# Refraction of matter waves

My question is a bit messy, so here is the background:

Normally the trajectory of a massive particle in the presence of a gravitational field is described in the context of general relativity.

Fernandez-Nunez has shown that Maxwell's equations in the presence of a gravitational field can be formulated as if space is flat and the electric permittivity and magnetic permeability, respectively, depend on time dilation and spatial curvature. The trajectories of electromagnetic waves predicted in that formulation are exactly the same as the trajectories predicted by general relativity.

If an uncharged massive particle is represented as a matter wave, it's got a frequency and a wavelength. Gravitational time dilation changes the frequency and wavelength of the particle as it traverses a nonuniform gravitational field. It seems reasonable to draw an analogy between the matter wave and an electromagnetic wave: the trajectory of the wave (that is, the propagation of the wave) might be modeled as due to refraction.

My question is:

Is it possible to describe the dynamics of an uncharged test particle in a gravitational field as if the particle's wavefunction were moving in a flat space whose "permittivity and permeability" depend on the time dilation and curvature corresponding to the gravitational field? I imagine that the dynamic equations, if this is possible, would resemble Maxwell's equations; and that the "effective matter-wave permittivity and permeability" might be higher-rank tensors than the "effective electromagnetic permittivity and permeability" derived by Fernandez-Nunez. However, if this can be done -- and if it can be done in the same flat Minkowski space Fernandez-Nunez uses to describe electromagnetism, it could lead to interesting new perspectives on the physics of gravitation.

Edit 1/18/19: I found this publication which, at least at first glance, provides an unequivocal answer "yes" to the question.

@BenCrowell commented that "the notion of time dilation, i.e., a gravitational potential, doesn't in general make sense in GR. It only makes sense for a static spacetime." I checked with one of the authors, who stated that they did not attempt to generalize to spacetimes beyond static & isometric because it would be much more complicated to do so: that "it would be analogous to doing geometrical optics in crystals, with speeds of propagation depending on direction and polarization". A paper by Tamm, dealing with the electrodynamics of moving media, is referenced in many of the papers dealing with this "geometrical optics" treatment of general relativity.

Edit 1/22/19: This paper seems to go straight to the heart of my question, and seems to provide an unambiguous affirmative answer (though it will take a while for me to slog through the math):

In general relativity (GR), the metric tensor of spacetime is essential since it represents the gravitational potential. In other gauge theories (such as electromagnetism), the so-called premetric approach succeeds in separating the purely topological field equation from the metric-dependent constitutive law. We show here that GR allows for a premetric formulation, too. For this purpose, we apply the teleparallel approach of gravity, which represents GR as a gauge theory based on the translation group. We formulate the metric-free topological field equation and a general linear constitutive law between the basic field variables. The requirement of local Lorentz invariance turns the model into a full equivalent of GR. Our approach opens a way for a natural extension of GR to diverse geometrical structures of spacetime.

Note that the "constitutive law" referred to there amounts to field-dependent "effective permittivity and permeability" of the vacuum.

• The link isn't to a paper by Tull, it's to a paper by Fernandez-Nunez and Bulashenko. I doubt that this has any interesting implications for general relativity, because the notion of time dilation, i.e., a gravitational potential, doesn't in general make sense in GR. It only makes sense for a static spacetime. – Ben Crowell Jan 16 at 23:56
• Thanks for pointing out my error on the reference. I have corrected it. I am hoping for comments that refer to the paper. The effective electromagnetic "constants" depend on the components of the metric tensor. I did not see anything in the paper that limited the result to static spacetime. – S. McGrew Jan 17 at 2:28
• Are you sure that you want to emphasize the uncharged rather than massive property? Because, simulating massive scalar particle seems much more difficult than massless. – A.V.S. Jan 18 at 4:58
• Yes, I want to consider the case where there are no electromagnetic field interactions (that is, no electromagnetic fields, just gravitational fields). I imagine that it should be possible to, e.g., apply Huygens principle to the propagation of matter waves in gravitational fields and end up with results equivalent to that of uncharged matter particles in the same gravitational fields. – S. McGrew Jan 18 at 12:04
• @S.McGrew: I did not see anything in the paper that limited the result to static spacetime. See p. 2, and this is also true whether or not they highlight it, because that's the way GR works. – Ben Crowell Jan 18 at 15:51