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 [253]; 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.
http://www.livingreviews.org/lrr-2011-3 |
Living Rev. Relativity 14, (2011), 3
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