Gravity and Strings

12.1 Dilaton black holes: the a-model 355
The next values of interest from the string-theory-supergravity point of view are a = 1;
and a = 1/ √ 3;
ds2 = H −1Wdt2 − H W −1dr2 + r 2d2
(2) ,
e−2ϕ = e−2ϕ0
H, ω= h 1 − (α/ √ 2) 2
,
ds2 = H − 3 2 Wdt2 − H 3
2 −1 2 2 2 W dr + r d(2) ,
e−2ϕ = e−2ϕ0 √
3
H 2 , ω= h 1 − α2 /3 .
(12.23)
(12.24)
d = 4 stringy solutions with these metrics will appear in the compactification of solutions
that describe the intersection of two and three extended objects, respectively, instead of just
one as in the previous case. We are going to see how this comes about in Section 20.1.
Finally, we have a = 0, the RN BH:
ds2 = H −2Wdt2 − H 2W −1dr2 + r 2d2
(2) ,
e −2ϕ = e −2ϕ0, ω= h 1 − (α/2) 2 .
(12.25)
This case will be seen to arise from the intersection of four extended objects in higher
dimensions.
In four dimensions, we can define the mass M, electric charge q, and “scalar charge”
more precisely by the asymptotic expansions
N M
gtt ∼ 1 − 2G(4)
r
, At ∼ 4G(4)
r
N e2aϕ0q
N
, ϕ ∼ ϕ0 + G(4)
r
. (12.26)
The dilaton-dependent factor e2aϕ0 in the definition of the electric charge is related to the
integral definition
q =
1
e −2aϕ⋆ F, (12.27)
16πG (4)
N
which is in turn related to the modification of the Gauss law introduced by the dilaton.
For a = 1 the integration constants in the solutions are given by
1 +
a2
h = G(4)
1 − a2 N M ± M2 − 4(1 − a2
)e2aϕ0q 2 ,
1 − a2
α =
1 + a2 4eaϕ0q M ± M2 − 4(1 − a2 ,
)e2aϕ0q 2
ω = 2a2
2
G(4)
1 − a2 N M ± G(4)
1 − a2 N M2 − 4(1 − a2 )e2aϕ0q 2 ,
S 2 ∞
(12.28)