Gravity and Strings

364 Dilaton and dilaton/axion black holes
k (n) :
where
ds2 = e2U W dt + Aϕdϕ 2 −2U −1 (3) − e W γijdxidx j ,
, τ = H1/H2.
A (n) t = 2e2U Re k (n)
H2 , Ã (n) t = 2e2U Re k (n) H1
e−2U = 2Im H1 ¯H2
,
Aϕ = 2N cos θ + α sin 2 θ e −2U W −1 − 1 .
(12.64)
(12.65)
The functions H1,2 take the form
H1 = 1 √ e
2 ϕ0
iβ
e τ0 + τ0M
+¯τ0ϒ
,
r + iα cos θ
H2 = 1
√ e
2 ϕ0
iβ M + ϒ
e 1 + ,
r + iα cos θ
(12.66)
and W and the background metric (3) γij take the forms
W = 1 −
r 2
0
r 2 + α 2 cos 2 θ ,
(3) γijdxidx j = r 2 + α2 cos2 θ − r 2 0
r 2 + α2 − r 2 dr
0
2 + r 2 + α2 cos2 θ − r 2
2
0 dθ
+ r 2 + α2 − r 2
2 2
0 sin θ dϕ .
The complex constants are given by
(12.67)
k (n) =− 1
√ e
2 −iβ MƔ(n) + ϒƔ (n)
|M| 2 −|ϒ| 2
. (12.68)
The metric can also be written in a more standard form:
ds 2 = − α2 sin 2 θ
dt
2 + 2α sin 2 θ + α2 sin 2 θ −
dtdϕ
−
dr2 − dθ 2
2 2 2 2 2
+ α sin θ − α sin θ
−
sin
2 θ dϕ 2 ,
= r 2 − R0 2 = r 2 + α 2 − r0 2 ,
= (r + M) 2 + (n + α cos θ) 2 −|ϒ| 2 .
(12.69)
We have expressed the functions that enter the solution in terms of physical constants
(charges and moduli). α = J/M is the angular momentum (J) per unit mass (M), and we
have combined the mass and NUT charge (N) into the complex “mass”
and the electric and magnetic charges into
Ɣ (n) ≡ Q (n) + iP (n) ,
M ≡ M + iN, (12.70)
Q (n) ≡ V −1
0 q(n) . (12.71)