Spacelike Cauchy Hypersurface Research Articles

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.

A family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry with Cauchy temporal functions

We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry that feature time coordinates, which are specific Cauchy temporal functions, i.e., the level sets of these time coordinates are smooth, asymptotically flat, spacelike Cauchy hypersurfaces. Coordinate systems of this kind are well suited for the study of the temporal evolution of matter and radiation fields in the joined exterior and interior regions of the Schwarzschild black hole geometry, whereas the associated foliations can be employed as initial data sets for the globally hyperbolic development under the Einstein flow. For their construction, we formulate an explicit method that utilizes the geometry of—and structures inherent in—the Penrose diagram of the Schwarzschild black hole geometry, thus relying on the corresponding metrical product structure. As an example, we consider an integrated algebraic sigmoid function as the basis for the determination of such a coordinate system. Finally, we generalize our results to the Reissner–Nordström black hole geometry up to the Cauchy horizon. The geometric construction procedure presented here can be adapted to yield similar coordinate systems for various other spacetimes with the same metrical product structure.

Lorentzian manifolds with causal Killing vector field: causality and geodesic connectedness

We prove that a compact Lorentzian manifold $$({\overline{M}},{\overline{g}})$$ admitting a causal Killing vector field is totally vicious or it contains a compact achronal Killing horizon. In particular a compact spacetime which satisfies the null generic condition and admits a causal Killing vector field is totally vicious. If in addition, its universal Lorentzian covering is globally hyperbolic then it is geodesically connected. In the non-compact case, we prove that a chronological spacetime admitting a complete causal Killing vector field, a smooth spacelike partial Cauchy hypersurface S and satisfying the null generic condition is stably causal. If additionally S is compact then the spacetime is globally hyperbolic. We also determine the geodesic connectedness in this case.

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.

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.

Dynamics of entanglement in expanding quantum fields

We develop a functional real-time approach to computing the entanglement between spatial regions for Gaussian states in quantum field theory. The entanglement entropy is characterized in terms of local correlation functions on space-like Cauchy hypersurfaces. The framework is applied to explore an expanding light cone geometry in the particular case of the Schwinger model for quantum electrodynamics in 1+1 space-time dimensions. We observe that the entanglement entropy becomes extensive in rapidity at early times and that the corresponding local reduced density matrix is a thermal density matrix for excitations around a coherent field with a time dependent temperature. Since the Schwinger model successfully describes many features of multiparticle production in e+e− collisions, our results provide an attractive explanation in this framework for the apparent thermal nature of multiparticle production even in the absence of significant final state scattering.

The global future stability of the FLRW solutions to the Dust-Einstein system with a positive cosmological constant

We study small perturbations of the Friedman–Lemaître–Robertson–Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the space-like Cauchy hypersurfaces are diffeomorphic to 𝕋3. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. In our analysis, we construct harmonic-type coordinates such that the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends those of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715; C. Lübbe and J. A. Valiente Kroon, A conformal approach for the analysis of the nonlinear stability of pure radiation cosmologies, Ann. Phys. 328 (2013) 1–25], where analogous results were proved for the Euler–Einstein system under the equations of state [Formula: see text], [Formula: see text]. The dust-Einstein system is the case cs = 0. The main difficulty that we overcome here is that the dust's energy density loses one degree of differentiability compared to the cases [Formula: see text], which introduces many obstacles for closing the estimates. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive elliptic estimates that complement the energy estimates of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715]. Our results apply in particular to small perturbations of the vanishing dust state containing vacuum regions.

Dirac equation with external potential and initial data on Cauchy surfaces

With this paper we provide a mathematical review on the initial-value problem of the one-particle Dirac equation on space-like Cauchy hypersurfaces for compactly supported external potentials. We, first, discuss the physically relevant spaces of solutions and initial values in position and mass shell representation; second, review the action of the Poincar\'e group as well as gauge transformations on those spaces; third, introduce generalized Fourier transforms between those spaces and prove convenient Paley-Wiener- and Sobolev-type estimates. These generalized Fourier transforms immediately allow the construction of a unitary evolution operator for the free Dirac equation between the Hilbert spaces of square-integrable wave functions of two respective Cauchy surfaces. With a Picard-Lindel\"of argument this evolution map is generalized to the Dirac evolution including the external potential. For the latter we introduce a convenient interaction picture on Cauchy surfaces. These tools immediately provide another proof of the well-known existence and uniqueness of classical solutions and their causal structure.

On the full holonomy group of Lorentzian manifolds

We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold and a properly discontinuous group of isometries. Most of our examples are quotients of pp-waves with disconnected holonomy and without parallel vector field. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor.

The gravity/CFT correspondence

General Relativity can be formulated in terms of a spatially Weyl invariant gauge theory called Shape Dynamics. Using this formulation, we establish a "bulk/bulk" duality between gravity and a Weyl invariant theory on spacelike Cauchy hypersurfaces. This duality has two immediate consequences: i) it leads trivially to a corresponding "bulk/boundary" duality between General Relativity and a boundary CFT, and ii) the boundary can be defined in a gauge-invariant way. Moreover, the corresponding bulk/boundary duality is sufficient to explain a large portion of the evidence in favor of gauge/gravity duality and provides independent evidence for the AdS/CFT correspondence.

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.

Further Results on the Smoothability of Cauchy Hypersurfaces and Cauchy Time Functions

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R \times S$ in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).

Proof of the Symmetry of the Off-Diagonal¶Hadamard/Seeley–deWitt's Coefficients in¶ C∞ Lorentzian Manifolds by a “Local Wick Rotation”

Completing the results obtained in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth $D$-dimensional Lorentzian manifolds. To this end, it is shown that, in any Lorentzian manifold, a sort of ``local Wick rotation'' of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinates are ``physical''. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point, or, more generally, in a neighborhood of a space-like (Cauchy) hypersurface, into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or K\"ahlerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to $C^\infty$ non analytic Lorentzian manifolds by approximating Lorentzian $C^{\infty}$ metrics by analytic metrics in common geodesically convex neighborhoods. The symmetry requirement plays a central r\^{o}le in the point-splitting renormalization procedure of the one-loop stress-energy tensor in curved spacetimes for Hadamard quantum states.

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF6, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincare conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT*TM of theTeichmuller space TMof conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, gV), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT*TM. For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.

The vacuum and Bach equations in terms of light cone cuts

Light cone cuts are the intersection of the light cones of space–time events with an initial data surface which is usually taken to be null infinity. These are determined by a ‘‘cut function,’’ a function of the space–time coordinates and the sphere of generators of null infinity. The Bach equations and the conformal vacuum equations are shown to lead to a single scalar ‘‘main’’ equation on the cut function. In simplified cases, i.e., when the Weyl curvature is either small or self-dual, this equation can be integrated to give a simple scalar conformally invariant elliptic equation on the sphere of null directions. It has the remarkable property that the general solution of the Einstein vacuum equations in these cases can be derived from the solution to this auxiliary equation in two dimensions. In general, it will not be possible to reduce it to an auxilliary equation on the sphere. Nevertheless, the main equation is still a necessary condition and it is conjectured that it is also sufficient to imply the Bach equations in general and supporting arguments are given. It is also shown how to extend the Kozameh–Newman framework to the case of light cone cuts of a spacelike Cauchy hypersurface. The analogous procedures for Yang–Mills are also discussed. A conformally invariant formalism for calculations on null infinity is described in an Appendix. This is used for all the calculations which are only described in brief form in the main text.