar curvature bound

A relative Lorentzian volume comparison estimate between spacelike hypersurfaces is studied with the integral curvature bound in terms of Ricci and Scalar curvature which generalize the Bishop–Gromov volume comparison theorem.

QUANTUM AMPLITUDES IN BLACK-HOLE EVAPORATION: COMPLEX APPROACH AND SPIN-0 AMPLITUDE

This paper is concerned with the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space–time and weak radiation at a very late time. We evaluate quantum amplitudes (not just probabilities) for transitions from initial to final states. This quantum description shows that no information is lost in collapse to a black hole. Boundary data for the gravitational field and (in this paper) a scalar field are posed on an initial space-like hypersurface ΣIand a final surface ΣF. These asymptotically flat three-surfaces are separated by a Lorentzian proper-time interval T (typically very large), as measured at spatial infinity. The boundary-value problem is made well-posed, both classically and quantum-mechanically, by a rotation of T into the lower-half complex plane: T → |T| exp (- iθ), with 0 < θ ≤ π/2. This corresponds to Feynman's +iϵ prescription. We consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action [Formula: see text] as a functional of the boundary data. Following Feynman, the Lorentzian quantum amplitude is recovered in the limit θ → 0+from the well-defined complex-T amplitude. Dirac's canonical approach to the quantisation of constrained systems shows that, for locally supersymmetric theories of gravity, the amplitude is exactly semi-classical, namely [Formula: see text] for weak perturbations, apart from delta functionals of the supersymmetry constraints. We treat such quantum amplitudes for weak scalar-field configurations on ΣF, taking (for simplicity) the weak final gravitational field to be spherically symmetric. The treatment involves adiabatic solutions to the scalar wave equation. This considerably extends work reported in previous papers, by giving explicit expressions for the real and imaginary parts of such quantum amplitudes.

Extremal black holes, gravitational entropy and nonstationary metric fields**Based on an essay selected for honorable mention in the 2010 Gravity Research Foundation Essay Competition. This work was partly completed at KITP, Santa Barbara, CA, USA.

We show that extremal black holes have zero entropy by pointing out a simple fact: they are time independent throughout the spacetime and correspond to a single classical microstate. We show that nonextremal black holes, including the Schwarzschild black hole, contain a region hidden behind the event horizon where all their Killing vectors are spacelike. This region is nonstationary and the time t labels a continuous set of classical microstates, the phase space [hab(t), Pab(t)], where hab is a three-metric induced on a spacelike hypersurface Σt and Pab is its momentum conjugate. We determine explicitly the phase space in the interior region of the Schwarzschild black hole. We identify its entropy as a measure of an outside observer's ignorance of the classical microstates in the interior since the parameter t which labels the states lies anywhere between 0 and 2M. We provide numerical evidence from recent simulations of gravitational collapse in isotropic coordinates that the entropy of the Schwarzschild black hole stems from the region inside and near the event horizon where the metric fields are nonstationary; the rest of the spacetime, which is static, makes no contribution. Extremal black holes have an event horizon but in contrast to nonextremal black holes, their extended spacetimes do not possess a bifurcate Killing horizon. This is consistent with the fact that extremal black holes are time independent and therefore have no distinct time reverse.

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

In this paper, we investigate complete spacelike hypersurfaces in the de Sitter space S 1 n + 1 ( c ) with constant k-th mean curvature and two distinct principal curvatures one of which is simple. We obtain some characterizations of the Riemannian product H 1 ( c 1 ) × S n − 1 ( c 2 ) or H n − 1 ( c 1 ) × S 1 ( c 2 ) in the de Sitter space S 1 n + 1 ( c ) .

Biharmonic Hypersurfaces of LP-Sasakian Manifolds

Biharmonic Hypersurfaces of LP-Sasakian ManifoldsIn this paper the biharmonic hypersurfaces of Lorentzian para-Sasakian manifolds are studied. We firstly find the biharmonic equation for a hypersurface which admits the characteristic vector field of the Lorentzian para-Sasakian as the normal vector field. We show that a biharmonic spacelike hypersurface of a Lorentzian para-Sasakian manifold with constant mean curvature is minimal. The biharmonicity condition for a hypersurface of a Lorentzian para-Sasakian manifold is investigated when the characteristic vector field belongs to the tangent hyperplane of the hypersurface. We find some necessary and sufficient conditions for a timelike hypersurface of a Lorentzian para-Sasakian manifold to be proper biharmonic. The nonexistence of proper biharmonic timelike hypersurfaces with constant mean curvature in a Ricci flat Lorentzian para-Sasakian manifold is proved.

Linear Weingarten spacelike hypersurfaces in de Sitter space

In this paper, we give a classification on spacelike linear Weingarten hypersurfaces in de Sitter space $S^{n+1}_{1}(1)$ according to the sectional curvature or the length of the second fundamental form.

Sharp Finite Speed for Hyperbolic Problems Well Posed in Gevrey Classes

For linear hyperbolic operators with coefficients a j, α(t, x) Gevrey-s in x and C k in t with 0 < k < 1 satisfying the conditions known to be sufficient to guarantee that the Cauchy problem with initial surface {t = 0} is well posed for Gevrey-s data we prove two things: 1) the domain of influence of any point x on t = 0 is contained in the union of influence curves through x, and 2) there is local uniqueness in the Cauchy problem for arbitrary spacelike hypersurfaces.