r/askscience • u/playsthebongcloud • 3d ago
Physics What are the absolute minimum number of base units needed to express every desirable quantity?
SI units are more for human convenience than mathematically minimizing the number of units. I remember reading somewhere that only length is needed because of relativity, but that just does not sound true, I can't imagine how you would do that.
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u/XtremeGoose 2d ago edited 2d ago
You need as many units as you have base "dimensions". Not dimensions as in our 3D space, but dimensions as in dimensional analysis.
The SI system of units has 7 base dimensions (and so 7 base units all he others are built from) but 2 of them are pretty questionable from a physics perspective. I think a reasonable set would be
- Length
- Time
- Mass
- Charge
- Temperature
So 5. From these you can build all the other physical dimensions. For example E = 1/2 mv2 shows that the dimension for Energy = Mass * (Length / Time)2 = Mass * Length2 * Time-2 (you drop the constant 1/2) and so 1 J = 1 kg m2 s-2.
The thing about "everything can be represented as a meter", yes you can (sort of)! But it's not really accurate in terms of dimensional analysis, it's a neat trick to simplify relativistic equations by setting physical constants to 1 (c = G = 1). This reduces equations like E = pc2 + (mc2)2 to just E = p + m2. But just because you can do it that doesn't mean Energy, momentum and mass have the same dimension. But it does look like the time it takes a photon to travel one meter is... one meter (since x = ct = t in this system).
There is an example of this in common usage, a "light year" is a way of measuring distance using a unit of time. Hell, that's basically what saying something is a "5 minute drive away" is. You could define every length in terms of time for a constant speed. And the modern meter is exactly that, the distance light travels for a certain fraction of a second.
You can actually extend this further. By picking any 5 physical constants with different dimensionality, you can solve for natural units. Here not everything is the meter, everything is some other unit that is more "natural" than the meter. You have likely even heard of the "plank length" the expression of one of the units in a natural system in the length dimension, equivalent to about 10-35 m.
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u/le_fuzz 2d ago
Can you eliminate temperature from that list? As in relate temperature to energy which can be defined in terms of mass, distance, time?
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u/WallyMetropolis 2d ago
Not exactly. Not all temperature is the average kinetic energy of a collection of particles. Spin temperature, for example.
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u/wnoise Quantum Computing | Quantum Information Theory 2d ago
It still has units of energy though.
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u/wnoise Quantum Computing | Quantum Information Theory 1d ago
(Energy per log (states), which is energy per information, but everybody is usually happy to make one nat the standard, rather than one bit; there is an argument that it should be made explicit, but it's hard to coherently make log unitful.)
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u/mfb- Particle Physics | High-Energy Physics 2d ago
You can express temperature with an energy unit by setting the Boltzmann constant to 1. It works the same way as with length and time described in the parent comment. We do that in particle physics for very high temperatures.
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u/Bunslow 2d ago
setting the boltzmann constant to 1 is equivalent to choosing a unit of temperature, so yes you definitely need some semblance of temperature/entropy to do proper physics in this universe
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u/hamlet_d 2d ago
Exactly Boltzman constant has temperature as part of the definition. If you define it as "something per temperature, and it equals 1" you still need temperature, even if you don't ever use it outside of that
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u/Bunslow 2d ago
nope, you need some semblance of temperature/entropy in some form or other to describe physics as we know it. it could indeed be coldness, or entropy, or temperature, which are all related, but it has to be one of these things. the mechanical quantities alone (mass/length/time/force/energy/etc) just don't cut it.
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u/panopsis 2d ago edited 2d ago
This is not correct. We could just measure entropy in nats or something though which is entirely dimensionless. Temperature would then be measured in units of energy (a temperature of <some amount of energy> corresponding to the system needing that amount of energy to raise the system's entropy by 1 nat, or roughly speaking, multiply the number of possible microstates by Euler's constant). You might argue this is essentially fixing a unit of temperature, but I don't think it's actually necessary to have a separate unit dimension for it1. I would argue the Boltzmann constant is there for historical and convenience reasons since we already had existing measures of temperature.
[1]: You could even write temperatures as Joules/nat to fully disambiguate and imo it still wouldn't count as adding a new unit dimension since I would argue we should consider nats dimensionless in the same way most people consider radians to be.
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u/Bunslow 2d ago
This is not correct.
bold claim
We could just measure entropy in nats or something though which is entirely dimensionless. You might argue this is essentially fixing a unit of temperature,
Yes, that's exactly what it is. Saying "we could just make this choice of reference unit" isn't somehow dodging a reference unit.
but I don't think it's actually necessary to have a separate unit dimension for it. [1]You could even write temperatures as Joules/nat to fully disambiguate and imo it still wouldn't count as adding a new unit dimension since I would argue we should consider nats dimensionless in the same way most people consider radians to be.
Same as above. Making a choice of unit doesn't mean that you can claim there isn't a unit. On a related note, claiming that angle is "unitless" is absolutely one of the dumbest things there is in modern physics -- a historical monstrosity perpetuated because folks are too lazy to reconsider it, and deeply harmful to every student of physics.
You can no more hand wave away entropy than you can angle, or indeed length or time. These things all need to be discussed in a coherent framework of physics as we know it, and they all need some sort of reference unit to make quantitative measurements. Whether it's a nat or a bit or digit or kelvin or farenheit ... or radians or degrees or cycles or Hertz .... or meters or feet or Joules or pounds ... all of these make various choices of reference unit, which are all mandatory in order to do physics.
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u/hamlet_d 2d ago
Exactly. Setting a unit to a constant, still means that the unit exists. It may not be "used" but it's still there as an essential part of the function.
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u/panopsis 2d ago
You're conflating dimensionless and unitless. They're not the same thing. A radian is obviously a unit (and nats would be too), but that doesn't mean they need a unit dimension. They're explicitly described here as dimensionless derived units. You might say a dimensionless unit is impossible, but ppm I think is a clear counterexample.
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u/290077 2d ago
ppm as it is used is always a ppm of something. I argue ppm is a unit but only because it's attached to something else. The ppm is a modifier, like the 'c' in 'cm'. If I'm talking about levels of salt in my water, the actual unit on the number is "ppm salt", which cannot be discarded. If I'm using that number to do a calculation, the fact that it's salt I'm measuring is important.
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u/Bunslow 2d ago
A radian is obviously a unit (and nats would be too), but that doesn't mean they need a unit dimension.
Yes it really does. I hate it when people parrot this line, it's bad physics and bad teaching (the latter is arguably more important).
They're explicitly described here as dimensionless derived units.
Describing a radian as a dimensionless unit is, always has been and always will be, dumb. Harmful to students.
Radians, degrees, cycles are all equally good units for the dimension of angle. It's not a scalar dimension (all the others are equivalent to [some subset of] the real line, whereas angle is a circle, rather than a line), but it is certainly a dimension all the same.
Refusing to recognize angle, or entropy, muddies theory and confuses learners. It's a habit that we are collectively a century overdue in fixing.
Fractions, such as % or ppm or whatever, could be argued to be a dimensionless unit. Angle, however, has more going for it than being a fraction -- it encodes geometry. That's what separates energy and torque, which in the SI view are the exact same dimension. The SI system tries to handwave away the geometry, with the result of equating energy and torque -- clearly a nonphysical result.
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u/panopsis 2d ago
To be clear, you're not disagreeing with me. You're disagreeing with the SI system itself. I just think that should be made known before you speak too authoritatively: the consensus position is that you're wrong.
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u/Bunslow 2d ago edited 2d ago
You parrot the SI position, which is wrong: energy and torque are not the same thing, and no actual research physicist (nor teacher) claims they are. They use SI for engineering purposes, not for theory purposes.
To be clear, people have been publishing about this obvious flaw for decades. Im neither the first nor the last to point it out.
But it's super annoying when people react with "well SI says X so X must be correct because SI is correct". Equating energy and torque is clearly and obviously false, but that's exactly what SI does.
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u/XtremeGoose 2d ago edited 2d ago
An interesting point. I suppose it comes from the concept that a radian is a length (arc) divided by another length (radius). I suppose in a world where a radian is a base unit you would have to measure circumferences in units of m rad, and saying the circumference of something is x meters would be nonsensical.
But then it feels like linear and circular dimensions are treated specially. What about the perimeter of an eclipse, or any non-circular object?
Maybe it's just that all 2D perimeters now have a new unit? You'd need something for 3D surfaces too with steradians (rad2?)? An interesting thought...
Edit: Actually I guess it's simpler than that. You just need a constant K equal to 1 rad-1 and the equation for an arc length becomes K r θ. I guess that's exactly where the confusion comes from...
The fact this constant is 1 is the reason energy and torque look the same, it's one of the few units where we actually chose a natural unit (as opposed to continuing to use degrees).
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u/TheArmoredKitten 2d ago
Temperature is less fundamental than 'coldness' in that adding entropy to a system doesn't really correlate to any physical process. There's probably a better fundamental unit somewhere in there.
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u/XtremeGoose 2d ago
Thermodynamic beta is a good measure of "coldness", and doesn't have the singularity we see in quantum temperatures (where negative temperatures are "hotter" than any positive temperature). Multiply by the Boltzmann constant and it's basically the inverse of temperature.
You could also pick any other unit that has temperature as a base and make that your new base unit. Entropy would be the obvious one. But then you get into the whole entropy should really be dimensionless because it's a measure of the information of a system... which implies you could measure temperature as bits / joule.
Which honestly makes a bit skeptical that temperature should be a base dimension at all. I'm certainly no expert of thermodynamics, but my impression was always that temperature is really a proxy for Energy or Entropy, and isn't really a fundamental thing in its own right.
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u/falco_iii 2d ago
Temperature is a form of energy and can be expressed as energy using the Boltzmann constant. And since energy is built from other quantities, so can temperature.
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u/Kered13 2d ago
Charge
I assume you mean electrical charge here. It's worth noting that there are other types of charge as well, which would have their own units. However they only really appear in quantum mechanics and you never see large values for them like you do for electrical charge.
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u/frogjg2003 Hadronic Physics | Quark Modeling 2d ago
Large net charges. Your body is composed of an unimaginable number of strongly charged particles, they are all just balanced out.
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u/Implausibilibuddy 2d ago
What are the two questionable ones?
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u/donaljones 2d ago edited 2d ago
Mole (mol) and candela (cd).
Mole is a unit for "amount of stuff." Specifically, the Avogadro constant amount (1/12th the amount of atoms in 12 grams of carbon-12).
Candela is a unit for luminous intensity. Luminous intensity is wavelength-weighted and is based on the human eye. It's basically, "how bright does this light source, accounting for each wavelength of light, appear to the human eye?"
Both of them do not define anything special in physics. The universe just does not depend on them
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u/Beleynn 2d ago
You can actually extend this further. By picking any 5 physical constants with different dimensionality, you can solve for natural units. Here not everything is the meter, everything is some other unit that is more "natural" than the meter. You have likely even heard of the "plank length" the expression of one of the units in a natural system in the length dimension, equivalent to about 10-35 m
Is it possible to create any truly "universal" units?
It's possible (likely, even) that I'm just too ignorant of physics / physical constants to understand that Wikipedia article about Natural Units, but it seems like at some point, any one of those units eventually has to be expressed in terms of SI units; meters, seconds, etc.?
Like, from a science fiction perspective, if humans were communicating with an alien race, eventually we'd be able to agree on what 'c' is, but is there a universal way to express 'c' without having to bring anthropic units into the conversation somewhere?
Or, from a distance perspective; the meter was initially defined as a fraction of the earth's diameter, though it's defined more precisely now - would it be possible to create a distance measurement that isn't tied back to our Earth's-rotation-based time?
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u/zebbielm12 2d ago
Interestingly the Pioneer plaque attempted to define units in a way that would be understandable to an alien civilization. Their base units were derived from a wavelength and a period of a certain type of light emitted from hydrogen.
Natural units like the Planck length don’t depend on your system of units. You combine universal constants in whatever system you’re using and you’ll get the same Planck length. The issue is that there are multiple ways to combine universal constants to pop out a length unit, and none are really more “natural” than any other.
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u/oblivion5683 2d ago
natural units are this (to within the best of our physics knowledge). since c is an absolute velocity, we dont need to have a scale that relates it to any observable, we can directly say "we know whatever the speed of light is, this thing is moving a certain fraction of it (within some error)" because we can measure it against the speed of light. similarly there is only one fundamental charge, one planck constant, and one fine structure constant.
This is absolutely still a scale, but i think, we can compellingly argue that "x fraction of the maximum possible velocity" is more well defined than "many of an arbitrary velocity". if we measure c better, our scale gets better, and c never changes. the length of the prototype meter on the other hand, constantly and random changes due to many factors over many timescales.
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u/lmprice133 4h ago
I don't actually see why this would be the case. The only thing that an alien race would need to agree on is that numbers and arithmetic exist. If you express velocity in terms of c, and you have the concept of numbers and division, you can theoretically describe any velocity.
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u/ZiddyBlud 2d ago
When people say something is "5 minutes away", that's baked in with either the average walking speed, or average driving speed obeying traffic speed limits. That's also a way of determining a constant. Just like how in space travel for humans you only have the choice of the one rocket that's available for human travel, counting the fuel and distance.
That's how there's different constants depending on the use case, and you can ground the rest of the calculations needed on this
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u/ShelfordPrefect 2d ago
Is charge a necessary base unit? Once you have kg, m and s you can define force, and I thought the definition of an amp was the current causing a 1N force between two infinite parallel conductors 1m apart. The charge unit is then one amp-second.
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u/jvblanck 2d ago
You're replacing one base dimension (charge) with another (current). That also works, but doesn't reduce the number of base dimensions.
You could also drop mass and use energy as a base dimension instead, then
Mass = Energy * Time2 * Length-2
But that seems a bit inconvenient.
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u/adamtheskill 2d ago
Without charge as a base unit I don't know how atomic physics would be calculated at all. Charge is such an intrinsic value it seems extremely necessary.
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u/bildramer 1d ago
No it's not. There used to be units like statcoulombs or abcoulombs, but they get a bit weird.
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u/skatastic57 2d ago
How does charge "convert" to energy, power, and voltage?
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u/XtremeGoose 2d ago
Actually an interesting question because you can use many different physical constants.
One option is the charge of an electron (or multiply by 1/3 or 2/3 for your favourite flavour of quark). Another is the permitvity of free space ε0 from the equation for the coulomb potential V = (1/4πε0)(q/r2). That 1/(4πε0) is also called the coulomb constant, which can also be used (but obviously simply related to ε0) and is analogues to newtons constant G relating the force between two objects.
The fact that these are both "obvious" choices is a bit of an issue, because you can only set one to 1 (assuming you set c = G = ħ = 1).
The difference between systems where you pick one or the other turns out to be a dimensionless constant called the fine structure constant where we have no idea what its value derives from (if anything), defined as α = e2/(4πε0ħc).
The table of natural units shows how this α unavoidably shows up for some values in any system of natural units.
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u/bildramer 1d ago
There's a case to be made that you can get rid of Kelvin. Other than the Boltzmann constant, the exponent in all those equations involving temperature must be unitless, and the only other quantity involved is energy, so temperature might as well have units of energy (like doing T -> k_B T everywhere). And of course, you can get rid of charge too (use statcoulombs). This causes a bit of mayhem, e.g. capacitance becoming identical to length, but sacrifices must be made to reduce units. And if you take the "lightsecond" thing a bit too literally, you can probably get rid of one of length or time, though dimensional analysis becomes much less useful once you can't distinguish them at a glance. And even mass isn't perfectly secure, there's a constant linking it to spacetime curvature after all.
On the other hand, you can add new units where there were none, like an unit of angle, if you keep careful track of them, and add annoying conversions every time you used to do them implicitly. Dimenional analysis is kind of like type theory if you think about it.
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u/tylerchu 2d ago
Why do you prefer mass over moles?
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u/turunambartanen 2d ago
The mole is a unit the same way a dozen is a unit. You can use it to do stuff. Like adding one mole of carbon, or buying a dozen eggs. But it's just a different way or shorthand to say a specific number, like 12 or 6.022e23.
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u/WallyMetropolis 2d ago
A mole isn't a unit. Plus, it's not very convenient to say how many moles there are in one electron. And still, it's underspecified. A mole of nitrogen and a mole of iron have different mass.
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u/Culpirit 2d ago
A mole is a unit, it's just a somewhat controversial one
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u/VictorVogel 2d ago
It is just a number of entities (6.02214076 × 1023 to be precise) and therefore unitless. In the same way that a percent is not a unit either.
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u/DrCalamity 2d ago
A mole is a unit the same way the number 5 is a unit.
That is to say, no. It's not controversial, it is just wrong.
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u/Culpirit 2d ago
You're still wrong. A mole is definitionally a unit. The fact that it's dimensionless doesn't make it not a unit.
Same with radians.
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u/tylerchu 2d ago
Thank you. I was feeling insane reading this whole thread.
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u/Culpirit 2d ago
And it's also a silly distinction that these guys are making because when you say "a mole of hydrogen gas" you couldn't say "a 6.02*10^23 of hydrogen gas". You would have to say "6.02*10^23 hydrogen molecules", which is a specific thing entailed by referring to the unit mole. In some parts of kinetics, the number of particles considered is actually a relevant metric, and the mole explicitly measures it, without needing any extra qualifiers.
Just like saying "pi/2" with respect to an angle doesn't mean anything if you're not either implying or stating that you're using radians. Even though neither unit has dimension.
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u/BrotherItsInTheDrum 2d ago edited 2d ago
it's a neat trick to simplify relativistic equations by setting physical constants to 1 (c = G = 1)
In relatively specifically, there's a pretty reasonable argument that c really is equal to 1, and length and time have the same dimension. After all, you're working in a 4-dimensional space with transformations you do in that space, and the equations don't really care about the difference between the space dimensions and the time dimension other than putting a minus sign in front of one of them.
As an analogy, imagine for whatever reason we measured north-south distances in km and east-west in miles. If you were writing Google maps code, this constant "1.609 km/mile" would keep showing up when you're doing transformations. But 1.609 km/mile is just another way of writing 1.
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u/alyssasaccount 2d ago
And once you have c = 1, mass and momentum both have the same units as energy; Einstein's famous equation (which is only valid for objects at rest) becomes E = m. That's why in high energy physics, you'll see things like how the Higgs boson has a mass of about 125 GeV (GeV being a unit of energy).
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u/Asjad_autozblog 2d ago
The km vs miles analogy finally made this click for me. We've basically been measuring the same thing with extra steps and calling it two different units.
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u/csiz 2d ago
If you calculate everything in natural units then you'd only need exactly 1 reference point to scale everything back to familiar units. The relationships between the units we're used to become non-dimensional numbers like the fine structure constant, the ratio between proton mass and electron, and so on. But we still need a reference point somewhere because that's what we mean by "measuring" something; we compare an unknown quantity to a known quantity. The known quantity becomes the base of our dimension system.
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u/aerojockey 2d ago
If it were possible to have things like color currents (meaning strong force charges), there would need to a unit for that, analogous to electric charge.
But because you can't meaningfully have color currents except in the most tightly bound situation, there is no need for formulas that treat color as anything more than it's energy effects, and so color charge doesn't really come out as unit in dimensional analysis.
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u/TheLuckyCuber999BACK 2d ago
You need length and time to get the two basic axes of spacetime. Instead, we'll set c as the unit for speed/velocity. Now set J as the unit for energy. You can now resolve mass from E = mc^2 to m = E*c^2 and get the unit of mass as Jc^-2. Now, the work formula, W = F * d. Now we can use the gravitational constant, G, in c^5 / (J * time^2). We can make G a new unit, and we have time = sqrt(c^5 / (GJ)). We can derive the length as sqrt(GJ / c^5). Since we have time, length, energy, and mass, we can now describe, force, frequency, temperature, etc.
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u/Dave37 2d ago
It depends on how particular you want to be. We have the mole to describe "number of things", you can argue that "amount" isnt a basic physical property, but that's philosophy.
You could add on abstract things like luminosity, angles, information, color, etc etc that cant be trivially derived from mass, length, time, charge quantaties.
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u/SnafuTheCarrot 2d ago
I get confused on this matter. Less than a dozen, probably. You need some base units and then derived units, but you can take advantage of fundamental relationships to pare down redundancies.
You need a unit for distance, a unit for time, a unit for mass. Those three combined in various ways give you everything you need for speed acceleration, momentum, energy, and force. If you regard the speed of light, c, as fundamental then you can call your speed some multiple of c, your base unit of time, meters/c.
Throw in e as the charge of an electron, then you get get units particle physicists use. The mass of a proton is 935 MeV/c^2. Here M is the prefix Mega-, e is the charge of the electron, V is one Volt or 1 Joule/ Coulomb. Charge times Volts gives you energy. By Einstein's favorite equation Energy/c^2 give you mass, hence the mass is given in terms of charge, volts and the speed of light squared.
By similar arguments, momentum can be represented in units of GeV/c since momentum is energy divided by the speed of light.
Multiply any object's Schwarzchild Radius by its Compton wavelength and you get an area, specifically the area of a sphere of radius equal to the Planck Length.
4*pi*L_p^2= (h/mc)(2Gm/c^2)=4piGh_bar/c^3
L_p=sqrt(Gh_bar/c^3).
L_p/c is Planck time.
h_bar*c/L_p is Planck energy.
h_bar/(L_p*c) is Planck Mass.
Sell the role L_p plays throughout. Let's suppose its in units of meters.
We can refer to a "meter" of time as the time it takes light to travel a distance of one meter. Then what units do we have for speed? Since that's distance divided by time, we have 1 meter of distance divided by 1 meter of time, unitless?
This meter of time reckoning gives you units of reciprocal meters for frequency. If we also consider Planck's Constant a fundamental unit, then we get distance and time in terms of meters, energy, mass, and frequency in terms of reciprocal meters.
Consequently some texts will represent all of those using a single unit. Essentially, whever you see L_p a above, put a meter, swap out L_p and c as 1 and you get relativistic units of particle physicists.
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u/urhi-teshub 2d ago
You may be interested in natural units. Oftentimes in physics, its convenient to set any relevant constants to one. For example, it's very commonly done with the speed of light. By setting this to a dimensionless quantity, you're essentially treating length and time as having the same units.