Quasiregular Singularities Research Articles

A topological drive for spacetime travel

We present a toy metric of spacetime travel from topological change. A bubble-like baby Universe is detached and re-attached from our Universe. Depending on where the bubble is re-attached, matter may travel superluminally or backwards-in-time through the bubble. Quasiregular singularities are formed at the detachment and re-attachment spacetime points. The spacetime is traversable and not covered by any horizons. Exotic matter violating energy conditions is required to realize such spacetimes.

Time machines with the compactly determined Cauchy horizon

The building of a time machine, if possible at all, requires the relevant regions of spacetime to be compact (that is, physically speaking, free from sources of unpredictability such as infinities and singularities). Motivated by this argument we consider the spacetimes with the compactly determined Cauchy horizons (CDCHs), the defining property of which is the compactness of $\overline{J^-(\EuScript U)}\cap J^+(\EuScript S_0)$, where $\EuScript U$ is an open subset of the Cauchy horizon and $\EuScript S_0$ is a Cauchy surface of the initial globally hyperbolic region $\ingh$. The following two facts are established: 1) $\ingh$ has no globally hyperbolic maximal extension. This means that by shaping appropriately a precompact portion of a globally hyperbolic region one can \emph{force} the Universe to produce either a closed causal curve, or a quasiregular singularity, whichever it abhors less; 2) Before a CDCH is formed a null geodesic appears which infinitely approaches the horizon returning again and again in the same --- arbitrarily small --- region. The energy of the photon moving on such a geodesic increases with each passage, or at least falls insufficiently fast. As a result, an observer located in the mentioned region would see a bunch of photons passing through his laboratory with the arbitrarily large total energy. We speculate that this phenomenon may have observable consequences.

Cracking the Taub-NUT

We present further analysis of an anisotropic, non-singular early universe model that leads to the viable cosmology presented in Dechant et al (2009 Phys. Rev. D 79, 043524). Although this model (the Dechant–Lasenby–Hobson (DLH) model) contains scalar field matter, it is reminiscent of the Taub-NUT vacuum solution in that it has biaxial Bianchi IX geometry and its evolution exhibits a dimensionality reduction at a quasi-regular singularity that one can identify with the Big Bang. We show that the DLH and Taub-NUT metrics are related by a coordinate transformation, in which the DLH time coordinate plays the role of conformal time for Taub-NUT. Since both models continue through the Big Bang, the coordinate transformation can become multivalued. In particular, in mapping from DLH to Taub-NUT, the Taub-NUT time can take only positive values. We present explicit maps between the DLH and Taub-NUT models, with and without a scalar field. In the vacuum DLH model, we find a periodic solution expressible in terms of elliptic integrals; this periodicity is broken in a natural manner as a scalar field is gradually introduced to recover the original DLH model. Mapping the vacuum solution over to Taub-NUT coordinates recovers the standard (non-periodic) Taub-NUT solution in the Taub region, where Taub-NUT time takes positive values, but does not exhibit the two NUT regions known in the standard Taub-NUT solution. Conversely, mapping the complete Taub-NUT solution to the DLH case reveals that the NUT regions correspond to imaginary time and space in DLH coordinates. We show that many of the well-known ‘pathologies’ of the Taub-NUT solution arise because the traditional coordinates are connected by a multivalued transformation to the physically more meaningful DLH coordinates. In particular, the ‘open-to-closed-to-open’ transition and the Taub and NUT regions of the (Lorentzian) Taub-NUT model are replaced by a closed pancaking universe with spacelike homogeneous sections at all times.

Repulsons in the Myers-Perry Family

In this paper, we show that curvature-regular asymptotically flat solitons with negative mass are contained in the Myers-Perry family of odd spacetime dimensions. These solitons do not have a horizon, but instead a conical singularity of quasi-regular nature surrounded by naked CTCs. This quasi-regular singularity can be made regular for a set of discrete values of angular momentum, at least for the Myers-Perry solutions with UG(N) symmetry. Although the time coordinate is required to have a periodicity at infinity and the spatial infinity becomes a lens space S^{D-2}/Z_n, the corresponding spacetime is simply connected, and the CTCs cannot be eliminated by taking a covering spacetime.

Anisotropic, nonsingular early universe model leading to a realistic cosmology

We present a novel cosmological model in which scalar field matter in a biaxial Bianchi IX geometry leads to a nonsingular “pancaking” solution: the hypersurface volume goes to zero instantaneously at the “big bang”, but all physical quantities, such as curvature invariants and the matter energy density remain finite, and continue smoothly through the big bang. We demonstrate that there exist geodesics extending through the big bang, but that there are also incomplete geodesics that spiral infinitely around a topologically closed spatial dimension at the big bang, rendering it, at worst, a quasiregular singularity. The model is thus reminiscent of the Taub-NUT vacuum solution in that it has biaxial Bianchi IX geometry and its evolution exhibits a dimensionality reduction at a quasiregular singularity; the two models are, however, rather different, as we will show in a future work. Here we concentrate on the cosmological implications of our model and show how the scalar field drives both isotropization and inflation, thus raising the question of whether structure on the largest scales was laid down at a time when the universe was still oblate (as also suggested by [ T.?S. Pereira, C. Pitrou and J.-P. Uzan J. Cosmol. Astropart. Phys. 2007 6 ()][ C. Pitrou, T.?S. Pereira and J.-P. Uzan J. Cosmol. Astropart. Phys. 2008 4 ()][ A. Gumrukcuoglu, C. Contaldi and M. Peloso J. Cosmol. Astropart. Phys. 2007 005 ()]). We also discuss the stability of our model to small perturbations around biaxiality and draw an analogy with cosmological perturbations. We conclude by presenting a separate, bouncing solution, which generalizes the known bouncing solution in closed FRW universes.

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.

Gravitational Aharonov-Bohm Effect

We study a purely gravitational Aharonov-Bohm effect. The space-time curvature is concentrated in the quasiregular singularity of a cosmic string, outside of which space-time is (locally) flat. The symmetries of this field configuration are described by the groupoid symmetries rather than by the usual group symmetries. The groupoid in question is formed by homotopy classes of piecewise smooth paths in the cosmic string region. A gravitational counterpart of the Aharonov-Bohm effect occurs if the symmetry of the system, with respect to the groupoid action, is broken down.

Unconventional stringlike singularities in flat spacetime

The conical singularity in flat spacetime is mostly known as a model of the cosmic string or the wedge disclination in solids. Another, equally important, function is to be a representative of quasiregular singularities. From all these points of view it seems interesting to find out whether there exist other similar singularities. To specify what ``similar'' means I introduce the notion of the stringlike singularity, which is, roughly speaking, an absolutely mild singularity concentrated on a curve or on a 2-surface $\mathcal{S}$ (depending on whether the space is three- or four-dimensional). A few such singularities are already known: the aforementioned conical singularity, its two Lorentzian versions, the ``spinning string,'' the ``screw dislocation,'' and Tod's spacetime. In all these spacetimes $\mathcal{S}$ is a straight line (or a plane) and one may wonder if this is an inherent property of the stringlike singularities. The aim of this paper is to construct stringlike singularities with less trivial $\mathcal{S}$. These include flat spacetimes in which $\mathcal{S}$ is a spiral, or even a loop. If such singularities exist in nature (in particular, as an approximation to gravitational field of strings), their cosmological and astrophysical manifestations must differ drastically from those of the conventional cosmic strings. Likewise, being realized as topological defects in crystals, such loops and spirals will probably also have rather unusual properties.

Quantum Singularity in Quasiregular Spacetimes, as Indicated by Klein-Gordon, Maxwell and Dirac Fields

Klein-Gordon, Maxwell and Dirac fields are studied in quasiregular spacetimes, those space-times containing a classical quasiregular singularity, the mildest true classical singularity [G. F. R. Ellis and B. G. Schmidt, Gen. Rel. Grav. 8, 915 (1977)]. A class of static quasiregular spacetimes possessing disclinations and dislocations [R. A. Puntigam and H. H. Soleng, Class. Quantum Grav. 14, 1129 (1997)] is shown to have field operators which are not essentially self-adjoint. This class of spacetimes includes an idealized cosmic string, i.e. a four-dimensional spacetime with a conical singularity [L. H. Ford and A. Vilenkin, J. Phys. A: Math. Gen. 14, 2353 (1981)] and a Gal'tsov/Letelier/Tod spacetime featuring a screw dislocation [K. P. Tod, Class. Quantum Grav. 11, 1331 (1994); D. V. Gal'tsov and P. S. Letelier, Phys. Rev. D 47, 4273 (1993)]. The definition of G. T. Horowitz and D. Marolf [Phys. Rev. D52, 5670, (1995)] for a quantum-mechanically singular spacetime is one in which the spatial-derivative operator in the Klein-Gordon equation for a massive scalar field is not essentially self-adjoint. The definition is extended here, in the case of quasiregular spacetimes, to include Maxwell and Dirac fields. It is shown that the class of static quasiregular spacetimes under consideration is quantum-mechanically singular independent of the type of field.

LETTER: Quantum Singularity of Quasiregular Spacetimes

Some of the mildest singularities in classical general relativity are shown to be singular quantum mechanically as well. A class of the mild, topological singularities known as quasiregular singularities remains singular when probed by quantum wave packets. These static spacetimes possessing dislocations and disclinations are quantum-mechanically singular since the spatial portion of the wave operator is not essentially self-adjoint and thus the evolution of a test quantum wave packet is not uniquely determined by the initial wave function.

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.

Classical singularities and the quantum origin of the Universe

The aim of the paper is to elucidate the question: what can be said about the quantum creation theories from the point of view of the classical singularity theory? To this end singularity theorems are critically examined, and the nature and classification of singularities briefly discussed. An observer approaching a quasi-regular singularity has no warning of the imminent danger, and he would never think to look for any quantum cosmology. For security reasons one could introduce a quantum censorship hypothesis which would forbid the existence of singularities not protected by the Planck threshold. The creation-from-nothing idea met both in classical and quantum theories is discussed at some length. It is emphasized that the nothingness these theories presuppose is a strongly theory-dependent concept.

Einstein algebras and general relativity

A purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebra but not vice versa. The Gelfand representation of Einstein algebras is defined, and two of its subrepresentations are discussed. One of them is equivalent to the global formulation of the standard theory of general relativity; the other one leads to a more general theory of gravitation which, in particular, includes so-called regular singularities. In order to include other types of singularities one must change to sheaves of Einstein algebras. They are defined and briefly discussed. As a test of the proposed method, the sheaf of Einstein algebras corresponding to the spacetime of a straight cosmic string with quasiregular singularity is constructed.

Stability of the quasiregular singularities in Bell-Szekeres spacetime.

The behavior of geodesics and test minimally coupled scalar waves on the Bell-Szekeres space-time is used to probe the nature of the quasiregular singularities present. Components of the stress-energy tensor in a parallel-propagated orthonormal frame diverge as does a stress-energy scalar at the quasiregular singularities. It is argued that this divergence makes the singularities unstable, converting them into scalar curvature singularities.

Quasi-regular singularities and cosmic strings

It is shown that a general two-dimensional timelike quasi-regular singularity has the same energy-momentum tensor as a straight conical cosmic string. Conversely any model of a cosmic string in which the energy-momentum is concentrated on a 2-surface and the curvature is sufficiently regular must be described by a quasi-regular singularity. It is shown that the direction of such a singularity is a fixed point of the holonomy group generated by parallel propagation around loops encircling the singularity and that the holonomy at two different points is the same. This is used to show that such singularities are totally geodesic. The implications for constraints on the dynamics of cosmic strings are discussed.

Electromagnetic fields in Khan-Penrose spacetime

The behavior of test electromagnetic waves on the Khan-Penrose colliding gravitational-wave spacetime is used to probe the nature of the quasiregular singularities present. It is argued that the divergence of stress-energy scalars for most wave modes makes these singularities unstable, converting them into scalar curvature singularities. However, a special subset of modes does not lead to divergence of stress-energy scalars at the singularities. In the presence of such modes the singularities should remain quasiregular in an exact back-reaction calculation, as confirmed in the colliding gravitational- and electromagnetic-wave spacetime of Chandrasekhar and Xanthopoulos.

Singularities in colliding gravitational plane-wave spacetimes

Singularities in colliding gravitational plane-wave spacetimes are studied. In addition to scalar curvature singularities, colliding delta -function gravitational plane-wave spacetimes and colliding sandwich plane-wave spacetimes are shown to possess quasiregular singularities while colliding thick gravitational plane-wave spacetimes are shown to possess non-scalar curvature singularities. It is argued that these singularities are unstable. Scalar wave perturbations of the impulsive wave spacetime support this result.

The importance of symmetries in spacetimes with quasiregular singularities

The process by which elementary quasingular singularities are constructed by cutting and pasting a regular spacetime after identifying points under some discrete group of isometries is reviewed. There is also a description of how the addition of “perturbing” test fields which retain the symmetry of the spacetime retains the quasiregular singularity while those which destroy the symmetry lead to stronger singularities.

Electromagnetic fields in a Taub-Nut-type cosmology

The behavior of a test electromagnetic field on R 1×T 3 and R 3×S 1 flat Kasner universes is used to probe the nature of the quasiregular singularity present. The stress-energy tensor for the field is calculated in coordinate and parallel-propagated orthonormal (PPON) frames. Theorems about the convergence properties of the field at the singularity are proven. A conjecture is made concerning the stability of the singularity structure.

Quasi-regular singularities based on null planes

We examine quasi-regular singularities that take the form of invariant two-dimensional null planes in Minkowski space-time, thus extending earlier studies of “conical” singularities based on timelike and spacelike planes. The result is described in terms of a deficit parameter. We also examine the form of the Riemann curvature tensor at the singularity.