Gravity and Strings

19.3 The masses and charges of the p-brane solutions 549
5. In d dimensions, the mass M of any static, asymptotically flat metric describing
a point-like object can be found from its asymptotic behavior at spatial infinity
[706, 877]. In the string-theory context, the mass ME associated with the “Einstein
metric” (i.e. with the “wrong” normalization factor g2 (16πG (d)
N )−1 in the action) can
be implicitly defined by
gE tt ∼ 1 − 16πG(d) N ME
g2 1
(d − 2)ω(d−2) |xd−1| d−3 , xd−1 = (x 1 ,...,x d−1 ), (19.92)
The mass M associated with the modified Einstein metric (i.e. with the “right” normalization
factor (16πG (d)
N )−1 )isdefined analogously by,
˜gE tt ∼ 1 − 16πG(d)
N M
(d − 2)ω(d−2)
1
. (19.93)
|xd−1|
d−3
If both the string metric and the “modified string metric” are asymptotically flat (as
we have assumed) then the Einstein metric is not, and it is necessary to rescale the
coordinates with factors of g 1 4 in order to be able to use Eq. (19.92). Taking this into
account, we find that the relation (in any dimension) between ME and M is given by
ME = g 1 4 M. (19.94)
6. Under IIB S duality, it is the (unmodified) Einstein-frame metric that is invariant.
Then ME is S-duality-invariant, which implies the following S-duality transformation
rule for M, which was already given in Eq. (19.3):
M ′ E = ME, ⇒ M ′ = g ′− 1 4
B g 1 4
B M = g 1 2
B M. (19.95)
We can apply these formulae to find the masses of any of the solutions we have studied.
They should coincide with the masses of the corresponding states in string/M theory.
Let us take, for example, the F1 solution given in Eq. (19.56), assuming that the string is
compactified on a circle and y is a compact dimension so we can dimensionally reduce the
above solution, and calculate the modified Einstein mass of the resulting point-like object
that lives in d = 9byusing Eq. (19.93).
First, we need the nine-dimensional dilaton (see Eq. (15.16)),
e −2(φ−φ0)
−2( ˆφ− ˆφ0)
= e
|ˆgyy|=H 1 2
F1
so, in this case, the relation between the nine-dimensional metrics is
˜gEµν = H 1 7
F1 gµν, ⇒˜gE tt = H − 6 7
F1
which, compared with Eq. (19.93), gives the right value,
MF1w = 6hF1ω(7)
16πG (9) =
N
12π R9hF1ω(7)
on account of Eqs. (19.87), (19.26), and (19.57).
16πG (10)
N
, (19.96)
6hF1
∼ 1 − , (19.97)
7ρ6 = R9
ℓ9 , (19.98)
s