Local clock variables for relativistic mechanics

Overview & Synopsis

Introductory mechanics is usually built around a single coordinate time $t$, an approximation that works well at low speeds but hides the relativistic question: which clock measures the interval? This article develops a compact local-clock formulation of one-dimensional relativistic mechanics using proper time $\tau$, proper velocity $w=\dd x/\dd\tau$, rapidity $\eta$, and proper acceleration $\alpha$. In these variables several Newtonian-looking relations remain valid at all speeds. Momentum is $p=mw$, the Lorentz factor is $\gamma=\sqrt{1+w^2/c^2}$, and rectilinear proper acceleration satisfies \begin{equation*} \alpha=\frac{\dd w}{\dd t} =c\frac{\dd\eta}{\dd\tau} =c^2\frac{\dd\gamma}{\dd x}. \end{equation*} Constant proper acceleration then gives transparent all-speed analogues of the familiar introductory formulas $v=at$, $x=at^2/2$, and $K=mv^2/2$. The formulation avoids some conceptual and numerical difficulties associated with coordinate velocities close to $c$, gives a simple way to compare accelerators and colliders, and clarifies what an onboard accelerometer measures. The same variables extend beyond free-float frames: on any flat patch of curved spacetime, or in any accelerated frame, Newton's second law survives in local form once one admits \emph{geometric forces} --- gravity, centrifugal, and inertial effects, which accelerometers cannot feel --- alongside the proper forces they can. Each geometric force is locally an aging gradient, $g=c^2\,\dd(\ln\gamma)/\dd h$, an identity derived here independently for Schwarzschild gravity, a rotating habitat, and a uniformly accelerated frame. The result extends Newton's laws to the non-free-float frames --- shell laboratories, habitats, rockets --- where measurements are actually made, bridging Newtonian mechanics, special relativity, and the equivalence-principle language of local frames at a level suitable for advanced introductory or intermediate undergraduate courses.