"Words, words, words. I get sick of words."

http://www.youtube.com/watch?v=H8zyF0ZOy3k

Guv = 8pi(index of refraction)^4GTuv/c^4 (8)

JW:What's wrong with all this? Well, several things are wrong. Working back through the argument, the first thing to note is that the substitution that leads to equation (8) is not valid.

JS: Give reasons. I think it's an empirical issue that must be settled by experiment.

JW:Neither is it supported by the facts of observation.

JS: I believe you have just made a false statement. What published experiments refute equation (8)? Please provide precise references.

JW: It's not valid because the polarizable vacuum model is not background independent, and any plausible theory that is physically equivalent to general relativity must be background independent.

JS: You have confounded my theory with Puthoff's theory. Yes, I was motivated by Puthoff's PV theory qualitatively. However, I agree with you that Puthoff's theory is not background independent. Indeed, that's the basic reason I rejected it - he does not use tensors. Indeed, the basic dynamical laws of nature should be background independent, but your argument about my equation (8) is fallacious because their solutions need not be! We can see why, with the simpler example of Maxwell's electromagnetic field theory. Maxwell's equations must be Lorentz invariant when written in terms of fundamental particle sources moving through the vacuum in the old microscopic Lorentz formulation with electric charges as Newton's hard massy marbles. However, when you are talking about electromagnetic fields in a material you use a lumped - parameter effective field theory version and the Lorentz invariance is spontaneously broken for a sub-class of interior measurements made inside the material (in principle). Another example is the formation of crystal lattices. The basic laws of nature are thought to be translationally invariant, but when a crystal forms the 3D translational continuous Lie group is spontaneously broken to a discrete crystal group say for the motions of electrons - as in Bloch band theory for example.

Therefore, as an effective low energy coarse-grained field theory (think statistical mechanics ---> thermodynamics), general coordinate invariance is spontaneously broken in a material for the same reason that Lorentz invariance is broken because the index of refraction changes the speed of light in the material. The material provides a preferred frame of reference for interior measurements. A particle can outrun the slowed speed of light inside the material at this coarse-grained level (Cerenkov radiation). Hence special relativity in the sense of solutions is violated in this limited sense. Similarly, general covariance is spontaneously broken in cosmology where we have a global cosmological time and a global rest frame of reference, i.e. temperature of the CMB and isotropy of the CMB Big Bang remnant respectively.

Remember, in spontaneous broken symmetry the ground state of the system does not respect a continuous symmetry of the total exact dynamical action S of the system and their Euler-Lagrange local equations from the Action Principle

&S = 0

in the sense of the calculus of variations generalized by the Feynman path integral for quantum theory.

Sure, if you look at the chunk of material as a unit, and look at its EM field outside it from two inertial frames outside it in free space - Lorentz transformations will describe the the fields outside the material as well as the effective currents inside the material. Indeed, Einstein was led to special relativity in 1905 by considering relative motions between a permanent magnet and a solenoid.

But if you are a tiny nano-observer inside the material it's a different story - very tricky see e.g. Chs 9 -11 Panofsky and Phillips "Classical Electricity and Magnetism" (convective currents with moving media). Also Landau & Lifshitz "Electrodynamics of Continuous Media" (missing from my library maybe packed away) I seem to remember Landau & Lifshitz being explicit about this as well as P.W. Anderson. Bye and bye I will try to find the quotes.

Remember what Maxwell's equations look like in vacuum in the elegant Cartan form notation.

F = dA

dF = 0

d*F = 0

where *F is the Hodge dual of F.

roughly F has the fields E and B in vacuum

*F has the fields D and H modified by the material's electric polarizability and magnetization from magnetic moments of particle constituents

D = (permittivity)E

B = (permeability)H

(it's possible I got B & H switched in my memory)

Maxwell's equations in a material are

F = dA purely topological independent of Lorentz group

dF = 0 (Faraday's EMF law + no magnetic monopoles)

d*F = *J (current densities) depends on constituitive relations of the material lumped parameters effective field theory

Rovelli gives similar equations for the gravity field in terms of tetrad forms (Ch 2 "Quantum Gravity")

So the question is, can one naively apply the Lorentz transformations inside the material where the speed of light is no longer invariant like it is in vacuum? For example, as I said above we have the Cerenkov effect when a charge moves faster than the speed of light in vacuum.

Nevertheless, we formally write Maxwell's equations inside the material in what looks like a Lorentz invariant form using D & H even though strictly speaking that is not the case for interior measurements (in principle) made by Maxwell Demons - the material provides an absolute rest frame for coarse grained lumped parameter effective field theory where we can use ideas like electric permittivity and magnetic permeability. The D and H fields are coarse-grained lumped parameter solutions that spontaneously break the Lorentz group symmetry of the fundamental Maxwell action written in terms of the fundamental charges moving in vacuum. The situation here for electromagnetism is analogous to the Hubble flow in the standard model of cosmology with inflation and a cosmological constant where the CMB provides a preferred global frame of reference even though the fundamental GR equations are background independent in the sense that they are generally covariant. What is true for the local field equations need not be true for the global solutions of those equations that also involve additional boundary constraints.

See also http://en.wikipedia.org/wiki/Background_independence

there is the Einstein hole problem which is solved by the proper understanding of gauge invariance in which all solutions connected by the gauge transformations of the theory form an equivalence class of "orbits" i.e. they correspond to the same objective physical reality seen from different sets of detectors in arbitrary subluminal motion. In the case of 1916 GR the gauge gravity field transformations are T4(x) analogous to U1(x) for the Maxwell electromagnetic field.

In summary, Woodward's vague appeal to "background independence" is not sufficient reason to reject my idea of eq (8) prematurely since the payoff would be so great - the survival of life on this planet is what we are talking about.

i.e. speed of light inside the material = 1/(permittivity permeability)^1/2

in the same sense, I write equation (8).

We simply don't know yet. It's an empirical issue. Woodward is wrong that experiments prove (8) wrong since no experiments have been done looking for anomalous gravity when the index of refraction >> 1 especially in capacitors filled with a proper kind of negative permittivity meta-material. There are no such experiments yet - I know of. If Woodward knows different let's see the references.

JW: The problem is that notwithstanding that the index of refraction of metamaterials is negative, the energy density of the electromagnetic field as it propagates through a metamaterial is not negative.

JS: I beg to differ since Woodward has just violated Maxwell's equations in material media with that remark. Again this is an empirical issue that requires experiments. Also Woodward makes another error talking about propagating far fields of real photons on the light cone. I am talking about non-propagating near fields both inside and outside the light cones made out of coherent Glauber states of virtual photons.

JW: Notice that the definition of the index of refraction in equation (1) above does not require that the signs of either or be negative for n to be negative. Negative n registers, rather, that the relationship between the group velocity of an electromagnetic wave and its constituent phase waves in a metamaterial is the reverse of that which normally obtains. Instead of the group and phase wave velocities being in the same direction as usual, in a metamaterial they are in opposite directions. When this is the case, the circular polarization of the wave is changed from right handed to left handed. So metamaterials are sometimes referred to as “left handed materials”.

JS: Woodward's argument is false. While what he says is true, he pulls it out of context and it does not at all with correct logic follow that the near field energy density is not negative. Note also that he keeps thinking of the group velocity of radiation that has nothing at all to do with the non-propagating near fields I am talking about. Indeed, Woodward's argument here is a Red Herring that is not even wrong in the context I mean. Again this is a matter for actual experiment.

JW: The practical reason why there is no reason to believe that metamaterials transform the energy densities of electromagnetic waves propagating through them from positive to negative is that were that the case, as already noted, serious violations of the law of local energy conservation would ensue.

JS: Woodward's argument here is also false as I have shown. There is no problem at all with conservation of energy. Start with the meta-material filled capacitor without any electric charge on the plates. It has internal energy Ui. We must do positive work Wi to charge the plates switching on the D field inside the metamaterial. The internal energy has now dropped to Uf where Uf < Ui because the capacitor is doing positive external work Wf on some load. Total energy is conserved

Ui + Wi(input) = Uf + Wf(output)

Uf < Ui

Wf > Wi

It's easy in principle to check this with experiments that have never been tried.

There is no negative energy instability here that is obvious. The effect will saturate. What will happen is that Ui will decrease to a minimum - the capacitor will cool down and it will be impossible to load more charge on the plates. Remember this requires a negative permittivity in the near field where the material response functions have independent k wave vectors and frequencies f not constrained by an equation f = f(k) as it is for real far EM fields that Woodward has confused it with.

JW: Consider an electromagnetic wave as it makes the transition from propagation in free space to propagation in a metamaterial. If the energy density of the wave propagating in the metamaterial is negative, the wave must divest itself of enough energy to change its state from positive to negative as it enters the metamaterial. That energy must go somewhere – presumably into the physical structure of the metamaterial. So, where the wave enters the metamaterial we should expect to see strong heating. And where the wave exits the metamaterial, we should expect to see the reverse process – strong cooling. No reports of this behavior, which could hardly be missed as it would be a pronounced effect, are to be found in the literature.

JS: Again Woodward confounds propagating EM far field waves with f = (c/n)|k| with my non-propagating EM near field modes in a completely different region of the f-k domain of support of the response susceptibility function Chi(k,f) of the metamaterial - in the sense of many-body Feynman propagators. Woodward is talking about apples while I am talking about oranges. All of his arguments are Red Herrings because he has not really understood my proposal - near fields not far fields.

The behavior of near fields can be counter-intuitive to the behavior of far fields. In terms of Feynman diagrams, Woodward is talking about the external lines while I am talking about the internal lines with c-number condensates like the Green's functions of Gorkov for superconductors and superfluids. Indeed, coherent near EM fields are analogous to superfluid order parameters and this will affect the many-body theory for the susceptibility response correlation functions of the metamaterial. Furthermore, Woodward says a "strong cooling" off the top of his head. Has he done an actual calculation of that number? I bet he has not done so. Also has anyone tried to measure temperature variations in this context? I think not. Show me. I predict anomalous cooling will be detected since no one looked for it yet. Also it may be a very weak effect for propagating waves. Remember I am talking about a capacitor of ordinary conductors sandwiched with metamaterial "meat" of the proper design for nearly static ELF near fields to begin with. Has anyone done any experiments of the kind I envision? I doubt it. If so, show me.