The scalarised Schwarzschild-NUT spacetime

TL;DR

This paper introduces a scalarised generalisation of the Schwarzschild-NUT spacetime considering Gauss-Bonnet and Chern-Simons invariants, revealing distinct solution structures and scalarisation conditions influenced by NUT charge and curvature couplings.

Contribution

It presents the first scalarised rotating black hole solutions based on the Schwarzschild-NUT spacetime with Gauss-Bonnet and Chern-Simons invariants, highlighting differences in solution space and scalarisation conditions.

Findings

Scalarisation occurs for NUT charge in GB case but not in CS case.

Double branch structure appears in CS solutions, affecting scalarisation.

Increasing horizon area requires stronger coupling in GB case, with opposite trend in CS case.

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-Nordstr\"om-NUT solutions.