Problem — Threshold laws

Problem

Suppose $F=ma$ did not hold for $v > 100$ m/s. How would physics organise itself? Argue the analogies and differences with how relativity relates to classical mechanics in the limit $v \ll c$.

Solution

Analogy: the correspondence principle. If $F=ma$ stopped holding above a certain speed, a more general theory would be needed, one that contains Newtonian mechanics as a limiting case for low speeds. This is exactly what happens with special relativity: the relativistic second law reduces to $F=ma$ when $v \ll c$. The correspondence principle requires that every new theory reproduce the old one in the regime where the old theory was already confirmed by experiment. In both cases physics would organise itself as a hierarchy of theories, each valid in a domain, with the more fundamental one reproducing the others in their respective limits.

Difference 1: gradual transition, not a threshold. In relativity there is no sharp threshold speed beyond which Newton “switches off”. The correction is governed by the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ which tends to $\gamma \to 1$ continuously at low speeds and grows smoothly. Deviations from Newton are always present, but become appreciable only gradually as $v$ grows. A rigid, sudden threshold at a fixed value (such as $100$ m/s) would be physically anomalous.

Difference 2: the reference-frame problem. A threshold at an absolute speed value is suspect because speed is not absolute: it depends on the reference frame. A body at $150$ m/s in one frame can be at rest in another. A law that changes form at $v=100$ m/s would violate the principle of relativity (the laws of physics must be the same in all inertial frames), unless those $100$ m/s referred to some privileged frame — a hypothesis modern physics has abandoned. The speed of light $c$, by contrast, is invariant: the same in every frame, and for this reason it can appear as a universal limit.

$\ev{\text{more general theory} \to \text{Newton (correspondence), but gradual, invariant transition, not an absolute threshold}}$

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Topics: Metodo sperimentale Skills: Analisi di casi limite e fantafisica