Abstract
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.
Highlights
A search of literature indicates that there is almost no concrete computations of entropy, laws of horizon dynamics and energy in the case of Minkowski spacetime E42
By similar argument volume of a unit vector field X is described as the volume of the submanifold in the unit tangent bundle defined by X(M ) [9]
We find energy on the particle in Frenet vector fields {T, N, B1, B2} as follows, respectively
Summary
Thanks to the arc-length, it is described Serret-Frenet frame, which allows us determining characterization of the intrinsic geometrical features of the regular curve This coordinate system is constructed by four pseudo-orthonormal vectors assuming the curve is sufficiently smooth at each point. Let K be a unit speed pseudo lightlike curve, Frenet equations are stated as the following [17]. Let K be a unit speed Cartan lightlike curve, Frenet equations are stated as the following [18, 19]. Let K be a unit speed partially lightlike curve K, Frenet equations are stated as the following [17].
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