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

[Dale]

Presumably, for any particular choice of units consistent with c=1 there is are corresponding values for various physical constants, like G for example.

Usually when people get comfortable with the idea of picking units where c=1 they also pick units where G=1 and h=1, etc. Such units are called natural units
https://en.m.wikipedia.org/wiki/Natural_units

The prototypical example of natural units is Planck units
https://en.m.wikipedia.org/wiki/Planck_units

Although my personal favorite is geometrized units
https://en.m.wikipedia.org/wiki/Geometrized_unit_system

 

[vanhees71]

Are there specific recommended standard units when c = 1? I would guess these units are seconds, kilo-grams, and light-seconds. Is that correct?

Usually this system of units, where $\hbar=c=1$, is used in high-energy particle and nuclear physics. Then the usual units used are GeV for masses, energies, momenta and fm for times and lengths. Of course, there's in principle only one independent base unit left. The only conversion factor you need to remember is $\hbar c \simeq 0.197 \;\text{GeV} \, \text{fm}$.

 

[Mister T]

Are there specific recommended standard units for c=1?
I would guess they are

time: seconds
mass: kilograms
distance: light-seconds​

Is that correct, making speed units light-seconds per second?

Strictly speaking, if you use seconds for time and light-seconds for distance, you get $c=1$ light-second per second.

If you really want a system where $c=1$, that is, a dimensionless quantity identically equal to $1$ then you must measure distance and time in the same units. So, for example, seconds of time and seconds of distance.

N. David Mermin proposes units of nanoseconds for time and the phoot for a unit of distance. Where the phoot is 0.299 792 458 meters (the foot is 0.3048 meters). In this system light speed is $1$ phoot per nanosecond. Not identically equal to the dimensionless $1$.

 

[vanhees71]

There's no need for new units. What should this be good for. You simply set $c=1$ and then measure lengths and times in some unit appropriate for your problem. In HEP it's fm (see my previous postings in this thread).

 

[PeterDonis]

If you really want a system where c=1c=1c=1, that is, a dimensionless quantity identically equal to 111 then you must measure distance and time in the same units. So, for example, seconds of time and seconds of distance.

How would you measure distance in seconds?

 

[MeJennifer]

How would you measure distance in seconds?

Easy, you express a unit of distance as the path traveled by light in vacuum for a given time interval. In fact, this is exactly how the meter is currently defined.

 

[The Bill]

How would you measure distance in seconds?

One second of distance is the same as 300,000,000 meters, or one light second. Likewise, there are 300,000,000 meters of time in one second of time. It's just a conversion factor of c or 1/c.

Essentially, this is why a relatively small curvature of spacetime can create the amount of gravitational effect we experience all the time. Throw a ball upward at about 4.9m/s, and it will go up about 1.2m, and back down to your hand about 1.2m, over a time of about 1s. The 300,000,000m of time the ball traverses is enough for the curvature of spacetime to cause it to fully curve back to about the same space coordinates as it's launch point. The total geodesic length is near enough to 300,000,000m as doesn't matter with the low precision I'm using.

 

[PeterDonis]

One second of distance is the same as 300,000,000 meters, or one light second. Likewise, there are 300,000,000 meters of time in one second of time. It's just a conversion factor of c or 1/c.

I already know all this (and so does everyone else in this thread--you should read through the entire thread before posting). I am asking Mister T because I want him to defend his contention (which I disagree with) that saying the speed of light is "1 light-second per second" is somehow different from saying that the speed of light is 1, a dimensionless number.

 

[Dale]

I am asking Mister T because I want him to defend his contention (which I disagree with) that saying the speed of light is "1 light-second per second" is somehow different from saying that the speed of light is 1, a dimensionless number.

I do agree with @Mister T on this point. Here you have to distinguish between the unit and the dimensionality of the unit.

For example, in SI units the Coulomb is the unit of charge. It is a base unit with dimensions of charge. In Gaussian units the statcoulomb is the unit of charge, but it is not a base unit and instead has dimensions of length^(3/2) mass^(1/2) time^(-1). So the dimensionality of a quantity depends on your system of units.

Thus, in Planck units c=1 Planck length/Planck time is a quantity with dimensions of length/time. In contrast, in geometrized units c=1 is a dimensionless quantity.

It is entirely a matter of convention, with no impact on the physics, but we are free to adopt a convention where length and time are different dimensions such that c is a dimensionful quantity whose magnitude is 1.

 

[Mister T]

If, for example, you want to define a timelike interval as $t^2-x^2$ you must measure $x$ and $t$ in the same units. In such a system $c=1$.

Measuring $x$ in light-seconds and $t$ in seconds won't do. You would instead have to write $(ct)^2-x^2$ where $c=1$ light-second per second. Otherwise the units won't work out.

 

[PeterDonis]

Measuring $x$ in light-seconds and $t$ in seconds won't do.

Why not? You are asserting that light-seconds and seconds are somehow different units; that you would have to measure distance in "seconds" to make $c$ dimensionless. I have asked you once already how you would measure distance in seconds. Do you have an answer?

 

[PeterDonis]

in Planck units c=1 Planck length/Planck time is a quantity with dimensions of length/time. In contrast, in geometrized units c=1 is a dimensionless quantity.

I understand the distinction you are making. I just don't think it's the distinction Mister T is making.

In geometrized units, c=1 is dimensionless because we define the units of length and time to be the same. For example, MTW uses centimeters for both. But we still measure centimeters of time with clocks, not rulers. We just calibrate our clocks so that one centimeter of time is the time it takes light to travel one centimeter of distance. So this centimeter of time could just as well be called a "light-centimeter".

However, Mister T, as I read him, would object to this. He would say that this "light-centimeter" of time is a different unit from a centimeter of distance, so he would not agree that c=1 is dimensionless in geometrized units. I don't understand why not, hence my questions to him to try to clarify his position.

 

[Dale]

You may be right, I cannot speak for @Mister T and may be assuming something wrong about his position.

But we still measure centimeters of time with clocks, not rulers

This is a very good point. I am sure that there are some people who, seeing that fact, would insist that therefore the units must have fundamentally different dimensions. That anything measured with a ruler must have dimensions different from anything measured with a clock.

Back in the sea faring days they measured vertical distances with a rope and horizontal distances with a sextant or with a combination of a rope and a clock.

 

[vanhees71]

Well, you can measure distances with a clock, as it is defined in the SI units. Quantities are not defined by one specific operational way to measure them but by an equivalence class of various ways (maybe even some future methods not yet developed or known).

 

[PeterDonis]

you can measure distances with a clock, as it is defined in the SI units.

SI units don't say you measure distances with a clock. They say you calibrate rulers with a clock. That's not quite the same thing.

That said, I agree that units are not defined by one particular operational measurement. I am perfectly fine with saying that seconds and light-seconds, for example, are the same unit so the speed of light in these units is the dimensionless number 1. I'm trying to understand from Mister T why he objects to that.

 

[Ken G]

Yet we should be aware that the speed of light is indeed a fundamental constant of nature. So this means that we can always choose a unit system that makes c=1, but we must not forget that we have indeed chosen a unit system to accomplish that. We must be consistent in choices like that-- somewhere in the backs of our minds we must keep track that this unit choice is in place. Put differently, any individual constant that has units can be made to have any particular value by choosing those units, but the unitless combinations of those constants must keep their same value in any self-consistent unit system. So the place where we need to include the actual speed of light is when c appears in unitless combinations with the other fundamental constants, to make sure we get the right value for those unitless combinations. Only quantities that do not have units have values that are fundamental to the physics. I believe that might be the objection of not explicitly calling c 1 light second per second, it can look like one is implying that c is one of those fundamentally unitless combinations of physical parameters-- which it is not. I think it would be fair to say that taking c=1 is really doing nothing more than deciding to drop all c's, out of convenience, knowing you can always recover them just by looking at the units of the expressions.

 

[Dale]

So the place where we need to include the actual speed of light is when c appears in unitless combinations with the other fundamental constants, to make sure we get the right value for those unitless combinations.

First, I have no idea what you could possibly mean by "the actual speed of light". Second, there is nothing whatsoever that you can do in choosing your system of units which will mess up or in any way alter any of the dimensionless fundamental constants.

Making c be unitless does not suddenly give units to the fine structure constant, and making c have a magnitude of 1 does not change its value. No matter what unit conventions you choose. It simply cannot happen.

 

[Ken G]

First, I have no idea what you could possibly mean by "the actual speed of light"].

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

Second, there is nothing whatsoever that you can do in choosing your system of units which will mess up or in any way alter any of the dimensionless fundamental constants.

Of course that's wrong as you stated it, because you included no provision for making the unit system internally consistent. Consider the quantity that we call the fine structure constant. This is one of those fundamental unitless combinations of which I spoke. I realize you know this, but its value is given by e² over h-bar c. 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, and perturbation theory doesn't work any more. What went wrong? A unit system like the one I just made cannot be made internally consistent. A general fact is that only unit systems that maintain the physically established values of fundamental unitless combinations of the physical constants can be internally consistent, and that's what I am talking about. Just saying c=1 leaves that rather unclear.

Of course, one can take c=1 self-consistently in part of what are called "natural units," where we also take h-bar = 1, but we cannot take e=1 in those units. We must take the value of e that gives the right result for the fine structure constant. So that's what I'm talking about, we always have to have an entire unit system in the backs of our minds, and it must be internally consistent, if we are setting c=1. Saying that the units of c is 1 light second per second is the way to keep track of that implicit unit system that we have in the backs of our minds, we are measuring distance in light seconds and time in seconds. We don't have to write that implicit choice in all our formulae, as it would get tedious to write 1 light second per second, but we do have to keep track of the fact that this is the unit system we are using.

Making c be unitless does not suddenly give units to the fine structure constant, and making c have a magnitude of 1 does not change its value. No matter what unit conventions you choose. It simply cannot happen.

One requires a consistent unit system, even if one says one is taking c=1. That statement by itself is not enough, you really do need a consistent system. Saying c is 1 light second per second is a way to keep track of the chosen unit system. One can not bother to explicitly keep track that way, but it is what one is doing, all the same, or one is risking an inconsistent unit system.

 

[MeJennifer]

Making c be unitless ...

c is unitless otherwise it would not even be a fundamental constant!

 

[robphy]

c is unitless otherwise it would not even be a fundamental constant!

Please back up this claim with some details and/or some definitions [which may be different than more standard definitions].
(Let's forget the "fundamental" aspect for now.)

If c is unitless [which I interpret as dimensionless], can you please provide its value?

A familiar dimensionless constant is the https://en.wikipedia.org/wiki/Fine-structure_constant
whose accepted value is 1/137.035... , independent of the system units used.

So, @MeJennifer, What is the value of c?

 

[MeJennifer]

Please back up this claim with some details and/or some definitions [which may be different than more standard definitions].
(Let's forget the "fundamental" aspect for now.)

If c is unitless [which I interpret as dimensionless], can you please provide its value?

A familiar dimensionless constant is the https://en.wikipedia.org/wiki/Fine-structure_constant
whose accepted value is 1/137.035... , independent of the system units used.

So, @MeJennifer, What is the value of c?

Will Schutz suffice?

https://books.google.com/books?id=V...l now do is adopt a new unit for time&f=false

 

[robphy]

Will Schutz suffice?

https://books.google.com/books?id=V1CGLi58W7wC&pg=PA4&lpg=PA4&dq=What+we+shall+now+do+is+adopt+a+new+unit+for+time&source=bl&ots=aIn_gpmSoI&sig=LaATax8b0FUymjo98gc8D3cpeMg&hl=en&sa=X&ved=0ahUKEwjPy9j6vsTNAhUSzWMKHTpRCZIQ6AEIHjAA#v=onepage&q=What we shall now do is adopt a new unit for time&f=false

Thanks for the clarification.
So, as Schutz says
"if we consistently measure time in meters, then c is not merely 1, it is also dimensionless!"

While I agree,
that assumption "if we consistently measure time in meters... [or some equivalent]" must accompany the statement that "c is dimensionless".
This is more restrictive that what needs to be said for the fine-structure constant... no analogous assumption is needed.

 

[MeJennifer]

I suppose alternatively we could set c at i. SR would work just fine but in GR we would end up with imaginary metric components, nobody does that (or can even handle that).

 

[MeJennifer]

Thanks for the clarification.
So, as Schutz says
"if we consistently measure time in meters, then c is not merely 1, it is also dimensionless!"

While I agree,
that assumption "if we consistently measure time in meters... [or some equivalent]" must accompany the statement that "c is dimensionless".
This is more restrictive that what needs to be said for the fine-structure constant... no analogous assumption is needed.

I would wonder how else would you do it?
How do you setup a line element, using a different unit of measure for x₀ and x₁,x₂, x₃?

 

[Mister T]

Why not? You are asserting that light-seconds and seconds are somehow different units;

Yes. Hence they have different names.

that you would have to measure distance in "seconds" to make $c$ dimensionless.

If you measure time in seconds, and you want to make $c=1$, then yes.

I have asked you once already how you would measure distance in seconds. Do you have an answer?

The distance light travels in a time of one second.

 

[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 $\alpha$.

 

[Mister T]

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.