# Toward a Mechanism for the Emergence of Gravity
## 1 Introduction  
The pursuit of a theory able to combine the principles of general relativity and quantum mechanics is one of the strongest driving forces in modern fundamental physics. Almost a century of active research on this problem has resulted in many insights [1], although a full comprehension of how both sets of principles might intertwine is still lacking. Approaches toward finding a theory of quantum gravity can be divided in emergent and non-emergent approaches. The first category corresponds to approaches in which the fundamental degrees of freedom are not taken to be the spacetime geometry itself, but other sort of microscopic degrees of freedom. The geometric structure characteristic of general relativity is assumed to be recovered in a suitable regime, for instance a weakly-coupled regime or a low-energy regime. String theory [2, 3] would fall into this category, as it is a specific proposal for these microscopic degrees of freedom. More generally, one can conceive systems having another substratum for the emergence, for instance condensed-matter systems [4, 5, 6, 7]. The second category corresponds to theories in which the geometric structure of general relativity is taken as fundamental. Different approaches in this category are set apart by the different quantization schemes applied to this structure. The most widely studied theory within this category is loop quantum gravity [8]. The distinction between the two categories above is nevertheless not a sharp one. In fact, the notion of emergence is inevitably tied up to every quantum gravitational theory, since one can show on generalgrounds that there are no local observables in the Dirac sense [9] (that is, diffeomorphism invariant) for any quantum gravity theory [10]. This work is devoted to analyzing a specific kind of system within the emergent category. The motivation for this study comes from the observation that it is quite common to find analogues of gravitational fields in many physical systems, especially in condensed-matter systems [6]. However, this analogy just holds at a kinematical level, meaning that the effective geometry emerging in those systems does not display the dynamical features characteristic of general relativity. Thus, a natural question that arises is whether these analogue systems can be designed to develop the correct dynamics. This is the notion of emergent gravity that we explore in this work. Our goal is finding the conditions that make such analogue systems develop fully relativistic dynamics in some regime. The emergence of analogue gravity usually comes associated with a low-energy regime in which the excitations can be typically described by a collection of weakly-coupled relativistic fields. They describe the collective excitations of this system [7]. In condensed-matter systems, this low-energy regime is usually characterized by the presence of Fermi points [4; 5]. Around these Fermi points, we know the field content relevant for describing the low-energy regime, and we know that the fields are weakly coupled. This low-energy Lorentz invariance becomes less exact as we increase the energy. It typically manifests itself in modified dispersion relations of the form  
\[E^{2}=k^{2}+\sum_{n}\xi_{n}\frac{k^{2n+2}}{E_{L}^{2n}}, \tag{1}\]  
for instance, but also on the very dissolution of the effective description. The dispersion relations by themselves offer a problem from an effective field theory rationale since they appear to generate a fine-tuning problem (see [11]). Here we do not delve into this problem, which we think arises because of the tacit assumption that the effective field description survives far in the dispersive regime, and just assume the existence of a low-energy Lorentzian regime. This ensures that we can apply the rules of effective field theory and write down the most general effective action compatible with the symmetries of the system in order to obtain a good description of the phenomena of interest [12; 13].