# Natural units and dimensional analysis

As far as I understand, a natural system of units is one in which the numerical values of ##c## and ##\hbar## are unity. However, they still have dimensions, indeed ##[c]=LT^{-1}## and ##[\hbar]=ML^{2}T^{-1}##. How is it the case then, that certain quantities, such as the action ##S##, can be treated as dimensionless in natural units? Action has the same dimensions as ##\hbar##, so how can it be dimensionless? Is it simply that we consider the quantity ##S/\hbar##, which is already dimensionless, and then in natural units we have that ##S/\hbar =S## and we refer to this as the action?

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Orodruin
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No, in natural units length and time have the same units. You leave only one independent dimensio L = T = 1/E. The action is dimensionless in those units.

No, in natural units length and time have the same units. You leave only one independent dimensio L = T = 1/E. The action is dimensionless in those units.
How is this the case though? Doesn't ##c## have to be dimensionless for this to be the case (is this something that is just stipulated)?

I can see how the numerical value of ##c## can be 1, as if we measure distance in lightyears (=distance travelled by light in 1 year) and time in years, then ##c=1\frac{\text{lightyear}}{\text{year}}##, but I can't immediately see how it can be made dimensionless?!

Orodruin
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How is this the case though? Doesn't cc have to be dimensionless for this to be the case (is this something that is just stipulated)?
But if c = 1, then it is dimensionless because 1 is. It is not a matter of using units where c has a numerical value of one in some units, it is a matter of using the same units for lengths and times.

But if c = 1, then it is dimensionless because 1 is. It is not a matter of using units where c has a numerical value of one in some units, it is a matter of using the same units for lengths and times.
So are we simply demanding that ##c## and ##\hbar## are dimensionless, i.e. that we can find a system of units in which length and time are measured in the same units? Does this follow from taking relativity into account in which we find that length and time are really two sides of the same coin (since in special relativity we find that temporal and spatial coordinates become interlinked under Lorentz transformations)?!

ShayanJ
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We can simply say that instead of calling t time, we call ##x^0=ct ## time!

Orodruin
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Does this follow from taking relativity into account in which we find that length and time are really two sides of the same coin (since in special relativity we find that temporal and spatial coordinates become interlinked under Lorentz transformations)?!
While it does not follow (it is a convention and you can just as well use a convention where you measure time and space using different units) it is certainly something which comes very natural in relativity.

While it does not follow (it is a convention and you can just as well use a convention where you measure time and space using different units) it is certainly something which comes very natural in relativity.
So is it a two step process then: choose a system of units in which the numerical values of ##c## and ##\hbar## are 1, and then furthermore choose this system such that time and length are measured in the same units and the unit of mass is related to that of length (time) by ##M=L^{-1}##? Is it natural to do this because in relativity space and time are not distinct concepts, but "woven" together into a spacetime continuum?

Orodruin
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I would say it is a one step process because of relativity. Without it, there is no invariant speed to set to one and so it would not make as much sense.

I would say it is a one step process because of relativity. Without it, there is no invariant speed to set to one and so it would not make as much sense.
So because c is an invariant, we can treat it as a fundamental conversion factor between spatial a temporal coordinates?!

Orodruin
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So because c is an invariant, we can treat it as a fundamental conversion factor between spatial a temporal coordinates?!
I would say that is a fair description of things.

I would say that is a fair description of things.
Ah OK, thanks for your help.

Is it simply the idea that systems of units are arbitrary (physics doesn't depend on what units you choose to measure physical quantities in) and so we can simply choose a unit system in which length and time are measured in the same units, thus they are no longer independent, but related to one another by a conversion factor, the speed of light. This is natural in the sense that we know from relativity that the speed of light is independent of the frame of reference and so this relation between length and time is frame independent (if one uses the same system of units)?!

You also have a choice as to which quantities you define as base units. If you arbitrarily make speed a base unit then length and/or time could be derived units.