Einstein's box in media and mass-energy equivalence

[sunroof]

Einstein proposed a very simple derivation to E=mc^2 in 1940s which is well-known as Einstein’s box and a brief introduction is in
http://galileo.phys.virginia.edu/classes/252/mass_and_energy.html . If this event occurs in media instead of vacuum, the light speed should be u=c/n rather than c. Thus, the momentum is Mv=E/u and time is t=L/u. The result is now E=mu^2 and will be reduced to E=mc^2 only in vacuum.

E=mc^2 is not universal?

 

[Dale]

No. The medium is not relevant.

 

[bcrowell]

The c appearing in all the formulas in relativity really has nothing to do with the speed of light. It's a property of spacetime, the maximum speed of cause and effect. Light just happens to move at that speed.

If you fill the box with a medium, it's equivalent to absorbing and reemitting the light many times, and this repeated absorption and reemission is in turn equivalent to simply putting a bunch of smaller boxes inside the big box. It doesn't change the result.

 

[sunroof]

No. The medium is not relevant.

In electrodynamics, the energy-mass relation of an electromagnetic field (photons) is related to media. The Poynthing vector S =energy density times velocity and momentum density g=mass density times velocity. Therefore, S/g=energy density/mass density=energy/mass=E/m.

On the other hand, S=EH=EB/magnetic permeability and g= dielectric constant times EB. Consequently, E/m=S/g=1/(dielectric constant times magnetic permeability)=square of light speed. In vacuum, it is c^2 which is consistent with the result of SR. But in media, the ratio is u^2. (u=c/n<c)

 

[Dale]

The point is that the equivalence of mass and energy is a result of the Lorentz transform and is derivable directly from the Lorentz transform. In the Lorentz transform the invariant speed c is 299,792,458 m/s whether we are talking about a system in vacuum or a system in some highly refractive transparent medium. The speed of light is special, not because it is the speed that light travels, but because even if there were no light it is the speed which is invariant under the Lorentz transform.

 

[bcrowell]

Sunroof, what's happening here is that you're making complicated arguments that have mistakes in them, when in fact the issue you're trying to address is much more simple and fundamental. We can keep on pointing out the specific mistakes in your arguments (e.g., you're neglecting momentum transfer between the light and the medium, as I pointed out in my previous post), but what you need to recognize is that you don't gain anything by taking a simple thought experiment and making it more complicated.

 

[sunroof]

The point is that the equivalence of mass and energy is a result of the Lorentz transform and is derivable directly from the Lorentz transform. In the Lorentz transform the invariant speed c is 299,792,458 m/s whether we are talking about a system in vacuum or a system in some highly refractive transparent medium. The speed of light is special, not because it is the speed that light travels, but because even if there were no light it is the speed which is invariant under the Lorentz transform.

You're right that mass-energy equivalence is related to the Lorentz transformation. But whether the transformation is tenable in media? If the Lorentz transformation is universal, the consequential Doppler effect should always depend on the relative velocity V,angle and light speed c, no matter this event occurs in vacuum or different media. This is unacceptable. In a medium,Doppler frequency should be related to the velocity V,angle and light speed u=C/n. Otherwise it is impossible to be reduced to the classical result f ' = f (1+V/u),u=C/n. When the light speed c is modified to C/n in the Lorentz transformation, we get a new Doppler formula which can be simplied to f' = f (1+V/u),u=C/n in media.

 

[sunroof]

If you fill the box with a medium, it's equivalent to absorbing and reemitting the light many times, and this repeated absorption and reemission is in turn equivalent to simply putting a bunch of smaller boxes inside the big box. It doesn't change the result.

I will use two examples to disprove.

In classical electrodynamics, energy density of an electromagnetic field (photons) is w=dielectric constant times squared electric field intensity and momentum density is g=dielectric constant times electric field intensity times magnetic induction intensity. So, E/p=w/g= electric field intensity/ magnetic induction intensity=phase velocity. In vacuum, the ratio is c. In media, w/g=c/n<c.

In quantum optics, E/p=hf/hk=f/k=u=c/n<c (k is 1/wavelength).

It changes the result.

 

[sunroof]

The c appearing in all the formulas in relativity really has nothing to do with the speed of light. It's a property of spacetime, the maximum speed of cause and effect. Light just happens to move at that speed.

_This is metaphysics.

In the advanced form of special relativity, the invariable speed should be introduced according to the group theory. But the theory cannot limit the number and value of the invariable speeds at all. They are only determined by experiments.

The light speed in vacuum c=299,792,458 m/s was confirmed that can play the role of the invariable speed in the last century. But the theory cannot exclude the existence of other velocities. For instance, in 2005 scientists of the University of Manchester discovered electrons in graphene behave as if they lose rest mass or neutrinos acquire electric charge. Especially, the relativistic Dirac equation is invalid to describe unless the light speed c in this equation is replaced by the Fermi velocity of condensed matter physics.

So, the Lorentz transformation is not unique. There might be lots of similar structures to the Lorentz transformation provided c is replaced by other velocities, e.g. the Fermi velocity in graphene.

 

[clamtrox]

Lorentz transformations apply to particles. If you have multiple particles, you have to use some sort of scheme to average over most of the degrees of freedom (and call the soup of particles "medium" or whatever). Just because your averaging scheme is not compatible with special relativity does not mean that SR does not work everywhere. In this case it means that your averaging scheme is bad.

 

[Dale]

You're right that mass-energy equivalence is related to the Lorentz transformation. But whether the transformation is tenable in media?

Yes, it is. And the c in the Lorentz transform in any medium is the invariant speed (aka the speed of light in vacuum) not the speed of light in the medium.

If the Lorentz transformation is universal, the consequential Doppler effect should always depend on the relative velocity V,angle and light speed c, no matter this event occurs in vacuum or different media. This is unacceptable.

I agree, this is unacceptable, it is also not what relativity says. Basically, if you use the medium's rest frame to analyze the Doppler effect and then boost the result into a frame where the medium is moving you can easily see that that you need to use the relativistic velocity addition to determine the Doppler effect. This works for EM waves in a refractive medium as well as sound waves and other very slow mechanical waves. This was confirmed by Fizeau even before Einstein and to higher precision by others later.