Timelike Curves Research Articles

Erratum: Axially symmetric Petrov type II general space–time and closed timelike curves

Erratum: Axially symmetric Petrov type II general space–time and closed timelike curves

New improved form of black holes of type D

We derive a new metric form of the complete family of black hole spacetimes (without a cosmological constant) presented by Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski and Demia\ifmmode \acute{n}\else \'{n}\fi{}ski in 1976. It further improves the convenient representation of this large family of exact black holes found in 2005 by Griffiths and Podolsk\'y. The main advantage of the new metric is that the key functions are considerably simplified, fully explicit, and factorized. All four horizons are thus clearly identified, and degenerate cases with extreme horizons can easily be discussed. Moreover, the new metric depends only on six parameters with direct geometrical and physical meaning, namely $m,a,l,\ensuremath{\alpha},e,g$ which characterize mass, Kerr-like rotation, Newman-Unti-Tamburino (NUT) parameter, acceleration, electric and magnetic charges of the black hole, respectively. This general metric reduces directly to the familiar forms of either (possibly accelerating) Kerr--Newman, charged Taub--NUT solution, or (possibly rotating and charged) $C$-metric by simply setting the corresponding parameters to zero, without the need of any further transformations. In addition, it shows that the Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski--Demia\ifmmode \acute{n}\else \'{n}\fi{}ski family does not involve accelerating black holes with just the NUT parameter, which were discovered by Chng, Mann and Stelea in 2006. It also enables us to investigate various physical properties, such as the character of singularities, horizons, ergoregions, global conformal structure including the Penrose diagrams, cosmic strings causing the acceleration of the black holes, their rotation, pathological regions with closed timelike curves, or explicit thermodynamic properties. It thus seems that our new metric is a useful representation of this important family of black hole spacetimes of algebraic type D in the asymptotically flat settings.

Special Ruled Surface in de-Sitter 3-Space

In this paper, timelike base curve and spacelike main geodesic with the timelike ruled surface are studied, which is a special class of ruled surface in de-Sitter space $S_{1}^{3}.$ A ruled surface in the de-Sitter space $S_{1}^{3}$ is obtained by moving a geodesic along a curve. So we will call these surfaces in the de-Sitter space as the geodesic ruled surface. Developable ruled surface, striction point, striction curve, dispersion parameter, and orthogonal trajectory concepts are investigated for the obtained geodesic ruled surface.

Feynman Propagator for Closed Timelike Curves in the Kerr Metric

We compute the Feynman propagator associated with closed timelike curves in the neighborhood of the ring singularity in the Kerr metric. The propagator is well defined outside r = 0, where it ceases to exist.

The D-CTC Condition is Generically Fulfilled in Classical (Non-quantum) Statistical Systems

The D-CTC condition, introduced by David Deutsch as a condition to be fulfilled by analogues for processes of quantum systems in the presence of closed timelike curves, is investigated for classical statistical (non-quantum) bi-partite systems. It is shown that the D-CTC condition can generically be fulfilled in classical statistical systems, under very general, model-independent conditions. The central property used is the convexity and completeness of the state space that allows it to generalize Deutsch’s original proof for q-bit systems to more general classes of statistically described systems. The results demonstrate that the D-CTC condition, or the conditions under which it can be fulfilled, is not characteristic of, or dependent on, the quantum nature of a bi-partite system.

Motion of a gyroscope on a closed timelike curve

We consider the motion of a gyroscope on a closed timelike curve (CTC). A gyroscope is identified with a unit-length spacelike vector - a spin-vector - orthogonal to the tangent to the CTC, and satisfying the equations of Fermi-Walker transport along the curve. We investigate the consequences of the periodicity of the coefficients of the transport equations, which arise from the periodicty of the CTC, which is assumed to be piecewise $C^2$. We show that every CTC with period $T$ admits at least one $T-$periodic spin-vector. Further, either every other spin-vector is $T-$periodic, or no others are. It follows that gyroscopes carried by CTCs are either all $T-$periodic, or are generically not $T-$periodic. We consider examples of spacetimes admitting CTCs, and address the question of whether $T-$periodicity of gyroscopic motion occurs generically or only on a negligible set for these CTCs. We discuss these results from the perspective of principles of consistency in spacetimes admitting CTCs.

Overspinning naked singularities in AdS3 spacetime

The Ba\~nados-Teitelboim-Zanelli (BTZ) black hole belongs to a family of locally three-dimensional anti-de Sitter (${\mathrm{AdS}}_{3}$) spacetimes labeled by their mass $M$ and angular momentum $J$. The case $M\ensuremath{\ell}\ensuremath{\ge}|J|$, where $\ensuremath{\ell}$ is the anti-de Sitter radius, provides the black hole. Extending the metric to other values of $M$ and $J$ leads to geometries with the same asymptotic behavior and global symmetries but containing a naked singularity at the origin. The case $M\ensuremath{\ell}\ensuremath{\le}\ensuremath{-}|J|$ corresponds to spinning conical singularities that are reasonably well understood. Here, we examine the remaining case, which is $\ensuremath{-}|J|<M\ensuremath{\ell}<|J|$. These naked singularities are mathematically acceptable solutions describing classical spacetimes. They are obtained by identifications of the covering pseudosphere in ${\mathbb{R}}^{2,2}$ and are free of closed timelike curves. Here, we study the causal structure and geodesics around these overspinning geometries. We present a review of the geodesics for the entire BTZ family. The geodesic equations are completely integrated, and the solutions are expressed in terms of elementary functions. Special attention is given to the determination of circular geodesics, where new results are found. According to the radial bounds, eight types of noncircular geodesics appear in the BTZ spacetimes. For the case of overspinning naked singularity, null, and spacelike geodesics can reach infinity passing by a point nearest to the singularity, others extend from the central singularity to infinity, and others still have a radial upper bound and terminate at the singularity. As expected for an anti-de Sitter spacetime, timelike geodesics cannot reach infinity; they either loop around the singularity or fall into it. The spatial projections of the geodesics (orbits) exhibit self-intersections, whose number is determined for null and spacelike geodesics, and it is found a special class of timelike geodesics whose spatial projections are closed.

Black hole microstates from the worldsheet

Recently an exact worldsheet description of strings propagating in certain black hole microstate geometries was constructed in terms of null-gauged WZW models. In this paper we consider a family of such coset models, in which the currents being gauged are specified by a set of parameters that a priori take arbitrary values. We show that consistency of the spectrum of the worldsheet CFT implies a set of quantisation conditions and parity restrictions on the gauging parameters. We also derive these constraints from an independent geometrical analysis of smoothness, absence of horizons and absence of closed timelike curves. This allows us to prove that the complete set of consistent backgrounds in this class of models is precisely the general family of (NS5-decoupled) non-BPS solutions known as the JMaRT solutions, together with their various (BPS and non-BPS) limits. We clarify several aspects of these backgrounds by expressing their six-dimensional solutions explicitly in terms of five non-negative integers and a single length-scale. Finally we study non-trivial two-charge limits, and exhibit a novel set of non-BPS supergravity solutions describing bound states of NS5 branes carrying momentum charge.

Caustic points of the timelike curve on the de Sitter 3-space

Caustic points of the timelike curve on the de Sitter 3-space

States of low energy in bouncing inflationary scenarios in loop quantum cosmology

In generic Friedmann-Lema\^itre-Robertson-Walker spacetimes, States of Low Energy (SLEs) are defined to minimize the regularized energy density smeared along the time-like curve of an isotropic observer, which is specified via a smearing function. For every smearing function, SLEs are unique (up to a phase) and are shown to be exact Hadamard states. In this work, we investigate the viability of SLEs as the vacuum for cosmological perturbations in hybrid Loop Quantum Cosmology, motivated by the fact that SLEs have been shown to provide suitable vacua in models where a period of kinetic dominance precedes inflation. We find that there are two classes of smearing functions that can be seen as natural choices within this context, for which the corresponding SLEs and the resulting power spectra at the end of inflation are quite insensitive to the exact shape and support of the smearing function. Furthermore, a preliminary analysis of the tensor-to-scalar ratio and of the spectral index indicates as good an agreement with observations as that of standard cosmology.

A substrate for brane shells from Toverline{T}

A solvable current-current deformation of the worldsheet theory of strings on AdS3 has been recently conjectured to be dual to an irrelevant deformation of the spacetime orbifold CFT, commonly referred to as single-trace Toverline{T} . These deformations give rise to a family of bulk geometries which realize a non-trivial flow towards the UV. For a particular sign of this deformation, the corresponding three-dimensional geometry approaches AdS3 in the interior, but has a curvature singularity at finite radius, beyond which there are closed timelike curves. It has been suggested that this singularity is due to the presence of “negative branes,” which are exotic objects that generically change the metric signature. We propose an alternative UV-completion for geometries displaying a similar singular behavior by cutting and gluing to a regular background which approaches a linear dilaton vacuum in the UV. In the S-dual picture, a singularity resolution mechanism known as the enhançon induces this transition by the formation of a shell of D5-branes at a fixed radial position near the singularity. The solutions involving negative branes gain a new interpretation in this context.

Pure Gauss\u2013Bonnet NUT black hole with and without non-central singularity

It is known that NUT solution has many interesting features and pathologies like being non-singular and having closed timelike curves. It turns out that in higher dimensions horizon topology cannot be spherical but it has instead to be product of 2-spheres so as to retain radial symmetry of spacetime. In this letter we wish to present a new solution of pure Gauss–Bonnet Lambda -vacuum equation describing a black hole with NUT charge. It has three interesting cases: (a) black hole with both event and cosmological horizons with singularity being hidden behind the former, (b) a regular spacetime free of both horizon and singularity, and (c) black hole with event horizon without singularity and cosmological horizon. Singularity here is always non-centric at r ne 0.

Time-traveling billiard-ball clocks: A quantum model

Closed timelike curves, along which an object can travel to its past, are interesting because they are predicted by general relativity and yet need to be reconciled with quantum mechanics. The authors consider a quantum model of a paradox where a billiard ball can travel along two trajectories: one unperturbed and the other where it can collide with its past.

An axially symmetric spacetime with causality violation

We present an axially symmetric spacetime which contains closed timelike curves, and hence violates the causality condition. The metric belongs to type III in the Petrov classification scheme with vanishing expansion, shear and twist. The matter-energy represents a pure radiation field with a negative cosmological constant. The spacetime is asymptotically anti-de Sitter space in the radial direction.

Analog time machine in a photonic system

Analogue physics has successfully tackled the problems of chromodynamics, event horizons, Big Bang and Universe expansion, and many others. Here, we suggest a photonic model system for a "time machine" based on the paraxial beam approximation. We demonstrate how the closed time-like curves and the well-known grandfather paradox can be studied experimentally in this system. We show how the Novikov's self-consistency principle is realized in quantum mechanics thanks to the Heisenberg's uncertainty principle.

Axially symmetric Petrov type II general space–time and closed timelike curves

We present a Petrov type II general space–time which violates causality in the sense that it allows for the formation of closed timelike curves that appear after a definite instant of time. The metric, which is axially symmetric, admits an expansion-free, twist-free and shear-free null geodesic congruence. From the general metric, we obtain two particular type II metrics. One is a vacuum solution while the other represents a Ricci flat solution with a negative cosmological constant.

Null hypersurface caustics, closed null curves, and super entropy

Recently it was discovered that null hypersurfaces can develop caustics outside the event horizon of super-entropic Kerr-AdS black holes, in contrast to the usual Kerr-AdS case. In this work we explore a few more examples of black hole spacetimes in which such exterior caustics can develop. If a closed null curve is present, e.g., in the case of Taub-NUT and the "transunital" Kerr-AdS spacetimes, then it coincides with a null hypersurface caustic (NHC) of a minimal separation parameter. Thus a spacetime on the verge of forming closed timelike curves could develop a caustic. Known examples of super-entropic black holes also have exterior NHC, although such spacetimes are free of closed null/timelike curves. Nevertheless the relationship between closed causal curves, NHC, and super-entropy is not straightforward. This is best illustrated with the BTZ black string, which for some choices of the warp factor in the extra dimension and the value of the charge, can be super-entropic. However, even those that are not super-entropic can admit NHC outside the horizon.

Closed Timelike Curves, Singularities and Causality: A Survey from Gödel to Chronological Protection

I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to “save” the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role.

Scalarized Nutty Wormholes

We construct scalarized wormholes with a NUT charge in higher curvature theories. We consider both Einstein-scalar-Gauss-Bonnet and Einstein-scalar-Chern-Simons theories, following Brihaye, Herdeiro and Radu, who recently studied spontaneously scalarised Schwarzschild-NUT solutions. By varying the coupling parameter and the scalar charge we determine the domain of existence of the scalarized nutty wormholes, and their dependence on the NUT charge. In the Gauss-Bonnet case the known set of scalarized wormholes is reached in the limit of vanishing NUT charge. In the Chern-Simons case, however, the limit is peculiar, since with vanishing NUT charge the coupling constant diverges. We focus on scalarized nutty wormholes with a single throat and study their properties. All these scalarized nutty wormholes feature a critical polar angle, beyond which closed timelike curves are present.

A new construction on the energy of space curves in unit vector fields in Minkowski space E₂⁴

In this paper, we firstly introduce kinematics properties of a moving particle lying in Minkowski space E₂⁴. We assume that particles corresponds to different type of space curves such that they are characterized by Frenet frame equations. Guided by these, we present geometrical understanding of an energy and pseudo angle on the particle in each Frenet vector fields depending on the particle corresponds to a spacelike, timelike or lightlike curve in E₂⁴. Then we also determine the bending elastic energy functional for the same particle in E₂⁴ by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy on the particle in each Frenet vector field.