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In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as fundamental and then understand time by dimensional analysis in terms of l/v (length divided by velocity)? If not, why not?

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You can work with any system of units you want, but some units are more convenient than others. In everyday physics we regularly have to calculate dependance on time, i.e. $df/dt$, and it's a great deal easier to do this if we express our function $f$ as a function of time than as a function of velocity. It's pretty rare we want to calculate $df/dv$.

So taking length and velocity as the fundamental units would just make our lives harder.

Having said this, and maybe this is what you meant, the SI unit the metre is currently defined as the distance travelled by light in a vacuum in 1/299,792,458 seconds. So we are using the velocity $c$ to define the metre. The second remains defined as a time i.e. as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

And just to add to the mix, in relativity we conventionally convert time to a distance by multiplying by the velocity $v$, though to be awkward we define $c$ as $1$.

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  • $\begingroup$ We can still work with time as a derived unit if it is possible to use velocity and length as fundamental units, so I do not see why it would make our lives harder on that count. If we think that a function $f$ depends upon time and want to express the idea that $df/dt$ we only set $df/d(l/v)$. Would that be a problem? $\endgroup$ – Sapiens Oct 30 '13 at 18:06
  • $\begingroup$ Sure you could use velocity and length instead of time and length, but since everybody else uses time, it's easier to just use time. $\endgroup$ – Pranav Hosangadi Oct 31 '13 at 4:44
  • $\begingroup$ @user24406: it is as Pranav says. The choice of units isn't down to some religious conviction. We choose the units that make our life easiest. In GR it's not uncommon to use highly esoteric units if they help. $\endgroup$ – John Rennie Oct 31 '13 at 6:51
  • $\begingroup$ Thanks. The advantages I look for are of a rather philosophical nature, but also relate to how to think about simultaneity. $\endgroup$ – Sapiens Nov 1 '13 at 12:15
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You need three quantities to fully define all mechanical units, like $({\rm Mass},\,{\rm Length},\,{\rm Time})$ or $({\rm Force},\,{\rm Length},\,{\rm Time})$.

If you want to chage instead a system using speed then you can easily work with units like

$$ \begin{aligned} {\rm Time} & = \frac{\rm Length}{\rm Speed} \\ {\rm Acceleration} & = \frac{\rm Speed^2}{\rm Length} \\ {\rm Force} & = \frac{\rm Mass\; Speed^2}{\rm Length} \\ {\rm Energy} & = {\rm Mass}\;{\rm Speed^2} \end{aligned}$$

but it won't change fundumentally the facts about the laws of motion and gravity.

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    $\begingroup$ Thanks. It would certainly be a problem with the suggestion if it changed the facts about the laws of motion and gravity. But it might perhaps change the way we think about time? $\endgroup$ – Sapiens Oct 30 '13 at 18:18
  • $\begingroup$ @user24406 I know of some problems where treating speed as an independent variable makes them solvable, so maybe you can proceed investigating this. Time is $$t=\int \frac{1}{a}\,{\rm d} v$$ and distance $$ x =\int \frac{v}{a}\,{\rm d} v$$ $\endgroup$ – ja72 Nov 1 '13 at 13:57
  • $\begingroup$ I do not understand. Can you explain, please? $\endgroup$ – Sapiens Nov 2 '13 at 3:32
  • $\begingroup$ These are direct integration of acceleration $a$ when the independent variable is speed $v$. $\endgroup$ – ja72 Nov 2 '13 at 20:00
  • $\begingroup$ So $dt/dv=1/a$ and $dx/dv=v/a$? What does "them" in your sentence "I know of some problems where treating speed as an independent variable makes them solvable, so maybe you can proceed investigating this." signify? $\endgroup$ – Sapiens Nov 2 '13 at 22:01
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There's two rational reasons why this would be a bad idea for everyday life:

  1. most speeds we encounter are significantly smaller than $c$.
  2. it would change our units system entirely

For the first point, I'd be traveling down my street at 0.00000037 instead of 25 mph (40 km/h). Airplanes would travel at about 0.0000002 instead of about 600 mph (965 km/h).

For the second point, for velocity to be unitless, we'd need to have time and distance with the same units. Accelerations would then have units of 1/s, so that forces are measured as kg/s. And so on.

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  • $\begingroup$ Your point 1 only adduces some practical concerns which we could easily adjust to by changing our numerical practice somehow, and does not concern matters of principle. In your point 2 you mistakenly suggest that the proposal was to take velocity as dimensionless. Velocity would instead have its own measure. $\endgroup$ – Sapiens Oct 30 '13 at 18:16
  • $\begingroup$ Your exact words: Could one instead take velocity, with c as its unit. With $c$ as the unit of velocity, then we need to scale all our velocities by $c$ (point 1) which means $v\to v/c$ which is unitless (point 2). Your comment here indicates that your true question ought to be, "Why isn't there a unit of velocity like there is a unit for length (meter)?" $\endgroup$ – Kyle Kanos Oct 30 '13 at 18:53
  • $\begingroup$ As you correctly display, I indicated that we could take c as a unit for velocity. Let $\gamma$ be the velocity of light as stated in m/s, and $v$ another velocity as stated in m/s. The correct transformation is $v \rightarrow (v/\gamma)c$, which is not unitless. $\endgroup$ – Sapiens Oct 30 '13 at 19:20
  • $\begingroup$ If $c$ is the unit for velocity, then photons travel at 1$c$, as opposed to $2.9979\times10^{8}$ m/s, and not $c^2/\gamma$ as your transformation gives. $\endgroup$ – Kyle Kanos Oct 30 '13 at 19:29
  • $\begingroup$ Actually, your transformation is mine, if you use $\gamma=c$, as you should. What you are improperly doing is giving $c$ a unit of m/s, rather than keeping it as the unit. Which, again, means your question should be "Why isn't there a unit of velocity like there is for length (meter)?" $\endgroup$ – Kyle Kanos Oct 30 '13 at 19:31
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From the point of view of dimensional analysis alone, this can be done. It would not give mathematically inconsistent results. Mathematically, any quantities $A$, $B$, $C$ can be chosen as 'fundamental', as long as no relationship of the form $f(A, B, C) = 0$ exists between them.

However, from the point of view of physics itself, such an approach needs to be questioned. In physics, we consider certain concepts to be primitive - meaning that they cannot be explained in terms of simpler ideas. Space and time are amongst such ideas. Other concepts are derived - meaning that they can be explained or defined in terms of existing ideas.

If we consider time and length as fundamental quantities, we should also be able to provide a simple prescription for measuring them. This can be easily done for time and length - for eg. - as the no. of oscillations of a particular pendulum, or as the no. of multiples of some unit length. In the case of velocity however, it would be quite difficult, or perhaps impossible to provide such a prescription which does not involve using the ideas of space or time. Hence, there does not appear to be a strong case for considering velocity as 'fundamental'. It does appear that certain quantities are really more fundamental than others, it does not appear to be just a conventional element.

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  • $\begingroup$ Thank you. It seems to me that taking velocity as fundamental has the conceptual advantage that we may postulate a natural unit in the speed of light. Moreover, I think that understanding time as really a construct from time and velocity may have conceptual advantages, e.g. when it concerns such topics as simultaneity and locality of time. Moreover, we could explore the point of view that all matter is travelling with great speed, usually accelerating, away from a priveleged origo in the Big Ban. We would continue to use clocks of course. $\endgroup$ – Sapiens Oct 30 '13 at 22:40
  • $\begingroup$ If you could come up with a way of directly measuring velocity in some way other than by measuring the distance travelled and dividing it by the time taken, then perhaps your approach may be justified. Can you think of such a way? $\endgroup$ – guru Oct 31 '13 at 8:48
  • $\begingroup$ If you want to treat speed of light as the natural you don't need to define velocity as fundamental in order to do that. For eg., see en.wikipedia.org/wiki/Geometrized_unit_system . $\endgroup$ – guru Oct 31 '13 at 8:50
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Personally, I don't see why not. And I too believe it has the potential to open up new lines of inquiry and understanding. I have long thought the concepts of velocity and change have a good chance of in fact being most fundamental. As an example, what if the reality of the fundamental nature of our universe is such that, at an event, information is emitted at an angle, with the angle 0 being the minimum possible angle, so that the speed of light (c) is actually the minimum velocital angle 0. Slower speeds in our current way of thinking about such things would simply be an angle greater than 0 so that it would take longer, hence more "time", for that information to arrive at and influence some distant point in space.

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protected by ACuriousMind Jun 20 '16 at 15:18

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