Sheaves of Einstein algebras

To include all types of singularities into a geometrically tractable theoretical scheme we change from Einstein algebras, an algebraic generalization of general relativity, to sheaves of Einstein algebras. The theory of such spaces, called Einstein structured spaces, is developed. Both quasiregular and curvature singularities are studied in some detail. Examples of the closed Friedmann world model and the Schwarzschild spacetime show that Schmidt'sb-boundary is a useful theoretical tool when considered in the category of structured spaces.     

Origin of Classical Singularities

We briefly review some results concerning theproblem of classical singularities in generalrelativity, obtained with the help of the theory ofdifferential spaces. In this theory one studies a givenspace in terms of functional algebras defined on it.Then we present a generalization of this methodconsisting in changing from functional (commutative)algebras to noncommutative algebras. By representingsuch an algebra as a space of operators on a Hilbertspace we study the existence and properties of variouskinds of singular space-times. The results obtainedsuggest that in the noncommutative regime, supposedly reigning in the Planck era, there is nodistinction between singular and non-singular states ofthe universe, and that classical singularities areproduced in the transition process from thenoncommutative geometry to the standard space-timephysics.     

Unwrapping Closed Timelike Curves

Closed timelike curves (CTCs) appear in many solutions of the Einstein equation, even with reasonable matter sources. These solutions appear to violate causality and so are considered problematic. Since CTCs reflect the global properties of a spacetime, one can attempt to extend a local CTC-free patch of such a spacetime in a way that does not give rise to CTCs. One such procedure is informally known as unwrapping. However, changes in global identifications tend to lead to local effects, and unwrapping is no exception, as it introduces a special kind of singularity, called quasi-regular. This “unwrapping” singularity is similar to the string singularities. We define an unwrapping of a (locally) axisymmetric spacetime as the universal cover of the spacetime after one or more of the local axes of symmetry is removed. We give two examples of unwrapping of essentially 2+1 dimensional spacetimes with CTCs, the Gott spacetime and the Gödel spacetime. We show that the unwrapped Gott spacetime, while singular, is at least devoid of CTCs. In contrast, the unwrapped Gödel spacetime still contains CTCs through every point. A “multiple unwrapping” procedure is devised to remove the remaining circular CTCs. We conclude that, based on the given examples, CTCs appearing in the solutions of the Einstein equation are not simply a mathematical artifact of coordinate identifications. Alternative extensions of spacetimes with CTCs tend to lead to other pathologies, such as naked quasi-regular singularities.     

Models of the universe

In this paper a theory of models of the universe is proposed. We refer to such models ascosmological models, where a cosmological model is defined as an Einstein-inextendible Einstein spacetime. A cosmological model isabsolute if it is a Lorentz-inextendible Einstein spacetime,predictive if it is globally hyperbolic, andnon-predictive if it is nonglobally-hyperbolic. We discuss several features of these models in the study of cosmology. As an example, any compact Einstein spacetime is always a non-predictive absolute cosmological model, whereas a noncompact complete Einstein spacetime is an absolute cosmological model which may be either predictive or non-predictive. We discuss the important role played by maximal Einstein spacetimes. In particular, we examine the possible proper Lorentz-extensions of such spacetimes, and show that a spatially compact maximal Einstein spacetime is exclusively either a predictive cosmological model or a proper sub-spacetime of a non-predictive cosmological model. Provided that the Strong Cosmic Censorship conjecture is true, a generic spatially compact maximal Einstein spacetime must be a predictive cosmological model. It isconjectured that the Strong Cosmic Censorship conjecture isnot true, and converting a vice to a virtue it is argued that the failure of the Strong Cosmic Censorship conjecture would point to what may be general relativity's greatest prediction of all, namely,that general relativity predicts that general relativity cannot predict the entire history of the universe.     

Star product geometries

We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows us to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. Principles of symmetry can be implemented in this way. Two examples are considered: (a) the general covariance in noncommutative spacetime, which leads to a noncommutative theory of gravity, and b) symplectomorphims of the algebra of observables associated with noncommutative configuration spaces, which leads to a geometric formulation of quantization on noncommutative spacetime (i.e., we establish a noncommutative correspondence principle from ⋆-Poisson brackets to ⋆-commutators).     

Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.     

Deformed field theory on $\kappa$-spacetime

A general formalism is developed that allows the construction of a field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime (the Poincaré group) is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable *-product. Fields are elements of this function algebra. The Dirac and Klein-Gordon equation are defined and an action is found from which they can be derived.Received: 17 July 2003, Published online: 26 September 2003     

Spatial Kasner Solution and an Infinite Slab with a Constant Energy Density

We study the solutions of the Einstein equations in the presence of a thick infinite slab with constant energy density. When there is an isotropy in the plane of the slab, we find an explicit exact solution that matches with the Rindler and Weyl-Levi-Civita spacetimes outside the slab. We also show that there are solutions that can be matched with general anisotropic Kasner spacetime outside the slab. In any case, it is impossible to avoid the presence of the Kasner type singularities in contrast to the well-known case of spherical symmetry, where by matching the internal Schwarzschild solution with the external one, the singularity in the center of coordinates can be eliminated.

     

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities.     

Return of the quantum cosmic censor

The influential theorems of Hawking and Penrose demonstrate that spacetime singularities are ubiquitous features of general relativity, Einstein's theory of gravity. The utility of classical general relativity in describing gravitational phenomena is maintained by the cosmic censorship principle. This conjecture, whose validity is still one of the most important open questions in general relativity, asserts that the undesirable spacetime singularities are always hidden inside of black holes. In this Letter we reanalyze extreme situations which have been considered as counterexamples to the cosmic censorship hypothesis. In particular, we consider the absorption of fermion particles by a spinning black hole. Ignoring quantum effects may lead one to conclude that an incident fermion wave may over spin the black hole, thereby exposing its inner singularity to distant observers. However, we show that when quantum effects are properly taken into account, the integrity of the black-hole event horizon is irrefutable. This observation suggests that the cosmic censorship principle is intrinsically a quantum phenomena.     

On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums

Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.     

An Einstein addition law for nonparallel boosts using the geometric algebra of space-time

The modern use of algebra to describe geometric ideas is discussed with particular reference to the constructions of Grassmann and Hamilton and the subsequent algebras due to Clifford. An Einstein addition law for nonparallel boosts is shown to follow naturally from the use of the representation-independent form of the geometric algebra of space-time.     

The structure of the space of solutions of Einstein’s equations coupled with scalar fields

Following the work of Arms, Fischer, Marsden and Moncrief, it is proved that the space of solutions of Einstein’s equations coupled with self-gravitating mass-less scalar fields has conical singularities at each spacetime possessing a compact Cauchy surface of constant mean curvature and a nontrivial set of simultaneous Killing fields, either all spacelike or including one (independent) timelike.     

Noncommutative closed Friedman universe

Heller et al. (J Math Phys 46:122501, 2005; Int J Theor Phys 46:2494, 2007) proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra defined on a groupoid having the frame bundle over space–time as its base space. The generalized Einstein equation is assumed in the form of the eigenvalue equation of the Einstein operator on a module of derivations of the algebra . No matter sources are assumed. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, generalized eigenvalues of the Einstein operator reproduce components of the perfect fluid energy-momentum tensor for the usual Friedman model together with the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space–time geometry. These results are briefly discussed.     

On the genericity of spacetime singularities

We consider here the genericity aspects of spacetime singularities that occur in cosmology and in gravitational collapse. The singularity theorems (that predict the occurrence of singularities in general relativity) allow the singularities of gravitational collapse to be either visible to external observers or covered by an event horizon of gravity. It is shown that the visible singularities that develop as final states of spherical collapse are generic. Some consequences of this fact are discussed.      

Gravity/CFT correspondence for three-dimensional Einstein gravity with a conformal scalar field

We study the three-dimensional Einstein gravity conformally coupled to a scalar field. Solutions of this theory are geometries with vanishing scalar curvature. We consider solutions with a constant scalar field which corresponds to an infinite Newton?s constant. There is a class of solutions with possible curvature singularities which asymptotic symmetries are given by two copies of the Virasoro algebra. We argue that the central charge of the corresponding CFT is infinite. Furthermore, we construct a family of Schwarzschild solutions which can be conformally mapped to the Martínez–Zanelli solution of Einstein?s equations with a negative cosmological constant coupled to conformal scalar field.     

Algebras of Random Operators Associated to Delone Dynamical Systems

We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C ^*-subalgebra to discuss a Shubin trace formula.     

A general method for constructing curved traversable wormholes in (2+1)-dimensional spacetime

We develop a general method for constructing curved traversable wormholes in (2+1)-dimensional spacetime, by generating surfaces of revolution around smooth curves. Application of this method to a straight line gives the usual spherically symmetric wormholes. The physics behind (2+1)-d curved traversable wormholes is discussed based on solutions to the Einstein field equations when the tidal force is zero. The Einstein field equations are found to reduce to one equation whereby the mass-energy density varies linearly with the Ricci scalar, which signifies that our (2+1)-d curved traversable wormholes are supported by dust of ordinary and exotic matter without radial tension nor lateral pressure. With this, two examples of (2+1)-d curved traversable wormholes: the helical wormhole and the catenary wormhole, are constructed and we show that there exist geodesics through them supported by non-exotic matter. This general method is applicable to our (3+1)-d spacetime.     

Pseudo-Effect Algebras as Total Algebras

It is shown that every pseudo-effect algebra can be organized as a total algebra. In general, this total algebra is not unique, but it always determines the original partial addition. Special attention is paid to lattice-ordered pseudo-effect algebras for which the total operations can be defined in a unique natural way and this allows one to characterize pairs of compatible elements.