# Toward a Mechanism for the Emergence of Gravity
## 3 Two Examples: Yang-Mills Theories and Linearized Gravity
### Linear Graviton Physics
Taking the trace of the equations of motion for the field \(h^{\mu\
u}\) and introducing the following notation \(T=\eta^{\mu\
u}T_{\mu\
u}\), we find  
\[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]6  
Taking a derivative, using the assumption of constant trace for the current, i.e., \(\
abla_{\mu}T=0\), and using also (50) we obtain the following condition (assuming that the parameters do not take specific values, which we discuss in Appendix A):  
\[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]7  
Again, for a generic system, and applying the argument that the lack of sources leads naturally to the trivial solution, we obtain the condition \(\
abla_{\
u}h=0\). Inserting this condition in Equation (52) we precisely see that the subspace \(\mathcal{U}\) introduced in Equation (42) is naturally recovered. Thus, under the assumption of coupling to a conserved current \(\
abla_{\mu}T^{\mu\
u}=0\) with constant trace, \(\
abla_{\mu}T=0\), we have found that the system develops emergent gauge symmetries that can either be understood as emergent diffeomorphisms or emergent Weyl-transverse diffeomorphisms. Let us summarize the discussion concerning the emergence of gauge symmetries for the two-index symmetric tensor field \(h^{\mu\
u}\) up to this point. We have managed to prove that for the most general Lorentz-invariant theory constructed from such tensor field, the mechanism presented in Section 2 works and provides us with emergent gauge symmetries. At the linear level, the gauge transformations that emerge can be either understood as the manifestation within this emergent framework of either linearized diffeomorphisms or linearized Weyl and transverse diffeomorphisms transformations. To put it explicitly, consider any generic Lorentz-invariant theory of \(h^{\mu\
u}\) which fulfills the following conditions:  
1. It describes massless excitations, namely the masses from the Lagrangian in Equation (20) are equal to zero (\(m_{1}=m_{2}=0\)). 2. It couples to a two-index symmetric source \(T^{\mu\
u}\) that is conserved, i.e., divergenceless \(\
abla_{\mu}T^{\mu\
u}=0\). 3. The source has constant trace, namely \(\
abla_{\mu}T=0\). Our analysis implies that at the linear level, such a theory has a natural truncation that is indistinguishable from either Fierz-Pauli theory or linearized WTDiff theory in the gauge described by (42). Notice that, instead of a projection to a given dynamical subspace of the theory, for these theories we understand the conditions \(\
abla_{\mu}h^{\mu\
u}=\
abla^{\
u}h=0\) as gauge fixing conditions. Actually, we emphasize that Fierz-Pauli theory and linearized WTDiff are indistinguishable in this specific gauge, in the sense that the residual gauge transformations that emerge are of the same form. This is the reason for understanding this emergent gauge theory as the manifestation of either WTDiff and Fierz-Pauli theory in the emergent framework: both theories are indistinguishable in this specific gauge. The extension to the non-linear regime of the gravitational case through a bootstrapping procedure is expected to be much more involved than for the vector-field case discussed in Section 3.1 and it is left for future work. It constitutes an ongoing project on which we expect to report soon. Here we simply advance that the source of this difficulty arises, apart from the obvious technical complications appearing due to the presence of much more terms in the Lagrangian, from the nature of the non-linear terms appearing in the series generated through the bootstrapping procedure. Whereas for vector fields there appears a finite series that finishes after two iterations, for gravity we need to deal with an infinite \"formal\" series to obtain the non-linear theory [22; 26].