# Toward a Mechanism for the Emergence of Gravity
## Appendix A Special Cases of Parameters for Linear Graviton Fields  
We have omitted the discussion of special cases of the parameters \((\xi_{1},\xi_{2},\xi_{3})\) for which some of the terms in the equations from Section 3.2 vanish. The unique special case that gives new insights is the case in which the three of them are equal: \(\xi=\xi_{1}=\xi_{2}=\xi_{3}\). For the remaining cases, for example the case for which some of the terms in Equation (53) vanish, namely \(-2+\xi_{2}-4\xi_{3}=0\) or \(1-\xi_{1}+2\xi_{2}=0\), one can see that the same arguments presented Section 3.2 for the selection of the subspace \(\mathcal{U}\) straightforwardly apply and hence we do not need to discuss them separately.  
The case \(\xi=\xi_{1}=\xi_{2}=\xi_{3}\) where the masses have been put to zero is interesting because the condition (32) that the generators \(\chi^{\mu}\) of the transformations (31) need to verify reduces to  
\[\
abla_{\mu}\square\chi_{\
u}=0. \tag{57}\]  
Thus, it seems that for this specific choice of parameters the transformations of the form  
\[h^{\mu\
u}\to h^{\mu\
u}+2\
abla^{(\mu}\chi^{\
u)},\qquad\square\chi^{\mu}=0, \tag{58}\]  
have a chance to become emergent gauge symmetries. Actually, for this choice of parameters the terms of the associated current can be recast in the form (36) without the \(M^{\prime\mu}\). This means that we have a current of the form  
\[J^{\prime\mu}=W^{\prime\mu}+2\
abla_{\mu}N^{\prime\mu\
u}+D^{\prime\mu}, \tag{59}\]with \(W^{\prime\mu}\) a term that vanishes on shell, \(N^{n\mu}\) a superpotential and \(D^{\prime\mu}\) a term that identically vanishes within the subspace \(\
abla_{\mu}h^{\mu\
u}=0\). Their explicit form is:  
\[W^{\prime\mu}=-\Box h^{\mu\alpha}\chi_{\alpha}-(1+\xi)\
abla_{ \
u}\
abla^{\mu}h\chi^{\
u}+(1+\xi)\Box h\chi^{\mu}+2T^{\mu\
u}\chi_{\
u}, \tag{100}\] \[N^{\prime\mu\
u}=\
abla^{\
u}h^{\mu\rho}\chi_{\rho}+h^{\
u\rho} \
abla^{\mu}\chi_{\rho}+\frac{1}{2}(1+\xi)h\
abla^{\
u}\chi^{\mu}+(1+\xi) \
abla^{\mu}h\chi^{\
u},\] (101) \[D^{\prime\mu}=-\
abla^{\alpha}h_{a\beta}\
abla^{\mu}\chi^{\beta} +\
abla^{\mu}\
abla^{\alpha}h_{a\beta}\chi^{\beta}+(1+\xi)\
abla_{\beta}h^{ \beta}_{\
u}\
abla^{\mu}\chi^{\
u}+(1+\xi)\
abla_{\beta}h^{\beta}_{\
u}\
abla^ {\
u}\chi^{\mu}. \tag{102}\]  
Thus, if we define a projection onto the subspace \(\tilde{\mathcal{U}}\) defined by the condition  
\[\
abla_{\mu}h^{\mu\
u}=0, \tag{103}\]  
the transformations of the form (100) become emergent gauge symmetries.  
The naturalness of such construction relies again on how natural it is to restrict the dynamics to the subspace \(\tilde{\mathcal{U}}\). We want to see if that subspace for this particular choice of parameters is selected dynamically. For this particular choice of parameters, the divergence of the equations of motion assuming we couple the theory to a conserved source automatically gives  
\[\Box\
abla_{\mu}h^{\mu\
u}=0, \tag{104}\]  
as it can be seen by putting all the \(\xi_{i}\) parameters equal to the same value \(\xi\) in Equation (48). Thus, for this choice of parameters, the transformations of the form (100) naturally emerge as gauge symmetries since the subspace onto which we need to project for them to become emergent gauge symmetries is naturally selected from a dynamical point of view.  
Although this construction seems to be closer to Fierz-Pauli theory since it is equivalent to Fierz-Pauli theory in the partial gauge fixing given by the only condition \(\
abla_{\mu}h^{\mu\
u}=0\), it does not seem to be natural due to the fine-tuning among the parameters that it requires. For instance, we expect that, at the quantum level, radiative corrections may tend to rapidly break the equality among parameters. However, it is possible that the emergent gauge symmetries themselves are enough to protect this equality among the parameters and render them radiatively stable. We leave this question for future work, when we extend this construction to the whole non-linear and quantum regime, in which these radiative corrections enter and may play a relevant role.