Gravity and Strings

11.3 KK reduction and oxidation of solutions 319
These are very useful formulae that we are going to use many times and they deserve to
be rewritten and framed. For n > 1andn = 1, respectively,
m=+∞
H = 1 + h
m=−∞
∼ 1 + hω(n−1)
2π Rzω(n−2)
m=+∞
H = h
m=−∞
∼− h
ln |x2|.
π Rz
1
[|xn+1| 2 + (z + 2πmRz) 2 ] n 2
1
,
|xn+1|
n−1
1
[|x2| 2 + (z + 2πmRz) 2 ] 1 2
m=+∞
− 2h
m=1
1
2πmRz
(11.124)
Using this approximated H (the zero mode of the periodic one) in the ˆd-dimensional MP
solution, we obtain a solution that does not depend on the periodic coordinate z and now
we can rewrite the solution in terms of the d = ( ˆd − 1)-dimensional fields: 18
ds 2 KK = H −2 dt 2 − H 2
d−2 d x 2
d−1 ,
ds2 E = H − 5 2 dt2 d−6
−
− H d−2 d x 2
d−1 ,
Vµ = δµtα(H −1 − 1), α =±2,
k = H 1
d−2 , V = V0. ∂i∂i H = 0,
(11.125)
where we have included a possible constant value for ˆVz. This form is valid for any
ˆd-dimensional MP solution with a z-independent harmonic function, and, in particular,
for the above H that corresponds to the zero mode of the ˆd-dimensional periodic ERN BH.
Why have we gone through the long procedure of finding periodic ERN BH solutions
and finding their zero modes when we could simply reduce the whole MP family assuming
independence of z? The reason is that, in the cases that will interest us, we will have a welldefined
ˆd-dimensional source that will determine the coefficient h of the ˆd-dimensional
harmonic function and only by going through all this procedure can we relate it to the
coefficient of the d-dimensional harmonic function.
The dimensionally reduced ERN solution does not have a regular horizon: near the origin
(the only place where the horizon could be placed), using spherical coordinates r =|xd−1|,
1 + h ′ /r d−3 − d−6
d−2
r 2 d 2 (d−2) ∼ h′ r (d−3)(d−6)
d−2 d 2 (d−2) , (11.126)
18 Since we have absorbed the asymptotic value of the KK scalar into the period of the coordinate z, k = ˜k and
there is no difference between the Einstein and modified Einstein frames.