Spacelike Hypersurfaces Research Articles

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.

Generalized Uncertainty Principle in Quantum Cosmology

The effects of gravity which manifest themselves when performing the simultaneous measurement of two non-commuting observables in the quantum theory are discussed. Matter and gravity are considered as quantum fields. The Schr¨odinger-type time equation is given for the case of a finite number of degrees of freedom: one for the matter field and one for geometry. For a spatially closed system filled with dust and radiation being in definite quantum states, the solutions to the quantum equations are found, and the existence of the minimum measurable length and the minimum momentum is shown. It appears that the simultaneous measurement of fluctuations of the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime cannot be performed with an accuracy exceeding the Planck constant. Unruh’s and Bronstein’s uncertainty relations are discussed.

Yang–Mills Theory of Gravity

The canonical formulation of general relativity (GR) is based on decomposition space–time manifold M into R × Σ , where R represents the time, and Ksi is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general coordinates, and local Lorentz transformations. These symmetries are associated with conserved currents that are coupled to gravity. These symmetries are studied on a three dimensional space-like hypersurface Σ embedded in a four-dimensional space–time manifold. This implies continuous symmetries and conserved currents by Noether’s theorem on that surface. We construct a three-form E i ∧ D A i (D represents covariant exterior derivative) in the phase space ( E i a , A a i ) on the surface Σ , and derive an equation of continuity on that surface, and search for canonical relations and a Lagrangian that correspond to the same equation of continuity according to the canonical field theory. We find that Σ i 0 a is a conjugate momentum of A a i and Σ i a b F a b i is its energy density. We show that there is conserved spin current that couples to A i , and show that we have to include the term F μ ν i F μ ν i in GR. Lagrangian, where F i = D A i , and A i is complex S O ( 3 ) connection. The term F μ ν i F μ ν i includes one variable, A i , similar to Yang–Mills gauge theory. Finally we couple the connection A i to a left-handed spinor field ψ , and find the corresponding beta function.

On the Cauchy Problem for the Linearised Einstein Equation

A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely solvable up to gauge solutions, given initial data on a spacelike Cauchy hypersurface. The solution map is an isomorphism between initial data (modulo gauge producing initial data) and solutions (modulo gauge solutions). In the first part of this work, we show that the solution map is actually an isomorphism of locally convex topological vector spaces. This implies that the equivalence class of solutions depends continuously on the equivalence class of initial data. We may therefore conclude well-posedness of the Cauchy problem. In the second part, we show that the linearised constraint equations can always be solved on a closed manifold with vanishing scalar curvature. This generalises the classical notion of TT-tensors on flat space used to produce models of gravitational waves. All our results are proven for smooth and distributional initial data of arbitrary real Sobolev regularity.

Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation

Abstract We prove that any sufficiently differentiable space-like hypersurface of ℝ 1 + N {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation ∂ t ⁢ t ⁡ u - Δ ⁢ u = | u | p - 1 ⁢ u {\partial_{tt}u-\Delta u=|u|^{p-1}u} on ℝ × ℝ N {{\mathbb{R}}\times{\mathbb{R}}^{N}} , for any 1 ≤ N ≤ 4 {1\leq N\leq 4} and 1 < p ≤ N + 2 N - 2 {1<p\leq\frac{N+2}{N-2}} . We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at t = 0 {t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at t = 0 {t=0} . To obtain a finite-energy solution of the original problem from trace arguments, we need to work with H 2 × H 1 {H^{2}\times H^{1}} solutions for the transformed problem.

First order perturbations of hypersurfaces of arbitrary causal character

In this work we study the problem of first order perturbations of a general hypersurface, i.e. with arbitrary causal character at each point. We extend the framework by Mars (2005 Class. Quantum Grav. 22 3325–48) where this problem was studied to second order for everywhere timelike or spacelike hypersurfaces, and we adapt it to cover the general case. We apply the formalism to the matching of spacetimes across a general hypersurface to first order in perturbation theory.

Uniform Boundedness and Continuity at the Cauchy Horizon for Linear Waves on Reissner\u2013Nordstr\xf6m\u2013AdS Black Holes

Motivated by the Strong Cosmic Censorship Conjecture for asymptotically Anti-de Sitter (AdS) spacetimes, we initiate the study of massive scalar waves satisfying Box _g psi - mu psi =0 on the interior of AdS black holes. We prescribe initial data on a spacelike hypersurface of a Reissner–Nordström–AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity. It was known previously that such waves only decay at a sharp logarithmic rate (in contrast to a polynomial rate as in the asymptotically flat regime) in the black hole exterior. In view of this slow decay, the question of uniform boundedness in the black hole interior and continuity at the Cauchy horizon has remained up to now open. We answer this question in the affirmative.

Spacelike hypersurfaces with constant conformal sectional curvature in ℝ1n+1

Spacelike hypersurfaces with constant conformal sectional curvature in ℝ1n+1

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordstr\"om-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon ("near" $i^+$), in such a way that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.

Extensivity, entropy current, area law, and Unruh effect

We present a general method to determine the entropy current of relativistic matter at local thermodynamic equilibrium in quantum statistical mechanics. Provided that the local equilibrium operator is bounded from below and its lowest lying eigenvector is non-degenerate, it is proved that, in general, the logarithm of the partition function is extensive, meaning that it can be expressed as the integral over a 3D space-like hypersurface of a vector current, and that an entropy current exists. We work out a specific calculation for a non-trivial case of global thermodynamic equilibrium, namely a system with constant comoving acceleration, whose limiting temperature is the Unruh temperature. We show that the integral of the entropy current in the right Rindler wedge is the entanglement entropy.

Defining a Relativity-Proof Notion of the Present via Spatio-temporal Indeterminism

In this paper we describe a novel approach to defining an ontologically fundamental notion of co-presentness that does not go against the tenets of relativity theory. We survey the possible reactions to the problem of the present in relativity theory, introducing a terminological distinction between a static role of the present, which is served by the relation of simultaneity, and a dynamic role of the present, with the corresponding relation of co-presentness. We argue that both of these relations need to be equivalence relations, but they need not coincide. Simultaneity, the sharing of a temporal coordinate, need not have fundamental ontological import, so that a relativizing strategy with respect to simultaneity seems promising. The notion of co-presentness, on the other hand, does have ontological import, and can therefore not be relativized to an observer or to an arbitrarily chosen frame. We argue that a formal representation of indeterminism can provide the structure needed to anchor the relation of co-presentness, and that this addition is in fact congenial to the notion of dynamic time as requiring real (indeterministic) change. The resulting picture is one of an extended dynamic present, implying a formal distinction between static (coordinate) simultaneity and dynamic co-presentness. After working out the basics of our approach in the simpler framework of branching time, we provide our full analysis in the framework of branching space-times, which allows for a formal definition of modal correlations. The spatial extension of the dynamic present can reach as far as the modal correlations do. In the limit, the dynamic present could extend across a maximal space-like hypersurface.

The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities

We consider the wave equation, [Formula: see text], in fixed flat Friedmann–Lemaître–Robertson–Walker and Kasner spacetimes with topology [Formula: see text]. We obtain generic blow up results for solutions to the wave equation toward the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to the solutions that blow up in an open set of the Big Bang hypersurface [Formula: see text]. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate [Formula: see text]-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the [Formula: see text] norms of the solutions blow up toward the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates, respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

The Localised Bounded $$L^2$$-Curvature Theorem

In this paper, we prove a localised version of the bounded $L^2$-curvature theorem of Klainerman-Rodnianski-Szeftel. More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface $\Sigma$ with boundary, and show that the time of existence of a classical solution depends only on an $L^2$-bound on the Ricci curvature, an $L^4$-bound on the second fundamental form of $\partial \Sigma \subset \Sigma$, an $H^1$-bound on the second fundamental form, and a lower bound on the volume radius at scale $1$ of $\Sigma$. Our localisation is achieved by first proving a localised bounded $L^2$-curvature theorem for small data posed on $B(0,1)$, and then using the scaling of the Einstein equations and a low regularity covering argument on $\Sigma$ to reduce from large data on $\Sigma$ to small data on $B(0,1)$. The proof uses the author's previous work, and the bounded $L^2$-curvature theorem as black boxes.

On the classification of MOTS in the de Sitter space

In this paper, we deal with marginally outer trapped surfaces (MOTS) immersed in the de Sitter space \(\mathbb {S}_1^{n+2}\). In this setting we are able to obtain a Simons formula for the null second fundamental form and under some appropriate constraints on the MOTS, we apply a weak maximum principle in order to guarantee that it must be either a totally geodesic submanifold or isometric to an open piece of an isoparametric submanifold with two distinct principal curvatures one of which is simple. In this last case, supposing that the initial data where the MOTS lying is a totally umbilical spacelike hypersurface of \(\mathbb {S}_1^{n+2}\), we conclude that it must be either isometric to a circular cylinder, a hyperbolic cylinder or a Clifford torus.

Is a topology change after a Big Rip possible?

Motivated by condensed matter physical systems, in which a finite-time singularity indicates that the topology of the system changes, we critically examine the possibility of the Universe’s topology change at a finite-time cosmological singularity. We emphasize on Big Rip and Type II and IV cosmological singularities, which we classify to future spacelike and timelike singularities. For the Type IV and Type II singularities, since no geodesics incompleteness occurs, no topological change is allowed, by using Geroch’s theorem arguments. However, for the Big Rip case, Tipler’s arguments allow a topology change, if the spacetime in which the topology change occurs is non-compact and the boundary of this region are two topologically distinct three-dimensional spacelike partial Cauchy hypersurfaces. Also, some additional requirements must hold true, among which the weak energy condition, which can be satisfied in a geometric way in the context of a modified gravity. We critically examine Tipler’s arguments for the Big Rip case, and we discuss the mathematical implications of such a topological change, with regard to the final hypersurface on which geodesics incompleteness occurs.

Spacelike Hypersurfaces in Spatially Parabolic Standard Static Spacetimes and Calabi–Bernstein-Type Problems

Complete spacelike hypersurfaces in spatially parabolic standard static spacetimes are studied. Under natural boundedness assumptions, we show how the parabolicity of the base is inherited by any spacelike hypersurface and vice versa. Moreover, we give new uniqueness and non-existence results for complete spacelike hypersurfaces in these ambient spacetimes as well as solve new Calabi–Bernstein-type problems.

Generalized Uncertainty Principle in Quantum Cosmology for the Maximally Symmetric Space

The new uncertainty relation is derived in the context of the canonical quantum theory with gravity in the case of the maximally symmetric space. This relation establishes a connection between fluctuations of the quantities, which determine the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime and introduces the uncertainty principle for quantum gravitational systems. The generalized time-energy uncertainty relation taking gravity into account gravity is proposed. It is shown that known Unruh’s uncertainty relation follows, as a particular case, from the new uncertainty relation. As an example, the sizes of fluctuations of the scale factor and its conjugate momentum are calculated within an exactly solvable model. All known modifications of the uncertainty principle deduced previously from different approaches in the theory of gravity and the string theory are obtained as particular cases of the proposed general expression.

Interior volume of Banados–Teitelboim–Zanelli black hole

We calculate the interior volume for the three-dimensional Banados–Teitelboim–Zanelli black hole in this paper. To that end, we obtain the volume functional and numerically find a path to extremize it, before choosing a spacelike hypersurface to get an asymptotic analytical expression for the interior volume based on the characteristics of the path.

Spacelike Möbius Hypersurfaces in Four Dimensional Lorentzian Space Form

In this paper, we first set up an alternative fundamental theory of Mobius geometry for any umbilic-free spacelike hypersurfaces in four dimensional Lorentzian space form, and prove the hypersurfaces can be determined completely by a system consisting of a function W and a tangent frame {Ei}. Then we give a complete classification for spacelike Mobius homogeneous hypersurfaces in four dimensional Lorentzian space form. They are either Mobius equivalent to spacelike Dupin hypersurfaces or to some cylinders constructed from logarithmic curves and hyperbolic logarithmic spirals. Some of them have parallel para-Blaschke tensors with non-vanishing Mobius form.

Spacelike hypersurfaces in standard static spacetimes

In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic angle or bounded height is maximal. Our conclusions follow from general mean curvature estimates for spacelike hypersurfaces. In case where the spacetime is a Lorentzian product with spatial factor of nonnegative Ricci curvature and sectional curvatures bounded below, we also show that a complete maximal hypersurface not intersecting a spacelike slice is itself a slice. This result is obtained from a gradient estimate for parametric maximal hypersurfaces.