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The Schwarzschild–de Sitter solution in the Einstein theory with a positive cosmological constant [math] becomes an exact solution to the de Rham–Gabadadze–Tolley (dRGT) nonlinear massive gravity theory with the mass parameter [math] when the theory parameters [math] and [math] satisfy the relation [math]. We study the perturbative behavior of this black hole solution in the nonlinear dRGT theory with [math]. We find that the linear perturbation equations become identical to those for the vacuum Einstein theory when they are expressed in terms of gauge-invariant variables. This implies that this black hole is stable in the dRGT theory as far as the spacetime structure is concerned, in contrast to the case of the bi-Schwarzschild solution in the bi-metric theory. However, we have also found a pathological feature that the general solution to the perturbation equations contain a single arbitrary function of spacetime coordinates. This implies a degeneracy of dynamics in the Stückelberg field sector at the linear perturbation level in this background. The physical significance of this degeneracy depends on how the Stückelberg fields couple observable fields.

*Keywords: *
E02;
E03;
E31

*Journal Article.*
*8320 words.*

*Subjects: *
Astronomy and Astrophysics
;
Relativity and Gravitation

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