Is the speed of light dictated by Vacuum Permittivity, Vice Versa or Neither?
Instinct, and my limited knowledge of Maxwell's Equations and the Wave Equation tell me that the first statement is true.
By my interpretation, the relationship between the frequencies and wavelengths of e.m. waves (and hence the speed of light) is dictated by the relationship between electric and magnetic fields, which is in turn dictated by Vacuum Permittivity, which I believe (possibly in error) to be an inherent property of our universe.
Is this right, or is the speed of light somehow dictating Vacuum Permittivity?
Or have I got something totally wrong?
electromagnetism speed-of-light maxwell-equations
Alan GeeAlan Gee
At first sight, I understand that it might be plausible that permeability and permittivity seem to be fundamental constants of spacetime which are together forming the constant of speed of light.
However, the speed of light is the more fundamental parameter. The speed of light c is not limited to electromagnetic waves, it is equally the speed of gravitational waves which are not at all electromagnetic. Thus it is easy to see that $c$ is more fundamental than $ε_0$ and $µ_0$.
For an intuitive model you can think of EM waves as only one form among others for the propagation at the universal speed limit, however with the particularity that they are based on two kinds of forces so that the speed limit has to be distributed among two kinds of forces (electric and magnetic), giving $ε_0$ and $µ_0$.
MoonrakerMoonraker
$\begingroup$ Thanks for that insight into the more general applicability of the speed of light. As you can probably tell, my levels of maths and physics knowledge are not what I would like them to be so the third paragraph helped too. $\endgroup$ – Alan Gee Aug 17 '16 at 16:30
$\begingroup$ I totaly agree that the speed of light is the more fundamental parameter. But I think your last point does not make much sense: there are not two forces only two fields and to describe an EM wave one does not need $\mu_0$ and $\epsilon_0$ only $c$. I think that from a modern point of view $\mu_0$ and $\epsilon_0$ have no fundamental meaning at all. $\endgroup$ – M. J. Steil Aug 18 '16 at 14:16
$\begingroup$ " ...it is equally the speed of gravitational waves ...". Has the speed of gravitational waves ever been properly measured? I mean, using the same, and only correct, way as the speed of light, i.e. bi-drectionally. $\endgroup$ – bright magus Aug 21 '16 at 8:44
I will limit my discussion to the vacuum but it can be expanded into medium as well: The vacuum permeability $\mu_0$ and vacuum permittivity $\epsilon_0$ are related to the speed of light by $$1/c^2=\mu_0\epsilon_0.$$ @Alexander Cska motivated that from the wave equations. This was not known when $\mu_0$ and $\epsilon_0$ first came up because the Maxwell equations where not known at that time.
With our modern knowlege this two constants are artifact of our definition of the electromagnetic units/force. In fact it is not necessary to even use a new dimension for electric current; see CGS-Units. In a CGS-System $\epsilon_0$ and $\mu_0$ can be expressed directly by a power of the speed of light and a dimensionless factor. In for example Lorentz–Heaviside CGS $\epsilon_0=1$ and $\mu_0=1/c^2$.
So in my opinion $\epsilon_0$ and $\mu_0$ have nothing to do with some universal properties of nature they are artifact of our definition of current (in SI). The speed of light is the constant related to electromagnetism.
M. J. SteilM. J. Steil
$\begingroup$ Still seems counterintuitive for a point of equilibrium of forces to be determined by a speed rather than a speed being determined by a point of equilibrium of forces. But I think I have to accept it. $\endgroup$ – Alan Gee Aug 18 '16 at 13:49
$\begingroup$ I get it that at first sight it is not very intuitive if you look at the classical equations. But from a fundamental point of view there are no magnetic and electric forces: there is only the electromagnetic force. $\endgroup$ – M. J. Steil Aug 18 '16 at 14:06
The constant speed of light for all reference frames is the fundamental building-block of Relativity. From Maxwell's equations in vacuum one can derive two wave equations $$ \left(\frac{1}{\mu_0\epsilon_0}{\bf\Delta} - \frac{\partial^2}{\partial t^2}\right){\bf E}=0 \\ \left(\frac{1}{\mu_0\epsilon_0}{\bf \Delta} - \frac{\partial^2}{\partial t^2}\right){\bf B}=0, $$ which tells us that the speed of light is equal to $\frac{1}{\sqrt{\mu_0\epsilon_0}}$. Therefore, the constancy of the speed of light tells you that the product $\mu_0\epsilon_0$ should have a particular fixed value. Note that both the permeability $\mu_0$ and the permittivity $\epsilon_0$ are not kinematic variables. They are medium properties. Or in the words of Wikipedia
In electromagnetism, permittivity or absolute permittivity is the measure of resistance that is encountered when forming an electric field in a medium https://en.wikipedia.org/wiki/Permeability_(electromagnetism) and
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. https://en.wikipedia.org/wiki/Permeability_(electromagnetism)
I would say that the constant speed of light for all observers dictates what the product of the permeability and the permittivity of vacuum should be equal to.
The constancy of the speed of light is a fundamental principle of physics. As axioms are defined in mathematics we have to use also some postulated principles in physics. For instance, one such principle tells us that the action integral has extremum along physical paths and this gives rise to the equations of motion.
The speed of light is a fundamental parameter and has far reaching consequences. For instance, special relativity tells us that this is the speed at which all massless fields propagate. The constancy of $\mu_0 \epsilon_0$ is just a by-product.
The Lorentz transformations are just one of the representations of the Lorentz group, which includes both boosts and rotations. By deriving the Lorentz transformations we arrive at a parameter which is the same for all frames. This is best seen from the wave equations, which I listed above. Since they are Lorentz invariant the factor $\frac{1}{c^2}$ must also be Lorentz invariant. If it was not, then Lorentz invariance will be spoiled since ${\bf \Delta}$ and $\frac{\partial^2}{\partial t^2}$ are invariant and this enforces also the multiplier $\frac{1}{c^2}$ to be the same for all frames. This is why I listed the wave equations. Non-constant $c$ will be a violation of the basic principle of SR.
Closer observation (setting $x'=0$ in the Lorentz transformations) tells us that this parameter scales the slope of the transformed coordinate axes. $$ x`=\gamma(x-vt)\\ x`=0 \\ x=\frac{v}{c}(ct) $$
and the same for the time axis $$ t`=\gamma(t-\frac{vx}{c^2})\\ t`=0 \\ x=\frac{c}{v}(ct), $$ which tells us that in the transformed frame $S'$, moving at speed ${\bf v}=(v,0,0)$ with respect to $S$, the coordinate axis have reciprocal slopes i.e. are symmetric. These equations tells us what is the unit of the unknown parameter $c$ i.e. the unit of velocity, since the slope has to be dimensionless.
Of course, as all other wave phenomenon $c$ is involved in all sorts of relations regarding the wave vector, frequency etc. that follows from our knowledge on the wave equation. By doing further analysis we see that massless particles sit exactly on the light cone (move with speed $c$) $$ p_{\mu}p^{\mu}=0\\ p_{\mu}=(\frac{E}{c},{\bf p}). $$
This tells us also that massless objects have momentum $$ |{\bf p}|=\frac{E}{c}. $$
Therefore, the speed of light $c$ is a fundamental parameter intertwined with the geometry of space-time. It is a fundamental speed limit, the velocity of massless objects and tells us how the transformed coordinate axes look like, among others. The speed of light is just a parameter, it can be also set equal to unity, a frequent choice in particle physics.
Alexander CskaAlexander Cska
$\begingroup$ I will try to understand what you have written better than I do now but I'm not sure how this distinguishes the cause from the effect. I.e. is that product fixed because the speed of light is constant or is the speed of light constant because the product is fixed? $\endgroup$ – Alan Gee Aug 17 '16 at 13:20
$\begingroup$ The speed of light is constant. That is a postulate. In physics we need some postulates, similar to the axioms in mathematics. Meaning, that we have a parameter, which is constant in all frames. Then the Maxwell equations (which are themselves Lorentz invariant) come into play, they tell you what this parameter is. Actually the wave solution and the wave propagation speed were found by Maxwell long before electromagnetic waves were discovered. It is a bit chicken and egg problem, but here $c$ constant orchestrates it all. $\endgroup$ – Alexander Cska Aug 17 '16 at 13:33
$\begingroup$ I have to admit it is contrary to my gut feeling but then so is the invariance of the speed of light to all observers. I'll try to learn a bit more and if after doing that I agree with you (and nobody comes up with a more intuitive one) I'll accept your answer. $\endgroup$ – Alan Gee Aug 17 '16 at 13:39
$\begingroup$ Try reading PANOFSKY AND PHILIPS, Classical Electricity and Magentism. I find this to be a really good book on EM and Realtivity. Google it, I found some online version. $\endgroup$ – Alexander Cska Aug 17 '16 at 13:42
Part of the answer to this question is on page 49 of the book "The Constants of Nature" by John D. Barrow (Vintage Books).
"The last important lesson we learn from the way that pure numbers like alpha define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by alpha is a combination of the electronic charge, e, the speed of light, c and Planck's constant, h. At first sight we might be tempted to think a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h and e were all changed ... but the value of alpha remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of the world are the values of the dimensionless constants of Nature."
Peter McNeallPeter McNeall
From a theoretical perspective, the vacuum permittivity and permeability are just unnecessary complications that only exist for historical reasons. To build up conceptual intuition for the special-relativistic aspects of E&M, I recommend working in either CGS or Lorentz-Heaviside units, in which $\epsilon_0$ and $\mu_0$ do not appear at all. In those units, it's very clear that all electromagnetic equations can be expressed only in terms of $c$ and not $\epsilon_0$ or $\mu_0$ individually, and so the speed of light is more fundamental.
tparkertparker
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