# Toward a Mechanism for the Emergence of Gravity
## 3 Two Examples: Yang-Mills Theories and Linearized Gravity
### Linear Graviton Physics
To understand this, we first notice that when we put the masses equal to zero \(m_{1}=m_{2}=0\), a field redefinition of the form  
\[h^{\mu\
u}\to h^{\mu\
u}+\lambda\eta^{\mu\
u}h, \tag{25}\]  
(with \(\lambda\
eq-1/4\) in order for it to be invertible) leaves the functional form of the Lagrangian invariant at the expense of changing the parameters \(\xi_{2}\) and \(\xi_{3}\) as [24]  
\[\xi_{2} \to\xi_{2}+4\lambda+\lambda(4\xi_{2}-2), \tag{26}\] \[\xi_{3} \to\xi_{3}+2\lambda(2+4\xi_{3}-\xi_{2})+\lambda^{2}(16\xi_{3}-8 \xi_{2}+6). \tag{27}\]  
Had we begun with the case \(\xi_{1}=\xi_{2}=\xi_{3}=0\), for which the theory is invariant under the whole set of diffeomorphisms, we would have ended up with new parameters \(\xi_{2}\) and \(\xi_{3}\)  
\[-\frac{1}{2}(1+\xi_{2})\
abla_{\mu}h^{\mu\
u}\
abla_{\
u}h+\frac {1}{4}(1+\xi_{3})\
abla_{\mu}h\
abla^{\mu}h-\frac{1}{4}m_{1}^{2}h_{\mu\
u}h^{ \mu\
u}-\frac{1}{4}m_{2}^{2}h^{2}\biggr{]}, \tag{20}\]0 \[-\frac{1}{2}(1+\xi_{2})\
abla_{\mu}h^{\mu\
u}\
abla_{\
u}h+\frac {1}{4}(1+\xi_{3})\
abla_{\mu}h\
abla^{\mu}h-\frac{1}{4}m_{1}^{2}h_{\mu\
u}h^{ \mu\
u}-\frac{1}{4}m_{2}^{2}h^{2}\biggr{]}, \tag{20}\]1  
Furthermore, notice that within this new parametrization of field space, the theory is not invariant under the standard transformations of the form (23) but transformations of the form (24) with \(k=2\lambda\). Thus, it is clear that the more general set of transformations of the form (24) is included in the set of transformations of the form of ordinary linear diffeomorphisms (23) upon, possibly, a field redefinition. Hence, we will focus only on the set of transformations (23) since the transformations (24) do not add anything new. On the other hand, if \(m_{1}=m_{2}=0\) and the parameters \(\xi_{i}\) satisfy the relations \(-2+\xi_{2}-4\xi_{3}=0\) and \(1-\xi_{1}+2\xi_{2}=0\), we have the following Weyl transformations, generated by a scalar field \(\phi\), as gauge transformations:  
\[-\frac{1}{2}(1+\xi_{2})\
abla_{\mu}h^{\mu\
u}\
abla_{\
u}h+\frac {1}{4}(1+\xi_{3})\
abla_{\mu}h\
abla^{\mu}h-\frac{1}{4}m_{1}^{2}h_{\mu\
u}h^{ \mu\
u}-\frac{1}{4}m_{2}^{2}h^{2}\biggr{]}, \tag{20}\]2  
This Weyl symmetry can be combined with transverse diffeomorphisms to build the so-called linearized WTDiff (Weyl transverse-diffeomorphism) invariant gravity by choosing \(\xi_{1}=0\), which guarantees that transverse diffeomorphisms are gauge symmetries as explained above. This forces \(\xi_{2}=-1/2\) and \(\xi_{3}=-5/8\) (this corresponds precisely to the case of considering the theory with \(\xi_{1}=0\) and performing a putative field redefinition of the form (25) with the limiting value \(\lambda=-1/4\) for which the transformation becomes singular. Thus, it is not a true field redefinition since one can not recover the trace of \(h\) from the transformed field [24]). The Fierz-Pauli theory (with the whole linearized group of diffeomorphisms) and WTDiff (with linearized Weyl and transverse diffeomorphism transformations) are the two largest possible gauge symmetry groups for a tensor field \(h^{\mu\
u}\)[24]. Once we have discussed the choices of parameters for which the theory displays gauge symmetries, let us go back to the discussion of emergent gauge symmetries. We will be assuming that the parameters \(\xi_{i}\) are such that the theory is _generic_, in the sense that the coefficients of the different terms that will appear from now in equations of motion, conserved currents, etc. do not vanish. Furthermore, we will assume that they take generic values different from the ones that endow action (20) with gauge symmetries from the beginning. The discussion of special choices of the parameters has been moved to Appendix A since it just involves the analysis of different particular cases and does not affect the main point of this section. Following the discussion in Section 2, the first step is to identify a set of physical symmetries in our system that can become potential gauge symmetries.