Undergrad Can c be set equal to 1 in certain systems of units?

[Ken G]

The point is, we are clearly missing something important if we only say that since c is a fundamental constant, it's value must in some sense be unity. What this is missing is something that Ole Romer, in 1676, did not miss. He did an experiment to measure the speed of light, and he did not say there would be no point in such an experiment since it was going to come out unity anyway! So we can set it to unity by choosing a unit system, but the speed of light is then simply embedded into that choice of units. This is what I was saying above-- one must choose an internally consistent unit system, so if one takes c=1, there is an implicit constraint on the other values in that system. We might not explicitly state that c has units of 1 light second per second, but that is going to be in there, in the unit system we choose-- or else our unit system will be in danger of inconsistency.

This is true with any equation, not just the expression for the relativistic line element. Take F=ma for example. We can always write this as:
F/F_o = m/m_o * a/a_o * k
where F_o, m_o, and a_o are completely arbitrary "unit" choices. But the value of k is not arbitrary, it comes from the combination of our units choices and the result of interrogating nature, analogous to Romer's experiment. Indeed, it is not even necessary that k be unitless, that already presupposes a constraint on our unit of force. Still, as with c, we can always choose our units such that k=1, as it is not one of the unitless combinations of parameters that is a fundamental physical constant, such as the fine structure constant, because it implicitly includes our choice of units in its value-- it is a kind of combination of physics and convention. The speed of light is like that-- not a fundamental unitless parameter of nature, but rather a combination of nature and convention. Because the convention is in there, we can take it to have any value we like, but there is an implied constraint on the unit system. This is like in the F=ma example, where we can take k=1 if we like, but that places a constraint on the units of F, m, and a. In particular, we must have that F_o acting on m_o produces an acceleration a_o. If that isn't true, our unit system is inconsistent with k=1, and we have no way of knowing that until we do the measurements. The bottom line is, when we set c=1, we must not pretend we have made a free choice with no consequences, nor should we pretend that nature had to make it that-- there is an implied constraint on our conventions when we make that choice, just like the constraint on the unit of force if we wish to use F=ma. Calling c 1 light second per second automatically embeds that constraint in our language, but if we just call it 1, we have to make sure to embed that constraint some other way.

 

[MeJennifer]

The point is, we are clearly missing something important if we only say that since c is a fundamental constant, it's value must in some sense be unity. What this is missing is something that Ole Romer, in 1676, did not miss. He did an experiment to measure the speed of light, and he did not say there would be no point in such an experiment since it was going to come out unity anyway! So we can set it to unity by choosing a unit system, but the speed of light is then simply embedded into that choice of units. This is what I was saying above-- one must choose an internally consistent unit system, so if one takes c=1, there is an implicit constraint on the other values in that system. We might not explicitly state that c has units of 1 light second per second, but that is going to be in there, in the unit system we choose-- or else our unit system will be in danger of inconsistency.

I think you are totally missing the point.

The speed of light is 42. That answer is just as valid as saying the speed of light is 1.

 

[Ken G]

I think you are totally missing the point.

The speed of light is 42. That answer is just as valid as saying the speed of light is 1.

I'd say what you are missing is if you choose c=42, or c=1, either way you have a constraint applied to the rest of your choices of units. That's the important thing, not the units of c. It is certainly not true that c has to be unitless, indeed most of physics is not framed so as to make c unitless. It all depends on the choice of units that is in place, so there is always a mixture of nature and convention in any value of c. This is not true of the fine structure constant, for example. My point is, taking c=1 and not saying anything about the implied constraint is an error, but taking c = 1 light second per second avoids any such error because there is no unstated constraint on the rest of the units there.

The situation is entirely analogous to the common statement of Kepler's law that P² = a³. We know this means that if you measure P in years you must measure a in AU, so we understand that taking the constant that would otherwise appear in that formula to be unity requires those choice of units. Taking c=1 is no different, there has been no demonstration that there is anything more special in expressions for the relativistic line element than in Kepler's law.

 

[robphy]

Enlightening:
http://stefangeens.com/2001-2013/20...-iv-and-what-a-fine-structure-constant-it-is/
https://arxiv.org/abs/1412.2040 "How fundamental are fundamental constants?" M. J. Duff
http://arxiv.org/abs/physics/0110060 "Trialogue on the number of fundamental constants" M. J. Duff, L. B. Okun, G. Veneziano

The point is... the value of the fine-structure constant is truly dimensionless and does not require specifying a set of units [(say) to communicate with a distant civilization]... this is not true of the speed of light.

 

[PeterDonis]

The distance light travels in a time of one second.

Which is also the definition of a light-second. So I still don't understand why you think "seconds of distance" and "light-seconds of distance" are different units, so that somehow we can magically make $c$ dimensionless by using "seconds" as the distance unit, but we can't by using "light-seconds", which have exactly the same definition, as the distance unit.

 

[PeterDonis]

How do you setup a line element, using a different unit of measure for x0 and x1,x2, x3?

You add appropriate coefficients. For example, in many SR textbooks you will see the line element written as $ds^2 = - c^2 dt^2 + dx^2 + dy^2 + dz^2$. What's the problem?

 

[MeJennifer]

You add appropriate coefficients. For example, in many SR textbooks you will see the line element written as $ds^2 = - c^2 dt^2 + dx^2 + dy^2 + dz^2$. What's the problem?

So what is c*t?

It is distance!

 

[PeterDonis]

So what is c*t?

It is distance!

It's in units of distance. That doesn't make it a distance; it's still a time. It's just a time expressed in "distance units", because those are the units of the other terms in the line element. That's a convention about unit choice, not a matter of physics.

For example, we could just as easily write the line element with all time units, this way:

$$
d\tau^2 = dt^2 - \frac{1}{c^2} \left( dx^2 + dy^2 + dz^2 \right)
$$

That would not make the $x$, $y$, and $z$ terms times instead of distances. It would just mean we were using "time units" because we wanted the final answer, $d\tau$, to be in those units. It would just be a unit convention; it wouldn't change the physics.

 

[Dale]

Then permit me to clarify my simple meaning: I mean the outcome of a measurement on light that we regard as a speed measurement.

So how is "the actual speed of light" any different from "the speed of light"? It seems like you are just saying the same thing that has been said before. Specifically, you seemed concerned that we could use c=1 as "the speed of light" in most situations but in calculating the fine structure constant, $\alpha$, we had to use "the actual speed of light".

I realize you know this, but its value is given by e2 over h-bar c.

No, this is incorrect and is, I think, the key to your misunderstanding. Physical formulas typically depend on the choice of units. The formula that you quoted $\alpha = e^2/\hbar c$ is only true in CGS units. In SI units the expression is $\alpha = k_e e^2/\hbar c$. In natural units it is $\alpha = e^2/4\pi$. Along with this change in the formula for $\alpha$ is a change in the expressions for Maxwell's equations and QED in each system of units.

So if we are free to choose any system of units we like, with no regard to internal consistency, we can measure all charges in units of e, such that e2=1, all actions in h-bar, such that h-bar = 1, and all speeds in c, such that c = 1. Voila, the fine structure constant is now unity,

No, in these units that you propose, then the formula for the fine structure constant would be different, it would be something rather uninformative like $\alpha = k/4\pi$ where k is a factor specific to this set of units. This factor k would show up throughout the Maxwell's and QED equations.

It may be easier to think of a simpler example. You could do Newtonian physics in units of lb for force, kg for mass, furlongs for distance, and fortnight for time. This is an inconsistent set of units. Newton's 2nd law would take the form $f=kma$, and k would be some universal physical constant which would show up all over our equations.

So even inconsistent units will still not alter the fundamental dimensionless physical constants, like $\alpha$.

 

[Herman Trivilino]

Which is also the definition of a light-second. So I still don't understand why you think "seconds of distance" and "light-seconds of distance" are different units, so that somehow we can magically make $c$ dimensionless by using "seconds" as the distance unit, but we can't by using "light-seconds", which have exactly the same definition, as the distance unit.

Well, I could be wrong, but here's my thinking. Let's look at the square of the timelike interval: $(ct)^2-x^2$.

For this expression to make sense $(ct)$ and $x$ must have the same dimensions.

Now, if instead we write this same quantity as $t^2-x^2$ the same restriction holds. $t$ and $x$ must have the same dimensions.

The only way the two expressions can be equivalent is if $c=1$.

As an example, let's look at Mermin's way of writing $c$ as $1$ phoot per nanosecond, where the phoot is defined as $0.299\ 792\ 458$ meters. Using that system of units we cannot write the square of the timelike interval as $t^2-x^2$ because $t$ and $x$ have different dimensions. $x$ is measured in pheet and $t$ is measured in nanoseconds. Thus, one must write $(ct)^2-x^2$ so that $(ct)$ and $x$ have the same dimensions.

 

[PeterDonis]

let's look at Mermin's way of writing $c$ as $1$ phoot per nanosecond, where the phoot is defined as $0.299\ 792\ 458$ meters

And how is a meter defined? If it is defined as the length of a particular stick somewhere, then you are correct that the phoot and the nanosecond are different units (since the nanosecond is defined in terms of the second, which is defined in terms of a particular atomic transition frequency).

But if the meter is defined in the SI manner, as the distance light travels in a certain fraction of a second, then the phoot and the nanosecond are the same unit, just with two different names for historical reasons. With the phoot defined in this way (using the SI definition of the meter), the speed of light has to be $1$, a dimensionless number, because the definitions of the meter and the definition of the second are not independent, therefore the definitions of the phoot and the second aren't either.

The same can be said about SI units; SI units are defined (currently) in such a way that the speed of light has a constant value of 299,792,458. We usually quote this as having units of "meters per second", but this does not express our best value for the speed of light in terms of a relationship between two independently defined physical standards, which is what "different units" would mean. It just expresses the fact that, again for historical reasons, our "standard" units make $c$ be a really wacky dimensionless number, 299,792,458, instead of the obvious dimensionless number 1.

 

[Herman Trivilino]

And how is a meter defined? If it is defined as the length of a particular stick somewhere, [...]

No, we'll use the modern $\mathrm {SI}$ definition. The distance light travels in a time of $\frac{1}{299\ 792\ 458}\ \mathrm s$.

But if the meter is defined in the SI manner, as the distance light travels in a certain fraction of a second, then the phoot and the nanosecond are the same unit, just with two different names for historical reasons.

Ahhh... I see now what you're getting at.

The same can be said about SI units; SI units are defined (currently) in such a way that the speed of light has a constant value of 299,792,458. We usually quote this as having units of "meters per second", but this does not express our best value for the speed of light in terms of a relationship between two independently defined physical standards, which is what "different units" would mean. It just expresses the fact that, again for historical reasons, our "standard" units make $c$ be a really wacky dimensionless number, 299,792,458, instead of the obvious dimensionless number 1.

Well, if I remember correctly, in a recent GCPM they passed a resolution to move the definitions of more SI units to the same scheme used to define the meter. That is, rather than basing the definition on an artifact and measuring the values of fundamental constants, the scheme will be to set the values of fundamental constants and use that to define the units. For example, we'll likely soon see Avagadro's Number set to a fixed value that will be used to define the kilogram, rather than relying on an artifact for the definition of the kilogram and then measuring Avadadro's Number.

But, anyway, I will have to think about how I'm going to explain dimensionless numbers to students. Currently I'm fond of remarking that they are special in the sense that their value is independent of the units used to measure them. Clearly, if the speed of light is, as a result of the way the meter is defined, a dimensionless number that can apparently take on any value, I can no longer say that..

 

[PeterDonis]

I will have to think about how I'm going to explain dimensionless numbers to students.

You just have to be clear about which numbers actually are dimensionless numbers whose value is truly independent of your choice of units. The speed of light is not such a number, as you remark. The fine structure constant, to give one example, is.

 

[vanhees71]

The same can be said about SI units; SI units are defined (currently) in such a way that the speed of light has a constant value of 299,792,458. We usually quote this as having units of "meters per second", but this does not express our best value for the speed of light in terms of a relationship between two independently defined physical standards, which is what "different units" would mean. It just expresses the fact that, again for historical reasons, our "standard" units make $c$ be a really wacky dimensionless number, 299,792,458, instead of the obvious dimensionless number 1.

Well, one should note that the SI units are defined to be convenient for engineering and trade in everyday-life and not for the beauty of theoretical physics. Usually SI units lead to ugly equations that hide the beauty and hinders physics intuition by a great deal. The worst example is electromagnetics in SI units, where all the beauty of the relativistic covariant (quantum) field theory is hidden under clumsy conversion factors $\epsilon_0$ and $\mu_0$. The only conversion factor that has physical relevance is the speed of light in vacuo, $c$, and this is in addition best set to $c=1$ and then measuring lengths and times in the same units, as just discussed here.

 

[Herman Trivilino]

You just have to be clear about which numbers actually are dimensionless numbers whose value is truly independent of your choice of units. The speed of light is not such a number, as you remark.

It's the only one, I think. For now at least, as I remarked above. According to official $\mathrm{SI}$ literature, the ratio $\mathrm{\frac{m}{s}}$ is formed by combining two of the seven independent base units. As such it is not considered by them to be dimensionless.

 

[PeterDonis]

According to official $\mathrm{SI}$ literature, the ratio $\mathrm{\frac{m}{s}}$ is formed by combining two of the seven independent base units.

Yes, but this is because they are using the word "independent" in a totally unusual way, since the definition of the meter depends on the definition of the second and so is not independent of it in any normal sense of that word.

 

[Herman Trivilino]

Yes, but this is because they are using the word "independent" in a totally unusual way, since the definition of the meter depends on the definition of the second and so is not independent of it in any normal sense of that word.

Right. So they will have to move forward from this antiquated way of speaking about the very units they are defining, especially as they continue with their efforts to no longer define the base units in terms of artifacts.

 

[burakumin]

May I propose a different point of view about units and physical constants? I could start an explanation by myself, but I guess a field medalist will give a better exposition:

https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/

However, as any student of physics is aware, most physical quantities are not represented purely by one or more numbers, but instead by a combination of a number and some sort of unit. For instance, it would be a category error to assert that the length of some object was a number such as

; instead, one has to say something like “the length of this object is

yards”, combining both a number

and a unit (in this case, the yard).

 

[ljagerman]

I use c "equal to 1" for convenience when I want to calculate a time in "how many YEARS," and/or I want to calculate a distance in "how many LIGHT YEARS." Remember that when c is set equal to 1, it means that c is a measure of speed wherein c = 1 lightyear/year. Of course: by definition a photon will go 1 light year in a year moving at the speed of light. (Thus you could say that the speed of a car is 1 when it goes 1 mile in 1 minute, and that would be an efficient way to calculate miles driven and/or minutes elapsed for a given car.)

This kind of calculation comes up for space vehicles when special relativity is used (gravitational fields and the expansion of the universe are ignored). E.g., calculate how far IN LIGHT YEARS ("d") a space vehicle will travel while accelerating for a given number of YEARS ("T," proper time on the vehicle) at a constant acceleration ("a') of 1.03 LIGHT YEARS per YEAR per YEAR. That particular acceleration conveniently happens to be Newton's g force, so that practically a = 1 g.

With c = 1, an easy way to solve for d in LIGHT YEARS is to use the trig function cosh (the trajectory is hyperbolic), and the equation is

d = (c squared/a)(cosh[aT/c] -1).

You can get cosh on most scientific calculators that have trig functions, and since c = 1, the solution requires only a few key strokes. (You should get d = 0.56 LIGHT YEARS after one YEAR.)

And if you wanted to know how many years "t" elapsed (in "coordinate time;" think twins paradox) meanwhile on earth, you can use sinh:

t = (c/a)(sinh[aT/c]), again a piece of cake when c = 1.

(You can avoid hyperbolic trig functions, but the algebra is much harder [uses forms of the Lorenz contraction]. But even this is easier with c = 1.)

 

[burakumin]

Actually I think I can hardly make sense of a paradigm where c=1 because it implies that physical quantities are mere numbers. But apparenlty many people here find this natural. Isn't this due to historical reasons? I seemed to me this view tended to become obsolete now and had a lot of drawbacks.

 

[Ibix]

Measure distance in feet anf time in nanoseconds. Or distance in light years and time in years. Or distance in light anything and time in the same anything. In all cases the speed of light is 1. We can, as long as we are confident in our algebra, simply ignore it. We can always put it back in by dimensional analysis if we wish to switch to units where it is not 1.

I don't think anyone is suggesting ignoring the dimensionality of c. We are simply selecting units where we can ignore it numerically.

 

[burakumin]

Measure distance in feet anf time in nanoseconds. Or distance in light years and time in years. Or distance in light anything and time in the same anything. In all cases the speed of light is 1.

I have a different opinion. It's not 1, it's 1 light year per year, or in other words 1 c. If you drop completely the unit then you're commiting a category error (the same kind of error as equating a number with a vector for example). If you don't then you're just stating a tautology. c is an object that is irreductible tot 1 or to any other number.

I don't think anyone is suggesting ignoring the dimensionality of c. We are simply selecting units where we can ignore it numerically.

"ignore it nummerically" implies that you consider "c" is a number in the first place. What precisely I don't agree with.

 

[Ibix]

I have a different opinion. It's not 1, it's 1 light year per year, or in other words 1 c. If you drop completely the unit then you're commiting a category error (the same kind of error as equating a number with a vector for example). If you don't then you're just stating a tautology. c is an object that is irreductible tot 1 or to any other number.

But a category error without consequence, as long as I pick units where the numerical value of c is 1. If I care about it I can always re-insert the c and G at any point because it can be uniquely determined by dimensional analysis (assuming I didn't make any mistake in the algebra) with no effect on the numbers.

If you prefer, you can consider the c taken into the units. So I measure distance in feet and time in feet/c, which has the correct dimensionality and and the numerical component of which is equal to the time measured in ns. A velocity in these units ends up having units of c.

"ignore it nummerically" implies that you consider "c" is a number in the first place. What precisely I don't agree with.

At some point I'm going to put the numbers into the calculator or computer, and I will only be putting in the numbers. I save myself some work if I pick units where I don't have to keep track of powers of c and G because I know their numerical value is 1.

 

[burakumin]

But a category error without consequence, as long as I pick units where the numerical value of c is 1.

A consequence is apparently the existence of endless debates and incompréhensions on the nature of physical quantities, objects and equations.

If you prefer, you can consider the c taken into the units. So I measure distance in feet and time in feet/c, which has the correct dimensionality and and the numerical component of which is equal to the time measured in ns. A velocity in these units ends up having units of c.

This is what I already do. I don't think we disagree here.

At some point I'm going to put the numbers into the calculator or computer, and I will only be putting in the numbers. I save myself some work if I pick units where I don't have to keep track of powers of c and G because I know their numerical value is 1.

Sure but there exists different perspectives. In your example (and in general) it seems you're mainly concerned with computational aspects. I'm more interested in conceptual ones. An image also needs to be encoded into numbers to be handled by a computer. There are various formats and encodings. But certainly you would not explain to someone (a child for example) that an image is a certain sequence of numbers according the jpeg format. This should be the same for physical quantities and it appears to me that several comments in this thread refer to the nature of physical concepts. So in the end I agree that choosing units such that c has numerical value 1 may simplify calculation. But I certainly do not agree with the statement already proposed here that it clarifies physics.

 

[Dale]

Actually I think I can hardly make sense of a paradigm where c=1 because it implies that physical quantities are mere numbers

I would instead say it implies that physical units are mere conventions, which is correct.

If you drop completely the unit then you're commiting a category error (the same kind of error as equating a number with a vector for example).

Not necessarily. It depends on your system of units. In some systems of units c is a dimensionful 1 (i.e. in Planck units c = 1 Planck length / Planck time), but in other systems of units it is dimensionless (e.g. in geometrized units c = 1).

The dimensionality of a quantity is not given by Nature, it is a man-made convention that we can change as desired within some logical limits.

 

[burakumin]

I would instead say it implies that physical units are mere conventions, which is correct.

I do not agree. Statements "physical units are mere conventions" and "physical quantities are numbers" are not equivalent. "c=1" implies at least that speeds can be identified absolutely to number which is certainly stronger than just asserting that "meter/second" is no more special than "inch/year" or "parsec/hour".

The dimensionality of a quantity is not given by Nature, it is a man-made convention that we can change as desired within some logical limits.

This is a strong philosophical statement. Now I think this sentence could be understood in different manners (from the weakest to the strongest):
- The exact dimensionality of certain quantities can be arbitrary chosen within some contraints (possibly as dimensionless in some cases)
- There are dimensionless and dimensionful quantities but distinctions of kind between dimensionful quantities is arbitrary (so a ratio between two of them can always be thought as a number).
- The whole notion of physical dimensionality is arbitrary so we could reduce any quantity to a number in an absolute manner.
Did you imply one of them (or something else I didn't think of) ?

 

[PeterDonis]

The dimensionality of a quantity is not given by Nature

Strictly speaking, this is only true of quantities that are not "natural" constants. Quantities that are, for example the fine structure constant, do have dimensionality that is "given by Nature" (namely that they are dimensionless and have the same value regardless of any human choice of units).

 

[SiennaTheGr8]

I do not agree. Statements "physical units are mere conventions" and "physical quantities are numbers" are not equivalent. "c=1" implies at least that speeds can be identified absolutely to number which is certainly stronger than just asserting that "meter/second" is no more special than "inch/year" or "parsec/hour".

But speeds can be identified absolutely by number: as a fraction of the universal speed limit. Hence the $v/c$ that we find everywhere in special relativity, no?

 

[Dale]

Strictly speaking, this is only true of quantities that are not "natural" constants. Quantities that are, for example the fine structure constant, do have dimensionality that is "given by Nature" (namely that they are dimensionless and have the same value regardless of any human choice of units).

I like that. I had never thought of that, but you are correct. No system of units can assign dimensions to the fine structure constant, etc.

 

[Dale]

Statements "physical units are mere conventions" and "physical quantities are numbers" are not equivalent.

I agree that they are not equivalent. That is why I would say the first one and not say the second one.

"c=1" implies at least that speeds can be identified absolutely to number which is certainly stronger than just asserting that "meter/second" is no more special than "inch/year" or "parsec/hour".

Yes. And while this is true for speeds it is not true for all physical quantities, which is why I would not say "physical quantities are mere numbers". At most "some physical quantities are mere numbers".

Did you imply one of them (or something else I didn't think of) ?

I am not sure, but perhaps it is easier to speak of concrete examples.

Are you aware of cgs units, which are often used in electromagnetics? In cgs units the unit of electric charge is the statcoulomb. The statcoulomb is not a base unit, but a derived unit equal to $1 cm^{3/2} g^{1/2} s^{-1}$. So the dimensionality of the statcoulomb is $L^{3/2} M^{1/2} T^{-1}$, which differs from the dimensionality of electric charge in SI. So the dimensionality of electric charge is a matter of convention.