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.


83A05 Special relativity
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
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[1] V.I. Arnold, Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1989.
[2] C. Mabenga, R. Tshelametse, Stopping oscillations of a simple harmonic oscillator using an impulse force, Int. J. Adv. Appl. Math. and Mech. 5(1) (2017) 1-6. · Zbl 1416.34012
[3] B.P. Shah, Bound state eigenfunctions of an anharmonic oscillator in one dimension: A Numerov method approach, Int. J. Adv. Appl. Math. and Mech. 2(2) (2014) 102-109. · Zbl 1359.65130
[4] G. Aruna, S. Vijayakumar Varma, R. Srinivasa Raju, Combined influence of Soret and Dufour effects on unsteady hydromagnetic mixed convective flow in an accelerated vertical wavy plate through a porous medium, Int. J. Adv. Appl. Math. and Mech. 3(1) (2015) 122-134. · Zbl 1359.76331
[5] A.A. Ungar, Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces. Kluwer, New York, 2002. · Zbl 0972.83002
[6] K. Re¸bilas, Lorentz-invariant Three-vectors and Alternative Formulation of Relativistic Dynamics, Am. J. Phys. 78 (2010) 294-299.
[7] L. Hong, Reestablishing Relativistic Dynamics Theory as a Logical System on the Basis of Newtonâ ˘A ´Zs Second Law and Relativity Principles, Phys. Essays 18(4) (2005) 467-476.
[8] R. U. Sexl, H. K. Urbantke, Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer, Wien, 2001. (p. 11) · Zbl 1057.83001
[9] A.A. Ungar, Extension of the Unit Disk Gyrogroup into the Unit Ball of any Real Inner Product Space, J. Math. Anal. Appl. 202(3) (1996) 1040-1057. · Zbl 0865.20055
[10] W. Rindler, Introduction to Special Relativity. Clarendon Press, Oxford, 1982. (p. 102). · Zbl 0507.70001
[11] M. Tsamparlis, Special Relativity. Springer-Verlag, Berlin-Heidelberg 2010. (Ch. 9.2, p. 281).
[12] E.F. Taylor, J.A Wheeler, Spacetime Physics. W.H. Freeman and Company, San Francisco, 1992. (Ch. 11, p. 108).
[13] B.F. Schutz, A First Course in General Relativity. Cambridge University Press, New York, 2009, (Ch. 2.4). · Zbl 1173.53002
[14] C. Møller, The theory of Relativity. Oxford University Press 3rd ed., Oxford, 1955. (Chapter III, p.74) · Zbl 0068.21901
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