Does the speed of light pop out of other theories than EM?

[greypilgrim]

Hi.

The numerical value of the speed of light $c$ pops out as propagation speed of the wave solutions to Maxwell's equations. It seems like everywhere else we need to plug in $c$ as a parameter. Why is that? Is there a way to "derive" the numerical value of $c$ in other theories, that e.g. describe neutrinos that don't interact electromagnetically at all?

Or am I approaching this the wrong way and SR including the numerical value of $c$ (and possibly QFT) is fundamental and Maxwell's equations (including $\varepsilon_0$ and $\mu_0$) are consequences?

 

[Orodruin]

Or am I approaching this the wrong way and SR including the numerical value of $c$ (and possibly QFT) is fundamental and Maxwell's equations (including $\varepsilon_0$ and $\mu_0$) are consequences?

Yes. The fundament of special relativity is the geometry of Minkowski space. The actual value of $c$ is a matter of unit convention. Working with relativity, it is common to simply set $c = 1$ (dimensionless) in order to work with the same units for time and lengths. This makes the geometrical structure much more apparent.

When you attempt to construct a field theory consistent with SR, the Maxwell equations drop out as one of the simpler types of field theory that one can write down - including the fact that you obtain a wave equation with wave speed $c$ in all inertial frames. The particular values of permittivity and permeability are also unit conventions (although once $c$ is chosen, you can only pick one of them independently - in SI units, this is effectively done when one defines the value of the elementary charge $e$).

Edit: To expand on the hierarchy of SI units, it goes like this:

• Define 1 s as 9192631770 periods of the radiation corresponding to the hyperfine splitting in caesium-133.
• Define 1 m as the distance travelled by light in 1/299792458 seconds.
• Define the kg by setting the value of Planck's constant to 6.62607015×10⁻³⁴ Js (the J contains the kg as well as meters and seconds) so this uniquely determines the kg.
• Define the elementary charge to be $e$ = 1.602176634×10⁻¹⁹ C.

With these definitions the values of $\epsilon_0$ and $\mu_0$ follow. For example, $\epsilon_0$ is the value that makes Coulomb's law
$$
F = \frac{Q_1 Q_2}{4\pi\epsilon_0 r^2}
$$
correctly normalised.

 

[Dale]

I would consider the numerical value of $c$ to be simply an artifact of your unit system. Therefore it is definable, but not derivable. It is a matter of convention.

The important thing is not its numerical value, but the fact that it is invariant.

 

[greypilgrim]

The particular values of permittivity and permeability are also unit conventions (although once is chosen, you can only pick one of them independently - in SI units, this is effectively done when one defines the value of the elementary charge ).

I don't quite understand that part. Assuming we work in SI units and have a value for $c$ obtained by direct measurement, now we build a field theory consistent with SR. So you're saying from this one measurement value $c$ we then can derive BOTH $\varepsilon_0$ AND $\mu_0$?

The equation $c=\frac{1}{\sqrt{\varepsilon_0\mu_0}}$ suggests that TWO constants need to be measured and only the third can be calculated from those two.

Or do we first get only a class of Maxwell's equations satisfying above equation and have yet to pick the correct ones by a different measurement?

 

[Orodruin]

I don't quite understand that part. Assuming we work in SI units and have a value for $c$ obtained by direct measurement,

You don’t measure $c$. The value of $c$ is defined in SI units (see also @Dale ’s post)

So you're saying from this one measurement value $c$ we then can derive BOTH $\varepsilon_0$ AND $\mu_0$?

Again, not a measurement of c, definition of c. With the definition of the second and the definition of c, the defined value of c defines what a meter is.

The equation $c=\frac{1}{\sqrt{\varepsilon_0\mu_0}}$ suggests that TWO constants need to be measured and only the third can be calculated from those two.

Not measured. You need two to determine the third.

In this case, c is defined. Once you define the elementary charge, you can measure either $\epsilon_0$ or $\mu_0$ and obtain a value of the other.

 

[Nugatory]

It seems like everywhere else we need to plug in as a parameter.

There are two lines of thought starting from Maxwell’s equations.
First, we note that the numerical value that emerges from them is equal to the measured speed of light (using old units of length and time so that the idea of a measured speed of light makes sense). That’s nature giving us a very strong hint that the phenomenon we knew as “light” is actually electromagnetic radiation.

But it is just as remarkable that any velocity at all pops out of Maxwell’s equations. That is nature giving us a strong hint that there is an invariant velocity and that the Minkowski model describes the relationship between time and space. Once we accept that, it follows that the invariant velocity is no more than the conversion factor between units of time and units of distance. Naturally this conversion factor appears in any equation that relate times and distances; it’s both less arbitrary and less important than inserting a parameter.

 

[Ibix]

I don't quite understand that part. Assuming we work in SI units and have a value for $c$ obtained by direct measurement, now we build a field theory consistent with SR. So you're saying from this one measurement value $c$ we then can derive BOTH $\varepsilon_0$ AND $\mu_0$?

As others have commented, if you look at SR as a geometric theory then $c$ is just a unit scale between units of distance and units of time. For an analogy, you might measure horizontal distance in nautical miles and depth in fathoms, and if you want a volume of water in cubic fathoms you need the formula $V=(N_f)^2whd$, where $N_f$ is the the number of fathoms per nautical mile. $c$ is a constant just like $N_f$ that tells you how your distance units and time units are related.

So the value of $c$ in relativity is chosen by deciding what you want your unit system to do. If you want to make it simple for maths you just say you'll use the same units for time and distance and $c$ becomes 1. If you want it to be a bit more practical you can use "about an arm's length" and "about one heart beat" and you get "about 300 million arms' lengths per heart beat".

The point about $c$ just being a unit conversion is that it pops up all over the place. Since it's the invariant speed it's the speed massless objects travel at, so it's the speed of all waves in massless field theories, including EM and gravitational waves.

 

[Sagittarius A-Star]

The equation $c=\frac{1}{\sqrt{\varepsilon_0\mu_0}}$ suggests that TWO constants need to be measured and only the third can be calculated from those two.

That suggestion is wrong. As others mentioned, the value of $c$ is indirectly defined by the SI-definition of "1 meter":

Since 2019 the metre has been defined as the length of the path travelled by light in vacuum during a time interval of $1/299792458$ of a second, where the second is defined by a hyperfine transition frequency of caesium.

Source:
https://en.wikipedia.org/wiki/Metre

The constants $\varepsilon_0$ and $\mu_0$ contain a redundancy. You could for example remove $\mu_0$ from all physics books and replace it by ${1\over c^2\varepsilon_0}$ without loosing anything.

 

[Vanadium 50]

The equation $c=\frac{1}{\sqrt{\varepsilon_0\mu_0}}$ suggests that TWO constants need to be measured and only the third can be calculated from those two.

One of those constants is π. Well, actually, it's 4π x 10p^-7, but still essentially π.

You have a history of blurring the lines between asking questions and proposing your own speculative and unorthodox ideas. This will not serve you well here. Those electromagnetic constants are there because of the way units were defined Way Back When. As @Ibix points out (and I have also discussed in the past), you'd have similar constants appear everywhere if we measured distances in different directions with different units.

 

[Orodruin]

One of those constants is π. Well, actually, it's 4π x 10p-7, but still essentially π.

In the undying words of inspector Clouseau:
- Not anymeure.

Its measured value is 1.25663706212(19)×10⁻⁶ H/m. The value $4\pi\cdot 10^{-7}$ H/m is remarkably close, but about 3.6 sigma off.

 

[Sagittarius A-Star]

One of those constants is π.

According to the following paper, published shortly before April 1^st, 2009, $\pi$ changes over time.
https://arxiv.org/abs/0903.5321

 

[hutchphd]

Love the citation list on that paper,

 

[Dale]

Assuming we work in SI units and have a value for c obtained by direct measurement

If you work in SI units then you have a value for $c$ given by definition. It is not obtained by measurement.

If you work in any other units then it may be that the speed of light is obtained by measurement. But even so, the numerical value of that measurement will depend on the units chosen.

 

[Vanadium 50]

Not anymeure.

Yes, I know. But a) it was intended to be 4π, and b) the OP is confused enough as it is.

 

[Orodruin]

Yes, I know. But a) it was intended to be 4π, and b) the OP is confused enough as it is.

I’m just not sure bringing pi into it solves (b) …

 

[Sagittarius A-Star]

Yes, I know. But a) it was intended to be 4π, and b) the OP is confused enough as it is.

The SI-definition of $\mu_0$ changed in 2019.

Since the redefinition of SI units in 2019 (when the values of e and h were fixed as defined quantities), $\mu_0$ is an experimentally determined constant, its value being proportional to the dimensionless fine-structure constant,

Source:
https://en.wikipedia.org/wiki/Vacuum_permeability

From 1948 until 2019 the ampere was defined as "that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to $2×10^{−7}$ newton per metre of length". This is equivalent to a definition of $\mu_0$ of exactly $4\pi×10^{−7}$ H/m.

Source:
https://en.wikipedia.org/wiki/Vacuum_permeability#Ampere-defined_vacuum_permeability

 

[Orodruin]

The SI-definition of $\mu_0$ changed in 2019.


Source:
https://en.wikipedia.org/wiki/Vacuum_permeability


Source:
https://en.wikipedia.org/wiki/Vacuum_permeability#Ampere-defined_vacuum_permeability

@Vanadium 50 knows this. Hence the ”Yes, I know”

The point he made was that (a) at the time of the redefinition , the value $4\pi\cdot 10^{-7}$ H/m was within the experimentally allowed region. The defined values of the natural constants were intended to reproduce the pre-2019 units within uncertainties. The vacuum permeability in the new definition is given by
$$
\mu_0 = \frac{2h}{e^2c}\alpha
$$
where everything in the fraction are defined values. The relative uncertainty in $\mu_0$ is therefore the same as that in the fine structure constant $\alpha$. After the redefinition, $\alpha$ has been more accurately measured with the result that the $\pi$ multiple now lies outside the experimentally allowed region.

This was of course bound to happen at some point if one assumes that we get better and better measurements. It just happened pretty fast in this case.