Spacelike character of the Minkowski four-momentum in analog gravity

Abstract

In analog gravity the recent experiment of Drori et al. [Phys. Rev. Lett. 122, 010404 (2019)] is impressive, as it shows how the emission of two Hawking quanta emitted in opposite directions lead to measurable consequences in the medium's rest system in a straightforward way. This result raises however the following problem: how can this experiment be explained in terms of classical electrodynamics? There must necessarily exist such an explanation (the experiment is after all classical); otherwise classical electrodynamics would be an incomplete theory. This is the main topic of the present paper. We propose that the measured effect is a demonstration of the spacelike character of the Minkowski four-momentum. Moreover, we extend the discussion by analyzing a Gedanken experiment (making use of the Kerr effect as a formal agency), to illustrate the transition from subluminal to superluminal phenomena in a straightforward way. Finally, we emphasize the close relationship that exists between the spacelike Minkowski momentum and the anomalous Doppler effect.

Classical and quantal aspects of Minkowski's four-momentum in analog gravity

“…In the next section we continue the classical analysis of the subluminal-superluminal ideal fluid model introduced in Ref. [11], where an electromagnetic wave propagates against the current. In particular, we calculate the electromagnetic pressure on the artificial black horizon.…”

“…We will start from the same Gedanken experiment as in Ref. [11], considering the inertial frame S in which a simple nondispersive fluid is flowing with constant velocity v in the negative z direction. We assume |v| large, but slightly less than the luminal limit c/n, where n = √ ε r µ r > 1 with ε r and µ r being the relative permittivity and the permeability of the medium when it is at rest.…”

“…We also need to observe the dispersion relation for a monochromatic wave with wave number k. From Ref. [11] we cite, for the quantity…”

“…From these, we obtain after some calculation the following expression for the relative transmitted field [11] E…”

“…The usual relationship g = S/c 2 , conventionally taken to express Planck's principle of inertia of energy, is thus broken. This characteristic property of Minkowski's momentum (or more generally Minkowski's energy-momentum tensor) does not seem to be very well known, although the present author has pointed it out repeatedly, also in an analog gravity context [11]. (The basic presentation given in Møller's book [12] is highly recommendable.)…”