Problem — SI without the second

Problem

SI without the second. Imagine removing the second from the SI base quantities. Which fundamental quantity should replace it, and why? Discuss which derived quantities would inevitably be affected by the disappearance of a time reference.

Solution

Step 1 — A “clock” is needed regardless. The second is today defined by fixing the frequency of the hyperfine transition of caesium-133. Frequency and time are each other’s inverse ($f = 1/T$): fixing a frequency is equivalent to fixing a time. If we remove the second, the most natural choice is to promote to fundamental a reference frequency — for instance precisely that of the caesium atomic transition (or another stable, reproducible atomic transition).

Step 2 — Why a frequency specifically. A good fundamental standard must be universal, stable and reproducible everywhere. Atomic transitions satisfy these requirements: every caesium atom in the universe emits at exactly the same frequency. Once that frequency $\nu_0$ is fixed, time becomes available again as a derived quantity: $t = N/\nu_0$, by counting $N$ oscillations. In practice this is exactly what an atomic clock does, just with the roles of fundamental and derived swapped.

Step 3 — Which derived quantities are affected. With a time reference gone, all quantities that contain time in their dimensions fall as well. In particular:

$[v] = \text{L}\,\text{T}^{-1}, \quad [a] = \text{L}\,\text{T}^{-2}, \quad [F] = \text{M}\,\text{L}\,\text{T}^{-2}, \quad [E] = \text{M}\,\text{L}^2\,\text{T}^{-2}, \quad [P] = \text{M}\,\text{L}^2\,\text{T}^{-3}$

Velocity, acceleration, force, energy, power, momentum, frequency itself: none would be definable any longer without a time standard. What remains unaffected are the few purely geometric or mass-based quantities (length, area, volume, density, provided the kilogram also remains fundamental).

$\ev{\text{replacement: a reference frequency (e.g. the caesium transition)}}$

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Topics: Grandezze fisiche e misura