Gravity and Strings

532 The extended objects of string theory
(Black) Dp-branes.
d ˜s 2 ˜p+1
−8
(d−2)
E = H 2 8
2 2
Dp Wdt − d y p − H p+1
(d−2) 2
2
Dp dz q + W −1dρ2 + ρ2d 2
(δ−2) ,
ds2 4
− 4 d−2 2 2 d−2 2
s = HDp Wdt − d y p − HDp dz q + W −1dρ2 + ρ2d 2
(δ−2) ,
e−2φ = e−2φ0 ˜p−p
−2 d−2
HDp , C (p+1) ty1
···y p = αe−φ0 H −1
Dp − 1 ,
HDp = 1 + hDp
ω
d − 2
, W = 1 + , ω= hDp 1 −
ρδ−3 ρδ−3 8 α2
.
(19.42)
19.2.1 Extreme p-brane solutions of string and M-theories and sources
The four families of solutions above contain subfamilies of extreme solutions with ω = 0in
which the H functions can be arbitrary functions of the transverse coordinates x(δ−1) (ρ =
|x(δ−1)|) forδ≥2. These are isotropic coordinates in which the metric of the transverse
space is conformally flat. As in the ERN BH case, in some cases they do not cover the whole
spacetime which can be analytically extended beyond ρ = 0, which is only a coordinate
singularity.
In Section 18.2.3 we saw that some of the extreme solutions of the p-brane a-model
(with q = 0 and a single-pole H) could be matched against some charged-p-brane sources
(obeying Eqs. (18.71) and (18.77)), which allowed us to determine h, the coefficient of the
pole of H, interms of the tension and charge of the source and in terms of the Newton
constant.
It turns out that the four families of objects that we are considering always satisfy
Eqs. (18.71) and (18.77) and we can use those results to determine h in the extreme solutions
with q = 0 and a single pole in terms of the tensions and Newton constants. Then,
for the ten- and 11-dimensional objects whose tensions we found in Section 19.1.1, we can
determine h as a function of ℓs, g, and ℓ (11)
Planck , using the values of G(10)
N and G (11)
N that we
determined there. For these d = 10, 11 objects α =±1isjust the relative sign between the
charge parameter µp and the tension T(p) which we consider positive.
All we have to do to find h for the families of extreme Mp-branes, Dp-branes etc. is to
substitute into Eqs. (18.78) the values of T(p) (which are unknown in general, except in the
cases studied in the previous sections) and α, determined by setting ω = 0inthe solutions;
α 2 Mp =
2(d − 2)
(p + 1)( ˜p + 1) , αFp = 2
p + 1 , αSp = 2
˜p + 1 , αDp = 8
. (19.43)
d − 2
Observe that, indeed, for the string/M-theory branes (M2 and M5 in d = 11 and F1, S5,
and Dp in d = 10) α 2 =+1.
Writing the four families of extreme solutions with the right values for h is straightforward,
but not very interesting, except in the case of the d = 11, 10 string/M-theory branes,
that we are going to write and study next.