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When comparing the Wikipedia articles on the International System of Units, the Planck unit system, and the geometrized unit system one question arises: can a physical quantity be of different physical dimension depending on the system of measurement?

What triggers this question is the table Geometrical Quantities in the geometrized unit system article: geometrical quantities

Contrarily to the Planck unit system in which a length stays a length, a "time" and a "length" in geometrical quantities have the same dimension [L].

Questions: Is it really the case? Or is that a "time" does not really exist in geometrical quantities? If that's the case then what would be a more rigorous phrasing corresponding to this table? What does this table actually says?

Any clarification is very welcomed.

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can a physical quantity be of different physical dimension depending on the system of measurement?

Yes, most definitely! The dimension of a physical quantity is a matter of convention which is established by the system of units. It is not a fundamental physical fact of the universe.

You have discovered this fact in the context of geometrized units which have only a single physical dimension, length. Geometrized units are the most extreme example of this, but is not commonly used so it is relatively obscure. However, the various “cgs” systems of units are commonly used but also have surprising variations in the dimensionality of electromagnetic quantities.

For example, the statcoulomb is the unit of charge in the cgs “Gaussian” units. Although the coulomb is the SI unit of charge, there is no direct conversion possible between the two. The Coulomb has dimensions of charge, Q, but the statcoulomb has dimensions of $L^{3/2} M^{1/2} T^{-1}$.

As a result the equations of electromagnetism are different in SI than in Gaussian units. In particular, Coulomb’s law in Gaussian units is $$F=\frac{q_1 q_2}{r^2}$$ in contrast to the usual expression in SI units $$F=\frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}$$

So the dimensionality of the physical quantity is a convention that is specified by the system of units used, and that convention will alter the mathematical form of the laws of physics when expressed in those units. At least regarding the presence of dimensionful constants.

Is it really the case? ... What does this table actually says?

Yes, that is really the case. That table actually says what it appears to be saying at face value. The physical dimensions of the geometrized units is different from the dimensionality of the corresponding SI quantities.

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  • $\begingroup$ Can one not view it conversely? That the dimensionality of the quantity determines the units $\endgroup$ – N. Steinle Nov 26 '18 at 2:35
  • $\begingroup$ I don’t think so. How would you determine the dimensionality of a quantity in the absence of the units? $\endgroup$ – Dale Nov 26 '18 at 2:39
  • $\begingroup$ $\dfrac{1}{\alpha}$ is 137 units of 1 (plus some fraction). $\endgroup$ – safesphere Nov 26 '18 at 3:31
  • $\begingroup$ Consider a quantity called "distance." This quantity has dimension of "length." I can choose to measure distance in whatever units I choose, no? This is why I cannot add quantities of different dimension, but I can add quantities of different units. Perhaps I'm missing your point entirely. $\endgroup$ – N. Steinle Nov 26 '18 at 4:06
  • $\begingroup$ How do you know that distance has dimensions of length? Or more relevantly, how do you know the dimensions of charge? In SI it has dimensions of $IT$, in Gaussian units $M^{1/2}L^{3/2}T^{-1}$, in Stoney units it has dimensions of $Q$. So how can you simply look at charge and say what dimensions it has? $\endgroup$ – Dale Nov 26 '18 at 4:16

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