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

[Herman Trivilino]

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

Hardly. If physical quantities were mere numbers there'd be no need for the $\mathrm{SI}$ and the attendant science of metrology. But it is the metrologists who have set things up so that $c$ is now dimensionless. The fact that it can be expressed as $1$ or as $299\ 792\ 458 \ \mathrm{m/s}$ or indeed as any number at all tells you that as a physical quantity it is far more than a mere number.

 

[Herman Trivilino]

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

If that's true then I've completely misunderstood Post #71.

 

[PAllen]

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?

This is traditionally called beta. It is dimensionless, and, for a given object, is the same in all systems of units. For precisely this reason, it is not a speed. On expects that a speed expressed in meters/second should be a different number from the same speed expressed in feet per hour. Thus, there is a difference between a speed expressed in units where c=1 (it still has units, e.g. light seconds per second), while beta has no units.

 

[PAllen]

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.

I don't completely agree with this. Distance is defined in terms of light speed and time, as a distance traveled; time is not defined in reference to light speed. To me, this makes it a separate unit. Lightspeed has a defined value, but this value has units, allowing it to act as conversion factor from, e.g. seconds to meters. [edit: one could have units where there is no distance unit, e.g. distances are measured as the time light takes to travel. Then there is only a base unit of seconds, rather than separate base units of meters and seconds. But that is not what SI does].

 

[PeterDonis]

Distance is defined in terms of light speed and time, as a distance traveled; time is not defined in reference to light speed. To me, this makes it a separate unit.

This is a question about how you want to use the expression "separate unit". My point is that the SI unit of distance is ultimately based on the same physical standard--cesium clocks--as the SI unit of time. Whereas, before that change was made, the SI unit of distance was based on a different physical standard--a standard meter stick--than the SI unit of time.

Lightspeed has a defined value, but this value has units, allowing it to act as conversion factor from, e.g. seconds to meters.

But this is just a matter of nomenclature. We only call the distance unit a "meter" instead of a "second" for historical reasons. There is no separate physical standard in SI for what a "meter" is; it's just a particular fraction (1/299792458) of a second. The denominator of that fraction has "units" of "meters per second", again, for historical reasons, because we are still using the term "meter" to denote 1/299792458 of a second, instead of just calling it a fraction of a second.

one could have units where there is no distance unit, e.g. distances are measured as the time light takes to travel. Then there is only a base unit of seconds, rather than separate base units of meters and seconds. But that is not what SI does

Yes, it is. It wasn't before the latest change to the SI meter was made, but it is now; a meter is 1/299792458 of a second, as above. It has to be, because there is only one physical standard in use: cesium clocks. The fact that our standard terminology obfuscates this fact does not mean it isn't a fact.

If you mean that, in practice, we don't use cesium clocks to measure distance, we use rulers, that's true, but it's irrelevant when we're talking about how SI units are defined. If I have a meter stick that claims to measure exactly one SI meter, that claim is strictly speaking unjustified unless I have a cesium clock and a way of timing light traveling from end to end of my meter stick to verify that it takes exactly 1/299792458 of a second according to the clock. Otherwise the stick is not measuring SI meters; it's measuring something that, for the practical purpose for which I'm using it, is equivalent to SI meters, but it's still not the same thing.

 

[SiennaTheGr8]

This is traditionally called beta. It is dimensionless, and, for a given object, is the same in all systems of units. For precisely this reason, it is not a speed. On expects that a speed expressed in meters/second should be a different number from the same speed expressed in feet per hour. Thus, there is a difference between a speed expressed in units where c=1 (it still has units, e.g. light seconds per second), while beta has no units.

I disagree with you about $\beta$ not being a "speed," but I'm sure we're just arguing semantics. Doubtless you'll agree that $v$ and $\beta$ measure the same physical quantity in different units, much like $E_0$ and $m$ do.

 

[SiennaTheGr8]

But I'd argue further that there's a conceptual benefit to conceiving of speeds as dimensionless fractions of the universal speed limit. I don't think of $\beta$ as "shorthand" for anything. It's $v = \beta c$ that's unnatural, an artifact from when we didn't know what we know now.

 

[PAllen]

I disagree with you about $\beta$ not being a "speed," but I'm sure we're just arguing semantics. Doubtless you'll agree that $v$ and $\beta$ measure the same physical quantity in different units, much like $E_0$ and $m$ do.

No, beta is dimensionless no matter what system of units you use, while E0 and m0 are measures with dimension in most systems of units. There is a family of units where beta and speed have the same value. You are free to insist that these units are conceptually preferred with all others being inferior. I am free to reject your insistence, in favor of the view that different units are convenient for different purposes.

 

[PAllen]

This is a question about how you want to use the expression "separate unit". My point is that the SI unit of distance is ultimately based on the same physical standard--cesium clocks--as the SI unit of time. Whereas, before that change was made, the SI unit of distance was based on a different physical standard--a standard meter stick--than the SI unit of time.

I do not see it as being defined just by the cesium clock. It is also defined by the physical speed of light. That it is defined so as to give this speed a particular value does not remove the extra element in its definition. I don't know if you are aware that there was an interim period when the meter was defined in terms of wavelengths of a standard light emission, with the second (as now) being defined in terms of period of a standard emission. IMO, the newest definition is just streamlining the wavelength definition by virtue of using speed and period in place of wavelength and period. Any two of these are equivalent to all three. Any one of these is NOT equivalent to all three.

But this is just a matter of nomenclature. We only call the distance unit a "meter" instead of a "second" for historical reasons. There is no separate physical standard in SI for what a "meter" is; it's just a particular fraction (1/299792458) of a second. The denominator of that fraction has "units" of "meters per second", again, for historical reasons, because we are still using the term "meter" to denote 1/299792458 of a second, instead of just calling it a fraction of a second.

I disagree. See above.

Yes, it is. It wasn't before the latest change to the SI meter was made, but it is now; a meter is 1/299792458 of a second, as above. It has to be, because there is only one physical standard in use: cesium clocks. The fact that our standard terminology obfuscates this fact does not mean it isn't a fact.

No, it is the distance traveled by light in a vacuum in 1/299792458 seconds. See the difference? I do. We have to actually use the physical speed of light to get the distance. If we didn't use light in a vacuum, we wouldn't be able to get the distance from the time (which comes from the cesium clock, which counts period, not speed).

 

[SiennaTheGr8]

No, beta is dimensionless no matter what system of units you use, while E and m0 are measures with dimension in most systems of units. There is a family of units where beta and speed have the same value. You are free to insist that these units are conceptually preferred with all others being inferior. I am free to reject your insistence, in favor of the view that different units are convenient for different purposes.

Setting $c=1$ likewise gives $E_0$ and $m$ the same value. I really don't think we're disagreeing on anything substantial here.

Cheers.

 

[PeterDonis]

I don't know if you are aware that there was an interim period when the meter was defined in terms of wavelengths of a standard light emission, with the second (as now) being defined in terms of period of a standard emission.

Yes, I'm aware of that.

IMO, the newest definition is just streamlining the wavelength definition by virtue of using speed and period in place of wavelength and period. Any two of these are equivalent to all three. Any one of these is NOT equivalent to all three.

Hmm. I see what you're saying. I still don't think it's the same as defining "separate units" by using, say, a standard meter stick for distance and a cesium clock for time, but I'll agree that the physical speed of light does provide a second standard.

 

[vanhees71]

Again, the fundamental constants $c$, $\hbar$, and $G$ are mere conversion factors between units. Setting them to 1, makes all quantities dimensionless, and everything is measured in "natural units". That's of course impractical to handle. Thus one defines various different systems of units depending on the application you are working on.

You can see this on the example of electromagnetics. There the only fundamental constant appearing in the equations is the speed of light, $c$ (i.e., the phase velocity of electromagnetic waves in a vacuum). However, due to practicality the SI has chosen to introduce an additional unit, the Ampere for electric currents (to be changed very soon by defining the elementary charge, but that doesn't matter here too much), which introduces additional conversion factors, namely $\epsilon_0$ and $\mu_0$. The relation to the physical units is $\mu_0 \epsilon_0=1/c^2$.

 

[Herman Trivilino]

This is traditionally called beta. It is dimensionless, and, for a given object, is the same in all systems of units. For precisely this reason, it is not a speed. On expects that a speed expressed in meters/second should be a different number from the same speed expressed in feet per hour. Thus, there is a difference between a speed expressed in units where c=1 (it still has units, e.g. light seconds per second), while beta has no units.

The central issue in this thread for me is whether it's possible, by a choice of systems of units, to set $c$ identically equal to the dimensionless $1$. Note, for example, that that is what (at least appears to me that) Taylor Wheeler do in Spacetime Physics.

For example, if we choose a system where distance is measured in meters and time in light meters (the time it takes light to travel a distance of one meter, then $c=1 \mathrm {\ meter\ per\ light\ meter}$. On the other hand, if we choose a system where distance is measured in meters and time in meters (the time it takes light to travel a distance of one meter) then $c=1$.

If I interpret what you're saying correctly, it can all be seen as a matter of semantics. If we insist as you do that $\beta$ is not a speed, then I need to give it another name, such as dimensionless speed, or whatever term seems suitable, to make my point. My central issue then becomes whether we have a system of units where the speed $v$ of light is $1 \mathrm{\ unit\ of\ distance\ per\ unit\ of\ time}$ or whether we have a system where the dimensionless speed $\beta$ of light is $1$.

Of course, it all seems silly when reduced to a matter of semantics. But it seems that's all there is to it!

 

[vanhees71]

The central issue in this thread for me is whether it's possible, by a choice of systems of units, to set $c$ identically equal to the dimensionless $1$. Note, for example, that that is what (at least appears to me that) Taylor Wheeler do in Spacetime Physics.

This is a no-brainer. We (in the high-energy heavy-ion community) use this natural system of units all the time, and it works: There we have $\hbar=c=k_{\text{B}}=1$. There's only one unit left, usually GeV. For convenience we also use fermi (fm) for lengths and times. The key to go from one to the other base unit is $\hbar c=0.197 \text{GeV} \text{fm}$. That's all you need in this field.

Of course the choice of the system of units is arbitrary and thus semantics. A change from one to another system doesn't change the physics.

 

[burakumin]

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.

No I didn't but as far as I understand this is equivalent to stating that the electric constant is a number. So I think this example and the debate about $c$ can be summed up with a single question: To what extent can we consider fundamental physical constants like $c, \varepsilon_0, h, G, e, m_e, \dots$ as naturally identifiable to (dimensionless) numbers (and when we can, to which numbers precisely)? The natural choice would be to set them to 1 (but by the way, which one exactly? Should we set $h = 1$ or $\hbar = 1$ ? $\varepsilon_0 = 1$ or $\mu_0 = 1$ ?).
I think @PeterDonis's remark about the fine structure constant has important consequences. This means in particular that $\frac{e^2}{\varepsilon_0 \cdot h}$ is a speed and it's entirely made up of fundamental constants. Now $\frac{G \cdot m_e^2}{h}$ is also one. And then $\frac{1}{h} \cdot \sqrt[n+m]{ \frac{G^m \cdot m_e^{2m} \cdot e^{2n}}{\varepsilon_0^n} }$ are also speeds for all $n, m$ in $\mathbb{Z}^*$.
So if we have to identifies fundamental constants to numbers, it suddently becomes unclear (at least to me) which one of these many non-equivalent combinations should be set to 1.

 

[Dale]

(and when we can, to which numbers precisely)?

(but by the way, which one exactly? Should we set h=1h=1h = 1 or ℏ=1ℏ=1\hbar = 1 ? ε0=1ε0=1\varepsilon_0 = 1 or μ0=1μ0=1\mu_0 = 1 ?).

it suddently becomes unclear (at least to me) which one of these many non-equivalent combinations should be set to 1.

In all of these, we choose whichever is most convenient for the application we have in mind. That is the great thing about a good convention: it makes things easier.

 

[PAllen]

The central issue in this thread for me is whether it's possible, by a choice of systems of units, to set $c$ identically equal to the dimensionless $1$. Note, for example, that that is what (at least appears to me that) Taylor Wheeler do in Spacetime Physics.

For example, if we choose a system where distance is measured in meters and time in light meters (the time it takes light to travel a distance of one meter, then $c=1 \mathrm {\ meter\ per\ light\ meter}$. On the other hand, if we choose a system where distance is measured in meters and time in meters (the time it takes light to travel a distance of one meter) then $c=1$.

If I interpret what you're saying correctly, it can all be seen as a matter of semantics. If we insist as you do that $\beta$ is not a speed, then I need to give it another name, such as dimensionless speed, or whatever term seems suitable, to make my point. My central issue then becomes whether we have a system of units where the speed $v$ of light is $1 \mathrm{\ unit\ of\ distance\ per\ unit\ of\ time}$ or whether we have a system where the dimensionless speed $\beta$ of light is $1$.

Of course, it all seems silly when reduced to a matter of semantics. But it seems that's all there is to it!

I think an earlier post by Peter put this best. Some fundamental constants are dimensionless no matter what units you use. Furthermore, their value is independent of units. Most physicists accept that these are the only true fundamental constants. For other constants, different systems of units determine both the value and the units of the constant. Thus, there are systems of units where c is a dimensionless 1, and others where it has a value of 1 with dimensions. In contrast, the fine structure constant is about 137 AND dimensionless in ALL systems of units. But c remains in the category of dimensionful constant, because its value and dimensions are not independent of unit choice.

Beta versus speed is similar: beta is dimensionless, and has the same value (for a given object), in all systems of units. Speed of that object will have different dimensions and values depending on system of units. You can construct systems of units where speed has the same value as beta and is dimensionless. However speed remains in the category of dimensionful parameter (because it has dimensions in many systems of units), while beta is a dimensionless parameter (because this feature is independent of units).