Timelike Curves Research Articles

Extremal Rotating BTZ Black Holes Cannot Be Dressed in (anti-)Self-Dual Maxwell Field

Abstract Under the (anti-)self-dual condition for orthonormal components of the Faraday tensor, the 3D Einstein–Maxwell system with a negative cosmological constant $\Lambda$ admits a solution obtained by Kamata and Koikawa and later by Cataldo and Salgado in the most general form. Actually, Clément first obtained this solution and interpreted it as a regular particle-like solution without horizon. Nevertheless, it has been erroneously stated in some literature that this Clément–Cataldo–Salgado (CCS) solution, locally characterized by a single parameter, describes a black hole even in the charged case as it reduces to the extremal rotating Bañados–Teitelboim–Zanelli (BTZ) solution in the vacuum limit and its curvature invariants are constant. In this paper, we supplement Clément’s interpretation by showing that there appears a parallelly propagated curvature singularity corresponding to an infinite affine parameter along spacelike geodesics at the location of the Killing horizon in the extremal rotating BTZ solution when the (anti-)self-dual Maxwell field is added. If the spatial coordinate $\theta$ is periodic, closed timelike curves exist near the singularity. It is also shown that the CCS solution is of Cotton type N (in contrast to charged rotating BTZ black holes which are of type I away from the horizon), and the energy-momentum tensor of the Maxwell field is of Hawking–Ellis type II. The metric solves the Einstein-$\Lambda$ equations also with a massless scalar field or a null dust fluid. We explicitly demonstrate that it belongs to the Kundt shear-free, nontwisting, and nonexpanding class of geometries, whereas extremal rotating BTZ black holes have expanding principal null directions. It means that the CCS metric represents the specific null (i.e. “radiative”) Maxwell field generated by a singular source, rather than an extremal rotating BTZ black hole dressed in an (anti-)self-dual Maxwell field.

Singularities of worldsheets along framed particle trajectories in Minkowski three-space

Modeling the worldsheets along the trajectory (a string) of a point particle (a 0-brane) as two timelike surfaces in 3-subspace of Minkowskian spacetime, respectively, the work aims at analyzing the topological structures of these two worldsheets along the trajectory of the point particle from the view point of singularity theory. Different from the regular curves, the traveling trajectory (modeled as framed timelike curve) of the particle is allowed to be singular, two worldsheets are generated by the traveling trajectories of the particle. As applications of singularity theory, we classify the singularities of these two worldsheets along the traveling trajectory of the particle. Using the approach of the unfolding theory in singularity theory, we find two new geometric invariants which are useful for characterizing the local topological structures of singularities of these two worldsheets along the particle. It is revealed that there exist cuspidal edge type and swallowtail type of singularities for these two worldsheets under the appropriate conditions of geometric invariants. Meanwhile, it is also pointed out that the types of singularities of these two worldsheets have a close relationships to the order of contacts between these two worldsheets and two timelike planes, respectively. Finally, some examples are presented to interpret our theoretical results.

New construction of a vacuum doubly rotating black ring by the Ehlers transformation

Using the Ehlers transformation, we derive an exact solution for a doubly rotating black ring in five-dimensional vacuum Einstein theory. It is well known that the vacuum Einstein theory with three commuting Killing vector fields can be reduced to a nonlinear sigma model with SL(3,R) target space symmetry. As shown previously by Giusto and Saxena, the SO(2,1) subgroup in the SL(3,R) can generate a rotating solution from a static solution while preserving asymptotic flatness. This so-called Ehlers transformation actually transforms the five-dimensional Schwarzschild black hole into the five-dimensional Myers-Perry black hole. However, unlike the case with the black hole, applying this method directly to the static black ring or the Emparan-Reall black ring, does not yield a regular rotating black ring due to the emergence of a Dirac-Misner string singularity. To solve this undesirable issue, we use a singular vacuum solution of a rotating black ring/lens that already possesses a Dirac-Misner string singularity as the seed solution for the Ehlers transformation. The resulting solution is regular, indicating the absence of curvature singularities, conical singularities, orbifold singularities, Dirac-Misner string singularities, and closed timelike curves both on and outside the horizon. We show that this solution obtained by the Ehlers transformation coincides precisely with the Pomeransky-Sen’kov solution. We expect that applying this method to other theories may lead to the finding of new exact solutions, such as solutions for black lenses and capped black holes, as well as black ring configurations. Published by the American Physical Society 2024

Time-machines construct in f(ℛ, 𝒜, Aμν Aμν ) and f(ℛ) modified gravity theories

In this paper, our objective is to explore a time-machine space-time formulated in general relativity, as introduced by Li (Phys. Rev. D 59, 084016 (1999)), within the context of modified gravity theories. We consider Ricci-inverse gravity of all Classes of models, i.e., (i) Class-I: f(ℛ, 𝒜) = (ℛ + κℛ2 + β 𝒜), (ii) Class-II: f(ℛ, Aμν Aμν ) = (ℛ + κℛ2 + γ Aμν Aμν ) model, and (iii) Class-III: f(ℛ, 𝒜, Aμν Aμν ) = (ℛ + κℛ2 + β𝒜 + δ𝒜2 + γ Aμν Aμν ) model, where Aμν is the anti-curvature tensor, the reciprocal of the Ricci tensor, Rμν , 𝒜 = gμν Aμν is its scalar, and β, κ, γ, δ are the coupling constants. Moreover, we consider f(ℛ) modified gravity theory and investigate the same time-machine space-time. In fact, we show that Li time-machine space-time serve as valid solutions both in Ricci-inverse and f(ℛ) modified gravity theories. Thus, both theory allows the formation of closed time-like curves analogue to general relativity, thereby representing a possible time-machine model in these gravity theories theoretically.

Axially symmetric solutions in Ricci-inverse modified gravity

We investigate a family of axial symmetry solution constructed in general relativity (GR) within the framework of Ricci-inverse (RI) gravity theory. In GR, these solutions admitted closed time-like curves at an instant of time from an initial spacelike hypersurface in a causally well-behaved manner, thus, violates the causality condition. Our aim is to examine these axial symmetry solutions within the context of Ricci-inverse gravity theory to determine whether closed time-like curves still appear in this new gravity theory. We consider two Classes of RI-gravity models: (i) Class-II models defined by a function f=f(R,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=f({\\mathcal {R}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document} gravity and (ii) Class-III models defined by f=f(R,A,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=f({\\mathcal {R}},{\\mathcal {A}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document}, where Aμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\mu \ u }$$\\end{document} is the anti-curvature tensor, A=gμνAμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}=g_{\\mu \ u }\\,A^{\\mu \ u }$$\\end{document} as its scalar, and Rμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\mu \ u }$$\\end{document} is the Ricci tensor. We are able solved the modified field equations considering these axial symmetry solutions as background in RI-gravity with null radiation as the matter content and the cosmological constant. This confirms that the chosen family of axial symmetry solutions are valid solutions in RI-gravity theory and, consequently, closed time-like curves is still form, analogous to their formation in GR.

Accelerating Kerr-Taub-NUT spacetime in the low energy limit of heterotic string theory

We construct a novel solution in the low energy limit of heterotic string theory describing accelerating charged and rotating black holes with NUT parameter. Some aspects of the new spacetime are discussed, such as locations of horizons, ergoregions, conic singularity, closed timelike curves, and area temperature product. We find some of the properties resemble the ones belong to accelerating Kerr-Newman-Taub-NUT spacetime.

Warp drives and closed timelike curves

Abstract It is commonly accepted that superluminal travel may be used to facilitate time travel. This is a purely special-relativistic argument, using the fact that for observers in two frames of reference, separated by a spacelike interval, the non-causal (spacelike) future of one observer includes part of the causal past of the other. In this paper we provide a concrete realization of this argument in a curved general-relativistic spacetime, using warp drives as the means of faster-than-light travel. By generalizing the usual warp drive metric to allow for a non-unit lapse function, we allow the warp drive to switch between reference frames in a purely geometric way. With an additional modification allowing the warp drive to have compact support, this permits us to glue two warp drives together to construct a closed timelike geodesic, such that a test particle following the geodesics of the two warp drives travels back to its own past. This provides a precise mathematical model for the connection between faster-than-light travel and time travel in general relativity, and the first such model to be explicitly formulated using two warp drives. We also give a detailed discussion of weak energy condition violations in the non-unit-lapse warp drive.

New characterization of Robertson–Walker geometries involving a single timelike curve

Abstract Our aim in this paper is two-fold. We establish a novel geometric characterization of the Robertson–Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the cosmological expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow.

Nonexistence of closed timelike geodesics in Kerr spacetimes

The Kerr-star spacetime is the extension over the horizons and in the negative radial region of the Kerr spacetime. Despite the presence of closed timelike curves below the inner horizon, we prove that the timelike geodesics cannot be closed in the Kerr-star spacetime. Since the existence of closed null geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)], this result shows the absence of closed causal geodesics in the Kerr-star spacetime.

Time-reversed information flow through a wormhole in scalar–tensor gravity

This Letter aims to advance unexplored properties of a new class of Closed Timelike Curves recently discovered in scalar–tensor gravity, reported in Nguyen and Lobo (2023) [31] and Nguyen and Azreg-Aïnou (2023) [30]. Therein, it was shown that when the Weak Energy Condition is violated, the topology of spacetime in scalar–tensor gravity is altered and directly enables the formation of two-way traversable wormholes. Furthermore, each of these wormholes acts a gateway between two time-mirrored worlds, where the two asymptotically flat sheets in the Kruskal–Szekeres diagram are glued antipodally along three directions—time t and the polar and azimuth angles (θ,φ) of the 2-sphere—to form a wormhole throat. This contrasts with the standard embedding diagram which typically glues the sheets only along the θ and φ directions. Crucially, due to the ‘gluing’ along the t direction, the wormhole becomes a portal connecting the two spacetime sheets with opposite physical time flows, enabling the emergence of closed timelike loops which straddle the throat. In this Letter, we shall point out that this portal mathematically permits the possibility of backward propagation of information against time. This feature is ubiquitous for wormholes in scalar–tensor theories. In addition, we formulate the Feynman sum for transition amplitudes of microscopic particles in the proximity of a wormhole throat in which we account for timelike paths that experience time reversal.

On the role of closed timelike curves and confinement structure around Kerr–Newman singularity

In this study, the particle motion around the naked singularity and black hole of Kerr–Newman spacetime is investigated with a special attention on the closed timelike orbits. It is found that both in the naked singularity (NS) and in black hole (BH), the singularity is concealed by causality violating regions, and the Cauchy surface consistently resides inside the inner horizon in nonextremal black holes. For neutral particles and particles with an identical charge to the source, only particles with positive angular momentum are permitted to traverse the closed timelike curves. Conversely, for particles with the opposite charge to the source, the strong Coulomb attraction draws all particles inside the Cauchy surface, allowing them to be present in the closed timelike curves irrespective of their angular momentum. However, in both the NS and BH (both extremal and nonextremal), test particles are confined at a considerable distance from the singular point such that there always exists an empty region surrounding the singularity which prevents particles from interacting with it. The radius of the empty surface that depends on the source parameters and the particle characteristics is investigated with an accurate expression.

Solution generation of a capped black hole

Utilizing the electric Harrison transformation developed in five-dimensional minimal supergravity, we construct an exact solution characterizing non-BPS (Bogomol’nyi-Prasad-Sommerfield) charged rotating black holes with a horizon cross section of a lens space L(n;1). Among these solutions, only the ones corresponding to n=0 and n=1 do not have any curvature singularities, conical singularities, Dirac-Misner string singularities, and orbifold singularities both on and outside the horizon; additionally, they are free from closed timelike curves. The solution for n=0 corresponds to the charged dipole black ring that we constructed in the previous paper. The specific solution for n=1, referred to as the “capped black hole,” was introduced in our previous article. This provides the first example of a non-BPS exact solution, representing an asymptotically flat, stationary spherical black hole with a domain of outer communication (DOC) having a nontrivial topology in five-dimensional minimal supergravity. We demonstrate that the DOC on a time slice has the topology of [R4#CP2]\B4. Differing from the well-known Myers-Perry and Cvetič-Youm black holes describing a spherical horizon topology and a DOC with a trivial topology of R4\B4 on a timeslice, the capped black hole’s horizon is capped by a disk-shaped bubble. We explicitly demonstrate that the capped black hole carries mass, two angular momenta, an electric charge, and a magnetic flux, with only three of these quantities being independent. Furthermore, we reveal that this black hole can possess identical conserved charges as the Cvetič-Youm black hole. The existence of this solution challenges black hole uniqueness beyond both the black ring and the BPS spherical black hole. Moreover, within specific parameter regions, the capped black hole can exhibit larger entropy than the Cvetič-Youm black hole. Published by the American Physical Society 2024

Particle detectors under chronological hazard

We analyze how the presence of closed timelike curves (CTCs) characterizing a time machine can be discerned by placing a local particle detector in a region of spacetime which is causally disconnected from the CTCs. Our study shows that not only can the detector tell if there are CTCs, but also that the detector can separate topological from geometrical information and distinguish periodic spacetimes without CTCs (like the Einstein cylinder), curvature, and spacetimes with topological identifications that enable time-machines.

Agnostic Phase Estimation.

The goal of quantum metrology is to improve measurements' sensitivities by harnessing quantum resources. Metrologists often aim to maximize the quantum Fisher information, which bounds the measurement setup's sensitivity. In studies of fundamental limits on metrology, a paradigmatic setup features a qubit (spin-half system) subject to an unknown rotation. One obtains the maximal quantum Fisher information about the rotation if the spin begins in a state that maximizes the variance of the rotation-inducing operator. If the rotation axis is unknown, however, no optimal single-qubit sensor can be prepared. Inspired by simulations of closed timelike curves, we circumvent this limitation. We obtain the maximum quantum Fisher information about a rotation angle, regardless of the unknown rotation axis. To achieve this result, we initially entangle the probe qubit with an ancilla qubit. Then, we measure the pair in an entangled basis, obtaining more information about the rotation angle than any single-qubit sensor can achieve. We demonstrate this metrological advantage using a two-qubit superconducting quantum processor. Our measurement approach achieves a quantum advantage, outperforming every entanglement-free strategy.

On causality violation in different classes of Ricci inverse gravity

In this paper, we investigate a causality-violating four-dimensional space-time, an extension of the two-dimensional Misner space within the framework of modified gravity theories, specifically in the context of Ricci-inverse gravity. To explore this, we focus on two classes of models denoted as f(R,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f({\\mathcal {R}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document} and f(R,A,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f({\\mathcal {R}}, {\\mathcal {A}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document}, and derive the modified field equations using the four-dimensional space-time under consideration. Here, Aμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\mu \ u }$$\\end{document} is the anti-curvature tensor defined by inverse of the Ricci tensor Rμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{\\mu \ u }$$\\end{document}, that is, Aμν=Rμν-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\mu \ u }=R^{-1}_{\\mu \ u }$$\\end{document}. Additionally, A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} denotes its scalar, defined as A=gμνAμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}=g_{\\mu \ u }\\,A^{\\mu \ u }$$\\end{document}. Our analysis shows that in both models, the modified field equations can be solved by considering the matter energy content as a pure radiation field including the cosmological constant. Furthermore, we demonstrate that the energy density of the pure radiation field and the cosmological constant undergo modifications due to the coupling parameters, and these modifications converge to the results of general relativity when these parameters are set to zero. This observation confirms the possibility of causality violation in the form of closed time-like curves within the framework of Ricci-inverse gravity theory.

Closed time-like curves in [formula omitted] modified gravity theory

In this manuscript, we investigate solutions that allow closed time-like curves (CTCs) in general relativity (GR) within the framework of a modified gravity theory. We consider here an alternative theory recently proposed in gravity theories known as f(R,A)=(R+αA) gravity or Ricci-inverse (RI) gravity, α is the coupling constant. This theory incorporates modifications into the Einstein-Hilbert action through a coupling with an anti-curvature scalar derived from the anti-curvature tensor. We investigate a family of Petrov type-N solutions that are non-vacuum solutions of the field equations, characterized by a negative cosmological constant. The matter-energy content is Vaidya radiation, and its energy density may either remain constant or vary with axial distance. One distinctive feature of the considered solutions is the formation of CTCs, which violate the causality condition. Another observation is that the determinant of the Ricci tensor, Rμν, differs from zero, confirming the existence of an anti-curvature tensor, Aμν, the inverse of the Ricci tensor, defined as Aμν=Rμν−1, and hence, the anti-curvature scalar A=gμνAμν. We demonstrate that the family of cosmological models serves as valid solutions within the framework of Ricci-inverse gravity model, with Vaidya radiation as the matter-energy content whose energy density is modified by the coupling parameter. Thus, our results affirm that the Ricci-inverse gravity theory allows for the existence of CTCs, similar to the case in general relativity.

Spinning particle geometries in AdS3/CFT2

We study spinning particle/defect geometries in the context of AdS3/CFT2. These solutions lie below the BTZ threshold, and can be obtained from identifications of AdS3. We construct the Feynman propagator by solving the bulk equation of motion in the spinning particle geometry, summing over the modes of the fields and passing to the boundary. The quantization of the scalar fields becomes challenging when confined to the regions that are causally well-behaved. If the region containing closed timelike curves (CTCs) is included, the normalization of the scalar fields enjoys an analytical simplification and the propagator can be expressed as an infinite sum over image geodesics. In the dual CFT2, the propagator can be recast as the HHLL four-point function, where by taking into account the PSL(2, ℤ) modular images, we recover the bulk computation. We comment on the casual behavior of bulk geometries associated with single-trace operators of spin scaling with the central charge below the BTZ threshold.

[formula omitted] gravity theory in Einstein space background and causality violation

In this paper, we exploration a Petrov type-N vacuum solution to Einstein’s field equations, while incorporating a negative cosmological constant (Λ<0) within the framework of modified gravity theories. This solution intriguingly accommodates closed time-like curves at a particular moment in time, effectively violates the causality condition, thus acts as a time-machine model. A key observation is that the determinant of the Ricci tensor Rμν for this particular Einstein space metric differs from zero. This noteworthy finding suggests to the existence of an anti-curvature tensor defined Aμν=Rμν−1 and hence, an anti-curvature scalar A=gμνAμν, which is introduced with the Lagrangian of the system, thereby giving rise to as Ricci-inverse gravity theory. We consider class-I models of Ricci-inverse gravity, where the function f=f(R,A)=(R+κA) with κ is the coupling constant. We demonstrate that this Einstein space metric serves as a vacuum solution with a negative modified cosmological constant within the framework of Ricci-inverse gravity. Consequently, the violation of causality persists within this new gravity theory as well. Moreover, we solve the modified field equations by considering matter content other than vacuum and demonstrate that the energy-density and isotropic pressure satisfies the equation ρ=−p=3κΛ.

Diffeomorphism invariance and quantum mechanical paradoxes

Paradoxes in gravitational physics, such as grandfather paradoxes with closed timelike curves or the AMPS paradox, are often constructed in a weak gravity regime but upon further examination engender strong gravitational responses such as unstable Cauchy horizons or firewalls. In contrast, some proposed paradoxes in nonrelativistic quantum mechanics ignore gravity completely. Such nonrelativistic proposed paradoxes are often not gauge invariant — they use local operators which should be forbidden by diffeomorphism-invariant quantum gravity. Ignoring this complication is reasonable if there is a consistent weak gravity regime where the paradox can be formulated with gauge invariant observables up to some order in Newton’s constant. We show that this approach remains inconsistent due to a lack of analyticity of the necessary solutions. As a consequence, there is no consistent weak gravity, diffeomorphism invariant embedding of such paradoxes—just as in other gravitational paradoxes they generate a strong gravitational response and are in principle sensitive to quantum gravity.

A SMARANDACHE CURVE ACCORDING TO THE DARBOUX FRAME AND THE KINEMATICS OF THIS CURVE IN MINKOWSKI SPACE

In this study, the curvatures and frames of the Smarandache curve formed by a timelike curve with unit velocity taken on a timelike surface according to the Darboux frame in Semi-Euclidean space were created. Then, the image curve on the screw surface was obtained by applying translational and rotational motion (screw motion) using the quaternion multiplication under the same frame of this curve..Finally, the curvature and torsion of the obtained image curve were calculated