When one is below both energy scales, one can describe the system as a set of Weyl spinors coupled to background electromagnetic and gravitational fields. For a particular Fermi point, the electromagnetic and gravitational fields encode, respectively, its position and its “light-cone” structure through space and time. Both electromagnetic and gravitational fields are built from bosonic degrees of freedom, which have condensed. Apart from any predetermined dynamics, these bosonic fields will acquire additional dynamical properties through the Sakharov-induced gravity mechanism. Integrating out the effect of quantum fluctuations in the Fermionic fields à la Sakharov, one obtains a one-loop effective action for the geometric field, to be added to the tree-level contribution (if any). This integration cannot be extended beyond , as at that energy scale the geometrical picture based on the bosonic condensate disappears. Thus, will be the cut-off of the integration.
Now, in order that the geometrical degrees of freedom follows an Einstein dynamics, we need three conditions (which we shall see immediately are really just two):
Unfortunately, what we have called special-relativity dominance is not implemented in helium three, nor in any known condensed-matter system. In helium three the opposite happens: . Therefore, the dynamics of the gravitational degrees of freedom is non-relativistic but of fluid-mechanical type. That is, the dynamics of the gravitational degrees of freedom is not Einstein, but of fluid-mechanical type. The possible emergence of gravitational dynamics in the context of a condensed-matter system has also been investigated for BECs [252, 571]. It has been shown that, for the simple gravitational dynamics in these systems, one obtains a modified Poisson equation, and so it is completely non-relativistic, giving place to a short range interaction (on the order of the healing length). However, starting from abstract systems of PDFs with a priori no geometrical information, the emergence of Nordström spin-0 gravity has been shown to be possible ; this is relativistic though not Einstein.
In counterpoint, in Hořava gravity the graviton appears to be fundamental, and need not be emergent [287, 288, 289]. Additionally, the Lorentz breaking scale and the Planck scale are in this class of models distinct and unconnected, with the possibility of driving the Lorentz breaking scale arbitrarily high [585, 584, 635, 681]. In this sense the Hořava models are a useful antidote to the usual feeling that Lorentz violation is typically Planck-scale.
Living Rev. Relativity 14, (2011), 3
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