This is a paradox I'm trying to understand. I'm not tackling relativity yet. I'm still working through Walter Lewin's lectures on electro magnetism. However, I understand base and derived units pretty well and in preparation, there's something that's picking on me.
Consider distance (aka length, in meters) is a base unit.
Time (in seconds) is also a base unit.
Speed (scalar form) or velocity (vector form) is a derived unit based on distance over time.
example: m/s
This is what we base so much of our physics on. But more than that, it logically makes sense.
Consider speed being held constant- that is light or radio waves travel at ~~300k km/s.
Wherefore if we try to travel faster than that with our Star Trek-nology, we can't. What happens is time slows down so that we don't travel faster than that. It's a universal speed limit.
But in the beginning of physics we measured speed as distance over time. It seems weird that we'd hold distance constant (I assume) and bend time to make distance/time a constant.
Basically in all of the rest of physics is there a case where we hold a derived value constant and bend it's constituent base units in order to keep the derived unit constant? It seems like a reflexive paradox, like either a not very smart or else a very very smart designer would do.
The derived unit – speed – rules over the base unit time. Is that right? I'm not ready to go too far into the deep end on this matter. I just want to stay as far in the shallow end and still have my question answered or at most pointed where to look in the deep end if that's necessary.
I can imagine base units as being flexible or mutable somehow, but just not in service of derived units which are based on the selfsame base units. Unless there's some intrinsic nature of speed and it should actually be the base unit and we got it all wrong, like seconds really equal meters over velocity, seconds or time is the derived unit and velocity should be the base!! Probably that's not right but that's where logic brings me.
Kinda reminds me of Ohm's law: V = IR, volts are equal to current * resistance. It's an expression of the relationship or proportionality, but in that form really doesn't express the physical dependence and independence as I understand it. Really the dependent variable is current (I) so it could be written more semantically (though not more usefully) I = V/R.
In the same way should the velocity equation be expressed not as v = distance/time or m/s but rather T = D/V? But then when in the real world do we access velocity as a base unit?
That is in no way workable or makes sense, but maybe it does make more sense inside the looking glass/down the rabbit hole of the quantum world whence reality is composed? Maybe we see an optical illusion and god views his velocity of light as the constant but it just so happens we never get close to bending the other constant, like really stiff guitar strings, so they both seem flexible.
If so, a simple yes to that last point would be enough to answer this question and close this thread, without needing the need to bring up gravitational field lines or other fancy concepts or details of implementation. But I have a feeling it is inevitable.
Best Answer
You actually are not too far off with your thoughts. There is a subtle issue of terminology.
In a system of units the choice of base units is arbitrary. For the SI there are seven base units: second, meter, kilogram, ampere, kelvin, candela, mole. In the SI system all other units are derived from some combination of these and are called defined units. In particular, the SI unit for speed is the m/s, a derived unit.
However, although the SI unit for speed in general is derived, the specific constant “the speed of light in vacuum” is defined in the SI. That means that in the SI system the speed of light is an exact number with no experimental uncertainty.
Together with the separately defined second, the defined speed of light defines the length of the meter. In other words, the second is first defined and then the meter is the distance that light travels in exactly 1/299792458 s.
Now, this may seem like a circular definition, but it is not. The meter is defined such that $c=299792458\text{ m/s}$ is true. It is perfectly valid to define a quantity as the solution to some equation. The BIPM is free to fix the length of the meter at any length they wish. They can certainly choose to fix it at the unique length that satisfies that equation. The designation of some units as “base” units and other units as “derived” units is immaterial to how the units are defined.