Mathematical model of relativistic 3-acceleration. (English) Zbl 1465.83003

Celakoska, Emilija; Chakmakov, Dushan

Int. J. Adv. Appl. Math. Mech. 6, No. 2, 14-19 (2018).

Summary: The standard Newtonian acceleration \(\overrightarrow{a}\) is one of the rare physical quantities that does not have corresponding relativistic analog. We introduce a relativistic 3-acceleration \(\overrightarrow{a}_{\text{rel}}\) derived directly from relativistic velocity addition law. Actually, \(\overrightarrow{a}_{\text{rel}}\) is shown to be a 3-acceleration in an instantaneously comoving rest frame represented in terms of coordinates of the corresponding 4-acceleration \(A\).
The relativistic 3-acceleration \(\overrightarrow{a}_{\text{rel}}\) possesses some interesting features which enable to express some of the relativistic dynamic quantities in a more convenient way. Particularly, relativistic 3-force takes convenient Newtonian form \(\overrightarrow{f}_{\text{rel}} = m\overrightarrow{a}_{\text{rel}}\) from which the relativistic formula for energy naturally arises. The general idea behind \(\overrightarrow{a}_{\text{rel}}\) has been implicitly exploited in other papers, commonly through the physical concept of 3-force and the corresponding equation of motions. However, we present a simple mathematical model which gives explicitly the true \(\overrightarrow{a}_{\text{rel}}\) origin and different ways of its derivation in a mathematically more compelling way.

MSC:

83A05	Special relativity
70H40	Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics

Keywords:

special relativity; 4-velocity; hyperbolic operations

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