Mass Density of Photons in Refractive Medium

[jeast]

The effective mass density of photons in a vacuum $\rho^{vac}_M$ is related to the photon energy density $\rho^{vac}_E$ by
$$\rho^{vac}_M=\frac{\rho^{vac}_E}{c^2}.$$
Is it true that the mass density of photons inside a medium of refractive index $n$, $\rho^n_M$, with phase velocity $v=c/n$, is related to the photon energy density $\rho^n_E$ by
$$\rho^n_M=\frac{\rho^n_E}{v^2}?$$

 

[PeterDonis]

effective mass density

What does this even mean?

the mass density of photons inside a medium

What does this even mean?

Where are you getting all this from?

 

[jeast]

In cosmology one works with the energy density of photons/radiation as a source of space-time curvature. In the low curvature, low speed limit of Newton’s law of gravitation it is mass that produces a gravitational acceleration. In order to use photons in Newtons law one has to convert photon energy to an effective mass.

 

[PeterDonis]

In cosmology one works with photons/radiation as a source of space time curvature.

Yes, and we describe this source in GR using a stress-energy tensor.

In the limit of Newton’s law of gravitation

Which is not a valid limit for photons/radiation.

In order to use photons in Newtons law

You can't. Newton's law does not work for photons/radiation. You have to use the correct relativistic equations with a stress-energy tensor.

 

[jeast]

Einstein’s equations and Newtons equations are consistent in the low space-time curvature limit of cosmology. The constant $\kappa=8\pi G/c^4$ in Einstein’s field equations contains one $c^2$ to convert between space-time curvature and Newtonian time acceleration and one $c^2$ to convert between energy density and mass density.

 

[PeterDonis]

Einsteins equations and Newtons equations are consistent in the low space time curvature limit of cosmology.

This is not correct. The Newtonian limit of GR is not the same as the low curvature limit.

The constant $\kappa=8\pi G/c^4$ in Einsteins field equations contain one $c^2$ to convert spacetime curvature to Newtonian time acceleration

I don't know where you are getting this from, but it's wrong.

Please be aware that personal theories and personal speculations are off limits here at PF. If you cannot give any references to back up your claims, this thread will be closed.

 

[vanhees71]

Of course you can calculate the polarization (self-energy) of photons in the medium. E.g., in a plasma there are the (longitudinal) plasmon modes which are indeed massive. For details, see, e.g.,

J. I. Kapusta and C. Gale, Finite-Temperature Field Theory;
Principles and Applications, Cambridge University Press, 2
edn. (2006).

 

[Ibix]

Einstein’s equations and Newtons equations are consistent in the low space-time curvature limit of cosmology.

I don't think this is generally true. Certainly not in a radiation-dominated universe where pressure is a significant source of gravity. And as far as I can see a Newtonian universe everywhere filled with a perfect fluid has no trouble being static, where a relativistic one cannot be static (absent careful and unstable fine tuning of the cosmological constant, which you can't have in Newtonian gravity).

 

[PeterDonis]

in a plasma there are the (longitudinal) plasmon modes which are indeed massive

"Massive" in the sense of not being on the normal photon mass shell of zero invariant mass. But the OP is using "mass" in a different (and not valid) sense.

 

[pervect]

In cosmology one works with the energy density of photons/radiation as a source of space-time curvature. In the low curvature, low speed limit of Newton’s law of gravitation it is mass that produces a gravitational acceleration. In order to use photons in Newtons law one has to convert photon energy to an effective mass.

As others have said, the Newtonian limit of GR is for low velocities. As a check on this, consider the Wikipedia entry on the Newtonian limit. The current version as of this post is https://en.wikipedia.org/w/index.php?title=Newtonian_limit&oldid=1154666633, and it cites Sean Carrol.

If you think differently, it would be good to try to recall why, and additionally cite a reference on the point.

In physics, the Newtonian limit is a mathematical approximation applicable to physical systems exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly changing (or completely static) gravitational fields.[1]

So, I would also encourage you to formulate your problem differently.

An approach to a good formulation of this issue has already been mentioned. What you want to find is the stress-energy tensor of the "photon" in the medium, not the effective mass. Finding the stress-energy tensor of a classical EM wave in a classical medium would be what I think you want. Other posters have suggested more specific sources.

Some more personal comments and examples related to the original point. The deflection of light according to GR is famously twice that of Newton's theory. It seems obvious, then, that is why the weak field limit ala Wiki and a Caroll specifically excludes fast moving particles from the limit - Newtonian theory fails to predict the "excess" deflection of light.

For instance, if you want your approach to be able to answer a question like "how does light deflect in a medium around a massive object", you want your approach to give the correct answer for how light deflects if the medium is a vacuum.

 

[Vanadium 50]

Which is not a valid limit for photons/radiation.

It is, however, a valid approximation for a "box of radiation" - i.e. light in a perfectly reflective cavity with E and no p.

However, I think the larger point that what is being written doesn't make much sense. "Photons" are not a "thing" in GR

Further, a photon in a medium is not a pre-existing particle that happens to enter a medium. It's a collective effect involving thousand of atoms. So the word "photon" makes this a poorly posed question.

However, the biggest problem is that n cannot be the correct factor relativistically. A moving magnetization gives rise to a polarization (even in SR) so n should not appear without a μsomewhere. Without calculating what the answer is (and I fear that will be the OPs response) I can tell what the answer isn't.

 

[PeterDonis]

It is, however, a valid approximation for a "box of radiation" - i.e. light in a perfectly reflective cavity with E and no p.

I'm not aware of any such model. The only model of a "box of radiation" that I'm aware of has $p = \rho / 3$, where $\rho$ is the energy density. Do you have a reference for the model of cavity radiation you are describing?

 

[pervect]

I'm not aware of any such model. The only model of a "box of radiation" that I'm aware of has $p = \rho / 3$, where $\rho$ is the energy density. Do you have a reference for the model of cavity radiation you are describing?

Steve Carlip has a paper relevant to the "light in a box" problem. See for instance https://arxiv.org/abs/gr-qc/9909014, "Kinetic energy an the equivalence principle".

According to the general theory of relativity, kinetic energy contributes to gravitational mass. Surprisingly, the observational evidence for this prediction does not seem to be discussed in the literature. I reanalyze existing experimental data to test the equivalence principle for the kinetic energy of atomic electrons, and show that fairly strong limits on possible violations can be obtained. I discuss the relationship of this result to the occasional claim that ``light falls with twice the acceleration of ordinary matter.''

Note that the majority of paper is about the effect of kinetic energy of massive particles, but Carlip also discusses the "light in a box" problem.

Also relevant is a paradox, initially introduced by Tolman. See Misner et al, "Active Gravitational Mass", https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1045. A quote from the abstract:

Tolman states that "...disordered radiation in the interior of a fluid sphere contributes roughly speaking twice as much to the gravitational field of the sphere as the same amount of energy in the form of matter." The gravitational pull exerted by a system on a distant test particle might therefore at first sight be expected to increase if within the system a pair of oppositely charged electrons annihilate to produce radiation. This apparent paradox is analyzed here in the case where gravitational effects internal to the system are unimportant. It is shown that tensions in the wall of the container compensate the effect mentioned by Tolman so that the net gravitational pull exerted by the system does not change.

To recap, if you have a very strong box, and inside the box you convert matter to radiation, the mass of the box doesn't change as a result of the conversion, as one would expect. There are different schemes for accounting for the mass - if we use the Komar scheme, we integrate $\rho + 3P$ over the box and the walls. The pressure term is positive in the interior of the box, and negative in the walls of the box, for a net result of zero for the box+wall system.

This is in principle experimentally testable - one can in principle measure the proper acceleration of a stationary test particle just inside the walls of the box and find that it's different from the proper acceleration of a test particle just outside the walls of the box. I've played with this in the past using the Schwarzschild metric - being unwilling to deal with junction conditions, I took the approach of making the model of the "walls" of the box of finite thickness so I didn't have to deal with learning about how to handle the junction conditions. I could try and dig out the posts where I did this with search, but I probably won't bother unless someone is interested enough to ask. I've long since forgotten the details.

Using the Newtonian ideas, if we have a massless box containing matter, and then convert the matter into radiation, we find the total mass of the system of box + contents unchanged, but the mass of the interior of the box (excluding the walls) is higher than we expect, in the relativistic limit the mass of the interior is doubled, the scale reading of a test mass is double E/c^2. But the scale reading outside the walls of the box is E/c^2 as we expect.

Of course the actual analysis does not use this Newtonian language. If you get past the paywall, you can read Misner's paper for his approach. The approach that I used when I was thinking about this issue was to use the Komar mass in the frame of a static observer, basically the integral of $\rho + 3P$, and a spherically symmetrical static metric, basically the Schwarzschild interior metric.

Going back to the light in the box problem for a bit - we can consider the walls + light to be an isolated system, and use E^2 - p^2 (in relativistic units where c=1) to find m^2, E being energy, p momentum, and m mass. However, if we try to find the mass of "just the light" and exclude the mass of the walls of the box, we find that the quantity is not covariant. If one checks Taylor's "space-time physics", one sees the fine print for E^2 - p^2 gives us the mass of an "isolated system", however, while the box + light is an isolated system, the light by itself is not isolated.

 

[PeterDonis]

Carlip also discusses the "light in a box" problem

Yes, and as you note, his resolution includes the fact that we cannot isolate just "the mass of the light" in an invariant way; we can only obtain an invariant for the total mass of the light + box system. This does indeed work out to the rest mass of the box plus the energy of the light (divided by $c^2$); and the reason, as Carlip shows, is the same as for any static system in hydrostatic equilibrium (such as a planet or star): the virial theorem requires the positive contribution of pressure to be exactly canceled by the negative contribution of whatever is holding the system together and keeping it static. In the case of the "box of light", this will be tension (negative pressure) in the box walls. In the case of a planet or star, it is (negative) gravitational binding energy.

 

[vanhees71]

"Massive" in the sense of not being on the normal photon mass shell of zero invariant mass. But the OP is using "mass" in a different (and not valid) sense.

To be very pedantic it's defined as the "pole mass", i.e., the real part of the retarded in-medium Green's function's poles. The imaginary part is (the square of) its inverse lifetime.

 

[Vanadium 50]

Steve Carlip has a paper relevant to the "light in a box" problem.

This goes back a long time, probably before Carlip was in graduate school. This is PPN β (which is one number, 1 in GR, but can be different numbers depending on what is being calculated in non-GR throies). It would deviate from 1 in Eotvos-type experiments, but I think the Nordtvedt effect (essentially an Earth-Moon-Sun Eotvos experiment) sets the tightest bounds.

 

[PeterDonis]

This is PPN β

Do you mean $\beta_1$ in the beta-delta notation described in the Wikipedia article?

https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism

 

[Vanadium 50]

In GR, $\beta_1 = \beta_2 =\beta_3 =\beta_4$ so it doesn't matter. In nonm-GR theories, it matters how a box of light differs from a box of marshmallows. Is it kinetic energy? Pressure? Somethg else?

More technically, in GR, the WEP ensures that the EOS doesn't matter. If you give up GR, in principle the EOS can matter. However, the fact that we cannot see any evidence of WEP violation means that it is premature to ask where the violation occurs.

 

[PeterDonis]

In GR, $\beta_1 = \beta_2 =\beta_3 =\beta_4$ so it doesn't matter.

Yes, agreed.

In nonm-GR theories, it matters how a box of light differs from a box of marshmallows. Is it kinetic energy? Pressure? Somethg else?

Yes, as I understand it, $\beta_1$ is the "kinetic energy" coefficient, which would be the one that would matter for light in a non-GR theory where the coefficients were different.

 

[Vanadium 50]

Think about a relativistic vs. non-relativsitic box of gas. Under GR, we agree that if the box is at rest and we measure its gravitational field (and if you want to get picky, it's acceleration field produced by gravity) we cannot tell what is in the box.

In non-GR, you may or may not be able to tell. That's PPN $\beta$, 1 in GR and possibly something else in an alternative theories.

If it is different, we don't know how it is different. If the difference is due to kinetic energy per unit mass, its $\beta_1$. If it is the pressure per unit mass - also different in the relativistic and non-relativistic case, it's $\beta_4$. So they would - or at least could - both vary.

$\beta_3$ deals with nonlinearities so is unlikely to matter in this case. $\beta_2$ probably doesn't matter for light, but does for a gemeric relativistic gas, as it would include vibrational and rotational modes on top of the kinetioc energy.