# Toward a Mechanism for the Emergence of Gravity
## 4 No-Go Theorems for Emergent Gravity Reconsidered
### Weinberg-Witten Theorem  
The Weinberg-Witten theorem [27] states that a theory that allows the construction of a conserved Lorentz-covariant energy-momentum tensor cannot contain gravitons, neither fundamental nor composite in the spectrum of asymptotic states (in fact, the theorem applies to particles with spin \(j>1\)). The content of the theorem can be rephrased as saying that gravitons are so special that one cannot build a Lorentz-covariant and gauge-invariant energy-momentum tensor for them. This distinctive feature already appears in the Fierz-Pauli theory for linear gravitons. One can certainly build different Lorentz-invariant energy-momentum tensors for \(h_{\mu\
u}\); however, none of them happens to be invariant under the (linear) gauge transformation generated by a vector field \(\chi^{\mu}\) that take the form \(\delta h^{\mu\
u}=2\
abla^{(\mu}\chi^{\
u)}\), with \(\
abla\) representing the connection compatible with flat spacetime metric, as before. From the perspective of emergent gravity the theorem implies that, if excitations with the properties of gravitons arise in a suitable regime, these excitations cannot have a Lorentz-invariant energy-momentum tensor that is gauge invariant. This is often taken as an indication that obtaining gravitons from systems akin to condensed-matter systems is not possible. However, our discussion of emergent gauge symmetries above indicates that such an interpretation is pulling too far, or misunderstanding, the significance of the theorem. Let us discuss this more explicitly. The key aspect is the emergent nature of gauge symmetries and what this implies for observables, in particular for the energy-momentum tensor. Our starting point is a system akin to a condensed-matter system which develops excitations with the properties of gravitons at low energies. We have seen that this requires certain conditions on the source to which the field \(h^{\mu\
u}\) couples which, when satisfied, are equivalent to the decoupling of some degrees of freedom leading to the natural restriction to the subspace \(\mathcal{U}\) described in previous sections. As the original high-energy theory has no gauge symmetries, all configurations are different and, therefore, in general will have different values for most observables (in this section, this word will always refer to the high-energy theory), which includes the fields themselves and the energy-momentum tensor as specific examples. However, the development of emergent gauge symmetries implies the emergence of equivalence classes for which the values of certain observables do not change, which are then denoted as gauge invariant. The specific observables that change or not within a given equivalence class are determined by the structure of the theory. The basic fields \(A^{\mu}_{a}\) and \(h^{\mu\
u}\) are examples of observables of the high-energy theory that change under the emergent gauge symmetries. On the other hand, for vector fields \(A^{\mu}_{a}\) the energy-momentum tensor is gauge-invariant [14; 15], while for gravity it is not. In summary, that certain observables of the high-energy theory turn out to be non gauge invariant does not pose a problem for emergence. From the perspective of the high-energy theory, all observables retain their meaning and have well-defined values regardless of the emergence of gauge symmetries. It is certainly not required and even unreasonable to assume that all observables of the high-energy theory become gauge invariant. In general, the process of emergence separates observables in two categories, depending on their behavior under gauge transformations. However, whether the set of gauge-invariant observables includes the energy-momentum tensor, or any other observable for that matter, is inconsequential for the rationale of emergence. We have discussed the Weinberg-Witten theorem from a purely conceptual point of view and highlighted that, although it points out a particular property of gravity, it does not forbid the emergence of gravity.