A rotation matrix can be generated by a unit timelike quaternion in the three dimensional Minkowski space. In this paper, we correspond columns of this rotation matrix to Frenet vectors of a nonlightlike curve and we find a reformulation of Frenet’s formulas of nonlightlike curves in terms of the timelike quaternions in the Minkowski 3-space. Moreover, we do same things for the parallel transpo...

The study of the flows curves is one most fascinating research areas in differential geometry. In this paper, we investigate geometry timelike according to quasi-frame Minkowski space R2,1 (In refer these as “quasi-timelike curves”). We evolution quasi-timelike using velocity functions and obtain necessary sufficient conditions for inextensibility. Additionally, explicit forms time equations qu...

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vecto...

The aim of this paper is to present some rigidity results for asymptotically simple spacetimes. By a spacetime we mean a smooth connected time oriented Lorentzian manifold (M, g) of dimension ≥ 3, having signature (−+ · · ·+). We use standard notation for causal theoretic notions, e.g., for A ⊂ M , I(A,M), the timelike future of A in M (resp., I(A,M), the timelike past of A in M), is the set of...

Fermi Transport is useful for describing the behaviour of spins or gyroscopes following nongeodesic, timelike world lines. However, Fermi Transport breaks down for null world lines. We introduce a transport law for polarisation vectors along non-geodesic null curves. We show how this law emerges naturally from the geometry of null directions by comparing polarisation vectors associated with two...

AbstractIntegrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2– 3– dimensional Minkowski space. The derivation uses a Lorentzian version geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation nonlinear Schrodinger (NLS) Euclidean space, respectively. In 2–dimensional timelike/spacelike curve shown defocusi...

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a time...

Closed timelike curves are among the most controversial features of modern physics. As legitimate solutions to Einstein's field equations, they allow for time travel, which instinctively seems paradoxical. However, in the quantum regime these paradoxes can be resolved, leaving closed timelike curves consistent with relativity. The study of these systems therefore provides valuable insight into ...