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Dimensional analysis in physics is based on the idea that only commensurable quantities can be added, subtracted, equated or otherwise compared. The five dimensions which appear in dimensional analysis are: length, time, mass, temperature and electric charge (or current, if you prefer).

I am well acquainted with how to use dimensional analysis. However, it recently occurred to me that I am not quite certain of the origin of the above five physical dimensions and why they are incommensurate.

What, then, determines the existence of a particular physical dimension? Is it an empirical question? It must be to some extent, because the dimension of say electric charge was not apparent before this phenomenon was discovered.

Is it purely an empirical question? Could we somehow discover that two quantities, say length and mass, which are thought to be incommensurate could somehow be commensurable? It seems obvious that they are not commensurable, but I do not know how to prove it.

It should be noted that there are quantities, energy and torque for example, which do have the same physical dimension, but which might superficially appear incommensurate since they refer to seemingly different quantities. Thus, I think that just because two quantities "obviously" seem to refer to different phenomena, one cannot necessarily conclude that they are incommensurate.

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  • $\begingroup$ What about electromagnetic waves (eg, light)? $\endgroup$
    – Barry Carter
    Commented Aug 11, 2022 at 13:57
  • $\begingroup$ @BarryCarter Well, what about them? I'm not sure what you're getting at. $\endgroup$
    – Quercus Robur
    Commented Aug 11, 2022 at 14:04
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    $\begingroup$ Oh, I was just adding another measureable thing to your list. If I were to answer this question, I'd talk about things like the speed of light connecting length and time, blackbodies connecting mass and "temperature" (which should really be measured as total heat capacity), and so on $\endgroup$
    – Barry Carter
    Commented Aug 11, 2022 at 14:10
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    $\begingroup$ You forgot two more things in your base SI units list - mole (amount of substance) and candela (luminous intensity). As about why they are as they are, it can be helpful to read SI units development history. It probably it is as it goes,- units were added and described naturally at that time. For example, metre was defined as one ten-millionth of the distance from the north pole to the equator. Another thing that base SI units set should be minimized, so that every other unit is derived from base units. $\endgroup$
    – Agnius Vasiliauskas
    Commented Aug 11, 2022 at 14:52
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    $\begingroup$ @BarryCarter I think the point by Agnius was rather that there are (infinitely) many more combinations than 42 of the SI units. In terms of e.g. mass [M], length [L] and time [T], a general quantity would have the dimension M^aL^bT^c, for arbitrary integer constants a,b, and c. The corresponding SI unit would simply be (kg)^a * (m)^b * (s)^c. There is no limit to how many such combinations you could make. $\endgroup$
    – Quercus Robur
    Commented Aug 11, 2022 at 15:21

2 Answers 2

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The number of physical dimensions is a matter of definition and personal choice in my opinion, and can be increased or diminished. Maxwell, following Gauss, believed that the number of dimensions of quantities was definitely three, (temperature not included) and Electromagnetism was defined to make this work: the force law between charges was $$F=\frac{q_1q_2}{r^2},$$ so unit charge was defined to be that charge which acts on an equal charge with unit of force at unit distance. Compare the situation in gravity, where Newton defined $$F=\frac{Gm_1m_2}{r^2},$$ where the units of mass, length and time are already defined, and the cost is a new constant of nature, $G$. Later we did the same in electromagnetism, putting $4\pi\epsilon_0$ into the denominator, a new constant of nature, and defining charge independently. So you can increase the number of dimensions by introducing new constants. Consider for example modelling the atmosphere. The horizontal length scale is thousands of kilometres, and the vertical just a few kilometres. So you will use rectangular grid boxes with very small height compared to horizontal size, effectively defining different units of distance in the vertical and horizontal. All equations in the model must agree in both length dimensions. If you need the distance between two grid boxes it will be, in horizontal units, $\sqrt{\delta x^2+\delta y^2+\delta z^2/k^2}$, where $\delta x$, $\delta y$ and $\delta z$ are the number of grid boxes separating the points in each direction, and $k$ is a defined constant, horizontal distance per unit vertical distance.

Conversely, you could view the defined value of the speed of light as a reduction in the number of dimensions. The most recent change in the definition of SI base units has had a complex effect on how many dimensions there really are any more: you could view time as the only remaining dimension, because the speed of light, Boltzmann's constant, Planck's constant and the charge of the electron all have defined values.

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  • $\begingroup$ Thank you for your interesting answer! I will hold off a bit on accepting it to see if there will be further answers. The example of the force law is very nice and illustrates the arbitrariness in whether or not we introduce new dimensions or news constants of nature. It would seem, however, that one or the other would have to be introduced. I guess then that the number of dimensions (or fundamental constants), is still an empirical question determined by the kind of phenomena we have discovered. $\endgroup$
    – Quercus Robur
    Commented Aug 11, 2022 at 15:14
  • $\begingroup$ Everything is good up until your last paragraph. The number of dimensions in the SI system is, as you said earlier, a matter of definition. They have defined the SI system as having 7 base dimensions, one for each base unit. See section 2.3.3 of the SI brochure bipm.org/documents/20126/41483022/SI-Brochure-9.pdf/… Now, you could make a new unit system that was identical to the SI system except that it had only one base dimension, but it would not be SI, by definition $\endgroup$
    – Dale
    Commented Aug 11, 2022 at 16:50
  • $\begingroup$ I agree about SI, but physics is wider than that. Particle physicists use units in which $c=\hbar=1$ which gives a single mechanical dimension. I think one of the defined constants now is Avogadro's number, so a mole is just a shorthand for a particular multiplier. The concept of dimension is slippery! $\endgroup$
    – CWPP
    Commented Aug 11, 2022 at 18:20
  • $\begingroup$ Similarly what's the difference a charge and a number of electrons if $e$ has a defined value? $\endgroup$
    – CWPP
    Commented Aug 11, 2022 at 23:01
  • $\begingroup$ @CWPP yes, physics is wider than the SI, but the SI is not wider than the SI. If you want to use non-SI units that is fine. You can even use personal units that are similar to SI except for a change in the dimensions (for me I frequently use Dale-SI units which are not SI because the unit “rad” is dimensionful with a dimension of angle). But when you choose to use SI units then the number of dimensions is crystal clear and not slippery at all. I merely object to using the term “SI” to apply to non-SI units, including units that are non-SI due to a non-SI dimensionality choice $\endgroup$
    – Dale
    Commented Aug 12, 2022 at 11:28
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To address a separate point in the question, units of angle do not have to balance in an equation so it doesn't work like a dimension for checking purposes. Take $\omega$ (radians per second). That doesn't balance in the case of SHM: $\omega^2=k/m$.

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