  We revisit the problem of perturbations of Schwarzschild-AdS$_4$ black holes
by using a combination of the Martel-Poisson formalism for perturbations of
four-dimensional spherically symmetric spacetimes and the Kodama-Ishibashi
formalism. We clarify the relationship between both formalisms and express the
Brown-York-Balasubramanian-Krauss boundary stress-energy tensor,
$\bar{T}_{\mu\nu}$, on a finite-$r$ surface purely in terms of the even and odd
master functions. Then, on these surfaces we find that the spacelike components
of the conservation equation $\bar{\mathcal{D}}^\mu \bar{T}_{\mu\nu} =0$ are
equivalent to the wave equations for the master functions. The renormalized
stress-energy tensor at the boundary $\displaystyle \frac{r}{L} \lim_{r
\rightarrow \infty} \bar{T}_{\mu\nu}$ is calculated directly in terms of the
master functions.
