# I Why use units in which c=1?

#### PeterDonis

Mentor
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.

#### Mister T

Gold Member
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/

a short extract said:
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.

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#### 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 completly 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 completly 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

Mentor
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 completly 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

Mentor
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

Mentor
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

Mentor
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.

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#### Mister T

Gold Member
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.

#### Mister T

Gold Member
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

Mentor
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.

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#### 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.

"Why use units in which c=1?"

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