Master equation as a radial constraint

We revisit the problem of perturbations of Schwarzschild-AdS4 black holes by using a combination of the Martel-Poisson formalism for perturbations of four-dimensional spherically symmetric spacetimes [K. Martel and E. Poisson, Phys. Rev. D 71, 104003 (2005).] and the Kodama-Ishibashi formalism [H. Kodama and A. Ishibashi, Prog. Theor. Phys. 110, 701 (2003).]. We clarify the relationship between both formalisms and express the Brown-York-Balasubramanian-Krauss boundary stress-energy tensor, T¯μν, 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 D¯μT¯μν=0 are equivalent to the wave equations for the master functions. The renormalized stress-energy tensor at the boundary rLlimr→∞T¯μν is calculated directly in terms of the master functions.