Gravity and Strings

644 Appendix F
and the Ricci scalar is
R =∇ 2 ln λR d−2 + (d − 2)(d − 3) 1 d − 2
−
R2 R (λµ) 1
2 R ′
λ
µ
1 ′
− 2
. (F.33)
However, if we are interested in finding singular contributions to the curvature, these
formulae are not completely appropriate, because in obtaining them we have performed
operations in which singular contributions are ignored. The unsimplified formulae are
Rqr =− 1
d − 2 gqr
⎧
⎨
⎩ ∇2 ln µ + 1
Rd−2 1
− ′
1
λ 2 λ 2 d−2 ′
R µ
µ µ
(d − 2)(d − 3)
−
R2
, (F.34)
R =∇ 2
λ
ln −
µ
1
Rd−2 1
− ′
1
λ 2 λ 2 d−2 ′
R µ
µ µ
d − 2
−
R (λµ) 1
2 R ′
′ 1
λ 2 (d − 2)(d − 3)
−
µ
R2 . (F.35)
F.2.2 A general metric for (single, black) p-branes
This metric can be understood as a generalization of the previous one with translational
isometries in p dimensions and it is adequate for describing the gravitational fields of pbranes.
Therefore, in general, it is not asymptotically flat in those p dimensions. It is,
roughly speaking, the result of adding those p dimensions to the general, static, spherically
symmetric (d − p)-dimensional metric of the previous section. Thus, it has the general
form
ds 2 = λ(ρ)dt 2 − f (ρ)d y 2 p − µ−1 (ρ)dr 2 − R 2 (ρ)d 2 ( ˜p+2) , (F.36)
where yp = (y 1 p ,...,y p p ) are the coordinates on the p-brane that we denote with the indices
i, j, k = 1,...,p, ρ 2 = (x p+1 ) 2 + ···+(x d−1 ) 2 is the radial coordinate in the (d − p −
1)-dimensional, asymptotically flat space transverse to the p-brane, the d − p − 2 angular
coordinates are labeled by q, r, s = 1,...,d − p − 2, and ˜p ≡ d − p − 4isthe dimension
of the object that is the electric–magnetic dual to the p-brane.
The non-vanishing components of the Levi-Cività connection are
Ɣtt ρ = 1
2 µλ′ , Ɣtρ t = 1
2 λ−1 λ ′ , Ɣρρ ρ =− 1
2 µ−1 µ ′ ,
Ɣρq r = δq r (ln R) ′ , Ɣij ρ =− 1
2 δijµf ′ , Ɣiρ j = 1
2 δi j f −1 f ′ ,
Ɣqr ρ =− 1
2δqrµ(R2 ) ′ q(r)/R 2 ,
Ɣqr s
= θrqδs (r) cot ψ(q) + θqrδs (q) cot ψ(r) − θqsδ(q)r cot ψ(s) q −1
(s) q(q)
.
(F.37)