Refringence, field theory and normal modes

Abstract

In a previous paper [gr-qc/0104001; Class. Quant. Grav. 18 (2001) [3595][3596][3597][3598][3599][3600][3601][3602][3603][3604][3605][3606][3607][3608][3609][3610] we have shown that the occurrence of curved spacetime "effective Lorentzian geometries" is a generic result of linearizing an arbitrary classical field theory around some non-trivial background configuration. This observation explains the ubiquitous nature of the "analog models" for general relativity that have recently been developed based on condensed matter physics. In the simple (single scalar field) situation analyzed in our previous paper, there is a single unique effective metric; more complicated situations can lead to bi-metric and multi-metric theories. In the present paper we will investigate the conditions required to keep the situation under control and compatible with experimenteither by enforcing a unique effective metric (as would be required to be strictly compatible with the Einstein Equivalence Principle), or at the worst by arranging things so that there are multiple metrics that are all "close" to each other (in order to be compatible with the Eötvös experiment). The algebraically most general situation leads to a physical model whose mathematical description requires an extension of the usual notion of Finsler geometry to a Lorentzian-signature pseudoFinsler geometry; while this is possibly of some interest in its own right, this particular case does not seem to be immediately relevant for either particle physics or gravitation. The key result is that wide classes of theories lend themselves to an effective metric description. This observation provides further evidence that the notion of "analog gravity" is rather generic.

Analogue Gravity

“…Closer to the heart of the analogue programme were the development of a “normal mode” analysis in [15, 16, 398], the development of dielectric analogues in [342], speculations regarding the possibly emergent nature of Einstein gravity [20, 398], and further developments regarding the use of 3 He − A [106] as an analogue for electromagnetism. Experimental proposals were considered in [19, 398, 331].…”

“…As in the single-field case we write (here we will follow the notation and conventions of [16]) Now use this to expand the Lagrangian Consider the action Doing so allows us to integrate by parts. As in the single-field case we can use the Euler-Lagrange equations to discard the linear terms (since we are linearizing around a solution of the equations of motion) and so get Because the fields now carry indices ( AB ) we cannot cast the action into quite as simple a form as was possible in the single-field case.…”

“…In this sense systems characterised by a high degree of quantum coherence, very cold temperatures, and low speeds of sound offer the best test field. Hence it is not surprising that in recent years Bose-Einstein condensates (BECs) have become the subject of extensive study as possible analogue models of general relativity [136, 137, 16, 19, 18, 115, 114]. …”

“…Analogue models are of particular interest in that the analogue spacetimes that emerge often violate, to some extent, the Einstein equivalence principle [16, 399]. Since the Einstein equivalence principle (or more precisely the universality of free fall) is experimentally tested to the accuracy of about 1 part in 10 13 , it is important to build this principle into realistic models of gravitation — most likely at a fundamental level.…”

“…Only if this algebraic constraint is satisfied can one assign a “metric” to each of the quadratic factors. If one further wishes to impose mono-metricity then the Fresnel equation must be some integer power of some single quadratic expression, an even stronger algebraic statement [16, 399]. …”