Gravity and Strings

352 Dilaton and dilaton/axion black holes
difference as the presence of secondary scalar hair in the b = 0family and of primary
scalar hair in the b =−1family. We are primarily interested in regular BHs and, therefore,
we only write the b = 0family of dilaton-BH solutions in its final form:
ds2 = e−2a(ϕ−ϕ0)
−2 2 −2a(ϕ−ϕ0) −2 H Wdt − e H − 1
d−3 W −1dr2 + r 2d2
(d−2) ,
Aµ = αe aϕ0δµt(H −1 − 1), e −2aϕ = e −2aϕ0 H 2x ,
H = 1 + h
ω
, W = 1 +
r d−3 r
x = (a2 /2)c
1 + (a2 d − 2
, c =
/2)c d − 3 .
d−3 , ω= h
1 − a2
4x α2
,
(12.10)
On adding the corresponding factors of W to this solution, one obtains the b =−1family.
Here a and d are given parameters that determine our theory and α, ϕ0 (the value of the dilaton
at infinity), and h (the coefficient of r −(d−3) in H) are the independent parameters. The
relation between ω and h is valid only for h = 0. If h = 0, then ω is an arbitrary constant,
there is no electromagnetic field, and we recover the solutions (8.216) which have primary
scalar hair (the scalar charge is unrelated to the conserved charges) and are singular except
for b = 0orfor a = 0 (which implies that x = 0), which is the Schwarzschild solution. For
a = 0werecover the RN solution, as we wanted.
Furthermore, for all values of d, a, and b, when ω = 0(extreme dilaton BHs) H can
be any arbitrary harmonic function in the transverse (d − 2)-dimensional Euclidean space.
This allows us to construct multi-BH (in general multicenter) solutions, as in the MP family
(which is included in this one with a = 0).
The b = 0 solutions were first obtained and studied in [432, 447]. The d = 4 solutions
were rediscovered from a string-theory point of view in [416] and those with arbitrary a
were also studied in [539]. The multicenter solutions were found in [853] (see also [743])
and the solutions with b =−1 are presented for the first time here.
Let us now study the properties and the geometry of the b = 0family. First, we want
to relate the integration constants to the physical parameters: mass, electric charge, and
“scalar charge.” Only the first two are independent. For the sake of clarity we omit most
numerical factors and define these charges by the asymptotic expansions of the fields:
gtt ∼ 1 − M
r d−3 , At ∼− Q
r d−3 , ϕ ∼ ϕ0 − S
. (12.11)
r d−3
These charges are related to the integration constants by
M = 2(1 − x)h − ω, Q = αe aϕ0 h, S = xh. (12.12)