Refringence, field theory, and normal modes

In a previous paper [gr-qc/0104001; Class. Quant. Grav. 18 (2001) 3595-3610] 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 experiment -- either 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 {\Eotvos} 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 pseudo-Finsler 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.