Hairy black holes in DHOST theories: exploring disformal transformation as a solution generating method

Abstract

Solutions-generating methods based on field redefinitions, such as conformal mapping, play an important role in investigating exact solutions in modified gravity. In this work, we explore the possibility to use disformal field redefinitions to investigate new regions of the solution space of DHOST theories, and present new hairy black holes solutions beyond the stealth sector. The crucial ingredient is to find suitable seed solutions to generate new exact ones for DHOST theories. We first consider a seed solution of the Einstein-Scalar system describing a naked singularity. Under suitable assumptions, we derive a no-go result showing that no black hole solution can be constructed from such a seed GR solution. Then, taking into account the stability of each three degeneracy classes of quadratic DHOST theories under a general disformal mapping, we consider two kinds of known black hole solutions as seed configurations: the Schwarzschild stealth solution and the non-stealth Reissner-Nordstrom like solution. Restricting our considerations to invertible disformal transformations, we show that building new hairy black hole solutions from the stealth solution associated to a constant kinetic term is also quite constrained. We obtain new solutions which either are stealth or describe asymptotically locally flat black holes with a deficit solid angle. However, starting from the non-stealth seed solution associated to a non constant kinetic term, we show that a disformal transformation can introduce rather general modifications of the exterior geometry. Finally, we consider the construction of minimally modified hairy black hole solutions using a small disformal transformation. Further applications of this solution generating method should allow to provide new hairy rotating black hole solutions beyond the stealth sector, as well as minimally modified GR solutions useful for phenomenological investigations of compact objects beyond GR.

Disforming the Kerr metric

“…In fact it has been shown that in the static case, for X constant, the disformal version of a Schwarzschild type solution is again a mass rescaled Schwarzschild black hole. With a basic disentangling of coordinates, and given that the disformal factor B = B(X) is a function of X (which is constant on-shell), the resulting metric is geometrically identical with a rescaled mass (and cosmological constant) [44,45] (see also [46] for interesting extensions and questions concerning the regularity of solutions). As we will see in the present paper, once rotation is present the static picture is completely changed.…”

“…Note that a similar observation has been noted in the case of Schwarzschild-de-Sitter metric in refs. [39,44,45] (see also [49]).…”

Degeneracy, matter coupling, and disformal transformations in scalar-tensor theories

“…Although this story is by now well understood, a rather unexplored question concerns the coupling to matter in DHOST. Applications of DHOST coupled to certain matter fields have of course been considered, particularly in the contexts of cosmology and astrophysics (see e.g., [25][26][27][28][29][30][31][32][33]), yet a general understanding of the consistency of matter coupling is currently lacking. This is in fact rather surprising: indeed, besides its obvious relevance for phenomenology, this question is crucial for the claim that the considered theory propagates fewer degrees of freedom than naively expected.…”

Hairy black-holes in shift-symmetric theories

“…Therefore the only DHOST theories with hair are the ones obtained via (4.3) starting from a sGB Horndeski theory, since this is the only Horndeski theory with hair. (Here we are not considering the possibility that a black-hole solution is mapped into a solution with a naked singularity, as discussed in [18]. )…”