Whenever two things are to be added together, one typically needs to check whether this actually makes sense, and an addition is said to make sense, in principle, when the units match up.
Yet, matching units clearly aren't enough:
- Both action (like $\hbar$) and angular momentum $L$ share the SI-units $Js$.
- Though adding such quantities can quickly be dismissed by stating that $\vec{L}$ is a directed quantity and vectors and scalars can't (usually) be added together either.
- Setting $c=1$, as commonly done in relativistic mechanics, various units become one and the same:
- $\left[\text{length}\right]=\left[\text{time}\right]$, $\left[\text{mass}\right]=\left[\text{energy}\right]=\left[\text{momentum}\right]$, $\left[\text{acceleration}\right]=\left[\text{frequency}\right]$ etc.
- those do mostly make sense with $4$-vectors, recognizing that, say, time is orthogonal to space, although I haven't seen $\left[\text{acceleration}\right]=\left[\text{frequency}\right]$ used before.
- going a step further, introducing Planck units derived from natural constants, all units become powers of $\left[\text{length}\right]$: (done by hand, errors possible)
- $\left[\text{1}\right]=\left[\text{#particles}\right]=\left[\text{%}\right]=\left[\text{odds}\right]=\left[\text{velocity}\right]=\left[\text{entropy}\right]=\left[\text{action}\right]=\left[\text{angular momentum}\right]=\left[\text{charge}\right]=\left[\text{resistance}\right]$
- $\left[\text{length}\right]=\left[\text{time}\right]=\left[\text{inductance}\right]$
- $\left[\text{length}^{-1}\right]=\left[\text{energy}\right]=\left[\text{mass}\right]=\left[\text{momentum}\right]=\left[\text{frequency}\right]=\left[\text{acceleration}\right]=\left[\text{torque}\right]=\left[\text{capacitance}\right]=\left[\text{current}\right]=\left[\text{voltage}\right]=\left[\text{temperature}\right]=\left[\text{chemical potential}\right]$
- $\left[\text{length}^{-2}\right]=\left[\text{force}\right]=\left[\text{magnetic flux density}\right]$
- $\left[\text{length}^{-4}\right]=\left[\text{pressure}\right]$
- probably many many more (which are actually in use somewhere – in principle there would of course be infinitely many such relations)
So for many of those I can clearly see why they couldn't possibly be added together in a consistent manner:
Some of these quantities are fundamentally scalars, others are axial vectors and yet other ones are polar vectors. – in normal vector calculus, those three can't just be added together.
Particle count and relative quantities are also different in that they are defined on different domains; respectively $\mathbb{N}$ and $\left[0,1\right]$. All the other above quantities are defined either on $\mathbb{R}$ or on $\left[0,\infty\right[$, although experimental data and theories that conform to them may limit some of them, like electric charge, further.
So the necessary conditions for a properly defined addition I found thus far are:
- matching units
- matching domains
- matching dimension
- matching transformational behavior or covariance (? – I am not well versed enough in GR to know that this always is a problem. I am vaguely aware of co- and contra-variant tensors and such, but I can't recall whether a co- and a contra-variant tensor from the same space can usually be added together. What I do not count, though, is addition as it might happen in Geometric Algebra, where you can have multivectors. In that context, I'm specifically asking about addition of like blades. In that case the fact that axial and polar vectors can't be added together becomes the fact that they correspond to vectors or bivectors.)
Though there must be more than that, right?
As far as I am aware the above rules wouldn't rule out, say, adding together a capacitance and a temperature. Or am I mistaken? (i.e. does the above suggested addition either violate one of the above four conditions or does it, in fact make physical sense in certain situations, or can you somehow reasonably think of them as different components of a shared vector?)
Considering all this, is there a clear mathematical or physical rule (or set of rules) which fully defines when two values actually can be added together in a physically consistent manner? What are the necessary and sufficient conditions?
EDIT 1: Choice of natural units
I made an error, or had an oversight, in my choice of units above: I set $c=\hbar=k_B=k_e=1$ but forgot to also set $G=1$. If I see this right, then $G=c=\hbar=1$ implies something rather weird:
$\left(c=1\right)$ implies $\left[\text{length}\right]=\left[\text{time}\right]$ and $\left[\text{energy}\right]=\left[\text{mass}\right]$
$\left(\hbar=1\right)$ implies $\left[\text{energy}\right]=\left[\text{time}^{-1}\right]$
$\left(G=1\right)$ implies $\left[\text{mass}\right]=\left[\text{length}^3\text{time}^{-2}\right]$
So together:
$\left[\text{length}^{-1}\right]=\left[\text{time}^{-1}\right]=\left[\text{energy}\right]=\left[\text{mass}\right]=\left[\text{length}^3\text{time}^{-2}\right]=\left[\text{length}\right]$
But $\left[\text{length}^{-1}\right]=\left[\text{length}\right]$ can only be fulfilled if $\left[\text{length}\right]=\left[1\right]$. So if you go full natural, all the units are, in fact, gone.
Related to this is the following comment by jwimberley to an answer of another question: What justifies dimensional analysis?
In that case my question becomes "Being in the natural unit system where $c=\hbar=G=k_e=k_B=1$, how can you tell whether two quantities can be added together?". I already identified a couple requirements above. Now it remains to say whether those are sufficient or there are some extra required conditions. I mean, you could conceivably just (perhaps as vaguely suggested per the above linked comment) put all quantities that are decidedly not the same into different components of a large vector containing all the units separately (as well as copies of units for when those exist, like, say, three separate spacial coordinates), as long as all the included units' domains match up (they all should be defined, each individually, on the entirety of $\mathbb{R}$ or what ever fits).
I don't see a reason why this wouldn't work but at the same time I'm unsure as to whether that'd actually be a good solution. But how else to keep track of all these types of values if their corresponding units all are just $1$? There are pretty good reasons for why we have $4$-vectors. Could a similar case be made for larger vectors or would the underlying structure connecting different units not work well with that?
And how large would this vector have to be? – For instance, as mentioned, $c=1$ implies $\left[\text{energy}\right]=\left[\text{mass}\right]$ as per $E^2=m^2c^4+p^2c^2$, so you probably wouldn't need separate entries for energy and mass but you would need three extra entries for momentum because that's a directed quantity.
I'm throwing in a lot of extra questions here but really they are all just consequences of asking when (not) to add two things. Feel free to ignore those extras as long as the original question is answered.
EDIT 2: Concerning Potential Duplicates (continued with $G\neq 1$)
While related, my question is quite a bit different from What justifies dimensional analysis?: I'm not asking about directly adding something like "$5m+10s$" – this is clearly undefined. However, you can add them using a 4-vector: $10\vec{X} = s \ \vec{e}_0 + \left( 3 m \ \vec{e}_2 + 4 m \ \vec{e}_3 \right) = \left(\begin{matrix} 10s & 0m & 3m & 4m\end{matrix}\right)$ or, using the same units throughout, $\vec{X}=\left(\begin{matrix} 2997924580 & 0 & 3 & 4\end{matrix}\right)m$. – not that this choice of units would be particular reasonable for such values.
This vector has a spacial part $\vec{x}=\left( 3 m \ \vec{e}_2 + 4 m \ \vec{e}_3 \right), \left|x\right| =5m$ and a temporal part $t=2997924580m=10s$ and, if I'm not mistaken (and I might very well be), given that the vector has a Minkowski metric, has a "length" $\left|\vec{X}\right|=5 \sqrt{359502071494727055}m \approx 10s $. (those 5m pretty much don't matter)
And there are more quantities within special relativity which admit such a form. For instance, the energy-momentum-4-vector, or the electromagnetic field.
Though as mentioned, $c=1$ implies more correspondences. For instance, is there a frequency-acceleration-4-vector? – Of course, there is 4-acceleration, but is the temporal part of it some kind of frequency, as the unit would suggest?
Of course, the simplest answer looking only at Newtonian mechanics which (to my knowledge), for problems in $\mathbb{R}^3$, is strictly limited to 3-vectors and scalars, would be that not even $\left[\text{length}\right]=\left[\text{time}\right]$ holds. However, we (that is, physicists within the past century+) have established a, by now, pretty clear correspondence between space and time that makes both physical and mathematical sense by accepting $c$ as a natural unit.
And other natural units suggest even more such correspondences. My question is: Are all those correspondences actually physical in some way perhaps similar to 4-vectors in SR? (Are even just all those suggested by $c=1$ alone reasonable within physical intuition? – at least a decent chunk of them apparently are.) And if not, what, precisely, goes wrong? (Because they do have matching units as far as natural units are concerned)
Or in short, what can or can't be added together under which circumstances?
Now, the accepted answer for the above-linked question, https://physics.stackexchange.com/a/98257/16568, may come pretty close to a partial answer:
Physics is independent of our choice of units
but as far as I can see this only rules out that you can add quantities with different units in the same space. That's again avoided within SR by changing over to 4-vectors, if I understand that right.
Finally, Fundamental question about dimensional analysis was given as another potential duplicate or at least something that might help. Though as said, it's clear to me why more complex functions than products or sums won't make sense with any quantity with units.
I'm aware that products and integer powers are the only functions that do not care about units (they'll work no matter what) and linear combinations will work if all terms have the same unit.
Best Answer
There is really just one underlying principle here: Any equation we write down should not depend on any arbitrary choices we made in order to define the quantities. All the examples you can discuss can be understood in this principle.
Can't add a vector and a scalar. Well, of course a vector is three numbers and a scalar is one number, so, for example, $\mathbf{v} + v$, , where $v$ is a speed, i.e. a scalar with units of velocity, doesn't even make mathematical sense. But we could add imagine adding one component of a vector to a scalar, i.e. $v_z + v$. But, this quantity shouldn't appear in a fundamental law of physics because our choice of what axis to call the $z$ axis is completely arbitrary, and if we made a different choice our equations would look different. But this is situation-dependent. For example, if we were discussing physics in a background uniform gravitational field, then we can use a convention where $z$ points along the direction of gravitational field. This is not arbitrary because the gravitational field sets a preferred direction. By declaring that we are going to call that particular direction the "z-direction'', it makes sense that any equations we then write down will only hold for that particular choice of $z$-axis. That is why the equation for the gravitational potential energy in a gravitational field, $U = mgz$, is valid even though $z$ is a component of the displacement vector $\textbf{r}$. However, you can still translate this equation into one that is valid for any choice of axes, namely $U = m \textbf{g} \cdot \textbf{r}$, where $\textbf{g}$ is the gravitational field vector.
Can't add numbers with different units. The point is that we normally work in units of physics which are chosen completely arbitrarily. If time $t$ is measured in seconds and position $x$ is measured in meters, then it makes no sense to write down an equation involving $x+t$ because this equation would depend on our definition of "second" and "meter", and there is no reason why the laws of physics should depend on the second being defined to be 9,192,631,770 times the period of some radiation mode of a cesium atom. But, if we choose to work in natural units, then this is not an arbitrary choice because, as the name suggests, natural units are uniquely determined given fundamental constants of physics. In natural units, there is nothing wrong with writing an equation involving $x+t$, because we remember that we have made a special choice of units, and the equation will hold only in those units.
Of course, any equation that you can write in natural units can still be translated into arbitrary units. Take Einstein's famous mass-energy equivalence. In natural units ($c=1$) it states that $E = m$. Obviously, in arbitrary units, this is a bad equation because if $E$ is measured in Joules, and $m$ is measured in kg, then it would depend on the definitions of Joules and kg. But that's fine, because this equation only holds in natural units. Its translation into arbitrary units is $E = mc^2$, and the units now match up.
Can't add covariant 4-vectors and contravariant 4-vectors. Again, this is because in special relativity, in order to write down components of vectors we have to make an arbitrary choice of coordinate directions in space-time. Equations we write down in special relativity shouldn't depend on this choice, and this prevents us from adding covariant 4-vectors and contravariant 4-vectors because they transform differently when you change coordinate directions in space-time.
Can't add things in different vector spaces. This is just because, if $\mathbf{v}$ is in one vector space and $\mathbf{w}$ is in a totally different vector space, then you wrote an equation involving $\mathbf{v} + \mathbf{w}$ then it would depend on how you relate the bases between the two vector spaces, which -- given that they are totally different spaces -- there is no way to do non-arbitrarily.