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

**Authors:** Steven Kenneth Kauffmann

In Newtonian particle dynamics, time is invariant under inertial transformations, and speed has no upper bound. In special relativity, it is the observed particle's proper time, rather than the arbitrary observer's time, which is inertial-transformation invariant, and it is the particle's proper-time speed which has no upper bound. It is thus perhaps not surprising the Newton's Second Law is also valid in relativistic particle dynamics presented in terms of the particle's proper time. This follows from the special-relativistic dynamical principle that the time derivative of special-relativistic momentum is equal to the applied force, provided proper force is defined to have an additional factor of gamma. The four-vector fully Lorentz-covariant completion of proper force is obliged to have zero contraction with the particle's proper four-velocity. A special-relativistic particle in either a scalar or a four-vector (electromagnetic) potential is shown to adhere to the proper-time Newton's Second Law, and when it is consistently taken into account that a metric (gravitational) potential modifies the rate of change of a particle's proper time with observer time, that adherence is shown to hold for metric potentials as well.

**Comments:** 6 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

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