The space-time cut locus

Let (M, g) be a space-time with Lorentzian distance functiond. If (M, g) is distinguishing andd is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metricsg conformal tog. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut pointq of p along the geodesicc must be either the first conjugate point ofp or else there must be at least two maximal timelike (respectively, null) geodesics fromp toq. Ifq is a closest cut point ofp in a globally hyperbolic space-time, then eitherq is conjugate top or elseq is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point.     

A Domain of Spacetime Intervals in General Relativity

We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. From this one can show that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains. We obtain a mathematical setting in which one can study causality independently of geometry and differentiable structure, and which also suggests that spacetime emerges from something discrete.     

On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem

Given a globally hyperbolic spacetime M, we show the existence of a smooth spacelike Cauchy hypersurface S and, thus, a global diffeomorphism between M and ×S.The second-named author has been partially supported by a MCyT-FEDER Grant BFM2001-2871-C04-01.     

Null cut loci of spacelike surfaces

The null cut locus of a spacelike submanifold of codimension 2 in a space-time is defined. In globally hyperbolic space-times, it is shown that the future (past) null cut locusc _n ⁺ (H) [c _n ^- (H)] of a compact, acausal, spacelike submanifoldH of codimension 2 is a closed subset of the space-time, and each pointx c _n ⁺ (H) is either a focal point ofH along some future-directed null geodesic meetingH orthogonally or there exist at least two null geodesics fromH tox, realizing the distance betweenH andx or both. Also, it can be shown that the assumptions of the Penrose's singularity theorem for open globally hyperbolic space-times may be weakened to the space-times which are conformal to an open subset of an open globally hyperbolic space-time.This study is based on Chapter 3 of the author's Ph.D. thesis.     

On conformal Killing 2-form of the electromagnetic field

Let D:C^∞Λ^pM→C^∞(T^*MΛ^pM) be the first order linear differential operator on an n-dimensional (1≤p≤n−1) pseudo-Riemannian manifold (M,g). We have by the representation theory of orthogonal group, that the tangent bundle of this operation decomposes into the orthogonal and irreducible sum of forms of degree p+1 (which gives the exterior differential d), the forms of degree p−1 (defining the codifferential d^*) and the trace-free part of the partial symmetrization (the corresponding first order operator is denoted by D). The general forms in the kernel of D are closely related to conformal Killing vector fields, called conformal Killing p-forms, while those in kernel of d are called closed conformal Killing p-forms or, according to another terminology, planar p-forms. In particular an arbitrary planar 1-form ω is dual (by g) to the special concircular vector field ξ. We consider some local properties for the closed conformal Killing p-forms. As an application we present examples of decomposition into irreducible components for the electromagnetic field 2-form ω and its covariant derivative in four-dimensional space–time. In particular, we prove that the energy–momentum tensor T of the electromagnetic field is a symmetric conformal Killing tensor if the electromagnetic field 2-form ω is a conformal Killing form.     

Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, , with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's ‘Chronology Protection Conjecture’, according to which the laws of physics prevent one from manufacturing a ’time machine‘. Specifically, we prove: Theorem 1. There is no extension to of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2. The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M × M) of (x,x). In consequence of Theorem 2, quantities such as the renormalized expectation value of φ² or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the ‘Propagation of Singularities’ theorems of Duistermaat and H?rmander. Received: 14 March 1996/Accepted: 11 June 1996     

Letter: On the Global Hyperbolicity of Lorentz 2-Step Nilpotent Lie Groups

This work is concerned with the existence of Lorentz 2-step nilpotent Lie groups having a timelike center and which are not globally hyperbolic. Namely, we prove that any left invariant Lorentz metric with a timelike center on the Heisenberg group H ²ⁿ⁺¹ is not globally hyperbolic.     

A new bundle completion for parallelizable space-times

The Schmidt [9]b-boundaryM, for completing a space-timeM, has several desirable features. It is uniquely determined by the space-time metric in an elegant geometrical manner. The completed space-time is¯M=M M, where¯M= ⁺ M/O ⁺ and ⁺ M is the Cauchy completion (with respect to a toplogical metric induced by the Levi-Cività connection) of a component of the orthonormal frame bundle having structure groupO ⁺. ThenM consists of the endpoints of incomplete curves inM that have finite horizontal lifts in ⁺ M, and ifM= we say thatM isb-complete. It turns out thatM isb-complete if and only ifO ⁺ M is complete. This criterion for space-time completeness is stronger than geodesic completeness and Beem [1] has shown that this remains so even for the restricted class of globally hyperbolic space-times. Clarice [3] has shown that for such space-times the curvature becomes unbounded as theb-boundary is approached.Now ifM, then ⁺ M may contain degenerate fibers; thus the quotient topology for¯M is non-Hausdorff and precludes a manifold structure. Precisely this has been demonstrated by Bosshard [2] for Friedmann space-time, casting doubt on the physical significance of the completion. The only neighborhood of the Friedmann singularity is the whole of¯M, and in the closed model initial and final singularities are identified inM. Similarly, Johnson [7] showed that the completion of Schwarzschild space-time is non-Hausdorff because of degenerate ibers in¯O ⁺ M.Here we introduce a modification of the Schmidt procedure that appears to be useful in avoiding fiber degeneracy and in promoting a Hausdorff completion. The modification is to introduce an explicit vertical component into the metric forO ⁺ M by reference to a standard section, that is, to a parallelizationpMO ⁺ M We prove some general properties of thisp-completion and examine the particular case of a Friedmann space-time where there is a fairly natural choice of parallelization.     

Micro-local approach to the Hadamard condition in quantum field theory on curved space-time

For the two-point distribution of a quasi-free Klein-Gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space-time we prove the equivalence of (1) the global Hadamard condition, (2) the property that the Feynman propagator is a distinguished parametrix in the sense of Duistermaat and Hörmander, and (3) a new property referred to as the wave front set spectral condition (WFSSC), because it is reminiscent of the spectral condition in axiomatic quantum field theory on Minkowski space. Results in micro-local analysis such as the propagation of singularities theorem and the uniqueness up toC of distinguished parametrices are employed in the proof. We include a review of Kay and Wald's rigorous definition of the global Hadamard condition and the theory of distinguished parametrices, specializing to the case of the Klein-Gordon operator on a globally hyperbolic space-time. As an alternative to a recent computation of the wave front set of a globally Hadamard two-point distribution on a globally hyperbolic curved space-time, given elsewhere by Köhler (to correct an incomplete computation in [32]), we present a version of this computation that does not use a deformation argument such as that used in Fulling, Narcowich and Wald and is independent of the Cauchy evolution argument of Fulling, Sweeny and Wald (both of which are relied upon in Köhler's proof). This leads to a simple micro-local proof of the preservation of Hadamard form under Cauchy evolution (first shown by Fulling, Sweeny and Wald) relying only on the propagation of singularities theorem. In another paper [33], the equivalence theorem is used to prove a conjecture by Kay that a locally Hadamard quasi-free Klein-Gordon state on any globally hyperbolic curved space-time must be globally Hadamard.To my parents     

Spherically symmetric solution in higher-dimensional teleparallel equivalent of general relativity

A theory of(N+1)-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity(TEGR).The fundamental gravitational field variables are the(N+1)-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution(Schwarzschild(4+1)-dimensions).Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.     

Initial Data Engineering

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.Partially supported by a Polish Research Committee grant 2 P03B 073 24Partially supported by the NSF under Grants PHY-0099373 and PHY-0354659Partially supported by the NSF under Grant DMS-0305048 and the UW Royalty Research Fund     

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 toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré 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 ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . 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.     

Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property

This paper continues the analysis of the quantum states introduced in previous works and determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state λ _M , of a massless conformally coupled scalar field propagating in M. λ _M is “holographically” induced in the bulk by the universal BMS-invariant state λ defined on the future null infinity of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at future null infinity discussed in previous papers. This induction is possible when some requirements are fulfilled, in particular whenever the spacetime M and the associated unphysical one, M͂, are globally hyperbolic and M admits future time infinity i ⁺. λ _M coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case of a curved spacetime M, the state λ _M enjoys the following further remarkable properties:

Homoclinic points in symplectic and volume-preserving diffeomorphisms

LetM ⁿ be a compactn-dimensional manifold and ω be a symplectic or volume form onM ⁿ. Let ? be aC ¹ diffeomorphism onM ⁿ that preserves ω and letp be a hyperbolic periodic point of Φ. We show that genericallyp has a homoclinic point, and moreover, the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp. Takens [11] obtained the same result forn=2.     

Special Temporal Functions on Globally Hyperbolic Manifolds

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings in Minkowski spaces.     

The existence of constant mean curvature foliations of Gowdy 3-torus spacetimes

We consider the class of smooth, maximally extended, globally hyperbolic, vacuum, Gowdy spacetimes onT ³×R and prove that these spacetimes are globally foliated by space-like, constant mean curvature hypersurfaces. Our results can easily be extended to cover electrovac solutions of the same symmetry type and can probably be extended to cover other spacetime topologies as well.Research supported in part by NSF grant No. PHY79-16482 at Yale and No. PHY79-13146 at Berkeley     

Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems

Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.     

Flows on Quaternionic-Kähler and Very Special Real Manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M×N of a quaternionic-Kähler manifold M of negative scalar curvature and a very special real manifold N of dimension n0. Such gradient flows are generated by the ``energy function' f=P<sup>2</sup>, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kähler manifolds. For the homogeneous quaternionic-Kähler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point pM such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kähler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kähler manifolds we find degenerate local minima. This work was supported by the priority programme ``String Theory'of the Deutsche Forschungsgemeinschaft.     

Punctured Haag Duality in Locally Covariant Quantum Field Theories

We investigate a new property of nets of local algebras over 4-dimensional globally hyperbolic spacetimes, called punctured Haag duality. This property consists in the usual Haag duality for the restriction of the net to the causal complement of a point p of the spacetime. Punctured Haag duality implies Haag duality and local definiteness. Our main result is that, if we deal with a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, then also the converse holds. The free Klein-Gordon field provides an example in which this property is verified.     

Cosmological application on five-dimensional teleparallel theory equivalent to general relativity

A theory of(4+1)-dimensional gravity has been developed on the basis of which equivalent to the theory of general relativity by teleparallel.The fundamental gravitational field variables are the 5-dimensional(5D) vector fields(pentad),defined globally on a manifold M,and gravity is attributed to the torsion.The Lagrangian density is quadratic in the torsion tensor.We then apply the field equations to two different homogenous and isotropic geometric structures which give the same line element,i.e.,FRW in five dimensions.The cosmological parameters are calculated and some cosmological problems are discussed.