Four-Dimensional Formulations of Newtonian Mechanics

First we reproduce section 2 from Michael Friedman’s “Simultaneity in Newtonian Mechanics and Special Relativity” in *Foundations of Space-Time Theories* (ed. Earman et al., UMinn, 1977), p.405-407. Then we provide the dual.

According to the space-time point of view, the basic object of both our theories is a four-dimensional manifold. I shall use R^{4}, the set of quadruples of real numbers, to represent the space-time manifold. Both theories agree that there is a natural system of straight lines defined on this manifold. If (*a*_{0}, *a*_{1}, *a*_{2}, *a*_{3}), (*b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}) are two fixed points in R^{4}, then a straight line is a subset of R^{4} consisting of elements (*x*_{0}, *x*_{1}, *x*_{2}, *x*_{3}) of the form

(1) *x _{0} = a_{0}r + b_{0}*

*x*

_{1}= a_{1}r + b_{1}*x*

_{2}= a_{2}r + b_{2}*x*

_{3}= a_{3}r + b_{3}where *r* ranges through the real numbers. A curve on R^{4} is a (suitably continuous and differentiable) map σ: R → R^{4}. Such a curve σ(*u*) is a geodesic if and only if it satisfies

(2) *x _{0} = a_{0}u + b_{0}*

*x*

_{1}= a_{1}u + b_{1}*x*

_{2}= a_{2}u + b_{2}*x*

_{3}= a_{3}u + b_{3}where (*x _{0}, x_{1}, x_{2}, x_{3}*) = σ(

*u*) and the

*a*

_{i}and

*b*

_{i}are constants. So if a curve is a geodesic its range is a straight line. Note that the geodesies are just the curves that satisfy

(3) *d ^{2}x_{i}*/

*du*= 0

^{2}*i*= 0, 1, 2, 3.

The importance of straight lines and geodesies is due to the fact that both theories agree that the trajectories of free particles are straight lines in space-time. So we can represent such trajectories as geodesies in R^{4}.