Physics Letters B 788 (2019) 295–301
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Physics Letters B
www.elsevier.com/locate/physletb
The scalarised Schwarzschild-NUT spacetime
Yves Brihaye
a
, Carlos Herdeiro
b
, Eugen Radu
c,∗
a
Physique-Mathématique, Universite de Mons-Hainaut, Mons, Belgium
b
Centro de Astrofísica e Gravitação – CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1 ,
1049-001, Portugal
c
Departamento de Física da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal
a r t i c l e i n f o a b s t r a c t
Article history:
Received
31 October 2018
Accepted
13 November 2018
Available
online 15 November 2018
Editor:
M. Cveti
ˇ
c
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.
© 2018 Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
It has long been known that violations of the strong equiva-
lence
principle, via the inclusion of non-minimal couplings, can
lead to asymptotically flat black hole (BH) scalar “hair” — see
[1–7] and [8–10]for recent reviews. More recently [11–13], 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 (see also [14–16]). This
led to the conjecture that a phenomenon of “spontaneous scalar-
isation”
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. Indeed, for
some choices of the function defining the non-minimal coupling,
scalarised BHs are thermodynamically preferred over the GR solu-
*
Corresponding author.
E-mail
address: eugen.radu@ua.pt (E. Radu).
tions and linearly stable [18], suggesting such spontaneous scalari-
sation
occurs dynamically.
Black
hole spontaneous scalarisation was indeed confirmed to
occur dynamically in a different class of model [19], albeit within
the same universality class related to non-minimal couplings. In
this case, the scalar field non-minimally couples to the Maxwell
invariant, rather than to higher curvature invariants, leading to a
phenomenon of spontaneous scalarisation for electro-vacuum GR
BHs. The technical simplification resulting from the absence of the
higher curvature terms, allowed, besides showing dynamically the
occurrence of spontaneous scalarisation, to explore the space of
non-spherical static solutions that the model contains.
1
In all cases mentioned above, only static scalarised solu-
tions
have been considered, which connect to the Schwarzschild
(or Reissner–Nordström) solution in the vanishing scalar field limit.
It would be of interest to include rotation in this analysis, by
considering the scalarisation of the Kerr solution. This is, how-
1
Charged scalarised BHs were also recently considered in models with a scalar
field non-minimally coupled to higher curvature corrections [20].
https://doi.org/10.1016/j.physletb.2018.11.022
0370-2693/
© 2018 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
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