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.     

Homothetic maps of the space-time distance function and differentiability

Let (M ₁,g ₁) and (M ₂,g ₁) be time-oriented space-times. Letd _i(p,q) be the supremum of lengths of future directed causal curves inM _i fromp toq. Ifq is not in the future ofp, thend _i (p, q)=0. A distance homothetic mapf is a function fromM ₁ ontoM ₂ which is not assumed to be continuous, but which satisfiesd ₂(f(p),f(q))=cd ₁(p,q) for allp,q M ₁. IfM ₁ is strongly causal, then the distance homothetic mapf is a diffeomorphism which mapsg ₁ to a scalar multiple ofg ₂. Thus for strongly causal space-times, distance homothetic maps are homothetic in the usual sense. WhenM ₁ is not strongly causal, distance homothetic maps are not necessarily differentiable nor even continuous. An example is given of a space-time which has discontinuous maps which are one to one, onto, and distance preserving.     

Morse theory on timelike and causal curves

It is shown that the set of timelike curves in a globally hyperbolic space-time manifold can be given the structure of a Hubert manifold under a suitable definition of timelike. The causal curves are the topological closure of this manifold. The Lorentzian energy (corresponding to Milnor's energy, except that the Lorentzian inner product is used) is shown to be a Morse function for the space of causal curves. A fixed end point index theorem is obtained in which a lower bound for the index of the Hessian of the Lorentzian energy is given in terms of the sum of the orders of the conjugate points between the end points.     

Stability of geodesic incompleteness for Robertson-Walker space-times

Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = –dt ² fh, where –a<b+, (H, h) is a Riemannian manifold andf: (a, b)(0, ) is a smooth function. We show that ifa>– and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC ⁰ perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC ¹ perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia and NSF grant No. MCS77-18723(02).     

The global structure of simple space-times

According to a standard definition of Penrose, a space-time admitting well-defined future and past null infinitiesI ⁺ andI ^– is asymptotically simple if it has no closed timelike curves, and all its endless null geodesics originate fromI ^– and terminate atI ⁺. The global structure of such space-times has previously been successfully investigated only in the presence of additional constraints. The present paper deals with the general case. It is shown thatI ⁺ is diffeomorphic to the complement of a point in some contractible open 3-manifold, the strongly causal regionI ₀ ⁺ ofI ⁺ is diffeomorphic to , and every compact connected spacelike 2-surface inI ⁺ is contained inI ₀ ⁺ and is a strong deformation retract of bothI ₀ ⁺ andI ⁺. Moreover the space-time must be globally hyperbolic with Cauchy surfaces which, subject to the truth of the Poincaré conjecture, are diffeomorphic to ³.     

Globally hyperbolic space-times which are timelike Cauchy complete

LetM be a globally hyperbolic manifold. Among the many forms of completeness that may be imposed onM are timelike Cauchy completeness and finite compactness. These two forms of completeness are shown to be equivalent for globally hyperbolic manifolds. They are also equivalent to the statement that every inextendible future-directed (past-directed) geodesic starting in the chronological future (past) ofp has points at arbitrarily large distance fromp.     

Some examples of incomplete space-times

An example is given of a space-time which is timelike and spacelike complete but null incomplete. An example is also given of a space-time which is geodesically complete but contains an inextendible timelike curve of bounded acceleration and finite length. These two examples may be modified so that in each case they become globally hyperbolic and retain the stated properties. All of the examples are conformally equivalent to open subsets of the two-dimensional Minkowski space.     

Singularities in globally hyperbolic space-time

A singularity reached on a timelike curve in a globally hyperbolic space-time must be a point at which the Riemann tensor becomes infinite (as a curvature or intermediate singularity) or is of typeD and electrovac.     

Conformal changes and geodesic completeness

Let (M, g) be a causal spacetime. ConditionN will be satisfied if for each compact subsetK ofM there is no future inextendible nonspacelike curve which is totally future imprisoned inK. IfM satisfies conditionN, then wheneverE is an open and relatively compact subset ofM the spacetimeE with the metricg restricted toE is stably causal. Furthermore, there is a conformal factor such that (M, ² g) is both null and timelike geodesically complete. IfM is an open subset of two dimensional Minkowskian space, thenM is conformal to a geodesically complete spacetime.     

Cauchy boundary andb-incompleteness of space-time

It is shown that if a space-time (M, g) is time-orientable and its Levi-Civita connection [in the bundle of orthonormal frames over (M, g)] is reducible to anO(3) structure, one can naturally select a nonvanishing timelike vector field and a Riemann metricg ⁺ onM. The Cauchy boundary of the Riemann space (M, g ⁺) consists of endpoints ofb-incomplete curves in (M, g); we call it theCauchy singular boundary. We use the space-time of a cosmic string with a conic singularity to test our method. The Cauchy singular boundary of this space-time is explicitly constructed. It turns out to consist of what should be expected.     

Colored null flags and para-Fermi fields

An inverse problem of deriving the concept of quantized fields from a certain observable conserved current is investigated. It is found that a natural framework in which to attack the problem is provided for by what we shall call Green's ansatz of null decomposition of the current. The null decomposition naturally yields a set ofcolored null flags hoisted at each space-time point, a null flag comprizing a real null vector and an associated real null six-vector, and is invariant under all permutations of colors. From the fact that to any null flag there corresponds a two-component spinor it follows that the color permutation group is extended tocolor groups O(p) orU(p), wherep is the number of null flags considered. It is shown that para-Weyl (para-Fermi) fields of orderp2 can be deduced from the (chiral) set ofp colored null flags, and that the color groupU(p) is singled out that functions as the gauge group of para-Fermi theory.     

Fermat principle in Finsler spacetimes

It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point q (emission event) to a timelike curve γ (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here “arrival time” refers to a parametrization of the timelike curve γ. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background.     

Connecting Solutions of the Lorentz Force Equation do Exist

Recent results on the maximization of the charged-particle action in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of over a given causal homotopy class of curves connecting two causally related events x ₀≤x ₁. Action is proved to admit a maximum on , and also one in the adherence of each timelike homotopy class C. Moreover, the maximum σ ₀ on is timelike if contains a timelike curve (and the degree of differentiability of all the elements is at least C ²). In particular, this last result yields a complete Avez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x ₀≪x ₁ can be connected by means of a timelike solution of the Lorentz force equation corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m≠0) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied. An erratum to this article is available at .     

Vacuum type N space-times admitting homothetic vector fields with isolated fixed points

We consider vacuum space-times (M, g) which are of Petrov type N on an open dense subset ofM, and which admit (proper) homothetic vector fields with isolated fixed points. We prove that if such is the case then, at the fixed point, (M,g) is flat and the homothetic bivector,X _[a;b] , is necessarily simple-timelike. Furthermore, we prove that if the homothetic bivector remains simple-timelike in some neighbourhood of the fixed point then, around the fixed point, the space-time in question is a pp-wave. The paper ends with a local characterization and some examples of space-tunes satisfying these conditions.     

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.     

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     

Some global properties of closed spatially homogeneous space-times

We consider some properties of the space-times which contain a spatially homogeneous domain of dependenceD(V), whereF is acompact achronal spatial hypersurface of homogeneity. For example, it is shown that ifV has a nonempty future Cauchy horizon then the timelike geodesies orthogonal toV are future incomplete and there is strong causality failure onH ⁺(V). Also, conditions for the global hyperbolicity of such space-times are obtained.     

Causality in noncommutative two-sheeted space-times

We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in detail when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.     

Symmetries of cosmological Cauchy horizons

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions). For the special case of a null surface diffeomorphic toT ³ we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT ³ and prove the generality of the previously constructed non-degenerate solutions. We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einstein's equations are essentially an artefact of symmetry.     

deSitter space-time and rotation

A metric containing a parameterb is presented. It represents two distinct families of space-times, the Taub-nut family and the deSitter family, according asb=1 andb=4 respectively. The metric of the deSitter family of space-times contains a further parameterm. Whenm=0, the space-time is the usual homogeneous and isotropic deSitter space-time. But ifm≠0, the metric represents a space-time which is homogeneous but not isotropic satisfyingR _ik =Λg _ik . In this space-time, the 4-velocity of an observer at rest will have non-zero twist. The metric withb=4,m≠0 is interpreted as a metric representing a “rotating deSitter space-time”.