JHEP10(2019)078
where M is the BH ADM mass. For a more general f(φ) the RN BH may or may not
solve (2.2)–(2.4). This naturally leads to two classes of EMS models. (Note that, in
this classification, we assume, without any loss of generality, that the scalar field vanishes
asymptotically, φ(r)
r→∞
−→ 0.)
Class I or dilatonic-type. In this class of EMS models φ(r) = 0 does not solve the
field equations.
3
Thus RN is not a solution. Then, the scalar field equation (2.3)
implies that
f
,φ
(0) ≡
df(φ)
dφ
φ=0
6= 0 . (2.11)
A representative example of coupling for this class is the standard dilatonic coupling
f(φ) = e
2αφ
, (2.12)
in which case we refer to φ is a dilaton field. The arbitrary nonzero constant α is
taken to be positive without any loss of generality. Indeed, the solutions remain
invariant under the simultaneous sign change (α, φ) → −(α, φ). Thus, flipping the
sign of α simply corresponds to flipping the sign of φ. The coupling (2.12) appears
naturally in Kaluza-Klein models and supergravity/low-energy string theory models.
Three reference values for the coupling constant α in (2.12) are:
α = 0 (EM theory) α = 1 (low energy strings) α =
√
3 (KK theory) .
(2.13)
Some exact, closed form BH solutions of (2.2)–(2.4) with (2.12) are known and
presented in appendix A. Other exact solution examples in this class (with a non-
dilatonic coupling) are given in [47].
Class II or scalarised-type. In this class of EMS models φ(r) = 0 solves the field equa-
tions. Thus RN is a solution. This demands that
f
,φ
(0) ≡
df(φ)
dφ
φ=0
= 0 . (2.14)
This condition is naturally implemented, for instance, if one requires the model to be
Z
2
-invariant under φ → −φ. The RN solution, however, is (in general) not unique.
These EMS models may contain a second set of BH solutions, with a nontrivial
scalar field profile — the scalarised BHs. Below some conditions for this to occur are
discussed. Such second set of BH solutions may, or may not, continuously connect
with RN BHs. This leads to two subclasses.
• Subclass IIA or scalarised-connected-type. In this subclass of EMS models, the
scalarised BHs bifurcate from RN BHs, and reduce to the latter for φ = 0. This
bifurcation moreover, may be associated to a tachyonic instability, against scalar
3
There is an exceptional case: if Q = P , φ = 0 solves this class, so that the dyonic, equal charges RN
BH is a solution.
– 5 –
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