and the Weyl tensors in this coordinate system be bounded at the horizon.

### Tensor perturbations

For a tensor perturbation, the metric components in the $(U,V,z^i)$ coordinate system are bounded only when $H_T$ is bounded at the horizon. Under this condition, from and , we see that a perturbation of the Weyl tensor, $\delta C_{****}$, in this coordinate system is bounded if and only if $H_T'$ and $H_T''$ are bounded at $r=r_h$. Thus, the regularity condition at horizon is given by

### Vector perturbations {#vector-perturbations}

For a stationary vector perturbation, $F_t=f_t$ is the only non-vanishing gauge-invariant variable. Because we have the metric components in the $(U,V,z^i)$ coordinate system are bounded only when $F^t=f^t$ is bounded at the horizon. Under this condition, from , it follows that the Weyl tensor in this coordinate system is bounded at the horizon if and only if $F^{(1)}$ is bounded there. To summarise, the regularity condition is given by

### Scalar perturbations {#scalar-perturbations}

For a scalar perturbation, the determination of the boundary condition at the horizon is not so simple, because all metric components can be non-vanishing. First, from the relations we obtain the conditions at the horizon. From this, we have $X^t=\Order{1}$. Next, from 