Varying the Vacuum Index of Refraction

After developing special relativity, one of Einstein’s first attempts to devise a theory of gravitation was based on a simple scalar speed of light whose value varies with position and time. In effect this represented a varying index of refraction for the vacuum. A scalar theory of gravity doesn’t satisfy the equivalence principle, so Einstein soon gave up on this approach, but in 1957 Robert Dicke proposed (in “Gravitation without a Principle of Equivalence”) a similar theory based on a variable index of refraction for the vacuum, which he attributed to variations in the dielectric constant and permeability of the vacuum. His main focus was on a theory of local gravitation in flat spacetime, involving an index of refraction that varies with spatial position (i.e., proximity to gravitating masses), but he also described a secular cosmological variation in the vacuum index of refraction as a function of time, and described how an increasing index (decreasing light speed) this would result in cosmological redshift without seeming to invoke a spatial expansion of the universe. However, he recognized that this would necessitate corresponding changes to the local physical measures of space and time (based on, e.g., the fine structure of atoms and their energy levels) such that the speed of light would always have the fixed value c in terms of local coordinates, and in terms of this effective physical metric the usual kinematic descriptions of general relativity would still apply. His motivation for considering units of measure in which the metric of spacetime is flat and the universe is not expanding was that, in terms of this system of measurement, gravitation more closely resembles the electromagnetic and nuclear forces, so he hoped it might be conducive to developing a unified quantum field theory of gravity. So far that program has not met with success.

In any case, it’s interesting to review the generic possibility of a continuous transparent non-dispersive medium of refractive index n(t) = n(0)e^kt where k is a small positive constant, so the index is gradually increasing with time. (Dicke’s proposed function was slightly more complicated than this.) The most natural hypothesis would be that this applies uniformly at every location in the limit of reducing density of material substances, but we can also consider other variants. For example, suppose that within some spherical regions the medium has undergone a phase shift and has condensed into small particles, leaving vacuum “bubbles” in the medium. (The bubbles contain the condensed particulates, but are otherwise empty.) The speed of light inside each bubble is c, but outside the bubbles it is C(t) = c/n(t) = C(0)e^–kt. 

Now consider a pulse of light emitted from inside one bubble at time t₀ and received inside another at time t₃, as depicted in the figure below.

The pulse is emitted with wavelength λ₀ and period T₀ inside the bubble on the left, and it propagates at speed c to the boundary, which it reaches at time t₁. The speed of the light at this spatial boundary changes abruptly from c to C(t₁), so the period is unchanged, T₁ = T₀, but the wavelength becomes λ₁ = λ₀C(t₁)/c, as is obvious from considering the paths of two consecutive wave crests shown below:

The wave then propagates through the refractive medium from time t₁ to time t₂, during which it travels a distance

As the speed of each wave crest drops at equal times during this interval, the wavelength is unchanged, but the period is increased by the ratio of the speeds, as illustrated below for a speed change at a particular instant.

Clearly we get the same result if the speed change occurs at multiple steps, or even continuously, provided the changes in the speeds of the wave crests all occur at equal times. Therefore, we have λ₂ = λ₁ = λ₀C(t₁)/c, and the period has increased by the factor of speeds, i.e.,

Lastly, the wave passes through the spatial boundary from the outer region with speed C(t₂) to the inner region of the receiving bubble with speed c. At this boundary, the period is unchanged and the wavelength is changed by the ratio of speeds c/C(t₂), so we have inside the receiving bubble

Thus the signal is received inside the bubble at speed c and redshifted by the factor

This redshift is independent of C(0) because it depends only on the ratio of beginning and ending outer light speeds, not on the absolute value of the speed, but of course the distance traveled during the interval from t₁ to t₂ time does depend on the absolute value of C during the trip. Thus to determine the relationship between distance and redshift we must evaluate the distance traveled. If we assume the size of the bubbles is negligibly small compared with the distances between them, we can just focus on the distance traveled by the light through the external medium. From the equation above we have

Thus [k/C(t₂)] L = λ₃/λ₀ – 1. Equivalently λ₃/λ₀ = 1 + [k/C(t₂)] L. By comparison, if a transmitter and receiver are in empty space and the distance between them is increasing at the rate v, then the usual linear “Doppler” shift would be 1 + v/c, so if we were to interpret the redshift as being caused by a changing distance, we would identify [k/C(t₂)] L with v/c and write, by analogy with Hubble’s Law in cosmology, the relation H(t₂)L = v where we have defined the parameter

Knowing the redshift λ₃/λ₀ and (by some independent means) the distance L between bubbles, we can infer the value of the ratio k/C(t₂), but this doesn’t determine the individual values of k and C(t₂), so we could not extrapolate the value of H(t₂) to other times. If we also knew the current rate of change of H with respect to time, we would know k = (dH/dt)/H, which would enable us to know the current speed of light C(t₂) in the outer medium.

One might wonder if this situation could serve as a viable “explanation” of cosmological redshift for a stationary system of galaxies, i.e., without cosmological expansion. (It is distinct from the “tired light” concept of Zwicky, because that proposal just involved reducing the energy, not the speed, of the photons.) First, we should note that the optical path length between galaxies in this model actually is increasing, so to some extent the distinction with the expansion interpretation is semantic. Indeed Dicke argued for an effective metric for material particle dynamics that can be made indistinguishable from the electromagnetic metric. One might consider a mechanical metric that differs from the optical metric, but this would not be empirically viable, since all evidence shows the optical and mechanical measures of space and time are consistent.

Further, the premise of a continuous and perfectly non-dispersive refractive medium is unphysical, since the only such thing we observe (to the available precision) is the vacuum itself. Any realistic material refractive medium would be dispersive (contrary to the observed equality of light speed in inter-galactic space for all frequencies), and would need to have exponentially increasing density to yield the increasing index of refraction. This would seem to require continual creation of new mass-energy.  (Ironically, a different kind of “steady-state” model for cosmology, championed by Bondi, Gold, and Hoyle, also entailed continuous creation of new matter.) It goes without saying that simply dispersing the matter comprising the galaxies would not significantly affect the refractive index of the universe, because of the extremely low mass density of the universe.

Another approach would be to argue that the external regions have a false vacuum, that has dropped to the true vacuum around the galaxies, but in that case the physics within those regions would be fundamentally different, and one could still argue that the dynamical metric needs to match the optical metric. Observations of hydrogen atoms (about 10 atoms per cubic meter in inter-galactic space) indicate that physics “works the same way” even in inter-galactic space.

Rather than the index of refraction being higher in the external regions, one might try to argue that it is lower, i.e., below unity, resulting in a speed of light in those regions exceeding c (but still exponentially decreasing). This doesn’t affect the above calculations, but it suggests that the vacuum within galaxies must be a false vacuum. Now, as Maxwell noted, the speed of light in a region where the density of material substances is arbitrarily low approaches the ratio of electromagnetic to electrostatic units, i.e., the value 1/√(μ₀ε₀) where μ₀ is the permeability (whose value is set by defining units) and ε₀ is the permittivity. The value of the permittivity also appears in Coulomb’s law F = q₁q₂/(4π ε₀ r²) for the electrostatic force between two changes q₁ and q₂ at a distance r from each other. Hence to posit that light propagates at a substantially different speed in some other region in the low-density limit, we would need the force between two charges at a given distance to be significantly different. In that case, the equilibrium configurations of solid objects such as measuring rods would presumably be affected, since they are comprised of charged particles held together by electric forces. In view of this, we could hardly expect the mechanical metric to be unaffected. Essentially to match the observed consistency of atomic behavior throughout the universe, the effective mechanical and electromagnetic metrics must be consistent (e.g., all forms of energy have inertia).  For any local observer, anywhere in the observable universe, physics must be the same in terms of the locally defined measures of space and time. It then follows that any model that accounts for the cosmological redshift must be effectively isomorphic to an expanding universe.

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