Abstract
We investigate the static, spherically symmetric black hole solutions in the quasi-dilaton model and its generalizations, which are scalar extended dRGT massive gravity with a shift symmetry. We show that, unlike generic scalar extended massive gravity models, these theories do not admit static, spherically symmetric black hole solutions until the theory parameters in the dRGT potential is fine-tuned. When fine-tuned, the geometry of the static, spherically symmetric black hole is necessarily that of general relativity and the quasi-dilaton field is constant across the spacetime. The fine-tuning and the no hair theorem apply to black holes with flat, anti-de Sitter or de Sitter asymptotics.
Highlights
Generic massive gravity theories are known to suffer from theoretical pathologies such as the Boulware-Deser ghost [1], van Dam-Veltman-Zakharov discontinuity [2, 3] and low strong coupling scale (Λ5 = (MP m2)1/5) [4]
It has been realized that the de Rham-Gabadadze-Tolley (dRGT) model has near-flat backgrounds that are totally free of the van Dam-Veltman-Zakharov (vDVZ) discontinuity and where the strong coupling scale can be raised to a quasi-Λ2 = (MP m)1/2 scale [11, 12]
Various aspects of the black holes in the dRGT model and its extensions have been investigated in the literature [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]
Summary
(Note that while in general relativity a coordinate transform of a solution is just the same solution in a different gauge, the coordinated transformed solution in massive gravity (in unitary gauge) is a physically different solution.) the quasi-dilaton field does not endow any extra hair for the static, spherically symmetric black holes. These results hold for black holes with flat, anti-de Sitter or de Sitter asymptotics.
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