Gravity and Strings

646 Appendix F
The non-vanishing components of the Ricci tensor are
Rtt = R (δ)
tt − 1
Rin jm =−1
2 δijδnm
Rρρ = R (δ) 1 N ρρ + 2 n=1 rn(µf(n)) − 1 2
Rqr = R (δ)
qr
4 µ N n=1 rn(ln fn) ′ λ ′ ,
∇2
fn + µf(n) (ln fn) ′2
,
(µf(n)) 1
′
′
2 ln f(n) ,
N
rn(ln fn) ′ ,
− 1
2 gqrµ(ln R) ′
n=1
(F.43)
where we have indicated with the superscript (δ) the components of the curvature of the
δ-dimensional metric that one obtains if the coordinates yn are suppressed.
The Ricci scalar is
R = R (δ) + 1
2
N
n=1
d-dimensional metrics of the general form
rn
∇ 2 (δ) ln f(n) + f −1
(n) ∇2 f(n) + µ (ln f(n)) ′ 2
. (F.44)
F.2.4 A general metric for extreme p-branes
ds 2 = H 2x ηijdy i dy j + H −2y ηmndx m dx n , (F.45)
where i, j = 0, 1,...,p and m, n = p + 1,...,d − 1 and H is a function solely of the
x m s often occur in the study of p-branes. The coordinates y i correspond to the p-brane
worldvolume and the coordinates x m are transverse to the p-brane. Observe that, with our
conventions, ηmn =−δmn. The non-vanishing components of the Levi-Cività connection
are
Ɣi j m = xηijH 2(x+y)−1 ∂m H, Ɣim j = xδi j H −1 ∂m H,
Ɣmn p =−yH −1
δpm∂n H + δpn∂m H − δmn∂p H
The non-vanishing components of the Ricci tensor are
Ri j = gi j∇ 2 ln H x ,
Rmn = gmn∇ 2 ln H −y + zH −1 ∂m∂n H
.
+ H −2 ∂m H∂n H x 2 (p + 1) + y 2 ( ˜p + 1) + (2y − 1)z ,
(F.46)
(F.47)
where we have used the fact that
|g|=H z−2y , z = x(p + 1) − y( ˜p + 1), (F.48)
and the fact that, for a scalar function of the x m sinthis metric,
∇ 2 f (x m ) =−H 2y−1 z∂m H∂m f + H∂ 2 f , ∂ 2 ≡+∂m∂m,
∇ 2 ln H = (1 − z)H 2y−2 (∂ H) 2 − H 2y−1 ∂ 2 H.
(F.49)