7.8 One specific route to the Einstein equations?

In fact, there is a specific route to reproduce Einstein equations within a Fermi-liquid-like system advocated by G. Volovik [660Jump To The Next Citation Point]. In a Fermi liquid like 3He-A phase, there exist two important energy scales. One is the energy scale E B at which bosonization in the system start to develop. This scale marks the onset of the superfluid behaviour of helium three. At energies below EB, the different bosons appearing in the systems condense so that they can exhibit collective behaviour. The other energy scale, EL, is the Lorentz scale at which the quasiparticles of the system start to behave relativistically (as Weyl spinors). As we mention in Section 4.2.2 discussing the “heliocentric universe”, this occurs in the 3He-A phase because the vacuum has Fermi points. It is in the immediate surroundings of these Fermi points where the relativistic behaviour shows up. There are additional relevant scales in these systems, but in this section we are going to talk exclusively of these two.

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 EB, as at that energy scale the geometrical picture based on the bosonic condensate disappears. Thus, EB 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):

  1. EL > EB: For the induction mechanics to give rise to an Einstein–Hilbert term, √ --- − gR, in the effective Lagrangian we have to be sure that the fluctuating Fermionic field “feels” the geometry (fulfilling a locally-Lorentz-invariant equation) at all scales up to the cut-off. The term √ --- − gR will appear multiplied by a constant proportional to 2 EB. That is why from now on we can called EB alternatively the Planck energy scale EP ≡ EB.
  2. Special relativity dominance or EL ≫ EP: The E2P dependence of the gravitational coupling constant tells us that the fluctuations that are more relevant in producing the Einstein–Hilbert term are those with energies close to the cut-off, that is, around the Planck scale. Therefore, to assure the induction of an Einstein–Hilbert term one needs the Fermionic fluctuation with energies close to the Planck scale to be perfectly Lorentzian to a high degree. This can only be assured if EL ≫ EB.
  3. Sakharov one-loop dominance: Finally, one also needs the induced dynamical term to dominate over the pre-existing tree-level contribution (if any).

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: EB ≫ EL. 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 [252Jump To The Next Citation Point, 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 [287Jump To The Next Citation Point, 288Jump To The Next Citation Point, 289Jump To The Next Citation Point]. 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 [585Jump To The Next Citation Point, 584Jump To The Next Citation Point, 635Jump To The Next Citation Point, 681Jump To The Next Citation Point]. In this sense the Hořava models are a useful antidote to the usual feeling that Lorentz violation is typically Planck-scale.

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