Gravity and Strings

12.2 Dilaton/axion black holes 365
ϒ, the (complex) axion/dilaton charge, and τ0, its asymptotic value, are defined by
2ϒ −2ϕ0 τ ∼ τ0 − ie . (12.72)
r
In these solutions ϒ depends on the conserved charges in this fixed way:
ϒ =− 1
2
n
Finally, the “non-extremality” parameter r0 is given by
r 2
0 =|M|2 +|ϒ| 2 −
( ¯Ɣ (n) ) 2
. (12.73)
M
n
|Ɣ (n) | 2 . (12.74)
In non-static cases when r0 = 0 the solution is supersymmetric,butfor α = 0itisnotan extreme
BH. A more appropriate name is supersymmetry parameter. The extremality parameter
will be R 2 0 = r 2 0 − α2 . When it is positive, we have two horizons placed at r± = M ± R0.
The area of the event horizon (the one at r+) isgiven, for BH solutions with zero NUT
charge, by
A = 4π r 2
+ + α2 −|ϒ| 2 . (12.75)
12.2.2 Supersymmetric SWIP solutions
When r0 = 0 W = 1 the general SWIP solution has special properties. First, the background
metric (3) γij is nothing but the metric of Euclidean three-dimensional space in oblate
spheroidal coordinates, which are related to the ordinary Cartesian ones by
x = √ r 2 + α 2 sin θ cos ϕ,
y = √ r 2 + α 2 sin θ sin ϕ,
z = r cos θ.
On rewriting the solution Eqs. (12.64) in Cartesian coordinates, we find the solutions
ds2 = 2Im(H1 ¯H2) (dt + A) 2 − [2 Im(H1 ¯H2)] −1d x 2
3 ,
, Ã (n) t = 2e2U Re k (n)
H1 , τ = H1/H2 .
,
A (n) t = 2e 2U Re k (n) H2
A = Aidx i , ɛijk∂i A j =±Re H1∂k ¯H2 − ¯H2∂kH1
∂i∂iH1,2 = 0,
N
(k (n) ) 2 = 0,
n=1
N
n=1
|k (n) | 2 = 1
2 .
(12.76)
(12.77)
That is, for any arbitrary pair of complex harmonic functions H1,2(x3) in the threedimensional
Euclidean space, it is clear that we can construct multi-BH solutions and that
r0 = 0can be reinterpreted as a no-force condition between the BHs.