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First Principles of PhysicsLimits & Uncertainty · Rule 35

Limits & Uncertainty

Rule 35Chapter V

Every approximation has a domain

Why this rule exists

Every physical law is an approximation valid within a range of conditions, because each is built on idealizations — point masses, weak fields, low speeds, linear response — that hold only over some domain. Outside that domain the neglected terms grow and the law breaks. A theory is therefore incomplete without a statement of where it applies.

First Principles of Physics

In practice

For any formula, write down its domain of validity alongside it: the small parameter assumed tiny, the regime it was tested in. Before applying it, check that parameter is actually small here; if not, keep the next term or switch theories. Newtonian mechanics for v ≪ c, geometric optics when wavelength ≪ aperture — know the boundary before you cross it.

When it doesn't apply

Even our deepest theories are effective theories with unknown domains of breakdown, so no formula's range is guaranteed complete. Near a domain's edge the transition is gradual, not sharp; corrections grow smoothly rather than switching on.

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