Abstract
It has recently been suggested that vacuum black holes of General Relativity (GR) can become spontaneously scalarised when appropriate non-minimal couplings to curvature invariants are considered. These models circumvent the standard black hole no scalar hair theorems of GR, allowing both the standard GR solutions and new scalarised (a.k.a. hairy) solutions, which in some cases are thermodynamically preferred. Up to now, however, only (static and spherically symmetric) scalarised Schwarzschild solutions have been considered. It would be desirable to take into account the effect of rotation; however, the higher curvature invariants introduce a considerable challenge in obtaining the corresponding scalarised rotating black holes. As a toy model for rotation, we present here the scalarised generalisation of the Schwarzschild-NUT solution, taking either the Gauss–Bonnet (GB) or the Chern–Simons (CS) curvature invariant. The NUT charge n endows spacetime with “rotation”, but the angular dependence of the corresponding scalarised solutions factorises, leading to a considerable technical simplification. For GB, but not for CS, scalarisation occurs for n=0. This basic difference leads to a distinct space of solutions in the CS case, in particular exhibiting a double branch structure. In the GB case, increasing the horizon area demands a stronger non-minimal coupling for scalarisation; in the CS case, due to the double branch structure, both this and the opposite trend are found. We briefly comment also on the scalarised Reissner–Nordström-NUT solutions.
Highlights
It has long been known that violations of the strong equivalence principle, via the inclusion of non-minimal couplings, can lead to asymptotically flat black hole (BH) scalar “hair” - see [1,2,3,4,5,6,7] and [8,9,10] for recent reviews
More recently [11,12,13], it has been appreciated that for a wide class of non-minimal couplings, the model accommodates both scalarised BHs and the standard vacuum General Relativity (GR) solutions. This led to the conjecture that a phenomenon of “spontaneous scalarisation” occurs in these models [11, 12], akin to the spontaneous scalarisation of neutron stars first discussed in [17], within scalar tensor theories, but with the key difference that the phenomenon is triggered by strong gravity rather than by matter
For some choices of the function defining the non-minimal coupling, scalarised BHs are thermodynamically preferred over the GR solutions and linearly stable [18], suggesting such spontaneous scalarisation occurs dynamically
Summary
It has long been known that violations of the strong equivalence principle, via the inclusion of non-minimal couplings, can lead to asymptotically flat black hole (BH) scalar “hair” - see [1,2,3,4,5,6,7] and [8,9,10] for recent reviews. In this paper we analyse the scalarisation of a different generalisation of the Schwarzschild solution, the Taub-Newman-Tamburino-Unti solution (Taub-NUT) [21, 22] This solution of the Einstein vacuum field equations can be regarded as the Schwarzschild solution with a NUT “charge” n (hereafter referred to as the Schwarzschild-NUT solution) which endows the spacetime with rotation in the sense of promoting dragging of inertial frames. We shall exhibit some basic properties of scalarised Schwarzschild-NUT solution This scalarisation will have a geometric origin, occurring due to a non-minimal coupling of a scalar field and a higher curvature invariant. The Kerr-like form of the Schwarzschild-NUT spacetime allows, considering a second case, the Einstein-Chern-Simons-scalar (ECSs) model, wherein the geometric scalarisation occurs due to a coupling between the scalar field and the Potryagin density. Summarise our results and provide some further remarks together with possible avenues for future research
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