Gravity and Strings

178 Conserved charges in general relativity
where the Minkowski spacetime Killing vectors ξ (µ) ν = η µν that generate constant translations
are used to obtain the P µ s and those which generate Lorentz transformations
ξ (µα) ν =−2x [µ ηα]ν are used to obtain the M µαs. The different definition of t µν is responsible
for the extra factor of √ |g| in this formula compared with Abbott and Deser’s. On the
other hand, in the Landau–Lifshitz approach we are forced not only to work with asymptotically
flat spacetimes, but also to use Cartesian coordinates. The Abbott–Deser approach
can be used for any spacetime in any coordinate system.
The main problem with Eq. (6.26) is also that the expression for t µν is very complicated;
it is in fact, an infinite series in h. The solution is, again, to use the equation of motion
Eq. (3.292) to rewrite it. The new integrand is, as we argued it would in general be, just
the covariantized Fierz–Pauli wave operator ¯D µν (h) contracted with a background Killing
vector, that is,
E(¯ξ)= 2
χ 2
d
d−1 µ ¯D µν (h)¯ξν. (6.29)
At this point we notice that the integrand of this expression is nothing but the conserved
Noether current j µ
N (¯ξ)in Eq. (3.314) and we can use the results of Section 3.4.1 to rewrite it
as a total derivative and then use Stokes’ theorem to rewrite it as a (d − 2)-surface integral,
E(¯ξ)=− 2
χ 2
∂=Sd−2 d
∞
d−2
µα ¯∇β K µανβ ¯ξν − K µβνα ¯∇β ¯ξν
, (6.30)
where
d d−2 1
µα =
(d − 2)! √ |¯g| ɛµαρ1···ρd−2dxρ1 ρd−2 ∧ ···∧dx . (6.31)
This is essentially Abbott and Deser’s final result, although one can massage the above
expression further to make it useful in specific situations. For instance, the following alternative
expression is noteworthy. We first observe the identity
¯∇β K µανβ = 3 ¯g λµα, ν ρσγλ ρσ . (6.32)
We can replace γµν ρ by Ɣµν ρ = Ɣµν ρ − ¯Ɣµν ρ because the difference is quadratic and
higher in hµν, which is assumed to go to zero at infinity fast enough. Then
E(¯ξ)=− 2
χ 2
d d−2 λµα, ν
µα 3 ¯g ρσ Ɣλ ρσ ¯ξν − K µβνα ¯∇β ¯ξν
. (6.33)
S d−2
∞
Furthermore, in Minkowski spacetime in Cartesian coordinates the generators of translations
are covariantly constant and the second term can be dropped, so we obtain for any
component of the momentum (and, in particular, for the energy) of asymptotically flat
spacetimes the expression
E(¯ξ)=− 2
χ 2
d
∂
d−2 µα3 ¯g λµα, ν ρσ Ɣλ ρσ ¯ξν, (6.34)