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Abstract
On the uniqueness of the static black hole with conformal scalar hair
Highlights
It is well-known that static black holes do not have the scalar hair with non-negative potential in asymptotically flat spacetimes [1]
We could prove the uniqueness of the photon surface of the BBMB solution, that is, the outside region of the photon surface is unique to be the BBMB solution in the Einstein gravity with a conformally coupled scalar field
The proof in Ref. [7] cannot be applied to the current situation. This is because the photon surface of the BBMB black hole in the Jordan frame is singular in the Einstein frame
Summary
It is well-known that static black holes do not have the scalar hair with non-negative potential in asymptotically flat spacetimes [1] (see Ref. [2] for a review). In the Einstein frame, the system is reduced to the Einstein-massless scalar field system In this system, the uniqueness of the outiside region of the photon surface has been proven in Ref. [7] cannot be applied to the current situation This is because the photon surface of the BBMB black hole in the Jordan frame is singular in the Einstein frame. Note that it will be difficult to prove the black hole uniqueness for cases except for vacuum/electrovacuum or well-motivated systems by string theory or so [8] In this sense, the current result may encourage us to try to prove the uniqueness of static black holes with a hair we know that the BBMB black hole itself is not stable [2].
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