Physics "Friday" #11 [Opinion]: Okay so I have a few more problems with SI Units
Preamble: Now what Missy?
Education Level: Primary School (Y5/6)
Topic: Measuring Systems (Metrology)
Ok so this was going to be my necessary follow up to my first opinion post, Is Fahrenheit the better temperature scale?
I was meant to save it for the next time I had too much stuff to deal with on Friday and needed to write something easier. In fact, that's going to be how I do opinion posts - when I don't need to do as much a-thunkin'.
Regardless, there are of course other problems with the SI units system that many people point out. This is basically me doing it myself.
What are the SI Units?
During the french revolution, there were several attempts to metricise the existing unit system. Generating standardised units on length, volume, weight, temperature ... they even tried to decimalise time.
These units were based off physical constants:
The metre was one 10 millionth the distance from the north pole to the equator
The litre was the volume taken up by one kilogram of water
Celsius was based on the boiling/melting points of water
A kilogram was based on the weight of a specific Platinum-Iridium Alloy called the IPK (International Prototype Kilogram)
Now obviously, these weren't perfect constants. The earth's radius changed, the boiling point of water depends on the pressure of the environment, and the IPK can vary based on surrounding conditions.
This was fixed in 2019, when every SI measurement unit was defined using a universal constant rather than physical objects like a mass or the earth:
Hyperfine transition frequency of Caesium
Speed of light
Planck Constant
Charge of the Electron
Boltzmann Constant
Avogadro's Constant
Luminous Efficacy of 540 THz radiation
This gives our seven SI units:
Time (Second)
Distance (Meter)
Mass (Kilogram)
Charge (Columb)
Temperature (Kelvin)
Amount (Mole)
Luminous Intensity (Candela)
Problem #1: Unit-less units
There are several units that come from mathematics that appear to be unitless, like:
Angle (Radians)
Solid Angle (Steradians)
Amount (Number count)
We use this all the time like in lumens (candela steradians), angular velocity (radians per second), and number density (particles per cubic metre).
Wait ... amount was already listed as an SI quanity - it was a mole wasn't it?
That's exactly it. A mole is just defined as 6.022×10²³ objects of something. This is where we experience our first problem.
Why is radian and steradian not considered a unit in it's own right but a mole is?
Radians are defined as being one metre per metre. The angle required to create an arc with a length of 1 metre given a radius of one metre.
Image Credit: BYJU's
This comes from the formula S=Rθ where circular arc length (S) is equal to the radius (R) multiplied by angle in radians (θ).
The problem with this, is that the radian is still an arbitrary dimension. We could've chosen our standard angle unit to be a degree, and all we need to do is change the definition of the sine function and the above arc length definition:
S=kθR were kθ is dimensionless and k = 1 rad⁻¹
It's almost as if, while the radian is dimensionless, it still behaves and acts like a unit. Much like how we can redefine everything and make the unit for length be a foot instead, we can do the same with angle.
You may object, as S=Rθ looks a lot more fundamental than adding an extra unneeded constant. But it's like if we were to use natural units, where the speed of light is equal to 1, making E = mc² actually E = m.
This doesn't mean that m is somehow 'fundamental' when we define it in terms of energy. All we've done is redefine our unit system in a way that makes a few constants equal to one.
The same thing goes for radians.
A little side-note on Intensity
You could argue that this also applies to relative intensity units like Magnitude or Bels. However, I'd argue that they don't count as their unit systems are defined as being relative. The strength of a signal or luminosity is just choosing a unit system of power.
For example, relative magnitude is based on the luminosity of the sun. Sound/Signal amplitude is based on watts or volts.
Problem #2: Redundant Units
Many of the quantities in the SI unit system are technically redundant. Temperature can just be redefined as energy density, mass can be redefined as energy, etc.
What I'm concerned with is one unit in particular: the Candela - what exactly is it?
I often find that a lot of people attempting to explain the SI unit system often brush over this unit, even though it feels rather important given it's called 'Luminous Intensity'.
But hold on, isn't luminosity just about energy production? After all, radiation is just energy, and it's emitted as a form of energy.
A candela is equal to one lumen per steradian, where a lumen is the total luminous intensity of an object that emits light everywhere. Candelas concern luminous intensity from a specific solid angle view of the object.
Watts, the measurement of power and bolometric luminosity, measure the total (or bolometric) luminosity of an object at all wavelengths of light.
Lumens, on the other hand, represent the power produced by an object after each wavelength is passed through some filtering function that accounts for how the human eye sees light. This weighting function is determined by the ISO.
Of course, our eyes only see light within a particular range, and inside of that range, different wavelengths appear more intense to us because we pick it up more.
But this fails to recognise that this is still just a glorified unit of power. Just because it's weighted based on some function, shouldn't change what unit is necessarily uses. Hence it is technically redundant as it can be defined as a combination of other SI units - similar to radians.
Problem #3: When the unit system isn't used
I mentioned it in the last post, but there are a lot of occasions where the standard SI units aren't used. The worst culprit I feel is Astronomy, with Physics only having issues in lieu of SI vs. Natural units vs. Electron/Atomic Units. Again here's the list:
My beloved SI units
CGS Units
Whatever the fuck a Jansky is
Don't even start with natural units I can't live without big G
"Ampere in CGS units is g1/2 cm3/2 s−2"
Solar Luminosity/Mass of Sun
Angstroms (like please can we just use nanometers?)
How many Jupiters or Earths fit into this cloud of gas?
The vomit of parallax units i.e. AU, pc, Mpc, arcseconds, radians
Steradians (Solid angles can be finicky)
Logarithms, logarithms everywhere!
Hubble's constant being in km/s/Mpc but then having to turn that into Hz or per year - like can someone please acknowledged how cursed this is?
When you do Kepler's 3rd law on Mercury and realise it doesn't work (because you forgot Einstein existed) ... so no units end up working
ADUs and/or whatever you get when you deal with telescope outputs
Sidereal time, J2000, etc.
Sky Coordinates (it always depends on the telescope mount)
(the last two are new entries into the list I forgot!)
A lot of these units are very, very, annoying to handle. Often at times because they are just so unnecessary. We have scientific notation for a reason - so why are we using Jansky? Why ever use the mass of planets unless if we are talking about specifically exoplanets?
It can especially be annoying when in astronomy course subjects, unit conversions make up like 50% of the work and 50% of the error in calculations.
And the biggest problem, of course, is the CGS units system. I hate it. So much. From what I know CGS is used simply because it appears as more correctly "scaled" for a lot of astronomical processes.
However, the problem is that it just adds extra conversion factors into every equation. Now I have to remember big G in both CGS units as well as in SI or solar units etc. And it doesn't jive with a lot of other astronomical units.
J2000 is also rather annoying - why are we using the Julian calendar at all? Shouldn't we try and strive for using a more accurate year instead? Because what happens is that every four years the calendar shifts by a day.
The "Ampere in CGS units is g1/2 cm3/2 s-2" is a direct quote from one of my professors. And it also makes very little sense - because it says that the coulomb and ampere units are actually redundant.
However, CGS units aren't usually the only thing we use. Sometimes we put things in Gaussian CGS units, where we define the coulomb's constant (kₑ) as being 1. Thus, similar to natural units, we can define a statCoulomb in terms of our three base units:
1 statC = 1 g1/2⋅cm3/2⋅s−1.
Now this is where we find our ampere definition! But wait ... this is materially different to the real ampere. In fact, we can determine the difference between coulombs and statcoulombs:
So they are different! And thus, after several paragraphs, my lecturer is wrong and they shouldn't've used the Ampere but a different stat ampere.
So what would I do?
Well, the big problem with all unit systems is that they are just equivalent to eachother dimensionally.
Because of all the seperate equations surrounding each of the SI units, we can define any SI unit in terms of any other SI unit combination just by setting certain constants to one.
So there's quite a lot of units that we could me missing.
But here's how I would do it ...
There are five unique SI units:
Distance - Metre
Time - Second
Mass - Kilogram
Charge - Coulomb
Temperature - Kelvin
There are also three dimensionless/special units:
Amount - Mole
Angle - Radian
Luminosity - Lumen
These three units are special as they can be expressed as dimensionless constants or in terms of other units, however we want them to be expressed in our own specific units.
Solid Angle is just determined from square radians where 1 sr = π/4 rad² (ratio of square and circle of same diameter).
The constants which define these unique units are:
Hyperfine transition frequency of Caesium
Speed of light
Planck Constant
Charge of the Electron
Boltzmann Constant
For the non-unique units we have:
Avogadro's Constant
Pi
Luminous Efficacy of 540 THz radiation
Conclusion
This post did end up taking longer than I expected ... much longer. Nevertheless it was fun to do!
As stated in the start, opinion posts are for those Fridays where things aren't as easy as spending 2-3 hrs writing a post. Because when you want to argue a point you have a little more passion behind it.
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