# Toward a Mechanism for the Emergence of Gravity
## 4 No-Go Theorems for Emergent Gravity Reconsidered
### Marolf's Theorem
Given the theorem, he then concludes that it is difficult to obtain an emergent gravitational description starting from microscopic kinematically local theories, without changing abruptly the notion of locality. Otherwise, the emergence would entail a rather strange behaviour on the non-gravitational description. Thus, he interprets the theorem as saying that one would rather start looking for emergence from microscopic theories which are already kinematically non-local at a fundamental level, as it is done for instance in the context of gauge-gravity duality [38; 39] which, to some extent, can be interpreted as emergent theories for the bulk gravity theory. However, this conclusion is based on the additional assumption that all observables in the UV completion of the theory must freeze at low energies or, equivalently, that all these observables must be gauge invariant, which may not be the case in an emergent framework. If at least one observable of the UV completion is not gauge invariant, then the notion of locality can be preserved. This would be generally the case if gauge symmetries are emergent. If we take our discussion of the linear setting above and assume the existence of a suitable non-linear extension along the lines realized in the discussed [PERSON] case, we can make the following observation. When an effective gauge invariance is developed in the low-energy regime of the theory, the observables of the high-energy theory can be separated into those that are gauge invariant from the perspective of the emergent gauge symmetry and those that are not. As an intermediate stage of emergence, we started with a generic massless Lorentz-covariant system described by action (20) for an object \(h^{\mu\
u}\). Without further considerations, this system does not describe gravity, as for instance it contains additional degrees of freedom, and, therefore, does not satisfy the conditions of [PERSON]'s theorem. However, we have discussed that, if the field \(h^{\mu\
u}\) couples to a conserved and constant-trace two-index symmetric current, there is a natural decoupling of certain degrees of freedom and a simultaneous emergence of gauge symmetries. Interestingly, this decoupling makes the construction compatible with [PERSON]'s conditions for the theorem. Then, it is true that all the local gauge-invariant observables away from the boundary need to freeze out. However, non-gauge-invariant observables, which are still meaningful from the point of view of the high-energy UV-completion of the theory, do not need to freeze out. In fact, our example shows that actually they (i.e., the \(h_{\mu\
u}\)) do not freeze out. As we have pointed out above, these observables are not constrained to be frozen by the theorem. The freeze out only affects a subset of the observables of the high-energy theory, namely those observables that are invariant under the emergent gauge transformations. In this way, the high-energy theory retains an \"interesting enough\" bulk dynamics. Another aspect that we think is worth keeping in mind is the effect of having a background metric. As our underlying theory has a (flat) background metric the degrees of freedom are defined with respect to this background and are perfectly local, as for electrodynamics and other quantum field theories. In the same vein, our theory has a well-defined local Hamiltonian density and does not have a fundamental gauge invariance. However, one could decide--and this is a choice--to analyze the theory from a gauge perspective, arguing that the background metric is not operationally measurable. Then, for consistency one has to change the notion of degree of freedom as now one cannot use the background metric as an external reference. The degrees of freedom must now be gauge-invariant variables defined with respect to the metric, variables which in general relativity are highly non-local. Now, the former Hamiltonian density is not gauge invariant and therefore one can no longer think of it as locally defining an energy density.