the traditional Regge–Wheeler [@ReWh] decomposition. One starts with a background metric written as, $$g^{(0)}_{\mu \nu} dx^{\mu} dx^{\nu} = -(1 -2M/r) \; dt^2 + (1 -2M
/r)^{-1} dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d \varphi^2.$$

For axisymmetric perturbations, the general $\ell = 1$ even parity terms take the form [@ReWh] $$\begin{aligned}
 h^{(1)}_{tt} & = & (1 -2 M/r ) H_0(t,r) \cos \theta  \nonumber \\
 h^{(1)}_{tr} & = & H_1(t,r) \cos \theta  \nonumber \\
 h^{(1)}_{rr} & = & (1 -2 M/r )^{-1} H_2(t,r) \cos \theta  \nonumber \\
 h^{(1)}_{t \theta} & = &  - h_0(t,r) \sin \theta  \nonumber \\
 h^{(1)}_{r \theta} & = &  - h_1(t,r) \sin \theta  \nonumber \\
 h^{(1)}_{\theta \theta} & = &  r^2 K(t,r) \cos \theta  \nonumber \\
 h^{(1)}_{\phi \phi} & = &   r^2 \sin^2 \theta K(t,r) \cos \theta  \end{aligned}$$

The first order metric perturbation coefficients are not uniquely defined, but may be changed by “gauge transformations” of the form [@ReWh] $$\widetilde{h}^{(1)}_{\mu\nu}  = 
h^{(1)}_{\mu\nu} -g_{\mu \nu},_{\rho} \xi_{(1)}^{\rho} -
g_{\mu \rho} \xi_{(1)}^{\rho},_{\nu} - g_{\rho \nu}
\xi_{(1)}^{\rho},_{\mu}$$ where the gauge 4-vector $\xi_{(1)}^{\mu}$ is arbitrary, except for the requirement of axisymmetry.

In particular, the $\ell = 1$ even parity coefficients transform as $$\begin{aligned}
\label{gauge1}
\widetilde{H}_0(t,r) & = &    H_0(t,r)  +2 {\partial 