# Toward a Mechanism for the Emergence of Gravity
## 3 Two Examples: Yang-Mills Theories and Linearized Gravity
### Linear Graviton Physics
However, notice that these transformations do not preserve such subspace and hence we would be in the first situation described in Section 2, namely the non-emergence of gauge symmetry. **Second case:**\(B_{\mu}=0\). In this second case we can relax the condition on \(h\) to be \(\
abla_{\mu}h=0\) in order to have a trivial current. Notice that these transformations preserve the subspace \(\
abla_{\mu}h=0\) and hence this second option corresponds to the second situation described in Section 2, i.e., the emergence of gauge symmetry. More explicitly, to have WTDiff emergent transformations, we need to restrict ourselves to the same subspace \(\mathcal{U}\) introduced above. We recall that such space is defined by the conditions (42) and that we need to choose it because the emergent Weyl gauge symmetries preserve them and the Noether currents restricted to such subspace give identically vanishing charges. Hence, we conclude that it is possible to also find emergent Weyl symmetries with the mechanism introduced in Section 2. It is interesting to realize that the space \(\mathcal{U}\) in which we find the emergence of gauge symmetries is the same independently of whether we are trying to find the closest theory to Fierz-Pauli theory or to linearized WTDiff within this emergent paradigm. In fact, a (non-transverse) diffeomorphism generated by a vector field with constant divergence \(\
abla_{\alpha}\chi^{\alpha}=\kappa\in\mathbb{R}\), such as the ones we have considered above, is equivalent to the composition of a transverse diffeomorphisms generated by the transverse part of such vector field \(\chi^{\mu}\) and a Weyl transformation generated by a constant scalar field with \(\phi=4\kappa\). Both of them can be seen to produce constant shifts in \(h\) and hence preserve the subspace \(\mathcal{U}\). In other words, when we understand the conditions defining subspace \(\mathcal{U}\), i.e., \(\
abla_{\mu}h^{\mu\
u}=\
abla^{\
u}h=0\), as gauge fixing conditions for either Fierz-Pauli or linearized WTDiff, the residual gauge symmetries that leave such subspace invariant for both theories are exactly the same. Thus, whether one decides to understand the emergent gauge theory we have built as having a Fierz-Pauli flavor or a linearized WTDiff flavor is inconsequential for the discussion at the linear level. Now, let us pass to discuss under which conditions the space \(\mathcal{U}\) is selected naturally by the dynamics of the theory. If we assume that the tensor \(T^{\mu\
u}\) is conserved, taking the divergence of the equations of motion uncovers a structural constraint:  
\[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]1  
Taking an additional divergence we find  
\[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]2  
Given this structural equation, which is a wave equation without a source, we can argue as we have already done before that the following solutions are selected dynamically,  
\[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]3  
Plugging this condition into Equation (48) we obtain  
\[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]4and applying again the argument that the lack of sources selects naturally the trivial solution, we conclude that  
\[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]5  
Notice that this condition does not imply that the two terms in the previous expression are zero independently. For this to happen one needs to impose further constraints. One possibility is to further require that the trace of \(T^{\mu\
u}\) is a constant over spacetime.