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

[PeterDonis]

Are there specific recommended standard units for c=1?

No. Any units that match up correctly will work. You can use years and light-years, seconds and light-seconds, meters and "light-meters" (the time it takes light to travel 1 meter), feet and nanoseconds, etc.

Also, the unit of mass has nothing to do with this; choosing units in which $c =$ does not force you to choose any units for mass.

 

[Buzz Bloom]

Hi Peter:

Thank you for the clarification. Presumably, for any particular choice of units consistent with c=1 there is are corresponding values for various physical constants, like G for example. Are such physical constant values generally calculated and published somewhere so that someone trying to do a physical calculation, for example the amount of space distortion at a given distance from a black hole, would not have to recalculate G as part of this calculation?

Regards,
Buzz

 

[PeterDonis]

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

Not necessarily. Choosing units for which $c = 1$ doesn't fully specify a value for $G$, for example, because it doesn't set units for mass.

 

[Buzz Bloom]

Not necessarily. Choosing units for which c=1c = 1 doesn't fully specify a value for GG, for example, because it doesn't set units for mass.

Hi Peter:

Suppose I choose the following units:

time: 1 second
distance: 1 light-second:
mass: 1 kilogram​

Is there some reference source where I could find the published corresponding value for G?

Regards,
Buzz

 

[Nugatory]

Are such physical constant values generally calculated and published somewhere so that someone trying to do a physical calculation, for example the amount of space distortion at a given distance from a black hole, would not have to recalculate G as part of this calculation?

I'm sure there's a published list somewhere, but usually it's easiest to just put the units back in when you're done calculating.

We started this thread with the velocity addition formula (actually split off from a thread about velocity addition) so let's use that as an example... Say we've used the $c=1$ version of the formula to see how fast a bullet fired at one-half lightspeed from a spaceship also moving at one-half lightspeed is moving... I can do that in my head, and the answer is 4/5. Then if I want an answer in meters/sec I multiply 299792458 by 4/5; if I want an answer in miles/hr I multiply 670800000 by 4/5.

 

[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 $\hbar c \simeq 0.197 \;\text{GeV} \, \text{fm}$.

 

[Herman Trivilino]

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.

 

[Herman Trivilino]

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?

 

[Herman Trivilino]

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.