# Toward a Mechanism for the Emergence of Gravity
## 3 Two Examples: Yang-Mills Theories and Linearized Gravity
### Linear Graviton Physics  
The discussion of the case for a symmetric tensor field \(h^{\mu\
u}\) is completely parallel to the above section. The first step is to write down the most general quadratic Lorentz-invariant action for such a field. This means that the \(h^{\mu\
u}\)-field is a good variable to describe our effective field theory. For instance, it can represent the deformations of a given condensed-matter-system near a Fermi point, as we have explained in the introduction. Therefore, the resulting effective theory does not need be gauge-invariant: this is definitely asking too much since everything we know about it at this stage is that it can be described by a field \(h^{\mu\
u}\). Explicitly, this action has the form  
\[S=\int d^{4}x\sqrt{-\eta}\biggl{[}-\frac{1}{4}\
abla_{\mu}h^{ \alpha\beta}\
abla^{\mu}h_{\alpha\beta}+\frac{1}{2}(1+\xi_{1})\
abla_{\mu}h^{ \mu\
u}\
abla_{\
u}h_{\alpha}^{\
u}\] \[-\frac{1}{2}(1+\xi_{2})\
abla_{\mu}h^{\mu\
u}\
abla_{\
u}h+\frac {1}{4}(1+\xi_{3})\
abla_{\mu}h\
abla^{\mu}h-\frac{1}{4}m_{1}^{2}h_{\mu\
u}h^{ \mu\
u}-\frac{1}{4}m_{2}^{2}h^{2}\biggr{]}, \tag{20}\]  
where indices are raised and lowered with the flat spacetime metric \(\eta_{\mu\
u}\), we have introduced the trace of the field \(h^{\mu\
u}\) as \(h=h^{\mu\
u}\eta_{\mu\
u}\), and we have introduced the dimensionless parameters \(\xi_{i}\) and the mass parameters \(m_{i}\). We can also include a coupling to matter by adding a term of the form \(h^{\mu\
u}\Upsilon_{\mu\
u}\) to the previous Lagrangian, with the properties of \(T_{\mu\
u}\) being discussed in more detail below (for the moment we will just assume that it is a symmetric tensor). The equations of motion can be computed by performing a variational derivative, which results in  
\[\frac{1}{2}\Box h_{\mu\
u}-(1+\xi_{1})\eta_{\beta(\mu}\
abla_{ \
u)}\
abla_{\alpha}h^{\alpha\beta}+\frac{1}{2}(1+\xi_{2})\Bigl{(}\
abla_{\mu }\
abla_{\
u}h+\eta_{\mu\
u}\
abla^{2}\cdot h\Bigr{)}\] \[-\frac{1}{2}(1+\xi_{3})\eta_{\mu\
u}\Box h-\frac{1}{2}m_{1}^{2}h _{\mu\
u}-\frac{1}{2}m_{2}^{2}\eta_{\mu\
u}h=T_{\mu\
u}, \tag{21}\]  
where we have introduced the notation \(\
abla^{2}\cdot h=\
abla_{\mu}\
abla_{\
u}h^{\mu\
u}\) and we will keep using it throughout this section. Before moving forward, it is better to pause and discuss the values of the parameters \(\xi_{i}\) and \(m_{i}\) for which the system has a gauge symmetry from the beginning, in order to avoid those cases in the discussion of emergent gauge symmetries. We will follow the discussion of [24]. On the one hand, for \(\xi_{1}=m_{1}=0\), the equations of motion are invariant under the group of linearized transverse diffemorphisms, whose infinitesimal transformations are generated by a divergenceless vector field \(\chi^{\mu}\) as follows:  
\[h^{\mu\
u}\to h^{\mu\
u}+2\
abla^{(\mu}\chi^{\
u)},\qquad\
abla_{\mu}\chi^{ \mu}=0. \tag{22}\]  
In addition, for \(m_{2}=0\), \(\xi_{2}=\xi_{3}=0\) we can relax the condition on the divergence for the vector field \(\
abla_{\mu}\chi^{\mu}=0\) and recover the full group of linearized diffeomorphisms, i.e., one recovers the Fierz-Pauli theory [25], i.e., transformations generated by arbitrary vector fields \(\chi^{\mu}\) as  
\[h^{\mu\
u}\to h^{\mu\
u}+2\
abla^{(\mu}\chi^{\
u)}. \tag{23}\]  
We could have considered a set of transformations that is slightly more general than the ones introduced here, namely transformations generated also by a vector field \(\chi^{\mu}\) as  
\[h_{\mu\
u}\to h_{\mu\
u}+2\
abla_{(\mu}\chi_{\
u)}+k\eta^{\mu\
u}\
abla_{ \alpha}\chi^{\alpha} \tag{24}\]However, these transformations do not add anything new.