# Toward a Mechanism for the Emergence of Gravity
## 4 No-Go Theorems for Emergent Gravity Reconsidered
### Marolf's Theorem  
The very interesting work of [PERSON] [28] contains a theorem that seems to constrain the kind of theories that might give rise to an emergent gravitational description. According to [PERSON] [28], the only theories that might give rise to such a description are the ones that he calls _kinematically non-local_ (we will introduce their definition of kinematical non-locality in a moment). From our perspective, although [PERSON]'s theorem pinpoints the particularities of constructing an emergent theory of gravity from a _kinematically local_ theory, it does not automatically imply that such theories cannot have an effective gravitational description in some regime. Let us now introduce [PERSON]'s theorem and carefully analyze it as well as its implications. Quoting it literatelly, [PERSON]'s theorem states [28]:  
**[PERSON]'s theorem:**_\"Consider any limit where the effective description of a local theory is a gravitational theory with universal coupling to energy, the same notion of time evolution, and a compatible definition of locality. In this limit all local observables away from the boundary become independent of time.\"_  
The theorem above contains several definitions that need to be clarified to make it intelligible, which we review here for the benefit of the reader. On the one hand, [PERSON] restricts their study to what he calls gravitational theories with _universal coupling to the energy_, namely theories for which the Hamiltonian can be written as the integral over the boundary of space at each time of some local function, which he calls gravitational flux. Furthermore, such gravitational flux needs to be a gauge-invariant function of the gravitational field and its derivatives. All diffeomorphism-invariant theories fall into this category (non-Lorentz-invariant theories like for instance Hofava-Lifshitz gravity [35] do not in a strict sense, although [PERSON] suggests the existence of a straightforward generalization of this theorem that would apply to such theories). Massive gravity theories, like dRGT theory [36], and scalar gravities, like the so-called Nordstrom gravity [37], do not fall into this category since they do not have universal coupling to the energy. On the other hand, by locality he assumes a version of microcausality which he calls _kinematic locality_. According to their definition, a kinematically local theory is one for which every pair of local bosonic gauge-invariant Heisenberg operators evaluated at spacelike points commute. Then, he states that if there were a kinematically local (non-gravitational) theory that in some regime gave place to an effective description in terms of a gravitational theory with universal coupling to energy, all of its local bulk observables would freeze out unless there were a change on the notion of time evolution or locality between the gravitational and non-gravitational descriptions (these are notes of caution). Technically speaking, [PERSON]'s observation is that given the gravitational flux which is a gauge-invariant operator, it should be possible to write it in the microscopic underlying theory as an integral over gauge-invariant operators supported near the boundary. Otherwise, it would correspond to a theory in which the notions of locality between the microscopic and effective gravitational theory would be very different. Assuming this _compatible-locality property_ among both descriptions, [PERSON] reaches the conclusion that, given a kinematically local underlying theory and having expressed the Hamiltonian as an operator in that theory just supported near the boundary, all the local (or bulk) observables of the theory need to freeze out in the regime in which the effective gravitational description applies. This is because the Hamiltonian (as it is itself another observable) will commute with any local observable away from the boundary.