Does speed of light being constant make time a derived unit? 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.
 A: When you are defining units, you need to distinguish some physical process that you will call standard and then supply some method which tells you how to compare some new process to the standard one. (Process might not be the right word for it, but I cannot think of better one)
Take for example seconds. You have some periodic process, say rotation of Earth around its axis, and you call this standard one which defines a day. To count how long does one rotation of Earth around the Sun takes, you need to supply method telling you how to compare this process to rotation of Earth around its own axis. To measure how much days does one year has you need both of these processes running in parallel, i.e. while the Earth revolves around the Sun it also needs to revolve around its own axis and you will count how many revolutions does Earth do while the second process completes. Note, you need to start both process at the same time and end them at the same time (or at least start and stop counting the both processes at the same time), so this heavily depends on how to asses that the processes indeed started at the same time.
Among all the methods and processes you need to pick the right ones. For example Earths revolution around its axis is not perfectly periodical at all times. By definition though, it always takes the same amount of time - one day. But the resulting physics expressed in terms of these days will have additional terms that are due to nonperiodicity of the day. We either need to find out what kind of sources produce this additional forces, or realize that day is simply not as good measure of time as we thought. The thing is that we cannot know which one of the two is the correct answer until we actually find meaningful sources or new meaningful way to define time which will get rid of these additional terms. Another thing is choosing the correct way to asses simultaneity of two events. By choosing the incorrect one as Newton did a lot of weird experimental results followed. The choice of Newton demanded that velocity are added together. When constancy of speed of light was discovered, it was to an extent incomprehensible. Speeds must be added and yet they are not? The problem was, that Newton based his notion of simultaneity on some absolute time. This in principle could be good definition if not for the fact that whole universe seemed to conspire to make this absolute time unmeasurable. There was no physical process which you could use to asses simultaneity of two events with Newtons definition of simultaneity. Einstein resolved this problem by simply choosing better, more physical way of defining simultaneity.
So you need to choose the correct base units the correct way, to make physics meaningful. Now, in Newtonian physics there is no physical and meaningful process by which you could compare seconds and meters, so these are distinct units. There is no preferred speed in Newtonian universe. If you would still pick one at random, it would lead to weird physics with additional incomprehensible terms - at least from the point of view of Newton. Not so in special theory of relativity. There is invariant speed - the speed of light. You can use this speed of light to compare seconds and meters. Simply define meter as a distance light travels by some ratio of a second. If this is indeed special speed in our universe, it will lead to simpler physics and theories and experimental results that make very good sense in this framework as it indeed does. Just imagine if light would not travel with the same speed for all observers and yet we would still declare it does by redefining meter in terms of this speed and a second. Then switching frames between two different observers would be quite complicated. This is not what we want, we want to simplify physics, not overcomplicate it.
The point is, when you are comparing two processes you need to have another physical process by which you are making this comparison. And choice of this processes is governed by making physical theories and experimental results meaningful and as simple as possible. In real universe, it so happens that speed of light is indeed distinguished and it is thus a good idea to define meter from second by use of the speed of light. Not so in Newtonian universe, where such idea would not work very well. It would probably still work, but doing physics in such a framework would be a headache.
A: 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 derived 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.
A: It is better to say that time and space are not independent units, valid for any inertial frame.
Instead of the distance $\Delta d = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$ and $\Delta t$ being constant for any inertial frame, $\Delta \tau^2 = \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2$ is now the invariant quantity.
A: 
It seems weird that we'd hold distance constant (I assume) and bend time to make distance/time a constant.

No, distance changes too.

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.

I think you're focusing on the wrong concept, here. It's not the units that are flexible and changing, it's spacetime itself, the actual physical reality of the universe, that changes to keep the speed of light constant.
Define the second and the meter in whatever way you want, as long as it makes no reference to your current velocity and is experimentally measurable from just local observations using equipment and materials you bring with you.
Using whatever arbitrary definitions you picked, measure the speed of light relative to yourself. Then accelerate to half the speed of light and, using those same definitions but in your new reference frame, measure the speed of light relative to yourself again. You will get the same number both times.
This happens, not because your units change, but because space and time physically do weird things when high speeds are involved.
A: 
Consider speed being held constant- that is light or radio waves travel at ~~300k km/s.

I don't agree with 2 details in this statement:

*

*it is awkward to "Consider speed being held constant". The speed "just is". In some systems (e.g. car, train) it can be affected, but for physical / chemical processes, the speed just is.


*light or radio waves travel at ~~300k km/s.
Yes, but only in void. In other mediums, the speed is different.

You made another assumption later:

What happens is time slows down so that we don't travel faster than that. It's a universal speed limit.

To refer to that meme, it is a limit so far, to the extent that we understand the universe around us at the moment. Future theories and experiments might prove that speeds higher than the speed of light are possible.


Really the dependent variable is current (I) so it could be written more semantically (though not more usefully) I = V/R.

Remember what that guy said? Everything is relative.
It is very convenient to consider I = V / R when using constant voltage power sources (wall power supplies, batteries...). However, when using a constant current power supplies, I=V/R looks at least strange, even though it might be useful when in the following context: I=V1/R1=V2/R2.


In the same way should the velocity equation be expressed not as v = distance/time or m/s but rather T = D/V?

In the proper context, it makes sense to rewrite the equation in that way. It is actually the simple way to solve that problem for children:

Two trains are on the same track, a long distance between them, facing each other. At the same time, the following happen:

*

*train 1 starts traveling ahead with speed vt1;

*train 2 starts traveling ahead with speed vt3;

*a super bird starts flying from the train 1 towards train 2 with speed vb;

Every time the bird reaches one train, it turns around flying towards the other train.
Question: after how much time the two trains and the bird will become one big mess?

(note: the real problem used slightly different wording; infinite acceleration / deceleration is implied)
So this problem can be solved easily by using t=d/v instead d=v*t.
Note: the formula d=v*t is just a simplification, considering that it does not include the Lorentz factor. And even the Lorentz factor is subject to "optimization", depending on context. It can "dilate" the time, or it can "contract" the distances. I also have the feeling that future "factors" will be found, to better explain the universe - and to clarify your (and our) concerns.

At the and of the "discussion", it is just a matter of:

*

*What is the best way to look a the problem, considering the specific context;

*What do we really know, and what do we really know correctly, considering everything that exists (and most of which we still do not know about).


Many things which were impossible or paradoxical in the past, are common life things today. Imagine the power one could have gotten 1000 years ago just by using two paired basic AM walkie-talkies.
Only God / Allah / The Architect knows what understanding of the universe we will have in the future.
A: When humans set up the time units they were unaware of the speed of light.  gcr is correct.  Now that we are aware of the speed of light as fundamental we could make the speed of light the numeral one (1c) and all other speeds a fraction of 1c.  Then as gcr suggests time (second) becomes a derived unit expressed as a distance (that light travels in a second) divided by c (the speed of light).
