Space-time Defects Research Articles

Making the world from topological order.

Topology used to be a term confined to a branch of pure mathematics, where it referred to an invariant property of shape. The classic example was the way objects containing a single hole, like a torus and a coffee cup with handle, can be smoothly moulded into one another without tearing. But topological considerations have long played a role in the physics of matter, where for example they might dictate particular arrangements of component parts that can’t be erased from the system. The classic example here is the fact that a ‘hairy ball’ can’t be combed flat without having at least two pointy tufts. Such ‘defects’ in organization can be considered ‘topologically protected’, since they are robust against any recombing of the hair. They are universal features that don’t depend on the material specifics of the system: topological defects in liquid crystals are analogous to defects in spacetime called cosmic strings. In the past several decades in particular, properties of matter arising from topological considerations have become a major theme, reflected for example in the award of the 1985 and 1998 Nobel Prizes in Physics for discoveries involving the quantum Hall effect. Here the ‘Hall conductance’, quantifying the passage of electrical current in a 2D conductor in the presence of a transverse magnetic field, takes precise integral or fractional multiples of a particular quantized value related to the electron charge. This behaviour persists regardless of how we modify the material, for example by adding impurities. Topological phases and transitions were also a feature of the work that won the 2016 Nobel Prize. It has become recognized that the topological properties of the quantum-mechanical electronic structures of certain materials can give them unusual and perhaps useful properties. Some researchers think, for example, that ‘topological matter’ might supply quantum bits for quantum computation that resist the randomizing effects of noise. Xiao-Gang Wen of the Massachusetts Institute of Technology has been developing ideas about ‘topological order’ in fundamental physics for several decades. His notions of how topology in the underlying structure of spacetime might give rise to fundamental particles and forces make a connection to the topological phases recognized in condensed matter, revealing a new unifying principle in physics. National Science Review spoke to him about his work.

Definite photon deflections of\xa0topological\xa0defects in metasurfaces and symmetry-breaking phase transitions with material loss

Combination of topology and general relativity can lead to some profound and farsighted predictions. It is well known that symmetry breaking of the Higgs vacuum field in the early universe possibly induced topological defects in space-time, whose nontrivial effects can provide some clues about the universe’s origin. Here, by using an artificial waveguide bounded with rotational metasurface, the nontrivial effects of a topological defect of spacetime are experimentally emulated. The photon deflection in the topological waveguide has a robust definite angle that does not depend on the location and momentum of incident photons. This is remarkably different from the random optical scattering in trivial space. By including material loss such a topological effect can be well understood from the symmetry breaking of photonic modes. Our technique provides a platform to investigate topological gravity in optical systems. This method can also be extended to obtain many other novel topological photonic devices.

General relativity with space-time defects

General relativity is incomplete because it cannot describe quantum effects of space-time. The complete theory of quantum gravity is not yet known and to date no observational evidence exists that space-time is quantized. However, in most approaches to quantum gravity the space-time manifold of general relativity is only an effective limit that, among other things like higher curvature terms, should receive corrections stemming from space-time defects. We here develop a modification of general relativity that describes local space-time defects and solve the Friedmann equations. From this, we obtain the time-dependence of the average density of defects. It turns out that the defects’ average density dilutes quickly, somewhat faster even than radiation.

A stealth defect of spacetime

We discuss a special type of Skyrmion spacetime-defect solution, which has a positive energy density of the matter fields but a vanishing asymptotic gravitational mass. With a mass term for the matter field added to the action (corresponding to massive “pions” in the Skyrme model), this particular soliton-type solution has no long-range fields and can appropriately be called a “stealth defect”.

Antigravity from a spacetime defect

We argue that there may exist spacetime defects embedded in Minkowski spacetime, which have negative active gravitational mass. One such spacetime defect then repels a test particle, corresponding to what may be called "antigravity."

Mass generation by a Lorentz-invariant gas of spacetime defects

We present a simple model of defects embedded in flat spacetime, where the model is designed to maintain Lorentz invariance over large length scales. Even without remnant Lorentz violation, there are still effects from these spacetime defects on the propagation of physical fields, notably mass generation for scalars and Dirac fermions.

Space-Time Defects and Group Momentum Space

We study massive and massless conical defects in Minkowski and de Sitter spaces in various space-time dimensions. The energy momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its space-time metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects, respectively. In particular, if we fix the direction of propagation of a massless defect in n+1-dimensional Minkowski space, then its space of holonomies is a maximal Abelian subgroup of the AN(n-1) group, which corresponds to the well known momentum space associated with the n-dimensional κ-Minkowski noncommutative space-time and κ-deformed Poincaré algebra. We also conjecture that massless defects in n-dimensional de Sitter space can be analogously characterized by holonomies belonging to the same subgroup. This shows how group-valued momenta related to four-dimensional deformations of relativistic symmetries can arise in the description of motion of space-time defects.

Experimental model of topological defects in Minkowski space-time based on disordered ferrofluid: magnetic monopoles, cosmic strings and the space-time cloak.

In the presence of an external magnetic field, cobalt nanoparticle-based ferrofluid forms a self-assembled hyperbolic metamaterial. The wave equation, which describes propagation of extraordinary light inside the ferrofluid, exhibits 2+1 dimensional Lorentz symmetry. The role of time in the corresponding effective three-dimensional Minkowski space-time is played by the spatial coordinate directed along the periodic nanoparticle chains aligned by the magnetic field. Here, we present a microscopic study of point, linear, planar and volume defects of the nanoparticle chain structure and demonstrate that they may exhibit strong similarities with such Minkowski space-time defects as magnetic monopoles, cosmic strings and the recently proposed space-time cloaks. Experimental observations of such defects are described.

Holographic dual of a conical defect

We consider a moving conical defect in the pure AdS3 space-time and calculate two-point correlation functions of a corresponding two-dimensional boundary quantum field theory in the geodesic approximation. We show that the presence of the defect leads to a gravitational lensing of geodesics, and this results in a finite number of similar terms in the Green’s function that correspond to winding geodesics in the bulk around the conical singularity. We show that for the quantized deficit angle γ = π/2n, the lensing produces domain wall excitations in the spectrum of the boundary theory.

Comparison of spacetime defects which are homeomorphic but not diffeomorphic

Certain remnants of a quantum spacetime foam can be modeled by a distribution of defects embedded in a flat classical spacetime. The presence of such spacetime defects affects the propagation of elementary particles. In this article, we show explicitly that both topology and differential structure of the defects are important for the particle motion. Specifically, we consider three types of spacetime defects which are described by the same topological manifold \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}\times \big (\mathbb {R}P^3-\lbrace \text{point}\rbrace \big )$\end{document}R×RP3−{point} but which are not diffeomorphic to each other. We investigate the propagation of a massless scalar field over the three different manifolds and find different solutions of the Klein–Gordon equation.

Skyrmion spacetime defect

A finite-energy static classical solution is obtained for standard Einstein gravity coupled to an $SO(3)\times SO(3)$ chiral model of scalars [a Skyrme model]. This nonsingular localized solution has nontrivial topology for both the spacetime manifold and the $SO(3)$ matter fields. The solution corresponds to a single spacetime defect embedded in flat Minkowski spacetime.

A nonsingular spacetime defect

A nonsingular localized static classical solution is constructed for standard Einstein gravity coupled to an $SO(3)\ifmmode\times\else\texttimes\fi{}SO(3)$ chiral model of scalars (Skyrme model). This solution corresponds to a spacetime defect and its construction proceeds in three steps. First, an Ansatz is presented for a solution with nonsimply connected topology of the spacetime manifold. Second, an exact vacuum solution of the reduced field equations is obtained. Third, matter fields are included and a particular exact solution of the reduced field equations is found. The latter solution has a diverging total energy, but its existence at least demonstrates that a nonsingular defect-type solution having nonsimply connected topology is possible with nontrivial matter fields.

Theory and Phenomenology of Space-Time Defects

Whether or not space-time is fundamentally discrete is of central importance for the development of the theory of quantum gravity. If the fundamental description of spacetime is discrete, typically represented in terms of a graph or network, then the apparent smoothness of geometry on large scales should be imperfect—it should have defects. Here, we review a model for space-time defects and summarize the constraints on the prevalence of these defects that can be derived from observation.

Geometrical approach to the atomic spectra theory. The helium atom

New equations for helium spectra calculations are obtained within geometrical interpretation of quantum mechanics suggested by the author earlier. The main idea of the above interpretation is that atoms can be considered not as the systems with many electrons but as a microscopic topological defect of the physical space-time without any point-like particles inside. The groups of symmetry transformations of such defects are suggested to be isomorphic to the symmetry groups of atoms with many identical electrons (permutation group, for example). New equations were derived within approximation that is similar to the one in the self consistent field theory of Hartree-Fok, but these equations differ strongly from Hartree-Fock equations. Numerical calculations of ionization potentials for para—and orto—helium lead to results that are in a good agreement with experiment.

Phenomenology of space-time imperfection. II. Local defects

We propose a phenomenological model for the scattering of particles on space-time defects in a treatment that maintains Lorentz-invariance on the average. The local defects considered here cause a stochastic violation of momentum conservation. The scattering probability is parameterized in the density of defects and the distribution of the momentum that a particle can obtain when scattering on the defect. We identify the most promising observable consequences and derive constraints from existing data.

Simple model of pointlike spacetime defects and implications for photon propagation

A model in which pointlike defects are randomly embedded in Minkowski spacetime is considered. The distribution of spacetime defects is constructed to be Lorentz-invariant. It does not introduce a preferred reference frame, because it is based on a sprinkling process. A field-theoretic action for the photon and a fermion is set up, in which the photon is assumed not to couple to the defects directly, but via a scalar field. We are interested in signs for Lorentz violation caused by the spacetime defects, which are expected to reveal themselves in the photon sector. A modification of the photon dispersion relation may result as a quantum effect and we compute it at leading order perturbation theory. The outcome of the calculation is that the photon dispersion law remains conventional, if the defect distribution is dense, homogeneous, and isotropic. This result sheds some new light on Lorentz violation in the framework of a small-scale structure of spacetime. It shows that Lorentz invariance can be preserved even in the presence of a spacetime structure that is supposed to emerge at the Planck scale. This conclusion has already been drawn on general grounds in other publications, where the current paper delivers a demonstration by a direct computation in a simple model.

Violation of CPT Invariance in the Early Universe and Leptogenesis/Baryogenesis

In this talk, I review some plausible scenarios entailing violation of CPT symmetry in the early Universe, due to space-time backgrounds which do not respect some of the assumptions for the validity of the CPT theorem (here considered will be Lorentz invariance and/or Unitarity). The key point in all these models is that the background induces different populations of fermions as compared to antifermions, and hence CPT Violation (CPTV), already in thermal equilibrium. Such populations may freeze out at various conditions depending on the details of the underlying microscopic model, thereby leading to leptogenesis and baryogenesis. Among the considered scenarios is a stringy one, in which the CPTV is associated with a cosmological background with torsion provided by the Kalb-Ramond antisymmetric tensor field (axion) of the string gravitational multiplet. We also discuss briefly (Lorentz Violating) CPTV models that go beyond the local effective lagrangian framework, such as a stochastic Finsler metric and D-particle foam. In the latter, the CPTV is induced by the interactions of stringy matter with populations of stochastically fluctuating point-like space-time defects (D-particles) that can be encountered in string/brane Cosmologies. The preferential rôle of neutrinos is singled out in this latter scenario as a consequence of gauge symmetry (charge) conservation.

CPT-violating leptogenesis induced by gravitational backgrounds

We explore leptogenesis induced by the propagation of neutrinos in non-trivial gravitational backgrounds that may characterise the Early Universe epochs of various theories, including string theory. The key point in all these models is that the background induces different populations of fermions as compared to antifermions, and hence CPT Violation (CPTV), already in thermal equilibrium. Depending on the model, then, such populations may freeze out at various conditions leading to leptogenesis and baryogenesis. Among the considered scenarios, is a stringy one, in which the CPTV is associated with a cosmological background with torsion provided by the Kalb-Ramond antisymmetric tensor field (axion) of the string gravitational multiplet. We also discuss CPTV models that go beyond the local effective lagrangian framework, such as a stochastic (Lorentz Violating) Finsler metric and D-particle foam, where the CPTV is due to populations of stochastically fluctuating point-like space-time defects that can be encountered in string/brane theory (D0 branes).

Asymptotic analysis of the Boltzmann equation for dark matter relics in the presence of a running dilaton and space-time defects

The interplay of dilatonic effects in dilaton cosmology and stochastic quantum space-time defects within the framework of string/brane cosmologies is examined. The Boltzmann equation describes the physics of thermal dark-matter-relic abundances in the presence of rolling dilatons. These dilatons affect the coupling of stringy matter to D-particle defects, which are generic in string theory. This coupling leads to an additional source term in the Boltzmann equation. The techniques of asymptotic matching and boundary-layer theory, which were recently applied by two of the authors (CMB and SS) to a Boltzmann equation, are used here to find the detailed asymptotic relic abundances for all ranges of the expectation value of the dilaton field. The phenomenological implications for the search of supersymmetric dark matter in current colliders, such as the LHC, are discussed.

Stringy models of modified gravity: space-time defects and structure formation

Starting from microscopic models of space-time foam, based on braneuniverses propagating in bulk space-times populated by D0-branedefects (``D-particles''), we arrive at effective actions used by alow-energy observer on the brane world to describe his/herobservations of the Universe. These actions include, apart from themetric tensor field, also scalar (dilaton) and vector fields, thelatter describing the interactions of low-energy matter on the braneworld with the recoiling point-like space-time defect(D-particle). The vector field is proportional to the recoil velocityof the D-particle and as such it satisfies a certain constraint. Thevector breaks locally Lorentz invariance, which however is assumed tobe conserved on average in a space-time foam situation, involving theinteraction of matter with populations of D-particle defects. In thispaper we clarify the role of fluctuations of the vector field onstructure formation and galactic growth. In particular we demonstratethat, already at the end of the radiation era, the (constrained)vector field associated with the recoil of the defects provides theseeds for a growing mode in the evolution of the Universe. Such agrowing mode survives during the matter dominated era, provided thevariance of the D-particle recoil velocities on the brane is largerthan a critical value. We note that in this model, as a result ofspecific properties of D-brane dynamics in the bulk, there is no issueof overclosing the brane Universe for large defect densities. Thus, inthese models, the presence of defects may be associated withlarge-structure formation. Although our string inspired models dohave (conventional, from a particle physics point of view) dark mattercomponents, nevertheless it is interesting that the role of``extra'' dark matter is also provided by the population of massivedefects. This is consistent with the weakly interacting character ofthe D-particle defects, which predominantly interact onlygravitationally.