## Newton's Second Law is Valid in Relativity for Proper Time

**Authors:** Steven Kenneth Kauffmann

In Newtonian single-particle dynamics, time is invariant under inertial transformations, and the particle's speed has no upper bound. In special relativity, it is the particle's proper time, rather than an arbitrary observer's time, which is inertial-transformation invariant, and it is the particle's proper-time speed which has no upper bound. Thus it is reasonable to surmise that the proper-time version of Newton's Second Law is implicit in special-relativistic single-particle dynamics. In fact, gamma times the usual special-relativistic force on a particle equals its rest mass times its proper-time acceleration, and gamma of course goes to unity in the nonrelativistic limit. Furthermore, we show that the scalar potential, the four-vector (electromagnetic) potential and all analogous such tensor potentials, as well as the metric (gravitational) potential, each produces a proper force on the particle equal to its rest mass times its proper-time acceleration. It is to be noted that the Lorentz-covariant four-vector completion of proper-time acceleration, as well as of proper force, has only three components which are mutually independent.

**Comments:** 7 Pages.

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### Submission history

[v1] 2018-12-19 23:57:10

[v2] 2018-12-22 12:25:52 (removed)

[v3] 2018-12-25 13:35:30 (removed)

[v4] 2018-12-28 06:41:49 (removed)

[v5] 2018-12-29 15:21:59 (removed)

[v6] 2019-01-31 23:25:29 (removed)

[v7] 2019-02-03 14:18:35 (removed)

[v8] 2019-02-07 21:24:59

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