# Instantaneous and Asymptotic Conservation Laws for Classical Relativistic Mechanics of Interacting Point Particles

## Abstract

The present article consists of two parts. First, we assume that the conservation laws for energy and linear momentum are valid and that these quantities are the sums of the energies and linear momenta of the individual particles, i.e., that there is no interaction energy and no interaction momentum. We then repeat a familiar argument and show that there can then be no interaction between the particles, that is, their world lines are straight. In the second part of the paper the interaction quantities for energy, linear and angular momenta, and the center-of-mass law are derived for the equations of motion proposed in an earlier paper. We then study these interaction quantities in the asymptotic region of collision processes, in order to arrive at asymptotic conservation laws. we find, in agreement with the earlier paper, that the interaction energy and the linear interaction momenta vanish asymptotically. This, however, is not true in general for the interaction angular momenta and center-of-mass motion. Asymptotic interaction angular momentum is present in all theories, such as classical electrodynamics, which lead to inverse-square-law forces.

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## Reference

- 1.See, for instance, P. G. Bergmann,
*The Special Theory of Relativity*, Handbuch der Physik IV (Springer, Berlin, 1962), p. 147. The theorem discussed here has a purely kinematical basis. Several “no interaction” theorems with a dynamical origin have appeared in the recent literature: D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963); D. G. Currie, J. Math. Phys. 4, 1470 (1963); J. T. Cannon and T. F. Jordan,*ibid*. 5, 299 (1964); H. Ekstein, Consistence of Relativistic Particle Theories, Université d’Aix-Marseille, (1964, unpublished), H. Leutwyler, Nuovo Cimento 37, 556 (1965).Google Scholar - 2.That one must introduce interaction energy and linear momentum for classical relativistic mechanics has been pointed out also by L. Brillouin, Compt. Rend. 259, 2361 (1964). For a simplified derivation of the conservation laws from invariance postulates, see W. Macke, Forsch. Fortschr. 39, 193 (1965).Google Scholar
- 3.H. Van Dam and E. P. Wigner, Phys. Rev. 138, B1576 (1965). The status of the classical theory of interacting point particles, prior to this article was summarized by P. Havas, in Statistical Mechanics of Equilibrium and Non-equilibrium, edited by J. Meixner (North Holland Publishing Company, Amsterdam, 1965). Also see D. G. Currie, Phys. Rev. 142, 817 (1966).Google Scholar
- 4.An interaction of this type has been proposed by P. Havas and J. Plebanski, Bull. Am. Phys. Soc. 5, 433 (1960).Google Scholar
- 5.For
*c’,k(p)=e;ek(d/dp’)5*(p^{2}), one obtains the equations of motion proposed by J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 21, 425 (1949).Google Scholar - 6.The presence of interaction angular momentum in collision processes has been recognized by a number of authors. It seems, however, that one has, so far, assumed this angular momentum to vanish asymptotically, i.e., as the particles separate. See, for instance, H. Yilmaz, Introduction to the Theory of Relativity and the Principles of Modern Physics (Blaisdell Publishing Company, New York, 1965), pp. 56–57. Also, see J. W. Dettman and A. Schild, Phys. Rev. 95, 1059 (1954).Google Scholar