Timelike Curves Research Articles

Lorentz–Darboux rectifying surfaces and osculating surfaces along frontal curves on timelike surfaces

In this paper, we consider two kinds of developable surfaces along a timelike frontal curve lying in a timelike surface in Minkowski 3-space, the Lorentz–Darboux rectifying surfaces and the Lorentz–Darboux osculating surfaces. Meanwhile, we also consider two curves generated by such a timelike frontal curve. We give two new invariants of the frontal curve which characterize singularities of the Lorentz–Darboux developable surfaces and Lorentz–Darboux rectifying and osculating curves. Unlike the regular curves, the frontal curves may have singular points. Using the methods of the unfolding theory in singularity theory, we complete the classifications of the singular points of these two surfaces and two curves. The main results indicate that compared with developable surfaces along a regular curve, there exists a more complicated construction for the singularities of the developable surfaces along a timelike frontal curve, there will appear an extra locus, arising by the singular point of the timelike frontal curve, for the singularities of the Lorentz–Darboux rectifying surface, whereas the Lorentz–Darboux osculating surface did not. In addition, we investigate the geometric properties of the timelike frontal curve, it is shown that the timelike frontal curve can be regarded as the envelope of a family of timelike frontal rectifying curves. Finally, we provide several examples to illustrate the theoretical results.

4-Boyutlu Lorentz Minkowski Uzayında Null Olmayan Ortak İzoasimptotik Eğrili Hiperyüzey Aileleri

In this study, the parametric equation of hypersurfaces passing through the spacelike and timelike curves that has non-null Frenet vectors in E_1^4 Lorentz Minkowski space was expressed with the help of this curve is Frenet frame. Furthermore, hypersurface families were created by giving necessary and sufficient conditions so that the spacelike and timelike curves on E_1^4 is common isoasymptotic on the hypersurface. Then, these conditions obtained with the help of deviation marching-scale functions were simplified. Finally, the study was supported with examples and the graphics were drawn using certain projection methods.

Geometric phases for three cases of the electric field with new type Bishop frame in ℝ13

In this paper, we study geometric phases for an electric field in Minkowski 3-space. In particular, we consider Bishop 2-type frames of a spacelike and a timelike curve in Minkowski 3-space and investigate three geometric phases for three cases of electric fields with respect to the Bishop 2-type frames along a spacelike and a timelike curve. Also, we compute spatial and time evolutions of the complex frame connected with electric fields for Bishop 2-type in terms of modified Hasimoto transformations. As a result, we give a modified Fermi–Walker parallel derivative and a Rytov parallel transportation of electric fields. Finally, we discuss Lorentz force of a magnetic field as a Killing vector field along a spacelike and timelike curve in Minkowski 3-space.

TsT, black holes, and Toverline{T} + Joverline{T} + Toverline{J}

We generate a class of string backgrounds by a sequence of TsT transformations of the NS1-NS5 system that we argue are holographically dual to states in a single-trace Toverline{T} + Joverline{T} + Toverline{J} -deformed CFT2. The new string backgrounds include general rotating black hole solutions with two U(1) charges as well as a smooth solution without a horizon that is interpreted as the ground state. As evidence for the correspondence we (i) derive the long string spectrum and relate it to the spectrum of the dual field theory; (ii) show that the black hole thermodynamics can be reproduced from single-trace Toverline{T} + Joverline{T} + Toverline{J} -deformed CFTs; and (iii) show that the energy of the ground state matches the energy of the vacuum in the dual theory. We also study geometric properties of these new spacetimes and find that for some choices of the parameters the three-dimensional Ricci scalar in the Einstein frame can become positive in a region outside the horizon before reaching closed timelike curves and singularities.

Chronology protection implementation in analogue gravity

Analogue gravity systems offer many insights into gravitational phenomena, both at the classical and at the semiclassical level. The existence of an underlying Minkowskian structure (or Galilean in the non-relativistic limit) in the laboratory has been argued to directly forbid the simulation of geometries with Closed Timelike Curves (CTCs) within analogue systems. We will show that this is not strictly the case. In principle, it is possible to simulate spacetimes with CTCs whenever this does not entail the presence of a chronological horizon separating regions with CTCs from regions that do not have CTCs. We find an Analogue-gravity Chronology protection mechanism very similar in spirit to Hawking’s Chronology Protection hypothesis. We identify the universal behaviour of analogue systems near the formation of such horizons and discuss the further implications that this analysis has from an emergent gravity perspective. Furthermore, we build explicit geometries containing CTCs, for instance spacetimes constructed from two warp-drive configurations, that might be useful for future analysis, both from a theoretical and an experimental point of view.

Timelike V-Bertrand Curves in Minkowski 3-Space $E_{1}^{3}$

In this paper, the timelike V-Bertrand curve, a new type Bertrand curve in Minkowski 3-Space $E_{1}^{3}$, is characterized. Based on the timelike V-Bertrand curve, the properties of the timelike T, N, and B Bertrand curves are obtained. From the timelike V-Bertrand curve, f-Bertrand curves and Bertrand surfaces are defined. We support the existence of these new curves and surfaces with examples. Finally, we discuss the results for further research.

Geometry of Curves with Fractional Derivatives in Lorentz Plane

In this paper, the geometry of curves is discussed based on the Caputo fractional derivative in the Lorentz plane. Firstly, the tangent vector of a spacelike plane curve is defined in terms of the fractional derivative. Then, by considering a spacelike curve in the Lorentz plane, the arc length and fractional ordered frame of this curve are obtained. Later, the curvature and Frenet-Serret formulas are found for this fractional ordered frame. Finally, the relation between the fractional curvature and classical curvature of a spacelike plane curve is obtained. In the last part of the study, considering the timelike plane curve in the Lorentz plane, new results are obtained with the method in the previous section.

The Potential of a Thick Present through Undefined Causality and Non-Locality.

This paper elaborates on the interpretation of time and entanglement, offering insights into the possible ontological nature of information in the emergence of spacetime, towards a quantum description of gravity. We first investigate different perspectives on time and identify in the idea of a “thick present” the only element of reality needed to describe evolution, differences, and relations. The thick present is connected to a spacetime information “sampling rate”, and it is intended as a time symmetric potential bounded between a causal past of irreversible events and a still open future. From this potential, spacetime emerges in each instant as a space-like foliation (in a description based on imaginary paths). In the second part, we analyze undefined causal orders to understand how their potential could persist along the thick present instants. Thanks to a C-NOT logic and the concept of an imaginary time, we derive a description of entanglement as the potential of a logically consistent open choice among imaginary paths. We then conceptually map the imaginary paths identified in the entanglement of the undefined orders to Closed Time-like Curves (CTC) in the thick present. Considering a universe described through information, CTC are interpreted as “memory loops”, elementary structures encoding the information potential related to the entanglement in both time and space, manifested as undefined causality and non-locality in the emerging foliation. We conclude by suggesting a possible extension of the introduced concepts in a holographic perspective.

Holography of time machines

We use holography to examine the response of interacting quantum fields to the appearance of closed timelike curves in a dynamically evolving background that initially does not contain them. For this purpose, we study a family of two-dimensional spacetimes that model very broad classes of wormhole time machines. The behavior of strongly coupled conformal theories in these spacetimes is then holographically described by three-dimensional AdS bulk geometries that we explicitly construct. The dual bulk spacetime is free from any divergences, but splits into two disconnected components, without and with CTCs, which are joined only through the boundary; then, passages across the chronology horizon are impossible for any field excitations. In dual terms, the strong self-interaction of the CFT decouples the pathological part from the rest of the spacetime. We also find that entangling the CFTs in two separate time machines connects them through a traversable bulk wormhole. Nevertheless, any entanglement-assisted chronology violations will be prevented by quantum bulk corrections, i.e., subleading 1/N effects, again without needing any gravitational backreaction of the CFT. We are led to speculate that chronology may be protected without involving Planck scale physics.

Biharmonic Curves in a Strict Walker 3-Manifold

In this paper, we study the geometry of biharmonic curves in a strict Walker 3-manifold and we obtain explicit parametric equations for biharmonic curves and time-like biharmonic curves, respectively. We discuss the conditions for a speed curve to be a slant helix in a Walker manifold. We give an example of biharmonic curve for illustrating the main result.

Parallel curves in Minkowski 3-space

For a curve on the plane, there exists the curve at a fixed distance (except for some degenerate cases). The curve can be alternatively produced as an envelope of circles with the radius moving along the curve . As a result, when this structure is translated to three-dimensional space two parallel curves are obtained. In this study defining some time-like parallel curves on 3-dimensional Minkowski space, some important theorems about these are stated and proved.

Morse-theoretical analysis of lensing in wormhole spacetimes

We consider a class of stationary and axisymmetric wormhole spacetimes that is closely related to, but not identical with, the class of Teo wormholes. We fix a point $p$ (observation event) and a timelike curve $\ensuremath{\gamma}$ (worldline of a light source), and we characterize the set of all past-oriented lightlike geodesics from $p$ to $\ensuremath{\gamma}$. As any such geodesic corresponds to an image of the light source on the observer's sky, this allows us to investigate the lensing properties of the wormhole. As a main result, we prove with the help of Morse theory that, under very mild conditions on $\ensuremath{\gamma}$, the observer always sees infinitely many images of $\ensuremath{\gamma}$. Moreover, we study some qualitative features of the lightlike geodesics with the help of two potentials that determine the sum of centrifugal and Coriolis forces of observers in circular motion for the case that the observers' velocity approaches the velocity of light. We exemplify the general results with two specific wormhole spacetimes.

On the Counter-Rotation of Closed Time-like Curves

While it is tempting to think of closed time-like curves (CTCs) around rotating bodies, such as a black hole, as being “caused” by the rotation of the source, Andréka et al. pointed out that the underlying physics are not as straightforward as this, since such CTCs are “counter-rotating”, i.e., the time orientation (the opening of the local light cones) of the CTCs is opposite to the direction in which the singularity or the ergosphere rotates. It was also suggested that this is a generic phenomenon that calls for a deeper intuitive physical understanding. In this short note, we point out—with Kerr–Taub–NUT as an example—that CTCs are counter-rotating with respect to the local angular velocity of the spacetime, not the global angular momentum, nor the angular velocity of the black hole horizon, which makes the physical interpretation of CTCs being “caused” by a rotating source even more problematic.

Remark on “Reversible dynamics with closed time-like curve and freedom of choice” by Germain Tobar Fabio Costa

A new paper by Germain Tobar et al seem interesting, that they claim that reversible dynamics of CTC is possible. We comment this paper by pointing out at least 4 issues that make this paper problematic, not only practically but also conceptually.

Introducing a general time machine solution and analysis of a vacuum spacetime generating closed timelike curves

Introducing a general time machine solution and analysis of a vacuum spacetime generating closed timelike curves

Optical modeling of Hasimoto map for antiferromagnetic timelike optical fiber

In this study, we study optical hasimoto map corresponds to under the quasi-frame for timelike curves in 3D Minkowski space. We investigate the relationship of this transformation with some flow equations, which are known examples for nonlinear optical systems. Here, we first obtain the q-Hasimoto map as equivalent of the Hasimoto map in the q-frame for timelike space curve in 3D Minkowski Space. Then, we examine the effect on the q-Hasimoto map of the some flow equations such as vortex filament equation, Heisenberg antiferromagnetic flow. Finally, we present comments on some of the physical consequences of these effects that we have examined.

Multiworld motives by closed time-like curves

We propose a new model, entitled S-CTC, for description of quantum systems in the presence of CTC – closed time-like curves. The model is based on the viewpoint on any quantum state as an observer’s state of knowledge of the system preparation procedure. We compare and contrast our S-CTC model with D-CTC and P-CTC models and show that S-CTC shares special quantum features with both D-CTC and P-CTC. As far as the interaction of the quantum system with itself coming from the future concerns, S-CTC is formally equivalent to P-CTC. On the other hand, when calculating outcome probabilities for a measurement within the time interval between the entrance and exit of CTC, S-CTC becomes equivalent to D-CTC. Both these models require the concept of alternative realities (worlds) where different measurement outcomes are recorded and alternative connections of these realities by CTC.

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.