Gravity and Strings

6.1 The traditional approach 175
Having the total conserved energy–momentum pseudotensor obeying the local continuity
equation, we go on to define the conserved charges (momentum and angular momentum)
by the volume integrals7 P µ
=
M µα
=
dd−1 √
ν |g|(Tmatter µν + tLL µν ),
dd−1 √
[α
ν |g| 2x Tmatter µ]ν + tLL µ]ν ,
(6.11)
where it is assumed that one integrates over a timelike hypersurface .
One of the shortcomings of this approach is that it is not clear why these are the (only)
conserved charges and how we can generalize it to other spacetimes in which these are not
necessarily the conserved charges. The Abbott–Deser approach will solve this problem.
A second shortcoming is the large number of terms that have to be calculated in order
to find the conserved quantities. In practice, though, one uses Eq. (6.8) to rewrite, using
Stokes’ theorem,
P µ = 1
2
∂
d d−2 η
νρ
µνρ
√ |g| , (6.12)
and similarly for M µα . This is an interesting expression that has to be evaluated at the
boundary of the hypersurface , which is, typically, for asymptotically flat spacetimes a
(d − 2)-sphere at spatial infinity Sd−2 ∞ .Wecould integrate over the boundary of smaller
regions of the spacetime. However, the integrand is not a general-covariant tensor and the
result of the integral would be coordinate-dependent and the momentum would not be well
defined. Only when we integrate over Sd−2 ∞ in asymptotically flat spacetimes does the integral
transform as a Poincaré tensor. This is the common behavior of most superpotentials
used to defined conserved quantities in GR, except for Møller’s [702], which is a true tensor.
For asymptotically flat spaces, we can use the weak-field expansion8 gµν = ηµν + hµν.
In this limit, we see that
g µσ,νρ = K µσ νρ + O(h 2 ), (6.13)
and
η µνρ = 2η µνρ
LL + O(h2 ), 2∂ρη µνρ
LL = Dµν (h), (6.14)
where η µνρ
LL was defined in Eq. (3.90) and Dµν (h) is the Fierz–Pauli wave operator.
Thus, in practice, all we have to do is to integrate the Fierz–Pauli wave operator over
the volume or ηLL over the boundary ∂,ifthe asymptotic weak-field expansion of
the metric is well defined. Many different gravity energy–momentum pseudotensors have
been proposed in the literature but, in the end, one never uses them directly. Instead one
integrates over , using the equations of motion, an expression that, in the weak-field
limit, is equivalent to the Fierz–Pauli wave operator. Usually this expression is rewritten as
an integral over the boundary using Stokes’ theorem. This can be done in many different
ways, as we discussed in Chapter 3, and here is where the differences arise. 9
7 Observe the “extra” factors of √ |g|.
8 Observe that the hµν that we are using in this chapter is χhµν of Chapter 3.
9 Some expressions may be better suited for certain boundary conditions. When we compare the weak-field
limits of the various expressions proposed in the literature, we have to bear in mind that the expansions used
are valid only under certain asymptotic conditions.