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

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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### References:

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