Spacelike Hypersurfaces Research Articles

On the behavior at infinity of complete spacelike hypersurfaces in a Lorentzian space form

On the behavior at infinity of complete spacelike hypersurfaces in a Lorentzian space form

Axially symmetric solutions in Ricci-inverse modified gravity

We investigate a family of axial symmetry solution constructed in general relativity (GR) within the framework of Ricci-inverse (RI) gravity theory. In GR, these solutions admitted closed time-like curves at an instant of time from an initial spacelike hypersurface in a causally well-behaved manner, thus, violates the causality condition. Our aim is to examine these axial symmetry solutions within the context of Ricci-inverse gravity theory to determine whether closed time-like curves still appear in this new gravity theory. We consider two Classes of RI-gravity models: (i) Class-II models defined by a function f=f(R,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=f({\\mathcal {R}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document} gravity and (ii) Class-III models defined by f=f(R,A,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=f({\\mathcal {R}},{\\mathcal {A}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document}, where Aμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\mu \ u }$$\\end{document} is the anti-curvature tensor, A=gμνAμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}=g_{\\mu \ u }\\,A^{\\mu \ u }$$\\end{document} as its scalar, and Rμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\mu \ u }$$\\end{document} is the Ricci tensor. We are able solved the modified field equations considering these axial symmetry solutions as background in RI-gravity with null radiation as the matter content and the cosmological constant. This confirms that the chosen family of axial symmetry solutions are valid solutions in RI-gravity theory and, consequently, closed time-like curves is still form, analogous to their formation in GR.

On Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form (M¯,g¯), where M¯=R×fM and g¯=ϵdt2+f2(t)gM. We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where (M¯,g¯) is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field ∂t and the so-called support function θ to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below.

Constraints and time evolution in generic f(Riemann) gravity

Abstract We give a detailed canonical analysis of the n-dimensional f(Riemann)
gravity, correcting the earlier results in the literature. We also
write the field equations in the Fischer-Marsden form which is amenable
to identifying the non-stationary energy on a spacelike hypersurface.
We give pure R^2 and R_{\mu\nu}R^{\mu\nu} theories as examples.

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

The Convexity of Entire Spacelike Hypersurfaces with Constant $$\sigma _{n-1}$$ Curvature in Minkowski Space

The Convexity of Entire Spacelike Hypersurfaces with Constant $$\sigma _{n-1}$$ Curvature in Minkowski Space

Back-reaction of long-wavelength cosmological fluctuations as measured by a clock field

We consider the back-reaction of cosmological fluctuations on the local expansion rate averaged over a space-like hypersurface of constant value of a clock field. We show that in the infrared limit, the fluctuations lead to a decrease in the average expansion rate, measured at a fixed value of the clock field, compared to what would be obtained in a homogeneous universe. We work in the context of Einstein gravity coupled to perfect fluid matter.

Constant Mean Curvature Hypersurfaces in Anti-de Sitter Space

Abstract We study entire spacelike constant mean curvature hypersurfaces in Anti-de Sitter space of any dimension. First, we give a classification result with respect to their asymptotic boundary, namely we show that every admissible sphere $\Lambda $ is the boundary of a unique such hypersurface, for any given value $H$ of the mean curvature. We also demonstrate that, as $H$ varies in $\mathbb {R}$, these hypersurfaces analytically foliate the invisible domain of $\Lambda $. Finally, we extend Cheng-Yau Theorem to the Anti-de Sitter space, which establishes the completeness of any entire constant mean curvature hypersurface.

A classical firewall transformation as a canonical transformation

The firewall transformation put forward by ’t Hooft in recent years has made ambitious claims of solving the firewall problem and the black hole information paradox while maintaining unitary evolution. However, the theory has received limited attention from the community, especially in regards to its foundations in purely classical gravitational physics. This paper investigates the underlying assumptions of ’t Hooft’s firewall transformation before quantization. We find that the limiting procedure used by ’t Hooft in order to obtain an identification of the quantum operators for ingoing and outgoing particles near a black hole is not consistent. We propose a correction, which involves a more relaxed approximation regime. In the new approximation regime, we find a new classical analog for the firewall transformation for spherical shells, which allows evolving the spherical shells’ dynamics past their point of collision. In the classical theory, no firewall is removed, as both ingoing and outgoing matter is present on every spacelike hypersurface, and it does not appear that any firewalls will be removed after a canonical quantization.

Riemannian domains and complete spacelike hypersurfaces with one end in steady state type spacetimes

Riemannian domains and complete spacelike hypersurfaces with one end in steady state type spacetimes

Characterization results and a sharp integral inequality for LW spacelike hypersurfaces in locally symmetric Lorentzian spaces

Characterization results and a sharp integral inequality for LW spacelike hypersurfaces in locally symmetric Lorentzian spaces

Generalized Minkowski Type Integral Formulas for Compact Hypersurfaces in Pseudo-Riemannian Manifolds

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature.

Curvature properties of spacelike hypersurfaces in a RW spacetime

Curvature invariants of both intrinsic and extrinsic nature play a significant role in elucidating the geometry of a spacetime. In particular, these invariants are useful in detecting event horizon of black holes. Notable examples of spacetimes are provided by the generalized Robertson-Walker (GRW) models. An (m+1)-dimensional GRW spacetime is a Lorentzian warped product I×fN endowed with the warped metric gˆ=−dt2+f2(t)g, having as base an open real interval I equipped with the metric −dt2 and as fiber any Riemannian space (N,g) of dimension m, where f is a smooth positive-valued function on I. In this article, we focus our study on the GRW spacetimes having the fiber a Riemannian space with metric g=gk of constant sectional curvature k and denoted by L1m+1(k,f), i.e. RW spacetimes. Using an approach originally developed in Decu et al. (2008) [46], we obtain in this work the lower bound of the (generalized) normalized Casorati curvatures for a spacelike hypersurface H in the GRW spacetime L1m+1(k,f) in terms of the (normalized) scalar curvature of H, the constant sectional curvature k of the fiber, the normal hyperbolic angle of the hypersurface and the warping function f. We also derive the conditions under which this lower bound is reached. We finally apply the results to some basic cosmological models, namely Lorentz-Minkowski, de Sitter, Einstein-de Sitter, anti de Sitter and steady state spacetimes.

Semiclassical dynamics of Hawking radiation

We consider gravity in 3+1 spacetime dimensions coupled to N scalar matter fields in a semiclassical limit where . The dynamical evolution of a black hole including the back-reaction of the Hawking radiation on the metric is formulated as an initial-value problem. The quantum stress-energy tensor is evaluated using a point-splitting regularization along spacelike geodesics. To account for the quantum entanglement of the matter fields, they are treated as a set of bilocal collective fields defined on spacelike hypersurfaces. The resulting semiclassical field equations include terms up to fourth order in derivatives that can be treated in a perturbative expansion. The formulation we arrive at should be amenable to numerical simulation of time-dependent semiclassical spacetime.

On the rigidity of cosmological space-times

In this paper, we analyse a family of geometrically well-behaved cosmological space-times (V^{n+1},g), which are foliated by intrinsically isotropic space-like hypersurfaces {M_t}_{tin mathbb {R}}, which are orthogonal to a family of co-moving observers defined by a global time-like vector field U. In particular, this implies such space-times satisfy several of the well-known criteria for isotropic cosmological space-times, although, in the family in question, the simultaneity spaces (M_t,g_t) associated with U can have as sectional curvature a sign-changing function k(t). Being this clearly impossible in the FLRW family of standard cosmological space-times, it motivates us to revisit the geometric rigidity consequences of different definitions of isotropy available in the literature. In this analysis, we divide such definition according to whether the isometries involved are taken to be (local) space-time (STI space-times) or (local) space isometries (SI space-times) of (M_t,g_t) for each t. This subtlety will be shown to be critical, proving that only when space-time isometries are considered one obtains the well-known rigidity properties associated with isotropic cosmological space-times. In particular, SI space-times will be shown to be a strictly larger class than the STI ones, allowing a family of basic cosmological curvature change models which are not even locally isometric to any FLRW space-time.

Curvature estimates for spacelike graphic hypersurfaces in Lorentz–Minkowski space R1n+1$\mathbb {R}^{n+1}_{1}$

AbstractIn this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the ‐dimensional Lorentz–Minkowski space , and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed 2‐nd Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane of center at origin and radius 1, can be proven.

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb R}^{n+1}_1$

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb R}^{n+1}_1$

Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson–Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics.

Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi–Bernstein-type problems

In this article spacelike hypersurfaces immersed in twisted product spacetimes Itimes _f F with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, and a relation that needs to hold on each immersed spacelike hypersurface in Itimes _f F to be a simple warped product is also derived. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersurfaces and other for totally umbilic ones with a suitable restriction on their mean curvature are presented, and a full description of maximal hypersurfaces in twisted product spacetimes with a one-dimensional Lorentzian fiber is also included. Finally, the mean curvature equation for a spacelike graph on the fiber is computed and as an application, some Calabi–Bernstein-type results are proven.

On Calabi–Bernstein’s Problem for Uniqueness of Complete Spacelike Hypersurfaces in Weighted Generalized Robertson–Walker Spacetimes

In this paper, we study Calabi–Bernstein’s problem for the uniqueness of complete constant weighted mean curvature spacelike hypersurfaces Σn in weighted generalized Robertson–Walker spacetimes −I×ρMfn. Under appropriate geometric assumptions, we confirm that Σn is a spacelike slice.