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.     

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.     

Relativistic star solutions in higher-dimensional pseudospheroidal space-time

We obtain relativistic solutions of a class of compact stars in hydrostatic equilibrium in higher dimensions by assuming a pseudospheroidal geometry for the spacetime. The space-time geometry is assumed to be (D − 1) pseudospheroid immersed in a D-dimensional Euclidean space. The spheroidicity parameter (λ) plays an important role in determining the equation of state of the matter content and the maximum radius of such stars. It is found that the core density of compact objects is approximately proportional to the square of the space-time dimensions (D), i.e., core of the star is denser in higher dimensions than that in conventional four dimensions. The central density of a compact star is also found to depend on the parameter λ. One obtains a physically interesting solution satisfying the acoustic condition when λ lies in the range λ > (D + 1)/(D − 3) for the space-time dimensions ranging from D = 4 to 8 and (D + 1)/(D − 3) < λ < (D ² − 4D + 3)/(D ² − 8D − 1) for space-time dimensions ≥9. The non-negativity of the energy density (ρ) constrains the parameter with a lower limit (λ > 1). We note that in the case of a superdense compact object the number of space-time dimensions cannot be taken infinitely large, which is a different result from the braneworld model.     

Higher dimensional cosmologies

Einstein equations are derived for D-dimensional space-time that spontaneously compactify to the product M⁴ × Π_{i = 1}^α M^d_i in which the metric is taken to be of the generalized Robertson-Walker form. Cosmological solutions for these equations are studied with power law, oscillatory and exponential behaviour for the D-dimensional Einstein-Maxwell, N = 2, D = 10 and N = 1, D = 11 supergravity models. In the Einstein-Maxwell case the presence of a cosmological constant forces the extra dimensions to be static. Nevertheless, it is possible to find solutions with vanishing effective 4 dimensional cosmological constant with an expanding 4-dimensional space-time. In the supergravity models the requirement of having compact extra dimensions restricts the solutions to have expansion only in the 4-dimensional space-time. Matter contribution is added to the energy-momentum tensor in an attempt to find new solutions.     

Global spinor fields in space-time

This paper is concerned with space-time manifolds that are space- and time-oriented, causal, and possess spinor structures. Five propositions are proven: (1) If a connected, space- and time-oriented manifold is simply con-nected, then it is non-compact; (2) If such a manifold is simply connected, it admits a spinor structure, which, moreover, is unique; (3) If the space-like section of M is compact, then there exists a global system of orthonormal tetrads on M; (4) The necessary and sufficient condition for every space-time M whose space-like section is compact to admit a spinor structure is that M have a global system of orthonormal tetrads; (5) Every space-time M which can be imbedded in R⁶ admits a spinor structure. It is further suggested that in view of the fact that the existence of a spinor structure is related to homotopy properties, space-time manifolds may be classified in terms of their homotopy groups _i (M), i=1,2, 3,4. In a concluding section, some avenues for future research are discussed.Supported by the National Science Foundation.     

Causality violation in spatially closed space-times

Some results are obtained which establish conditions under which causality violation cannot occur in space-times admitting a compact, but not necessarily acausal, spacelike hypersurface. For example, it is shown that causality violation can occur in a space-time admitting a compact spacelike slice and obeying fairly reasonable energy conditions only if it is “singular” in the sense that some “causal interval”J ⁺(p) ∩J ^? (q) has noncompact closure.     

Two theorems on flat space-time gravitational theories

The first theorem states that all flat space-time gravitational theories must have a Lagrangian with a first term that is an homogeneous (degree-1) function of the 4-velocityu ⁱ , plus a functional of _ij u ⁱ u ^j . The second theorem states that all gravitational theories, that satisfy the strong equivalence principle have a Lagrangian with a first termg _ij (x)u ⁱ u ^j plus an irrelevant term. In both cases the theories must issue from a unique variational principle. Therefore, under this condition it is impossible to find a flat space-time theory that satisfies the strong equivalence principle.     

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     

Holonomy and Skyrme's Model

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L²_loc connection; the local developing maps for such connections need not be continuous.The first author was partially supported by NSF grant DMS-0204651.The second author was partially supported by NSF grants DMS-9970638, and DMS-0200670     

Riccion from higher-dimensional space-time with D-dimensional sphere as a compact manifold and one-loop renormalization

Lagrangian density of riccions is obtained with the quartic self-interacting potential using higher-derivative gravitational action in (4 +D)-dimensional space-time withS ^D as a compact manifold. It is found that the resulting four-dimensional theory for riccions is one-loop multiplicatively renormalizable. Renormalization group equations are solved and its solutions yield many interesting results such as (i) dependence of extra dimensions on the enegy mass scale showing that these dimensions increase with the increasing mass scale up toD = 6, (ii) phase transition at 3.05 × 10¹⁶ GeV and (iii) dependence of gravitational and other coupling constants on energy scale. Results also suggest that space-time above 3.05 × 10¹⁶ GeV should be fractal. Moreover, dimension of the compact manifold decreases with the decreasing energy mass scale such thatD = 1 at the scale of the phase transition. Results imply invisiblity of S¹ at this scale (which is 3.05 × 10¹⁶ GeV).     

A generalization of the concept of constant mean curvature and canonical time

Some compact spaces of achronal hypersurfaces are constructed in various types of space-time. A variational principle is introduced on these spaces, smooth extremals of which are spacelike hypersurfaces of constant mean curvature. The integrand of the variational principle is shown to be upper semicontinuous and the direct methods of the calculus of variations are applied to obtain aC ⁰ extremal, which is defined to be a spacelike hypersurface of generalized constant mean curvature. The family of such hypersurfaces generated by altering the value of the mean curvature is discussed and the mean curvature itself is shown to have many of the properties of a canonical time coordinate.     

Ergodicity of probabilistic cellular automata: A constructive criterion

We give a sequence of criteria (of increasing complexity) for the exponential ergodicity of discrete time interacting particle systems. Each criterion involves estimating the dependence on initial conditions of the process on finite space-time volumes. It generalizes and improves the existing single site condition and is the analog of the Dobrushin-ShlosmanC _v condition in equilibrium statistical mechanics. Our dynamic criterion may also be used to prove the uniqueness of Gibbs state in situations where theC _v condition fails. As a converse we prove that if there is a certain form of convergence to the stationary measure faster thann ^–d , wheren is the time andd is the dimension of the lattice, then our condition holds for some space-time volumes and hence the convergence must be exponentially fast.     

Geodesic focusing and space-time topology

Given a space-timeM and a pointp inM, it is shown that, if the locus of first conjugate points ofp along future-directed null geodesics consists of a single point, thenM admits a compact (S ³) spacelike hypersurface. If in addition the null geodesics do not intersect before focusing, then, in a simply connected space-time, the spacelike hypersurface is a partial Cauchy surface.     

Fermionic Quantization and Configuration Spaces for the Skyrme and Faddeev-Hopf Models

The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is S². It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given. The first author was partially supported by NSF grant DMS-0204651 The second author was partially supported by EPSRC grant GR/R66982/01     

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.     

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.     

Selection rules for topology change

It is shown that there are restrictions on the possible changes of topology of space sections of the universe if this topology change takes place in a compact region which has a Lorentzian metric and spinor structure. In particular, it is impossible to create a single wormhole or attach a single handle to a spacetime but it is kinematically possible to create such wormholes in pairs. Another way of saying this is that there is a ₂ invariant for a closed oriented 3-manifold which determines whether can be the spacelike boundary of a compact manifoldM which admits a Lorentzian metric and a spinor structure. We evaluate this invariant in terms of the homology groups of and find that it is the mod2 Kervaire semi-characteristic.     

Toward a universal Lagrangian

A gauge theory with the gauge groupU(1)×U(1)×SU(2)×SU(3)×SU(4) is shown to fit well into the generalized Kaluza scheme with eleven-dimensional space-time and its compact subspaceS ²×S ⁵. A unified theory is obtained which exhibits some broken super-symmetric features (N = 8). Our approach is dictated by phenomenological requirements. The appearance of three generations of leptons and six flavors of colored quarks follows naturally. Within our Lagrangian there appear several free parameters (coupling constants), but some relations between them may follow from the requirement of cancellation of divergencies.Temporarily at Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Rd., Dublin 4, Ireland.     

Integrable optical geometry

It is shown that a Lorentzian 4-manifold admitting a congruence of optical (null) geodesics without shear and twist defines an optical geometry which is integrable (locally flat) in the sense of the theory of G-structures. The existence of a symmetric linear connection compatible with the optical geometry is another condition equivalent to the integrability of the optical G-structure.     

Banach spaces of weights on quasimanuals

Section 1 is a brief introduction. Section 2 contains the basic definitions of quasimanuals, weights, and operational logics. The linear spaceW of all weights on a quasimanualA is introduced and given a norm.W with this norm is seen to be a Banach space. The subspaceV ofW generated by the positive cone ofW is given the base norm and is also shown to be an Archimedian ordered Banach space with an additive norm. In Section 3 normal linear functionals onV ^* are defined in analogy with normal linear functionals onw ^* algebras. The spaceV is shown to be the set of normal functionals onV ^* and we showV to be the unique partially ordered Banach space with a closed generating cone which is predual toV ^*. Next, weakly compact subsets ofW are characterized in terms of eventwise convergence. This is the Hahn-Vitali-Saks theorem of classical measure theory in this noncommutative setting; several weak compactness results are drawn from this and compared with their classical counterparts. Section 4 introduces the ultraweak topology forV ^* in analogy with the same for the trace class operators on Hubert space. Here the condition for a compact base for the cone ofV is examined and shown to be a poor and unnecessary hypothesis in many circumstances. Many connections with the existent literature are made and throughout the paper there are many examples and open questions.