Abraham's light momentum breaks special relativity?

[rpfeifer]

I'll just expand on my last comment a bit.

"I may have misunderstood you there: You may have wanted to discuss a system where the dielectric is initially in motion, then stops when the light pulse enters it."
Total momentum is also conserved in this instance, as the momentum transferred from the light to the block is sufficient to stop its initial motion. Again, however, you cannot consider the Abraham EM density in isolation, as you have to also consider how the momentum which is transferred to the block propagates. The Maxwell stress tensor tells us that it is not instantaneously spread over the entire block: It has to propagate as some sort of matter wave. Thus, in that frame the momentum of the block behaves like this:

Before the pulse enters it:
Total momentum: Nonzero. Locally: Uniformly ρv\not=0.

Once the pulse has entered it:
Total momentum: Zero. Locally: Takes the form of a matter wave, i.e. <ρv(x)>=0 but ρv(x)\not=0 for most x.
Here, v(x) represents the velocity of the medium at point x (where x is a point within the medium).
Note that for a bounded wave packet propagation of the associated matter wave is made more complicated as you must take into account ongoing interactions as the edges of the packet propagate through the material. Yet another explanation as to why you cannot consider the EM and material portions of the momentum density separately, and thus why it doesn't matter that the Abraham EM density (in isolation) doesn't transform covariantly: You never encounter it in isolation.

By the way, do you feel that we have answered your original question yet? I ask because we do now seem to be talking about something only tangentially related (i.e. whether the framework in RMP79 is consistent, rather than whether the Abraham EM expression is inconsistent with SR).

 

[sciencewatch]

#98:
...
"|ExH/c**2 in a medium| / |ExH/c**2 in vacuum| =1/n (refractive index)"

Think about this. You just said the density goes down by a factor of n. At the same time, the wave packet slows down by a factor of n, so the total momentum of the wave packet as described by ExH/c**2 goes down by a factor of n^2. Conservation of momentum says this has to go somewhere. That is, the material is placed in motion with total momentum
∫ |ExH/c**2 in vacuum| x (1-1/n**2) dV
unless the refractive index is equal to 1.
As you can see, you were mistaken in saying I had made a mistake.
...

Please pay attention to the model: Suppose that there is a plane wave propagating in an isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium. Observed in the medium-rest frame, the medium material momentum should be zero and the refractive index can be assumed to be >1. Observed in the medium-rest frame, the dielectric is not moving.

In your arguments, in such case you think the field momentum = ExH/c**2, the same as in vacuum because the material momentum is zero, resulting in refractive index n=1. However, there are two mistakes now in your reasoning:

(1) The field momentum should be DxB (instead of ExH/c**2) in a medium,
(2) Under the energy conservation, the EM energy density with a medium is the same as that without a medium (vacuum). Thus the ExH/c**2 in a medium is different from the ExH/c**2 in vacuum:

|ExH/c**2 in a medium| / |ExH/c**2 in vacuum| =1/n (refractive index) ----- (1)
---
Momentum flus conservation:

With a medium, momentum flux = DxB*(c/n)=n**2 *(ExH)/c**2 *(c/n) = n *(ExH)/c ----(2)
Without a medium (vacuum), momentum flux = ExH/c**2 * c = ExH/c -----(3)

Using Eq. (1), we find Eq. (2) = Eq. (3), the momentum-flux conservation.
----------------------
Why n =1 must hold?

 

[Dale]

Observed in the medium-rest frame, the dielectric is not moving.

The medium rest frame is non-inertial.

 

[rpfeifer]

"Please pay attention to the model: Suppose that there is a plane wave propagating in an isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium. Observed in the medium-rest frame, the medium material momentum should be zero and the refractive index can be assumed to be >1. Observed in the medium-rest frame, the dielectric is not moving."

OK, how did this EM wave get into this dielectric? When it entered, it set up a wave in the dielectric, which is still there. That is, <ρv(x)>=0 but ρv(x)\not=0. And this dipole matter wave couples to the EM field.

If you do not allow for the entry of the wave into the medium at some point in the past, then the situation you describe is unphysical (e.g. infinite dielectric). This even applies to a plane wave (for which it is necessary to assume that the beam is "turned on" gradually at t=-∞, but I will not attempt to explain details of boundary conditions at infinity to you here. If interested, look it up).

"In your arguments, in such case you think the field momentum = ExH/c**2, the same as in vacuum because the material momentum is zero, resulting in refractive index n=1. However, there are two mistakes now in your reasoning:"

*rolls eyes* Here we go again.

(1) You assert the Minkowski momentum. However, for conservation of angular momentum the total momentum multiplied by c must equal the Poynting vector plus the material energy flux, i.e.
c x (p_EM+p_mat) = ExH/c+ρcv.
Note that neither the Poynting vector nor the material energy flux have ever been under debate in the controversy. These are the S terms in T^{\mu\nu}.

If we use the Minkowski momentum, then we have to have a "material" momentum of
ρv+ExH/c^2-DxB.
Do you like having a material momentum density which contains E, H, D, and B? Does this make sense to you? Sure, it might not contribute to momentum transfer, but it doesn't describe the movement of matter any more.

(2) "Under the energy conservation, the EM energy density with a medium is the same as that without a medium (vacuum)."

Don't be ridiculous. The wave packet is packed into a smaller volume, so the energy density goes up. Exactly the same as happened with momentum. If you're thinking about a plane wave, which is infinite, it still has this compression. To see this, remember that a plane wave may be infinite, but it still propagates. Therefore, consider a "chunk" of wave just about to enter the dielectric. Work out what happens. You'll see it gets compressed.

"Using Eq. (1), we find Eq. (2) = Eq. (3), the momentum-flux conservation."

There are mistakes in your working. If you fix them, you might end up with a derivation of the Minkowski momentum density - which then violates conservation of angular momentum.

If you eventually managed to fix that, you would have the material given in RMP79.

You are retracing the steps of hundreds of physicists before you. This is why literature reviews exist: To show what has already been done. Go away and read the literature. Start with all the citations in RMP79. M∅ller is particularly relevant here.

----------------------
"Why n =1 must hold?"

Because the argument I gave above is valid, and the one you gave was based on two mistakes.

 

[rpfeifer]

The medium rest frame is non-inertial.

Thanks, Dale, very elegant. Yep, you can construct a co-ordinate frame using GR in which space--time as you know it is rippling gently back and forth rather than the atoms of the medium.

That would be very silly, but in a good way. (The physics would be 100% correct, and absolutely no practical use whatsoever!) Nice one!

Being a GR solution, the maths involved are, of course, way beyond those presented in RMP79 (and would involve a stress-energy tensor for spacetime in the adopted co-ordinate frame, which would (by definition of the frame) couple to the EM tensor).

I doubt that sciencewatch had such a solution in mind.

(Note for junior physicists: The phrase "co-ordinate frame" has a special mathematical meaning. It comes from differential geometry, and refers to a very specific type of mathematical object, which people usually first encounter as the set of objects e^a_\mu(x) in tetrad-formalism GR.)

 

[rpfeifer]

"Do you like having a material momentum density which contains E, H, D, and B? Does this make sense to you? Sure, it might not contribute to momentum transfer, but it doesn't describe the movement of matter any more."

Just wanted to say - there's nothing wrong with such a formulation, but you do have to be very careful in interpreting it. That is:
Even if DxB contains all the momentum, the medium still moves because its behaviour is governed by ρv, not ρv+ExH/c^2-DxB.

Thus the Minkowski formulation is good for describing momentum transfer to objects in a dielectric, but not so useful for describing the motion of the dielectric itself.

 

[sciencewatch]

The medium rest frame is non-inertial.

For a plane wave propagating in an isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium, the refractive index can be assumed to be >1. Observed in the medium-rest frame, the dielectric is not moving.

For such an ideal model, macro-electromagnetically speaking, there are no reasons to say the medium moves with accelerations. Don’t ask me how to set up the plane wave and how to get such a medium; I don’t know. But I do know there is such a solution to Maxwell equations, and this physical model is widely presented in physics textbooks:

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, 1999), 3rd Edition;
M. Born and E. Wolf, Principles of optics (5th edition) (Oxford, 1975);
J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941);
J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & Sons, NY, 1975);
W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, 1968), 3rd edition;
D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition;

to name a few.

 

[rpfeifer]

Error: The situation we are discussing is more complex than those addressed in the textbooks. Approximations valid in the problems they address may not be valid here.

 

[rpfeifer]

If you don't know how to set up a rigorous model of the wave and medium, and then take limits, that's fair enough. It is good to be able to admit your limitations.

This same limitation is why you are struggling with the A-M controversy. You need to learn how to set up this sort of more detailed model before you can understand the explanation.

Regards,
R. Pfeifer

 

[Dale]

For such an ideal model, macro-electromagnetically speaking, there are no reasons to say the medium moves with accelerations.

Conservation of momentum is a pretty good reason to say the medium moves with accelerations. The acceleration can often be neglected, as in the textbooks you cited, except when it cannot be neglected, as in the fiber bending experiment.

 

[sciencewatch]

Conservation of momentum is a pretty good reason to say the medium moves with accelerations. The acceleration can often be neglected, as in the textbooks you cited, except when it cannot be neglected, as in the fiber bending experiment.

You are talking about media with discontinuities.

Let me restate: For a plane wave propagating in an infinite isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium, the refractive index can be assumed to be >1. Observed in the medium-rest frame, the dielectric is not moving.

For such an ideal model, speaking in macro-electromagnetic theory , there are no reasons to say the medium moves with accelerations. Don’t ask me how to set up the plane wave and how to get such a medium; I don’t know. But I do know there is such a solution to Maxwell equations, and this physical model is widely presented in physics textbooks:

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, 1999), 3rd Edition;
M. Born and E. Wolf, Principles of optics (5th edition) (Oxford, 1975);
J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941);
J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & Sons, NY, 1975);
W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, 1968), 3rd edition;
D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition;

to name a few.
------
If there are any discontinuities in a medium, there is no strict uniform plane-wave solution to Maxwell equations; I think this implication is well-known in the community of electromagetic theory. I am sorry I did not strcitly state it.

 

[Dale]

a plane wave propagating in an infinite ...

Oops, I did miss that. I assume that you mean an infinite wave also, otherwise the rest frame would be non-inertial even for an infinite medium.

You will have to remind me: why are we considering an infinite dielectric? How is it relevant to the problem under consideration? And what is the concern for an infinite dielectric?

EDIT: actually, even for an infinite medium and an infinite plane wave a reference frame where the medium is everywhere at rest will be non-inertial. Parts of the medium accelerate wrt each other as the momentum flux at each location changes. I.e. You can consider conservation of momentum for a differential element of the medium as the fields vary across it.

 

[sciencewatch]

Oops, I did miss that. I assume that you mean an infinite wave also, otherwise the rest frame would be non-inertial even for an infinite medium.

You will have to remind me: why are we considering an infinite dielectric? How is it relevant to the problem under consideration? And what is the concern for an infinite dielectric?

EDIT: actually, even for an infinite medium and an infinite plane wave a reference frame where the medium is everywhere at rest will be non-inertial. Parts of the medium accelerate wrt each other as the momentum flux at each location changes. I.e. You can consider conservation of momentum for a differential element of the medium as the fields vary across it.

Let me restate: For an infinite uniform plane wave propagating in an infinite isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium, the refractive index can be assumed to be >1. Observed in the medium-rest frame, the dielectric is not moving.

For such an ideal model, speaking in macro-electromagnetic theory , there are no reasons to say the medium moves with accelerations. Don’t ask me how to set up the plane wave and how to get such a medium; I don’t know. But I do know there is such a solution to Maxwell equations, and this physical model is widely presented in physics textbooks:

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, 1999), 3rd Edition;
M. Born and E. Wolf, Principles of optics (5th edition) (Oxford, 1975);
J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941);
J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & Sons, NY, 1975);
W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, 1968), 3rd edition;
D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition;

to name a few.
------
If there are any discontinuities in a medium, there is no strict uniform plane-wave solution to Maxwell equations; this implication is a well-known common sense in the community of electromagetic theory. I apologize again that I did not strcitly state it.

PS:

"speaking in macro-electromagnetic theory..." --- means there is no motion for the medium as a whole because the medium is "rigid", and the total force is zero after taking average over light-wave period and space for all possible micro-forces.

"why are we considering an infinite dielectric?" ---- Thousands of examples cannot validate a theory; however, to negate it, one is enough. For example, when I write down a code for solving differential equations, I don't know if it is correct, and I usually use a simple analytically solvable equation to check first.

I can be sure that you know all these simple things. Since you ask me to say, I have to say.

 

[Dale]

"why are we considering an infinite dielectric?" ---- Thousands of examples cannot validate a theory; however, to negate it, one is enough. For example, when I write down a code for solving differential equations, I don't know if it is correct, and I usually use a simple analytically solvable equation to check first.

OK, so what theory does this example negate and how?

 

[Dale]

Thanks, Dale, very elegant. Yep, you can construct a co-ordinate frame using GR in which space--time as you know it is rippling gently back and forth rather than the atoms of the medium.

It took me a while, but I got what you said here. I wish I could take credit for such a subtle idea. I was thinking of a finite block of dielectric and just the fact that radiation pressure would accelerate it as the wave entered. But your idea applies even for the infinite dielectric.

 

[sciencewatch]

OK, so what theory does this example negate and how?

Suppose that an infinite uniform plane wave propagates in an infinite isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium with a refractive index >1. Speaking in macro-electromagnetic theory, the medium-rest frame is an inertial frame.

For such an ideal plane-wave model, the medium is assumed to be "rigid", and the total force acting on the whole medium is zero after taking average over time (in one light-wave period) and space (in one wavelength) for all possible micro-scale forces. In other words, there are no accelerations for the medium. Don’t ask me how to set up the plane wave and how to get such a medium; I don’t know. But I do know there is such a solution to Maxwell equations, and this physical model is widely presented in physics textbooks:

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, 1999), 3rd Edition;
M. Born and E. Wolf, Principles of optics (5th edition) (Oxford, 1975);
J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941);
J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & Sons, NY, 1975);
W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, 1968), 3rd edition;
D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition;

to name a few.

************************
“OK, so what theory does this example negate and how?”

The author [Rev. Mod. Phys.79:1197-1216 (2007), http://arxiv.org/abs/0710.0461 ] claims that the total energy-momentum tensor is “uniquely determined by consistency with special relativity”. However, if applying their total-tensor model to the ideal plane wave described above, we obtain the total momentum = ExH/c**2 (=Abraham’s momentum density vector) from their Eqs. (40)-(43), or Eq. (33) by setting the dielectric velocity v = 0, and ExH/c**2 cannot be used to constitute a Lorentz covariant momentum-energy 4-vector. Therefore, the total-tensor model does break the special relativity, unless they have strong arguments to refute the above ideal plane-wave model.

------------------
To the authors [Rev. Mod. Phys.79:1197-1216 (2007)]: Please check the last term of Eq. (33), “+ ExM/c**2”, and make sure if there is a sign typo: – ?. Seems not consistent with Eq. (31) and Eqs. (40)-(43).

************************
PS:

In post #70, the author of Rev. Mod. Phys.79:1197-1216 (2007) (henceforth RMP79) claims:

In fact, the main thrust of Sec. VIII of RMP79 is that once the material properties of the dielectric are specified, the total momentum tensor is uniquely determined by

(i) consistency with special relativity, and
(ii) conservation of linear and angular momentum.

This leads to two important conclusions:

(a) No valid combination of EM and material energy-momentum tensors can break special relativity. If you are using a combination of tensors which appears to break this, then your choice of tensors is incorrect (usually, the material tensor is incorrect or missing). Note that I have never yet seen a fully relativistic formulation of the material counterpart tensors written down anywhere in the literature - even those given in RMP79 are valid only for media moving at v<<c, though the full expressions could be obtained from Eqs. (33)-(34).

(b) As the _total_ energy-momentum tensor is uniquely fixed by (i) and (ii) above, any division into components necessarily yields the same total tensor, and thus the same physical behaviours. That is, Abraham and Minkowski correspond to the same T, and thus the same physics.

 

[Dale]

First, you are making a whole bunch of unphysical assumptions, so even if your conclusion is correct, it could serve as a reducto ad absurdum disproof of your assumptions rather than a disproof of the total tensor model. However, I don't think that you have reached that level yet:

However, if applying their total-tensor model to the ideal plane wave described above, we obtain the total momentum = ExH/c**2 (=Abraham’s momentum density vector) from their Eqs. (40)-(43), or Eq. (33) by setting the dielectric velocity v = 0, and ExH/c**2 cannot be used to constitute a Lorentz covariant momentum-energy 4-vector.

Why not? Can you form this into a proper total momentum tensor and show that boosting it to some other frame v<<c gives the wrong total momentum?

 

[PhilDSP]

The medium rest frame is non-inertial.

Just to make sure I'm following... The medium rest frame will be non-inertial while a force is being exchanged with any particle in the media, right? For that matter, the media rest frame will also be non-inertial, won't it?

However, once all forces are exchanged (or balanced) both the medium and media frames will be inertial and the relationship between the rest frames gives the classical Fizeau result (medium partially dragged by the media)

 

[Dale]

The medium rest frame will be non-inertial while a force is being exchanged with any particle in the media, right?

Yes.

However, once all forces are exchanged (or balanced) both the medium and media frames will be inertial

Yes, provided the material is perfectly rigid so that only the 0 total momentum (net force) is important and not the non-zero local momentum flux (internal forces).

 

[Dale]

But I do know there is such a solution to Maxwell equations, and this physical model is widely presented in physics textbooks:

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, 1999), 3rd Edition;
M. Born and E. Wolf, Principles of optics (5th edition) (Oxford, 1975);
J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941);
J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & Sons, NY, 1975);
W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, 1968), 3rd edition;
D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition;

to name a few.

FYI, although it is slightly off-topic, just in case you were unaware I wanted to let you know that ideal gasses are also not physical, although they are even more widely presented in textbooks.

Oh, also, rigid bodies aren't really rigid.

And ideal capacitors, inductors, and batteries don't actually exist.

And there aren't any perfect crystals.

There aren't any classical point particles either.

And Santa Claus and the Easter Bunny and the Tooth Fairy too.

 

[PeterDonis]

Whaddaya mean all that stuff isn't real? I can buy it on the web, right here:

http://www.lhup.edu/~dsimanek/ideal/ideal.htm

 

[Dale]

Whaddaya mean all that stuff isn't real? I can buy it on the web, right here:

http://www.lhup.edu/~dsimanek/ideal/ideal.htm

That is excellent! Thanks.

 

[jtbell]

But it doesn't list inertial reference frames.

 

[PeterDonis]

Yes, I think they need to expand their catalog:

NEW! Taylor-Wheeler ideal clocks! Guaranteed to tick off *exact* proper time regardless of acceleration or your money back! Never worry about the clock postulate again!

Special bundle offer: order a Taylor-Wheeler ideal clock and get a FREE Taylor-Wheeler ideal meter stick as well! Guaranteed to mark off exactly 9,192,631,770 / 299,792,458 cesium-133 hyperfine transition wavelengths!

 

[sciencewatch]

...please take a closer look at the first paper you cited: Tomás Ramos, Guillermo F. Rubilar, Yuri N. Obukhov, Phys. Lett. A 375, 1703 (2011); http://arxiv.org/abs/1103.1654
This is an excellent example of how to use the total energy-momentum tensor formalism I am talking about, to show equivalence of Abraham and Minkowski approaches for an Einstein Box....

In the paper [Phys. Lett. A 375, 1703 (2011); http://arxiv.org/abs/1103.1654 ], the total energy-momentum tensor T_mⁿ is given by Eq. (3), which is a Lorentz covariant second-rank 4-tensor.

To my knowledge, a row- or column-vector of a Lorentz covariant second-rank 4-tensor has 4 components, but it is not necessarily a Lorentz covariant 4-vector. Am I right?

But, why does Eq. (5) denote a Lorentz covariant 4-momentum? T_m⁰ is just a row- or column-vector of the total energy-momentum tensor T_mⁿ. Did I miss something?

To DaleSpam: Help me, please. You are a good expert.

 

[sciencewatch]

FYI, although it is slightly off-topic, just in case you were unaware I wanted to let you know that ideal gasses are also not physical, although they are even more widely presented in textbooks.

Oh, also, rigid bodies aren't really rigid.
...

I guess the Einstein box [Phys. Lett. A 375, 1703 (2011); http://arxiv.org/abs/1103.1654 ] is also assumed to be "rigid", made from isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal dielectric; otherwise the internal energy of Einstein box would change when the light pulse goes through the box, and the refractive index n would be a complex number and a function of frequency. Note: A light pulse occupies a finite-width spectrum of frequency.

 

[Dale]

To my knowledge, a row- or column-vector of a Lorentz covariant second-rank 4-tensor has 4 components, but it is not necessarily a Lorentz covariant 4-vector. Am I right?

But, why does Eq. (5) denote a Lorentz covariant 4-momentum? T_m⁰ is just a row- or column-vector of the total energy-momentum tensor T_mⁿ. Did I miss something?

I agree, I am not comfortable taking a certain column, it makes the equation no longer manifestly covariant. Particularly under different coordinate systems that may not be orthonormal or inertial.

 

[sciencewatch]

Suppose that an infinite uniform plane wave propagates in an infinite isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium with a refractive index >1. Speaking in macro-electromagnetic theory, the medium-rest frame is an inertial frame.

For such an ideal plane-wave model, the medium is assumed to be "rigid", and the total force acting on the whole medium is zero after taking average over time (in one light-wave period) and space (in one wavelength) for all possible micro-scale forces. In other words, there are no accelerations for the medium. Don’t ask me how to set up the plane wave and how to get such a medium; I don’t know. But I do know there is such a solution to Maxwell equations, and this physical model is widely presented in physics textbooks:

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, 1999), 3rd Edition;
M. Born and E. Wolf, Principles of optics (5th edition) (Oxford, 1975);
J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941);
J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & Sons, NY, 1975);
W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, 1968), 3rd edition;
D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition;

to name a few.

************************
“OK, so what theory does this example negate and how?”

The author [Rev. Mod. Phys.79:1197-1216 (2007), http://arxiv.org/abs/0710.0461 ] claims that the total energy-momentum tensor is “uniquely determined by consistency with special relativity”. However, if applying their total-tensor model to the ideal plane wave described above, we obtain the total momentum = ExH/c**2 (=Abraham’s momentum density vector) from their Eqs. (40)-(43), or Eq. (33) by setting the dielectric velocity v = 0, and ExH/c**2 cannot be used to constitute a Lorentz covariant momentum-energy 4-vector. Therefore, the total-tensor model does break the special relativity, unless they have strong arguments to refute the above ideal plane-wave model.

------------------
To the authors [Rev. Mod. Phys.79:1197-1216 (2007)]: Please check the last term of Eq. (33), “+ ExM/c**2”, and make sure if there is a sign typo: – ?. Seems not consistent with Eq. (31) and Eqs. (40)-(43).

************************
PS:

In post #70, the author of Rev. Mod. Phys.79:1197-1216 (2007) (henceforth RMP79) claims:

In fact, the main thrust of Sec. VIII of RMP79 is that once the material properties of the dielectric are specified, the total momentum tensor is uniquely determined by

(i) consistency with special relativity, and
(ii) conservation of linear and angular momentum.

This leads to two important conclusions:

(a) No valid combination of EM and material energy-momentum tensors can break special relativity. If you are using a combination of tensors which appears to break this, then your choice of tensors is incorrect (usually, the material tensor is incorrect or missing). Note that I have never yet seen a fully relativistic formulation of the material counterpart tensors written down anywhere in the literature - even those given in RMP79 are valid only for media moving at v<<c, though the full expressions could be obtained from Eqs. (33)-(34).

(b) As the _total_ energy-momentum tensor is uniquely fixed by (i) and (ii) above, any division into components necessarily yields the same total tensor, and thus the same physical behaviours. That is, Abraham and Minkowski correspond to the same T, and thus the same physics.

A misunderstanding of the special principle of relativity?

In the community of special theory of relativity, there is a well-recognized implicit assumption that a physical formulation represented by a Lorentz covariant 4-vector or tensor must be consistent with the principle of relativity. Some scientists even say everything is Lorentz transformation. In fact, this is a delusion. One example, as I indicated, is the total energy-momentum tensor model developed by Rev. Mod. Phys.79:1197-1216 (2007), http://arxiv.org/abs/0710.0461 , and verified by Phys. Lett. A 375, 1703 (2011); http://arxiv.org/abs/1103.1654 (confer: Posts #116, #126, #125, #127). The total energy-momentum tensor is Lorentz covariant, indeed; however, its physical implication is breaking the principle of relativity.

There is another simpler example to show this delusion, presented in the thread entitled “Are the principle of relativity and the Lorentz invariance equivalent?” [ https://www.physicsforums.com/showthread.php?t=551544 ], which is copied below:

The fundamental requirements of the special relativity on relativistic electrodynamics are that time-space coordinates and two electromagnetic (EM) field-strength tensors follow Lorentz transformations, resulting in the invariance of Maxwell equations in forms in any inertial frames.

Sometimes a formulation, which follows Lorentz transformation, might breaks the special principle of relativity. A typical example is the formulation for Fizeau running water experiment. Why?

The Fizeau experiment is usually used to illustrate the relativistic velocity addition rule in the textbooks. Observed in a frame which is fixed with a uniform medium with a refractive index of n, the photon's speed is c/n, and the photon's 4-velocity is gp'*(up',c), with |up'|=c/n and gp'=1/sqrt(1-up'**2/c**2). Suppose the medium moves at v, relatively to the lab frame. Observed in the lab frame, the photon 3D-velocity, up, is obtained from the Lorentz transformation of gp'*(up',c). However, the obtained-photon-velocity up is not parallel to the 3D-wave vectror k in the lab frame, unless the medium moves parallel to the wave vector k.

According to the principle of relativity, the photon's velocity must be parallel to the wave vector in any inertial frames. Thus from above analysis, the photon's 3D-velocity in a medium can not be used to constitute a Lorentz covariant 4-velocity. In other words, the photon's 4-velocity in a medium, which follows Lorentz transformation, breaks the special principle of relativity instead.

------

PS:

A slightly different total–momentum model seems to be first presented by Baxter et al. [Phys. Rev. A 47, 1278 (1993), http://pra.aps.org/abstract/PRA/v47/i2/p1278_1 ], and later by Leonhardt at a different angle of view [Phys. Rev. A 73, 032108 (2006), http://pra.aps.org/abstract/PRA/v73/i3/e032108 ]. The model of total momentum = dielectric kinetic momentum + Abraham’s momentum is further analyzed and identified by Barnett [see Eq. (7), Phys. Rev. Lett. 104, 070401 (2010), http://prl.aps.org/abstract/PRL/v104/i7/e070401] . When applying this total-momentum model to the ideal infinite uniform plane wave in an ideal infinite dielectric medium (Confer Post #116), the total momentum is Abraham’s momentum, which is the same as that obtained from the mentioned-above total momentum-energy tensor model [Rev. Mod. Phys.79:1197-1216 (2007), http://arxiv.org/abs/0710.0461] .

 

[rpfeifer]

Hello all!

It looks like a lot of interesting things have been said on this thread since I last checked in. In particular, sciencewatch's model is now much more thoroughly described.

First, though, I'd like to warn about using macroscopic EM theory to attempt to calculate momentum density. This is inappropriate as momentum density is a microscopic property of the system.

Now, on to the interesting question raised by this model:
Why should p be covariant?
In vacuum, this doesn't seem to be open to debate. In a material medium, however, v is essentially a free parameter, so this doesn't seem to be enforced.
I'll address my argument to steady state scenarios (e.g. laser beam passing through glass block), as the situation is more complex when not at steady state.

(1) In a system where the beam enters the medium from vacuum, p is a 4-vector just outside the medium, and conservation of momentum requires that it be a 4-vector inside the medium as well, constraining v accordingly.

(2) In a system where the dielectric is infinite and fills all of space and time, the EM pulse does not enter the system from vacuum. In the absence of external constraints, we may set v arbitrarily, as sciencewatch has done, even where this results in a T^{\mu\nu} which violates special relativity.

However, such a system is "unphysical": Our universe is largely empty space, and is believed to be described by physical models where the density of all fields (including matter fields) drops to zero at spatial infinity. Trace its trajectory back to t=-∞, and no matter how large the dielectric, that photon originally entered from vacuum.

Although the situation is a little more complicated when you consider that the EM wave may have been emitted by decay of an excited electron state in e.g. air rather than vacuum, the emitting matter itself also has a history, and this too must be consistent with the boundary conditions of the universe. To properly treat this goes beyond the limitations of classical EM, but ultimately, the implication is that only situations in which total momentum forms a 4-vector are compatible with the assumption of zero fields (of all sorts; bosons & fermions) at x→∞.

Hence SR is not compatible with situation (2), but situation (2) does not describe a situation compatible with the physical universe.

I guess sciencewatch and I must just rub each other up the wrong way - the sophistication of his arguments seem to have increased the moment I left the thread

Also:
PS - Thanks for the erratum! Fortunately the error doesn't carry forwards into any subsequent expressions. I'll recheck the explicit expression for T as well, and upload a correction. It just goes to show - you should always check what you read in the papers!

 

[keji8341]

...

(2) In a system where the dielectric is infinite and fills all of space and time, the EM pulse does not enter the system from vacuum. In the absence of external constraints, we may set v arbitrarily, as sciencewatch has done, even where this results in a T^{\mu\nu} which violates special relativity.

However, such a system is "unphysical": Our universe is largely empty space, and is believed to be described by physical models where the density of all fields (including matter fields) drops to zero at spatial infinity. Trace its trajectory back to t=-∞, and no matter how large the dielectric, that photon originally entered from vacuum.

...

Please note: Einstein used a unifrom plane wave in free space to derive Doppler effect [A. Einstein, Ann. Phys. Lpz. 17, 891 (1905), “On the Electrodynamics of Moving Bodies,” http://www.fourmilab.ch/etexts/einstein/specrel/www/ ].

Is there a real plane wave in practice?
Is there a real free space in practice?

 

[PeterDonis]

Our universe is largely empty space, and is believed to be described by physical models where the density of all fields (including matter fields) drops to zero at spatial infinity. Trace its trajectory back to t=-∞, and no matter how large the dielectric, that photon originally entered from vacuum.

For models that are just supposed to cover isolated systems, I agree; but for cosmological models, where we are trying to model the entire universe, this is not really true, is it? The universe's average density of mass-energy is nonzero. And if we trace a photon's trajectory back far enough (assuming it's a photon that has been traveling freely through the universe's entire history since "recombination", such as a CMB photon), we end up in a hot, dense plasma, not vacuum.

 

[rpfeifer]

Keji - regarding your question (which was subsequently deleted) as to whether I had "banned" sciencewatch: I have no moderation powers, and I had assumed sciencewatch had voluntarily unsubscribed from the forum.

I'm not sure if you were trying to insult me with the deleted comment, or whether you were just asking whether I had some sort of unquestioned authority here (I don't). I'll give you the benefit of the doubt on that one.Regarding plane waves and free space:
(i) Plane waves are generally recognised as a useful basis from which an arbitrary waveform may be constructed; however, in my experience most scientists agree that they can only be at best approximated in a real-world context. Thus: Any waveform may be considered a superposition of a possibly infinite number of plane waves - but you're unlikely to meet one on its own.
(ii) Free space: Good question.

Peter Donis: You also makes some good points regarding this.

One way to refine the argument is as follows:
It is in fact sufficient that the area of interest be enclosed by _any_ boundary B on its past light cone (not necessarily at infinity), provided all fields and their gradients vanish on this boundary, and that this boundary, or a smooth perturbation of it, persists over the duration that we monitor the system of interest. Thus we don't need to go to t=-∞.

I am not sure of the rigorous status of the "fields vanish at infinity" argument, except that it is widely used in quantum field theory. If it isn't proven, then it is at least a useful working assumption. Nor do I know how it is usually reconciled with the early history of the universe. Possibly it's just really hard to find a situation where you need to go that far back to construct a boundary B.

It's a good question, and one you'd need to ask a cosmologist or quantum cosmologist.

 

[rpfeifer]

One way to refine the argument is as follows:
It is in fact sufficient that the area of interest be enclosed by _any_ boundary B on its past light cone (not necessarily at infinity), provided all fields and their gradients vanish on this boundary, and that this boundary, or a smooth perturbation of it, persists over the duration that we monitor the system of interest. Thus we don't need to go to t=-∞.

Which is really just a paraphrase of Peter Donis' comment that this is fine for isolated systems :)

 

[PeterDonis]

Which is really just a paraphrase of Peter Donis' comment that this is fine for isolated systems :)

Yes, as long as the average density of mass-energy in the universe as a whole is small enough to ignore when constructing the model of the isolated system, so that you can find a boundary surface B on which the fields can be taken to be zero to within the accuracy desired.

 

[yoron]

Very nice reading. And I think you've done a very good job explaining your points in words too RP. I will have to read up on those tensors.

 

[Dale]

In the community of special theory of relativity, there is a well-recognized implicit assumption that a physical formulation represented by a Lorentz covariant 4-vector or tensor must be consistent with the principle of relativity.

This is correct. There are two transformations which are consistent with the principle of relativity, the Lorentz transform and the Galilean transform. If a law is covariant wrt arbitrary diffeomorphisms then it is necessarily covariant wrt both the Lorentz transform and the Galilean transform. Therefore any physical law expressed in a tensor form is mathematically guaranteed to be compatible with the principle of relativity.

Some scientists even say everything is Lorentz transformation. In fact, this is a delusion.

This language is completely inappropriate for this forum.

One example, as I indicated, is the total energy-momentum tensor model developed by Rev. Mod. Phys.79:1197-1216 (2007), http://arxiv.org/abs/0710.0461 , and verified by Phys. Lett. A 375, 1703 (2011); http://arxiv.org/abs/1103.1654 (confer: Posts #116, #126, #125, #127). The total energy-momentum tensor is Lorentz covariant, indeed; however, its physical implication is breaking the principle of relativity.

In post 117 I challenged you to justify this claim by showing that boosting the total momentum tensor into a different frame and showing that you get the wrong total momentum, which you did not do. Simply repeating a statement is not a justification.

 

[Dale]

(2) In a system where the dielectric is infinite and fills all of space and time, the EM pulse does not enter the system from vacuum. In the absence of external constraints, we may set v arbitrarily, as sciencewatch has done, even where this results in a T^{\mu\nu} which violates special relativity.

I am not convinced that even in the admittedly unphysical case of an infinite dielectric that special relativity is violated. It seems to me that you should be able to find a frame where the matter tensor corresponding to Abraham's light tensor is 0 (at least the spacelike components). If you boost that total momentum you should get the correct total momentum in any frame. The mere fact that the matter tensor is 0 in some frame doesn't seem to violate anything.

I haven't worked it out myself to verify, but neither did sciencewatch, so I am skeptical about the assertion.

 

[PhilDSP]

(2) In a system where the dielectric is infinite and fills all of space and time, the EM pulse does not enter the system from vacuum. In the absence of external constraints, we may set v arbitrarily, as sciencewatch has done, even where this results in a T^{\mu\nu} which violates special relativity.

However, such a system is "unphysical": Our universe is largely empty space, and is believed to be described by physical models where the density of all fields (including matter fields) drops to zero at spatial infinity. Trace its trajectory back to t=-∞, and no matter how large the dielectric, that photon originally entered from vacuum.

Although the situation is a little more complicated when you consider that the EM wave may have been emitted by decay of an excited electron state in e.g. air rather than vacuum, the emitting matter itself also has a history, and this too must be consistent with the boundary conditions of the universe. To properly treat this goes beyond the limitations of classical EM, but ultimately, the implication is that only situations in which total momentum forms a 4-vector are compatible with the assumption of zero fields (of all sorts; bosons & fermions) at x→∞.

Shouldn't the atom or molecule from which the EM wave was emitted be considered a micro-dielectric? Certainly \epsilon != \epsilon_0 and \mu != \mu_0 in the atom's near field.

 

[rpfeifer]

I am not convinced that even in the admittedly unphysical case of an infinite dielectric that special relativity is violated. It seems to me that you should be able to find a frame where the matter tensor corresponding to Abraham's light tensor is 0 (at least the spacelike components). If you boost that total momentum you should get the correct total momentum in any frame. The mere fact that the matter tensor is 0 in some frame doesn't seem to violate anything.

I haven't worked it out myself to verify, but neither did sciencewatch, so I am skeptical about the assertion.

You're right - I was wrong in considering the possibility of SR violation. The value of p has no effect on the transformation properties of p. I got dazzled by the unphysicality of the situation (which really isn't all that relevant to this part of the issue, after all).

Also, it should still be possible to model such a system consistent with the approach in RMP79. Earlier, I assumed (mistakenly) that without constraints from a spatial material boundary, the v field would be arbitrary. However, this is only true for the v you choose as initial conditions, on some initial slice of Minkowski space-time which is isochronous in the rest frame of the dielectric. That slice then acts as initial conditions for subsequent evolution, and even from a starting point with arbitrary v the evolution of the system is going to involve coupling between the EM wave and the dielectric. Two possibilities arise for v=0 on that initial slice:
(i) If the material is only instantaneously at rest, this is not incompatible with the Abraham material component. It just represents one extreme of the wave motion.
(ii) If the material is at rest and its acceleration is zero, this probably just represents an instantaneous superposition of an EM wave, its accompanying material excitation, and another material wave of some sort (pressure, phonons) that just happened to be passing through. Coupling to the EM wave will presumably still occur for any reasonable (i.e. physical) model of a dielectric.

So I'm going to change my position and say that I don't think this situation has the potential to violate SR or to cause problems for RMP79, either.

I think we'd need at least two separate Cauchy surfaces (sets of initial conditions on surfaces in Minkowski space that lie on the past or forward lightcone of everywhere) to create a contradiction with RMP79, and all this would disprove is the physical validity of the initial conditions, not of the model. I don't think it's possible to get conflict with SR at all, which is as it should be, since the mechanism used for time evolution (i.e. RMP79) respects SR.

(An example of such a non-physical set of initial conditions for n\not=1:
Surface 1: Wave present, v everywhere 0, at time t.
Surface 2: Wave present, v everywhere 0, at time t+dt where dt is infinitesimal, implying no coupling between wave and dielectric.)

 

[rpfeifer]

Shouldn't the atom or molecule from which the EM wave was emitted be considered a micro-dielectric? Certainly \epsilon != \epsilon_0 and \mu != \mu_0 in the atom's near field.

I'm not sure how safe the description of a dielectric in terms of \epsilon and \mu is in this regime. Given that the usual description of the origin of refractive index is in terms of multiple dipole scatterings retarding the propagation of the wavefront, I'd be very careful about using this model on the scale of a single atom.

That's not to say it doesn't work - just that based on the derivation I'm aware of, this situation is running right up against the limits of validity for this particular set of tools. I'd be much more comfortable here considering an ideal wave source rather than a source atom.

After writing this, I had some fun with the idea of an ideal wave source - e.g. put the atom (or a lot of hot atoms) in a box with a shutter, then open the shutter to let out the wave pulse. If you time it right, the wave can potentially completely avoid interacting with the box at all, but we can consider the opening in the box to be the source.

But then, if the wave doesn't interact with the box, we can remove the box entirely to guarantee no interaction... so really, a perfect source is just any bit of vacuum that a photon happens to be traveling through after leaving its matter source, and we can call anything a vacuum so long as the nearest atoms are far enough away...

No point to that bit of musing, really, other than as a reflection on the sorts of abstractions that go into these large-scale descriptions of physical systems.

 

[rpfeifer]

I haven't worked it out myself to verify, but neither did sciencewatch, so I am skeptical about the assertion.

I haven't formally set up the most general situation, evolved it, and boosted it, either, but I don't particularly feel the need to right now. Given the explicitly SR-invariant construction of the formalism, I believe the burden of proof is now on anyone who wishes to claim they have a situation which uses this formalism to violate SR.

 

[rpfeifer]

Very nice reading. And I think you've done a very good job explaining your points in words too RP. I will have to read up on those tensors.

Thanks yoron, it's nice to see my efforts (and those of other posters, such as DaleSpam) appreciated. That's why I stuck around in this thread - in case there was someone reading it who would find my explanations useful.

Best of luck with the tensors! They're not as fierce as they look.

 

[Dale]

I haven't formally set up the most general situation, evolved it, and boosted it, either, but I don't particularly feel the need to right now. Given the explicitly SR-invariant construction of the formalism, I believe the burden of proof is now on anyone who wishes to claim they have a situation which uses this formalism to violate SR.

I agree. Since the formalism is based on tensors and since tensor equations are manifestly covariant it seems impossible for the formalism to violate SR. That is, after all, the whole point of expressing physics in terms of tensors.

 

[PhilDSP]

I'm not sure how safe the description of a dielectric in terms of \epsilon and \mu is in this regime. Given that the usual description of the origin of refractive index is in terms of multiple dipole scatterings retarding the propagation of the wavefront, I'd be very careful about using this model on the scale of a single atom.

That's not to say it doesn't work - just that based on the derivation I'm aware of, this situation is running right up against the limits of validity for this particular set of tools. I'd be much more comfortable here considering an ideal wave source rather than a source atom.

Okay, thanks for your opinion. I imagine you were thinking of the derivation behind the Ewald-Oseen extinction theory. I agree with the need for care in deriving and using procedures more fundamental than that. I'm thinking that a rigorous derivation of a provisional more basic theory plus one or two or three dependent experimental facts would be enough to warrant some attention.

 

[yoron]

Well, Dale is also good to read. And uses clear approaches which makes it understandable for us laymen, on the whole this thread is one of the most interesting I've read so far. And there Science watch also have to get credit. After all, without his questions I wouldn't have gotten this far, learning how physics considered it mathematically.

 

[keji8341]

...any physical law expressed in a tensor form is mathematically guaranteed to be compatible with the principle of relativity.
...

You'd better carefully check your statement: What are your grouds? Mathematically or physically? Sufficiency or/and necessity?

I checked Sciencewatch's simple example copied below; it's true. You'd better check it to see what mistakes Sciencewatch made before refuting him (her).
--------------------
Sometimes a formulation, which follows Lorentz transformation, might breaks the special principle of relativity. A typical example is the formulation for Fizeau running water experiment. Why?

The Fizeau experiment is usually used to illustrate the relativistic velocity addition rule in the textbooks. Observed in a frame which is fixed with a uniform medium with a refractive index of n, the photon's speed is c/n, and the photon's 4-velocity is gp'*(up',c), with |up'|=c/n and gp'=1/sqrt(1-up'**2/c**2). Suppose the medium moves at v, relatively to the lab frame. Observed in the lab frame, the photon 3D-velocity, up, is obtained from the Lorentz transformation of gp'*(up',c). However, the obtained-photon-velocity up is not parallel to the 3D-wave vectror k in the lab frame, unless the medium moves parallel to the wave vector k.

According to the principle of relativity, the photon's velocity must be parallel to the wave vector in any inertial frames. Thus from above analysis, the photon's 3D-velocity in a medium can not be used to constitute a Lorentz covariant 4-velocity. In other words, the photon's 4-velocity in a medium, which follows Lorentz transformation, breaks the special principle of relativity instead.
---------------
Be sure to set the dielectric moving direction NOT parallel to the wave vector. In the dielectric-rest frame, the photon's 3D-velocity is parallel to the wave vector.

 

[Dale]

Well, Dale is also good to read. And uses clear approaches which makes it understandable for us laymen

Thanks, I appreciate that. Sometimes encouragement is difficult to come by.

 

[longlive]

Suppose that an infinite uniform plane wave propagates in an infinite isotropic, homogeneous, non-conducting, no-loss, non-dispersive, ideal medium with a refractive index >1. Speaking in macro-electromagnetic theory, the medium-rest frame is an inertial frame.

For such an ideal plane-wave model, the medium is assumed to be "rigid", and the total force acting on the whole medium is zero after taking average over time (in one light-wave period) and space (in one wavelength) for all possible micro-scale forces. In other words, there are no accelerations for the medium. Don’t ask me how to set up the plane wave and how to get such a medium; I don’t know. But I do know there is such a solution to Maxwell equations, and this physical model is widely presented in physics textbooks:

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, 1999), 3rd Edition;
M. Born and E. Wolf, Principles of optics (5th edition) (Oxford, 1975);
J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941);
J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & Sons, NY, 1975);
W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, 1968), 3rd edition;
D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition;

to name a few.

************************
“OK, so what theory does this example negate and how?”

The author [Rev. Mod. Phys.79:1197-1216 (2007), http://arxiv.org/abs/0710.0461 ] claims that the total energy-momentum tensor is “uniquely determined by consistency with special relativity”. However, if applying their total-tensor model to the ideal plane wave described above, we obtain the total momentum = ExH/c**2 (=Abraham’s momentum density vector) from their Eqs. (40)-(43), or Eq. (33) by setting the dielectric velocity v = 0, and ExH/c**2 cannot be used to constitute a Lorentz covariant momentum-energy 4-vector. Therefore, the total-tensor model does break the special relativity, unless they have strong arguments to refute the above ideal plane-wave model.

------------------
To the authors [Rev. Mod. Phys.79:1197-1216 (2007)]: Please check the last term of Eq. (33), “+ ExM/c**2”, and make sure if there is a sign typo: – ?. Seems not consistent with Eq. (31) and Eqs. (40)-(43).

************************
PS:

In post #70, the author of Rev. Mod. Phys.79:1197-1216 (2007) (henceforth RMP79) claims:

In fact, the main thrust of Sec. VIII of RMP79 is that once the material properties of the dielectric are specified, the total momentum tensor is uniquely determined by

(i) consistency with special relativity, and
(ii) conservation of linear and angular momentum.

This leads to two important conclusions:

(a) No valid combination of EM and material energy-momentum tensors can break special relativity. If you are using a combination of tensors which appears to break this, then your choice of tensors is incorrect (usually, the material tensor is incorrect or missing). Note that I have never yet seen a fully relativistic formulation of the material counterpart tensors written down anywhere in the literature - even those given in RMP79 are valid only for media moving at v<<c, though the full expressions could be obtained from Eqs. (33)-(34).

(b) As the _total_ energy-momentum tensor is uniquely fixed by (i) and (ii) above, any division into components necessarily yields the same total tensor, and thus the same physical behaviours. That is, Abraham and Minkowski correspond to the same T, and thus the same physics.

Seems right.

Time-space coordinates and electromagnetic field-strength tensors obey Lorentz transformations, and the Maxwell equations keep the same forms in all inertial frames.

An ideal uniform plane wave is a simplest solution to Maxwell equations, and observed in any inertial frames, it is always a plane wave and satisfies Maxwell equations.

If an electromagnetic expression or equation (in which all field quantities must satisfy Maxwell equations) is derived from Maxwell equations without finite-boundary conditions used, then this electromagnetic expression or equation should be applicable to a plane wave, because all field quantities of the plane wave satisfy Maxwell equations. If not, the first thing I would like to do is to check my derivations, including basic assumptions or physical models.

When light propagates in a block of uniform dielectric medium with its dimension much larger the wavelength (Einstein's box), is the light momentum problem beyond the macro-electromagnetic theory? I don’t think so.

 

[Dale]

You'd better check your statement: What are your grouds?

The statement is correct. The principle of relativity is simply that the form of the laws of physics is preserved under boosts. A boost is a diffeomorphism. The form of any tensor equation is preserved under any diffeomorphism. Therefore, any physical law which is written in tensor form is preserved under boosts, and so it is compatible with the principle of relativity.

 

[keji8341]

The statement is correct. The principle of relativity is simply that the form of the laws of physics is preserved under boosts. A boost is a diffeomorphism. The form of any tensor equation is preserved under any diffeomorphism. Therefore, any physical law which is written in tensor form is preserved under boosts, and so it is compatible with the principle of relativity.

Sciencewatch said "the photon's 4-velocity in a medium, which follows Lorentz transformation, breaks the special principle of relativity instead". That is true or not in your opinion?