Gravity and Strings

304 The Kaluza–Klein black hole
dimension does not have to go to −1 atinfinity but can be any real negative number. If the
metric is asymptotically flat in the non-compact directions then the dimensionally reduced
metric (assuming that the compact dimension is isometric) will be asymptotically flat in the
KK conformal frame and the value of k at infinity will be some positive real number k0.
When we rescale the metric to go to the Einstein conformal frame, the metric does not
look asymptotically flat any longer, but
and a change of coordinates is necessary:
lim
r→∞ gE µν = k 2
x µ → x ′µ = k 1
d−2
0 x µ , ⇒ gE µν → g ′ E µν
d−2
0 ηµν, (11.51)
2
d−2
= k−
0
gE µν
r ′ →∞
−→ ηµν. (11.52)
Thus, if we start with ˆd-dimensional metrics that are asymptotically flat in the noncompact
dimensions, we are forced to perform a rescaling of the coordinates, which is, at
the very least, quite unusual. Of course, this change of coordinates, does not modify the
action Eq. (11.45).
We could have decided to start with ˆd-dimensional metrics, which naturally lead to
asymptotically flat Einstein metrics with no need for changes of coordinates, but this looks
rather artificial.
As we pointed out before, a very interesting aspect of the massless sector of the KK theory
is that the truncated massive modes reappear as solitonic solutions. A further problem
of the standard Einstein conformal frame is that the masses one finds for solitons are not
the ones expected in the spectrum of Kaluza–Klein theories. We are going to check this
explicitly in Section 11.2.3.
The prescription we have used to go to the Einstein frame is not canonical, though. We
just wanted to eliminate the unconventional (local) factor of k in front of the curvature
scalar and the conformal factor that does the job is unique only up to an overall constant
1
factor. In particular, we could have rescaled the KK metric by the factor ˜
−
= (k/k0) d−2
which defines the modified Einstein conformal frame
2
−
gµν = (k/k0) d−2 ˜gE µν. (11.53)
One of the main characteristics of this metric is that it is invariant under the scale transformations
Eq. (11.29). It is appropriate to use with it fields that are also invariant under
those rescalings:
õ = k0 Aµ, ˜k = k/k0. (11.54)
In terms of these scale-invariant fields, the action takes the form
˜SE = 2πℓk0
16πG (
ˆd)
N
d d x
|˜gE| ˜RE +
d − 1
2 ˜k
−2
∂ ˜k
d − 2
− 1
d−1
˜k
2 d−2 ˜F 4 2
, (11.55)
which is identical to the action in the original “Einstein frame” Eq. (11.45) except for the
overall factor.