Conical Spacetime Research Articles

Nonminimally coupled gravitating vortex: Phase transition at critical couplingξcinAdS3

We consider the Nielsen-Olesen vortex non-minimally coupled to Einstein gravity with cosmological constant $\Lambda$. A non-minimal coupling term $\xi\,R\,|\phi|^2$ is natural to add to the vortex as it preserves gauge-invariance (here $R$ is the Ricci scalar and $\xi$ a dimensionless coupling constant). This term plays a dual role: it contributes to the potential of the scalar field and to the Einstein-Hilbert term for gravity. As a consequence, the vacuum expectation value (VEV) of the scalar field and the cosmological constant in the AdS$_3$ background depend on $\xi$. This leads to a novel feature: there is a critical coupling $\xi_c$ where the VEV is zero for $\xi\ge \xi_c$ but becomes non-zero when $\xi$ crosses below $\xi_c$ and the gauge symmetry is spontaneously broken. Moreover, we show that the VEV near the critical coupling has a power law behaviour proportional to $|\xi-\xi_c|^{1/2}$. Therefore $\xi_c$ can be viewed as the analog of the critical temperature $T_c$ in Ginzburg-Landau (GL) mean-field theory where a second-order phase transition occurs below $T_c$ and the order parameter has a similar power law behaviour $|T-T_c|^{1/2}$ near $T_c$. The critical coupling exists only in an AdS$_3$ background; it does not exist in asymptotically flat spacetime (topologically a cone) where the VEV remains at a fixed non-zero value independent of $\xi$. However, the deficit angle of the asymptotic conical spacetime depends on $\xi$ and is no longer determined solely by the mass; remarkably, a higher mass does not necessarily yield a higher deficit angle. The equations of motion are more complicated with the non-minimal coupling term present. However, via a convenient substitution one can reduce the number of equations and solve them numerically to obtain exact vortex solutions.

Motion of charged particles in spacetimes with magnetic fields of spherical and hyperbolic symmetry

The motion of charged particles in spacetimes containing a submanifold of constant positive or negative curvature is considered, with the electromagnetic tensor proportional to the volume two-form form of the submanifold. In the positive curvature case, this describes spherically symmetric spacetimes with a magnetic monopole, while in the negative curvature case, it is a hyperbolic spacetime with magnetic field uniform along hyperbolic surfaces. Constants of motion are found by considering Poisson brackets defined on a phase space with gauge-covariant momenta. In the spherically-symmetric case, we find a correspondence between the trajectories on the Poincar\'{e} cone with equatorial geodesics in a conical defect spacetime. In the hyperbolic case, the analogue of the Poincar\'{e} cone is defined as a surface in an auxiliary Minkowski spacetime. Explicit examples are solved for the Minkowski, $\mathrm{AdS}_4\times S^2$, and the hyperbolic AdS-Reissner--Nordstr\"{o}m spacetimes.

Thermal Casimir effect in a classical liquid in a quasiperiodically identified conical spacetime

In this paper, we study the finite-temperature quantum fluctuation of a classical liquid induced by the topology of an effective conical spacetime, as well as by a quasiperiodic boundary condition. The conical spacetime could be either a disclination or a cosmic string. In this context, we consider a phonon field representing quantum excitations of the liquid density, which obeys an effective Klein-Gordon equation with the sound velocity replaced by the light velocity. We obtain closed analytic expressions for the thermal Hadamard function, and consequently, the renormalized mean square density fluctuation of the liquid along with thermodynamics quantities such as internal energy, free energy, total energy, and entropy densities. We also discuss the limiting cases, including low- and high-temperature regimes, and the situations in which there is only either the conical spacetime or quasiperiodicity.

Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background

Non-relativistic conformally invariant systems in a rotating cosmic string (conical) spacetime are analyzed at the classical and quantum levels by means of the gravitoelectromagnetic interpretation of the background. Solutions of the equations of motion are found by employing a local canonical transformation, that leads to their natural interpretation in terms of Riemann surfaces. The cone parameter α and the angular velocity Ω of the background determine the existence of hidden symmetries. Globally defined higher order integrals associated with perihelion of geodesic orbits appear at rational values of α. For the harmonic oscillator system with frequency ω, the integrals responsible for the trajectory closure arise only for rational values of α and |γ| = |Ω/ω|, with |γ| = 1 corresponding to the Landau problem. We face a quantum anomaly problem since the hidden symmetry operators can only be constructed when α is integer. Such operators are non-local in the case of the free particle. For the harmonic oscillator, the symmetry generators are obtained with the help of the conformal bridge transformation. We also study a multi-particle version of the harmonic oscillator system with |γ| = 1 using the mean-field theory and find that the emerging vortex structure respects a singular point of the background.

Codimension- n holography for cones

We propose a novel codimension-n holography, called cone holography, between a gravitational theory in $(d+1)$-dimensional conical spacetime and a CFT on the $(d+1-n)$-dimensional defects. Similar to wedge holography, the cone holography can be obtained by taking the zero-volume limit of holographic defect CFT. Remarkably, it can be regarded as a holographic dual of the edge modes on the defects. For one class of solutions, we prove that the cone holography is equivalent to AdS/CFT, by showing that the classical gravitational action and thus the CFT partition function in large N limit are the same for the two theories. In general, cone holography and AdS/CFT are different due to the infinite towers of massive Kaluza-Klein modes on the branes. We test cone holography by studying Weyl anomaly, Entanglement/R\'enyi entropy and correlation functions, and find good agreements between the holographic and the CFT results. In particular, the c-theorem is obeyed by cone holography. These are strong supports for our proposal. We discuss two kinds of boundary conditions, the mixed boundary condition and Neumann boundary condition, and find that they both define a consistent theory of cone holography. We also analyze the mass spectrum on the brane and find that the larger the tension is, the more continuous the mass spectrum is. The cone holography can be regarded as a generalization of the wedge holography, and it is closely related to the defect CFT, entanglement/R\'enyi entropy and AdS/BCFT(dCFT). Thus it is expected to have a wide range of applications.

Non-singular vortices with positive mass in 2+1-dimensional Einstein gravity with AdS3 and Minkowski background

In previous work, black hole vortex solutions in Einstein gravity with AdS3 background were found where the scalar matter profile had a singularity at the origin r = 0. In this paper, we find numerically static vortex solutions where the scalar and gauge fields have a non-singular profile under Einstein gravity in an AdS3 background. Vortices with different winding numbers n, VEV v and cosmological constant Λ are obtained. These vortices have positive mass and are not BTZ black holes as they have no event horizon. The mass is determined in two ways: by subtracting the numerical values of two separate asymptotic metrics and via an integral that is purely over the matter fields. The mass of the vortex increases as the cosmological constant becomes more negative and this coincides with the core of the vortex becoming smaller (compressed). We then consider the vortex with gravity in asymptotically flat spacetime for different values of the coupling α = 1/(16πG). At the origin, the spacetime has its highest curvature and there is no singularity. It transitions to an asymptotic conical spacetime with angular deficit that increases significantly as α decreases. For comparison, we also consider the vortex without gravity in flat spacetime. For this case, one cannot obtain the mass by the first method (subtracting two metrics) but remarkably, via a limiting procedure, one can obtain an integral mass formula. In the absence of gauge fields, there is a well-known logarithmic divergence in the energy of the vortex. With gravity, we present this divergence in a new light. We show that the metric acquires a logarithmic term which is the 2 + 1 dimensional realization of the Newtonian gravitational potential when General Relativity is supplemented with a scalar field. This opens up novel possibilities which we discuss in the conclusion.

The first law of differential entropy and holographic complexity

We construct the CFT dual of the first law of spherical causal diamonds in three-dimensional AdS spacetime. A spherically symmetric causal diamond in AdS3 is the domain of dependence of a spatial circular disk with vanishing extrinsic curvature. The bulk first law relates the variations of the area of the boundary of the disk, the spatial volume of the disk, the cosmological constant and the matter Hamiltonian. In this paper we specialize to first-order metric variations from pure AdS to the conical defect spacetime, and the bulk first law is derived following a coordinate based approach. The AdS/CFT dictionary connects the area of the boundary of the disk to the differential entropy in CFT2, and assuming the ‘complexity=volume’ conjecture, the volume of the disk is considered to be dual to the complexity of a cutoff CFT. On the CFT side we explicitly compute the differential entropy and holographic complexity for the vacuum state and the excited state dual to conical AdS using the kinematic space formalism. As a result, the boundary dual of the bulk first law relates the first-order variations of differential entropy and complexity to the variation of the scaling dimension of the excited state, which corresponds to the matter Hamiltonian variation in the bulk. We also include the variation of the central charge with associated chemical potential in the boundary first law. Finally, we comment on the boundary dual of the first law for the Wheeler-deWitt patch of AdS, and we propose an extension of our CFT first law to higher dimensions.

The study of the generalized boson oscillator in a chiral conical space–time

Our work mainly study the relativistic generalized boson oscillator namely generalized Duffin–Kemmer–Petiau (DKP) oscillator with the function [Formula: see text] considered as the Cornell potential under the chiral conical space–time background. We obtain the wave function and energy spectrum of radial equation by using commonly used the Nikiforov–Uvarov method. It is shows that the energy spectrum of the generalized DKP oscillator depend explicitly on the angular deficit [Formula: see text], related rotation parameter [Formula: see text] and torsion parameter [Formula: see text], which characterize the global structure of the metric in the chiral conical space–time. In addition, the Cornell potential parameters [Formula: see text] have non-negligible influence on the energy spectrum of the studied systems.

Quantum vacuum fluctuation effects in a quasi-periodically identified conical spacetime

We consider a quasi-periodically identified conical spacetime, like the one of a cosmic string or disclination, to investigate nonzero averaged quantum vacuum fluctuations effects on the energy-momentum tensor and induced current density associated with a charged scalar field. We obtain exactly closed analytical expressions for the two-point Wightman function and the vacuum expectation value of the field squared, as well as for all components of the energy-momentum tensor. As to the induced current density, due to the quasi-periodic condition used, the only nonzero averaged component found is the azimuthal one. We also compare our results with previous ones found in literature.

Caustics bounding entanglement wedges

We study the caustics on the boundaries of entanglement wedges in the context of holography in asymptotically AdS3 spacetimes. These entanglement wedges play an important role in our understanding of the emergence of bulk locality. A procedure was proposed by Sanches and Weinberg for identifying boundary operators which are local in the bulk, which also applies to certain regions that lie beyond the reach of HRT surfaces by taking advantage of the lightsheets which bound entanglement wedges. We identify the caustics which terminate these lightsheets in conical deficit and BTZ black hole spacetimes and find that in some examples these caustics lead to a sharp corner in the entanglement wedge. The unexpected shape of these entanglement wedges leads, in those cases, to a breakdown of this procedure. Many of the properties of the rich variety of caustics possible in higher dimensions remains to be explored which, as this work demonstrates, could lead to more unexpected features in the shapes of entanglement wedges.

Finite temperature fermionic condensate in a conical space with a circular boundary and magnetic flux

We investigate the edge effects on the finite temperature fermionic condensate (FC) for a massive fermionic field in a (2+1)-dimensional conical spacetime with a magnetic flux located at the cone apex. The field obeys the bag boundary condition on a circle concentric with the apex. The analysis is presented for both the fields realizing two irreducible representations of the Clifford algebra and for general case of the chemical potential. In both the regions outside and inside the circular boundary, the FC is decomposed into the boundary-free and boundary-induced contributions. They are periodic functions of the magnetic flux with the period equal to the flux quantum and even functions under the simultaneous change of the signs for the magentic flux and the chemical potential. The dependence of the FC on the magnetic flux becomes weaker with decreasing planar angle deficit. For points near the boundary, the effects of finite temperature, of planar angle deficit and of magnetic flux are weak. For a fixed distance from the boundary and at high temperatures the FC is dominated by the Minkowskian part. The FC in parity and time-reversal symmetric (2+1)-dimensional fermionic models is discussed and applications are given to graphitic cones.

Induced vacuum magnetic flux in quantum spinor matter in the background of a topological defect in two-dimensional space

A topological defect in the form of the Abrikosov-Nielsen-Olesen vortex is considered as a gauge-flux-carrying tube that is impenetrable for quantum matter. The relativistic spinor matter field is quantized in the vortex background in $2+1$-dimensional conical space-time which is a section orthogonal to the vortex axis; the most general set of boundary conditions ensuring the impenetrability of the vortex core is employed. We find the induced vacuum current circulating around the vortex and the induced vacuum magnetic field strength pointing along the vortex axis. The requirement of finiteness and physical plausibility for the total induced vacuum magnetic flux allows us to restrict the variety of admissible boundary conditions. The dependence of the results on the transverse size of the vortex, as well as on the vortex flux and the parameter of conicity, is elucidated.

Arbitrarily coupled massive scalar field in conical thin-shell spacetimes

We study the massive scalar field of a charged particle held at rest in conical thin-shell spacetimes with one or two asymptotic regions. Resonant and stable scalar Green’s functions are characterized in terms of the coupling of the field to the trace of the extrinsic curvature jump at the shell. Stable coupling values, within the safety domain of the configuration parameters, are used to analyze the self-force of static point charges.

Kinematic space for conical defects

Kinematic space can be used as an intermediate step in the AdS/CFT dictionary and lends itself naturally to the description of diffeomorphism invariant quantities. From the bulk it has been defined as the space of boundary anchored geodesics, and from the boundary as the space of pairs of CFT points. When the bulk is not globally AdS3 the appearance of non-minimal geodesics leads to ambiguities in these definitions. In this work conical defect spacetimes are considered as an example where non-minimal geodesics are common. From the bulk it is found that the conical defect kinematic space can be obtained from the AdS3 kinematic space by the same quotient under which one obtains the defect from AdS3. The resulting kinematic space is one of many equivalent fundamental regions. From the boundary the conical defect kinematic space can be determined by breaking up OPE blocks into contributions from individual bulk geodesics. A duality is established between partial OPE blocks and bulk fields integrated over individual geodesics, minimal or non-minimal.

Wave diffraction by a cosmic string

We show that if a cosmic string exists, it may be identified through characteristic diffraction pattern in the energy spectrum of the observed signal. In particular, if the string is on the line of sight, the wave field is shown to fit the Cornu spiral. We suggest a simple procedure, based on Keller's geometrical theory of diffraction, which allows to explain wave effects in conical spacetime of a cosmic string in terms of interference of four characteristic rays. Our results are supposed to be valid for scalar massless waves, including gravitational waves, electromagnetic waves, or even sound in case of condensed matter systems with analogous topological defects.

Quantum dress for a naked singularity

We investigate semiclassical backreaction on a conical naked singularity space-time with a negative cosmological constant in (2+1)-dimensions. In particular, we calculate the renormalized quantum stress-energy tensor for a conformally coupled scalar field on such naked singularity space-time. We then obtain the backreacted metric via the semiclassical Einstein equations. We show that, in the regime where the semiclassical approximation can be trusted, backreaction dresses the naked singularity with an event horizon, thus enforcing cosmic censorship.

Finite temperature fermion condensate, charge and current densities in a ( $$2+1$$ 2 + 1 )-dimensional conical space

We evaluate the fermion condensate and the expectation values of the charge and current densities for a massive fermionic field in (2+1)-dimensional conical spacetime with a magnetic flux located at the cone apex. The consideration is done for both irreducible representations of the Clifford algebra. The expectation values are decomposed into the vacuum expectation values and contributions coming from particles and antiparticles. All these contributions are periodic functions of the magnetic flux with the period equal to the flux quantum. Related to the non-invariance of the model under the parity and time-reversal transformations, the fermion condensate and the charge density have indefinite parity with respect to the change of the signs of the magnetic flux and chemical potential. The expectation value of the radial current density vanishes. The azimuthal current density is the same for both the irreducible representations of the Clifford algebra. It is an odd function of the magnetic flux and an even function of the chemical potential. The behavior of the expectation values in various asymptotic regions of the parameters are discussed in detail. In particular, we show that for points near the cone apex the vacuum parts dominate. For a massless field with zero chemical potential the fermion condensate and charge density vanish. Simple expressions are derived for the part in the total charge induced by the planar angle deficit and magnetic flux. Combining the results for separate irreducible representations, we also consider the fermion condensate, charge and current densities in parity and time-reversal symmetric models. Possible applications to graphitic nanocones are discussed.

Conical spacetimes and global hyperbolicity

Vickers and Wilson ([26]) have shown global hyperbolicity of the conical spacetime in the sense of well-posedness of the initial value problem for the wave equation in generalized functions. We add the aspect of metric splitting and preliminary thoughts on Cauchy hypersurfaces and causal curves. AMS Mathematics Subject Classi cation (2010): 46F30, 83C75

On Closed Timelike Curves and Warped Brane World Models

At first glance, it seems possible to construct in general relativity theory causality violating solutions. The most striking one is the Gott spacetime. Two cosmic strings, approaching each other with high velocity, could produce closed timelike curves. It was quickly recognized that this solution violates physical boundary conditions. The effective one particle generator becomes hyperbolic, so the center of mass is tachyonic. On a 5-dimensional warped spacetime, it seems possible to get an elliptic generator, so no obstruction is encountered and the velocity of the center of mass of the effective particle has an overlap with the Gott region. So a CTC could, in principle, be constructed. However, from the effective 4D field equations on the brane, which are influenced by the projection of the bulk Weyl tensor on the brane, it follows that no asymptotic conical space time is found, so no angle deficit as in the 4D counterpart model. This could also explain why we do not observe cosmic strings.

Exploring three-dimensional higher-spin supergravity based on sl(N|N − 1) Chern-Simons theories

We investigate various aspects of higher-spin anti-de Sitter supergravity in three dimensions as described by Chern-Simons theory based on the finite-dimensional superalgebra sl(N |N - 1), with the particular case of N = 3 as our prime example. This class of theories serves as a natural supersymmetrization of the higher-spin gravity theory based on sl(N) Chern-Simons theories. We demonstrate explicitly that the asymptotic symmetry algebra contains the N = 2 superconformal algebra in each sector. The appropriate Killing spinor equations are derived and used to classify existing and new classical solutions. We also discuss holonomy conditions, higher-spin black holes and conical defect spacetimes in this class of theories.