Fundamental quantities in physics

I observed that the fundamental units like meter, kilogram, ampere, kelvin and candela are all indirectly dependent on one single fundamental value 'second' and each other. For example:

Meter:- 1 meter is the length that makes the speed of light in vacuum to be 299792458 when expressed in $$\rm m\,s^{-1}$$, where the second is defined in the terms of ground state hyperfine transition frequency of Cs 133

Here, Meter is indirectly dependent on second and does not really standout as a fundamental unit to me. Can anyone clarify on how it is a fundamental quantity?

• Although the definition of the metre mentions the second, this doesn't stop the former being "fundamental". In this context, that adjective is defined only by the following statement: the fundamental units are chosen so others are products of powers of them.
– J.G.
Commented Sep 16, 2021 at 16:24
• The SI units are pretty arbitrary and not necessarily independent from each other. Now they are based on fundamental constants, but remain still arbitrary (see mole and candela). Light intensity can be easily defined from the kg, m and s, but candela is defined in such a contrived way that is not even useful for most scientific endeavors. Commented Sep 16, 2021 at 17:20

As already pointed out by the quote in your questions the SI-Units are nowadays defined by fixing physical constants in order to avoid artifacts due to the reliance on real-world physical samples. For example, keeping a meter e.g. as some rod that is "one meter long" is imprecise as there are always measurement errors on the measurements of the rod and the rod could change its shape with time (e.g. through corrosion, etc.).

Now your question is what classifies the meter as a fundamental unit. The short answer is nothing. As already demonstrated within your question you could as well define the velocity to be "fundamental" and derive length from the fundamental "velocity" unit and the "time" unit.

This counts for all the SI-Units, the important thing is that you need a set of units by which you can express all other units.

In terms of the current SI-Units, you can write the unit $$[Q]$$ of every physical quantity $$Q$$ in terms of the SI-Units (m, s, kg, A, K, mol, cd)

$$[Q]=\text{m}^\alpha\ \text{s}^\beta\ \text{kg}^\gamma\ \text{A}^\delta\ \text{K}^\epsilon\ \text{mol}^\zeta\ \text{cd}^\eta\$$ with $$\alpha,\beta,\gamma,\delta,\epsilon,\zeta,\eta\in\mathbb{Z}$$.

It is now convention that we say we use the length instead of the velocity as a fundamental unit.

• Shouldn't the exponents be in $\mathbb{Q}$ if not $\mathbb{R}$?
– Ian
Commented Sep 17, 2021 at 2:23
• @Ian Do you have a particular example in mind? Would like no know your thoughts on that in more detail. I was thinking of all "derived" units e.g. for the force $F$ we would have $[F]=kg^1\ m^1 s^{-1}$ and therefore $\alpha=1$, $\beta=-1$, $\gamma=1$ and the rest is zero. In am currently not aware of some "squareroot" of a fundamental unit of an exponential. Commented Sep 17, 2021 at 15:13
• The first thing that came to mind from my own work is the square root of the ordinary diffusion coefficient, which has units $\mathrm{m} \mathrm{s}^{-1/2}$. You might argue that the meaningful quantity is that multiplied by the square root of a time, but if you don't already have the time scale in mind then this square root is a meaningful quantity in itself (since it tells you how the length scale depends on the unknown time scale). See also physics.stackexchange.com/questions/13245/… for even some irrational cases.
– Ian
Commented Sep 17, 2021 at 16:02

It's kind of like building blocks. They define a unit for time (second) based on an unchanging physical frequency. That gives us time. Then they add to that the speed of light (m/s), which gives us all kinematic units (m, m/s, m/s^2, etc). Then if we add Planck's constant, which has kg, we can define mass, force, energy, momentum, etc. Electric charge is a separate class of thing, so they need another physical constant (electron charge) to create a new unit, Coulomb. That gives you Amps (C/s), Volts (J/C), Tesla, Ohms, etc. This covers vast majority of situations, but Temperature is a new thing that we need a new constant for, Boltzmanns constantly (J/K), which defines Kelvin. Mole is not really a new physical quantity, just a number, but it's useful for chemists so they define it. Finally, candela is defined based on the light that a human eye can perceive. Not really a physical property of the universe, but a useful thing for calibrating lighting and sensors, which is after all why we created units in the first place. Hope that helps a little?

What you have stated is not a definition; it is a redefinition. In any unit system, we first need to assign values to some fundamental quantities which we know very well about, to begin with. Then we can work backward and assign values to all other quantities such that they are consistent with our fundamental quantities. Now the fundamental quantities need to be easily accessible and should be the same for any observers.

Meter was initially defined in terms of the length of the earth's quadrant (which was later engraved on a Pt-Ir rod and circulated across the globe). The Second was defined in terms of the time period of rotation of the earth. It was from this, that we assigned the value of, say, the velocity of light to be 299792458. Note that both of these are not permanent, and we cannot rely on them forever.

Now, since we found that the speed of light is invariant for all inertial frames, it's convenient to make it into a fundamental unit. We can make extremely accurate measurements of the speed of light, and still get the same value anywhere else in the universe (provided you are in an inertial frame, of course). We also have very accurate ways to express one second (in terms of the vibration of the Cesium-133 atom).

So now since we have sufficiently accurate as well as reliable measurements of both time and velocity, it's logical to express length in terms of both of them. But that doesn't mean that length is not a fundamental quantity-which it is. We just 'redefined' it using velocity and time.