Warping of the space-time surfaces and dielectric constant

The flat warped space-time surfaces are characterized by the reduced light-velocity β₀=c_#/c≤ 1. There is a criticality with respect to the variations of c_# (instability of metal plates illustrates this). Also the twisted Hamilton-Jacobi structures would be characterized by c_# (see this). The criticality of the warping could induce or accompany various kinds of quantum criticalities.  In the case of the Allais effect, this kind of  quantum criticality would explain the variation of the  pendulum frequency cannot be explained in terms of gravitation. Quite generally, one can write f_#= c_#/λ = f/n, where n= c/c_# is analogous to the refractive index appearing in electrodynamics in presence of matter.   In Maxwellian electrodynamics, refractive index relates to the relative dielectric constant ε_r via the formula n=c/c_#= (ε_r^1/2. Could reflective index and dielectric properties have a geometric description in  terms of the warping of the space-time surface? If so, the  warping of the space-time surface could be seen directly via the reflection of light! Refractive index depends on frequency. This can be understood in terms of quantum criticality implying the value of c_# associated with the massless extremal assignable to the photons depends on frequency. At resonance, at which ε_r diverges, the value c_# would in the ideal case vanish: there would be no propagation of signals. The standard interpretation would be in terms of absorption of the signal by atoms, which contribute to the resonance frequencies. How the criticality of warping could manifest itself in critical systems?

1. For a harmonic oscillator, the frequency is given in terms of force constant and mass as ω=(k/m)^1/2. A reasonable  dimensional guess is that the   force constant k  characterizing the electromagnetic force  is proportional to (c_#/c)². For instance, cyclotron frequency would be proportional to c_#. More generally,  the Coulomb force in a dielectric is  scaled from its vacuum value by 1/ε_r= (c_#/c)². Also capaticance of a capacitor would be propoertional to (c_#/c)². The variation of c_# at quantum criticality would make it possible to change the contribution of the electromagnetic force.
2.  Gravitational masses have always the same sign so that the notion of dielectret does not make sense and c_# is not expected to play any role: this conforms with the character of warping. For instance, the gravitational force created by a constant mass density ρ corresponds to potential energy proportional to Gmρ r², which is  harmonic oscillator potential energy. The force constant k∝ Gmρ does not depend on c_#.  
3. If the system is in an  equilibrium involving electromagnetic and gravitational forces, the variation of c_# appearing in the electromagnetic component of force could make possible  the loss of equilibrium. The tuning of c_#  could allow the field body to change the equilibrium point of a physical system and even destroy or create the equilibrium.  In biology the generation of nerve pulse, the splitting of DNA double strand preceding transcription and replication could serve as examples of this.

See the chapter M⁸ H duality viz. Hubble law, and gravitational Planck constant viz. Allais effect and warping or the article with the same title.