Gravity and Strings

11.3 KK reduction and oxidation of solutions 327
Generation of charged solutions by higher-dimensional boosts. The first example consists
of three steps.
1. Oxidation of the Schwarzschild solution to ˆd = d + 1 dimensions.
2. Lorentz-boosting the Schwarzschild black 1-brane solution in the compact direction.
3. Reduction in the same direction.
We have already performed the first operation in the previous section. The ˆd-dimensional
solution is
d ˆs 2 = Wdt 2 − dz 2 − W −1 dr 2 − r 2 d 2
ω
[ ˆd−3]
, W = 1 + , (11.151)
ˆd−4 r
and we are ready to perform a Lorentz boost in the positive- or negative-z direction, which
evidently transforms a solution of the ˆd-dimensional Einstein equations into another one:
t
z
→
cosh γ ± sinh γ
± sinh γ cosh γ
The new solution can be rewritten in the form
t
z
, γ >0. (11.152)
d ˆs 2 = H −1 Wdt 2 − H dz − α(H −1 − 1)dt 2 − W −1 dr 2 − r 2 d 2
[ ˆd−3] ,
W = 1 + ω
ˆd−4
, H = 1 +
h
r r ˆd−4 , ω= h(1 − α2 ),
(11.153)
if we parametrize α =±coth γ , which is a sort of “black pp-wave” metric. The nonextremality
function W disappears when we boost at the speed of light α =±1 and then
we recover exactly the pp-wave solutions Eq. (10.42), for which H can be any general
harmonic function in ( ˆd − 2)-dimensional Euclidean space.
Now, the third step gives a new d-dimensional class of solutions whose existence we
announced:
ds2 KK = H −1Wdt2 − W −1dr2 − r 2d2 (d−2) ,
d ˜s 2 d−3
−
E = H d−2 Wdt2 − H 1
(d−2) W −1dr2 − r 2d2
(d−2) ,
Ãt = α(H −1 − 1),
˜k = H 1 2 ,
W = 1 + ω
d−3 , H = 1 +
h
r r d−3 , ω= h(1 − α2 ).
(11.154)
These are the non-extreme electric KK BHs. They have regular event horizons and
Cauchy horizons (for negative ω) and, in the extreme limit ω = 0, they become the extreme
electric KK BHs, Eq. (11.131).
The same procedure can be used with higher-p branes and also with “charged p-branes.”