Timelike Curves Research Articles

Focusing conditions for extended teleparallel gravity theories

In the context of extended theories of teleparallel gravity f(T) we derive the focusing conditions for a one-parameter dependent congruence of timelike auto-parallels of the Levi-Civita connection. We also consider the f(T) field equations for a general metric tensor before moving on to consider a spatially flat Robertson-Walker space-time. Following this, we study the expansion rate for a one-parameter dependent congruence of timelike auto-parallel curves of the Levi-Civita connection. Given the fact that test particles follow auto-parallels of the Levi-Civita connection, the torsion-free Raychaudhuri equation is used in order to determine the desired focusing conditions. Finally we study the obtained focusing conditions for three f(T) paradigmatic cosmological models and discuss the satisfaction or violation of these conditions. Through this, we find f(T) models that allow for the weak and strong focusing conditions to be satisfied or violated. It is mentioned that this behaviour can also be found in the so-called f(R) and f(Q) theories.

Spinning up a time machine

We present a global metric describing closed timelike curves embedded in Minkowski spacetime. Physically, the metric represents an Alcubierre warp drive on a rotating platform. The physical realizability of such a metric is uncertain due to the exotic matter required to produce it. Never-the-less we suggest that this metric will have applications in more rigorously studying the behavior of quantum fields interacting with closed timelike curves.

Axial Symmetry Cosmological Constant Vacuum Solution of Field Equations with a Curvature Singularity, Closed Time-Like Curves, and Deviation of Geodesics

In this paper, we present a type D, nonvanishing cosmological constant, vacuum solution of Einstein’s field equations, extension of an axially symmetric, asymptotically flat vacuum metric with a curvature singularity. The space-time admits closed time-like curves (CTCs) that appear after a certain instant of time from an initial space-like hypersurface, indicating it represents a time-machine space-time. We wish to discuss the physical properties and show that this solution can be interpreted as gravitational waves of Coulomb-type propagate on anti-de Sitter space backgrounds. Our treatment focuses on the analysis of the equation of geodesic deviations.

Strings in irrelevant deformations of AdS3/CFT2

We generalize our recent analysis [1] of probe string dynamics to the case of general single-trace Toverline{T},Joverline{T} and Toverline{J} deformations. We show that in regions in coupling space where the bulk geometry is smooth, the classical trajectories of such strings are smooth and approach the linear dilaton boundary at either the far past or the far future. These trajectories give rise to quantum scattering states with arbitrarily high energies. When the bulk geometry has closed timelike curves (CTC’s), the trajectories are singular for energies above a critical value Ec. This singularity occurs in the region with CTC’s, and the value of Ec agrees with that read off from the dual boundary theory for all values of the couplings and charges.

The geometry of conformal timelike geodesics in the Einstein universe

This paper studies the geometry of the critical points of the simplest conformally invariant variational problem for timelike curves in the n-dimensional Einstein universe. Such critical curves are referred to as conformal timelike geodesics. The functional defining the variational problem is the Lorentz analogue of the conformal arclength functional in Möbius geometry. We compute the Euler–Lagrange equations and show that the trajectory of a conformal timelike geodesic is constrained into some totally umbilical Einstein universe of dimension 2, 3, or 4. The case of dimension 2 leads to orbits of 1-parameter groups of Lorentz Möbius transformations, while that of dimension 3 has been dealt with in [8]. In this paper, we discuss the case of conformal timelike geodesics in the 4-dimensional Einstein universe whose trajectories are not contained in any lower dimensional totally umbilical Einstein universe. It is shown that such curves can be explicitly integrated by quadratures and explicit expressions in terms of elliptic functions and integrals are provided.

Timelike Uzay Eğrisi boyunca q-çatı kullanılarak doğrultmanların gelişimiyle regle yüzeylerin yönlü gelişimleri

In this study, the ruled surfaces obtained by normal and binormal vectors along a timelike space curve in 3 dimensional Minkowski space by using q-frame is investigated. Directional evolutions of both quasi normal and quasi binormal ruled surfaces are studied by using their directrices. Then, we work on some geometric properties such as inextensibilty, developability and minimality of these ruled surfaces.

Rigidly rotating perfect fluid stars in 2+1 dimensions

Cataldo has found all rigidly rotating self-gravitating perfect fluid solutions in 2+1 dimensions with a negative cosmological constant $\Lambda$, for a density that is specified a priori as a function of a certain radial coordinate. We rewrite these solutions in standard polar-radial coordinates, for an arbitrary barotropic equation of state $p(\rho)$. For any given equation of state, we find the two-parameter family of solutions with a regular centre and finite total mass $M$ and angular momentum $J$ (rigidly rotating stars). For analytic equations of state, the solution is analytic except at the surface, but including at the centre. Defining the dimensionless spin $\tilde J:=\sqrt{-\Lambda}\,J$, there is precisely one solution for each $(\tilde J,M)$ in the region $|\tilde J|-1<M<|\tilde J|$, which consists of parts of the point particle region $M<-|\tilde J|$ and overspinning regions $|\tilde J|>|M|$. In an adjacent compact part of the black hole region $|\tilde J|<M$ (whose extent depends on the equation of state), there are precisely two solutions for each $(\tilde J,M)$. Hence exterior solutions exist in all three classes of BTZ solution (black hole, point particle and overspinning), but not all possible values of $(\tilde J,M)$ can be realised as stars. Regardless of the values of $\tilde J$ and $M$, the causal structure of all stars for all equations of state is that of anti-de Sitter space, without horizons or closed timelike curves.

Accelerating NUT black holes

We present and analyze a class of exact spacetimes which describe accelerating black holes with a Newman-Unti-Tamburino (NUT) parameter. First, by two independent methods we verify that the intricate metric found by Chng, Mann, and Stelea in 2006 indeed solves Einstein's vacuum field equations of general relativity. We explicitly calculate all components of the Weyl tensor and determine its algebraic structure. As it turns out, it is actually of algebraically general type I with four distinct principal null directions. It explains why this class of solutions has not been (and could not be) found within the large Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski--Demia\ifmmode \acute{n}\else \'{n}\fi{}ski family of type D spacetimes. Then we transform the solution into a much more convenient metric form which explicitly depends on three physical parameters: mass $m$, acceleration $\ensuremath{\alpha}$, and the NUT parameter $l$. These parameters can independently be set to zero, recovering thus the well-known spacetimes in standard coordinates, namely the $C$-metric, the Taub--NUT metric, the Schwarzschild metric, and flat Minkowski space in spherical coordinates. Using this new metric, we investigate main physical and geometrical properties of such accelerating NUT black holes. In particular, we localize and study four Killing horizons (two black-hole plus two acceleration horizons) and carefully investigate the curvature. Employing the scalar invariants we prove that there are no curvature singularities whenever the NUT parameter is nonzero. We identify asymptotically flat regions and relate them to conformal infinities. This leads to a complete understanding of the global structure of the spacetimes: each accelerating NUT black hole is a ``throat'' which connects ``our universe'' with a ``parallel universe.'' Moreover, the analytic extension of the boost-rotation metric form reveals that there is a pair of such black holes (with four asymptotically flat regions). They uniformly accelerate in opposite directions due to the action of rotating cosmic strings or struts located along the corresponding two axes. Rotation of these sources is directly related to the NUT parameter. In their vicinity there are pathological regions with closed timelike curves.

AN APPROXIMATE SOLUTION FOR LORENTZIAN SPHERICAL TIMELIKE CURVES

In this article, the differential equation of lorentzian spherical timelike curves is obtained in E14. It is seen that the differential equation characterizing Lorentzian spherical timelike curves is equivalent to a linear, third order, differential equation with variable coefficients. It is impossible to solve these equations analytically. In this article, a new numerical technique based on hermite polynomials is presented using the initial conditions for the approximate solution. This method is called the modified hermite matrix-collocation method. With this technique, the solution of the problem is reduced to the solution of an algebraic equation system and the approximate solution is obtained. In addition, the validity and applicability of the technique is explained by a sample application.

N-BISHOP DARBOUX VECTOR OF A TIMELIKE CURVE IN MINKOWSKI 3-SPACE

It is known that a Bishop frame of a curve is one of the effective alternative approach in the differential geometry. Recently, several important works have been done about the Bishop frames. The aim of our paper is to investigate the N-Bishop frame for timelike curves in Minkowski space. We define the N-Bishop frame for the timelike curve in Minkowski space. Then, we consider some properties of the frame. Moreover, we describe the N-Bishop Darboux frame for the first time. Additionally, we compute some geometrical characterizations for the N-Bishop Darboux axis and momentum rotation vector.

Mannheim B-curve couples in Minkowski 3-space

In this paper, we define null Cartan and pseudo null Mannheim curves in Minkowski 3-space according to their Bishop frames. We obtain the necessary and the sufficient conditions for pseudo null curves to be Mannheim B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Mannheim B-curves, by considering the cases when their Mannheim B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Mannheim B-curve pairs.

Reversible dynamics with closed time-like curves and freedom of choice

The theory of general relativity predicts the existence of closed time-like curves (CTCs), which theoretically would allow an observer to travel back in time and interact with their past self. This raises the question of whether this could create a grandfather paradox, in which the observer interacts in such a way to prevent their own time travel. Previous research has proposed a framework for deterministic, reversible, dynamics compatible with non-trivial time travel, where observers in distinct regions of spacetime can perform arbitrary local operations with no contradiction arising. However, only scenarios with up to three regions have been fully characterised, revealing only one type of process where the observers can verify to both be in the past and future of each other. Here we extend this characterisation to an arbitrary number of regions and find that there exist several inequivalent processes that can only arise due to non-trivial time travel. This supports the view that complex dynamics is possible in the presence of CTCs, compatible with free choice of local operations and free of inconsistencies.

Speedup of Grover’s search algorithm and closed timelike curves

The quadratic reduction of query complexity of Grover’s search algorithm (GA), while significant, would not be enough to enjoy exponentially fast data searching in large-scale quantum computation. One of the ways to enhance the speedup in the framework of Grover’s algorithm is to employ a novel quantum operation, i.e., inversion against an unknown state; however, this is not possible at least in quantum theory. We thus extend the Grover algorithm assisted by closed timelike curves (CTCs), in which the unknown-state inversion is achievable by combining the superposition of two unknown states with cloning. We dubbed this refined algorithm CTC-assisted Grover algorithm (CTC-GA). We show that the CTC-GA can vastly reduce the query complexity compared to the original algorithm; remarkably, from polynomial to poly-logarithmic. These results will provide a broader intuition for exponential quantum speedup of data searching problems.

Directional spherical indicatrices of timelike space curve

In this work, the directional spherical indicatrices of a timelike space curve using tangent, quasi-normal and quasi-binormal vectors with q-frame are introduced. Then we work on the condition, that a timelike space curve to be slant helix, by using the geodesic curvature of the directional normal spherical indicatrix. Finally, an application of the results is given.

On the properties of a deformed extension of the NUT space-time

We consider a class of space-times given by a stationary extension of the Zipoy–Voorhees metric that was found by Halilsoy. We show that the solutions do not describe rotating sources but must be interpreted, similarly to the NUT case, as deformed sources endowed with a gravitomagnetic charge. We show that the Halilsoy family is directly linked to the NUT space-time, which can be obtained in the limit of vanishing deformations. We investigate the motion of test particles and photons in this class of space-times, in particular the innermost stable circular orbits and photon capture radius. Finally we show that this class of solutions possesses a sub-manifold where closed time-like curves are allowed.

Non-lightlike constant angle ruled surfaces in Minkowski 3-space

In this paper, non-lightlike ruled surfaces of constant slope in Minkowski 3-space with a non-lightlike base curve c(s)=∫αt+βn+γbds, where t, n, b are tangent, principal normal and binormal vectors of an arbitrary timelike curve Γ(s) are investigated. Non-lightlike ruled surfaces of constant slope parallel to a tangent of a timelike general helix and normal of timelike slant helix are studied. It is shown that all non-lightlike ruled surfaces of constant slope are developable but not minimal surfaces.

Curves in the Lorentz-Minkowski plane with curvature depending on their position

Abstract Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in L 2 {{\mathbb{L}}}^{2} whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.

Axially Symmetric Type N Space-Time with Causality Violating Curves and the von Zeipel Cylinder

An axially symmetric nonvacuum solution of the Einstein field equations, regular everywhere and free from curvature divergence is presented. The matter-energy content is a the pure radiation field satisfying the energy conditions, and the metric is of type N in the Petrov classification scheme. The space-time develops circular closed timelike curves everywhere outside a finite region of space i.e., beyond a null curve. Furthermore, the physical interpretation of the solution based on the study of the equations of geodesic deviation is presented. Finally, the von Zeipel cylinders with respect to the Zero Angular Momentum Observers (ZAMOs) is discussed. In addition, circular null and timelike geodesic of pace-time are also presented.

Gravitational Wu-Yang monopoles

We show that in an asymptotically flat space where an S-Matrix can be defined, dual supertranslations leave all its matrix elements invariant and the Hilbert space of asymptotic states factorizes into distinct super-selection sectors, labeled by their dual supertranslation charges. These results suggest that dual supertranslation may be interpreted as a redundant gauge symmetry of asymptotically flat spacetimes. This would allow to recast general relativity as a theory of diffeomorphisms possessing an additional asymptotic gauge symmetry. We then use the conjectured dual supertranslation gauge symmetry to construct a gravitational equivalent of the Wu-Yang monopole solution. The metric describing the solution is defined using two overlapping patches on the celestial sphere. The solution is regular on each one of the patches separately and differentiable in the overlap region, where the two descriptions are identical by virtue of a dual supertranslation gauge transformation. Our construction provides an alternative to Misner's interpretation of the Taub-NUT metric. In particular, we find that using our approach the Taub-NUT metric can be made regular everywhere on the celestial sphere and at the same time it is devoid of closed timelike curves, provided that the bound $\frac{m}{\ell} \leq \sqrt{\frac{5}{27}}$ on the ratio of mass to NUT charge is obeyed.

Analysis for Lorentzian conformal field theories through sine-square deformation

Abstract We reexamine two-dimensional Lorentzian conformal field theory using the formalism previously developed in a study of sine-square deformation of Euclidean conformal field theory. We construct three types of Virasoro algebra. One of them reproduces the result by Lüscher and Mack, while another type exhibits divergence in the central charge term. The third leads to a continuous spectrum and contains no closed time-like curve in the system.