Kiefer C. Quantum gravity

THE CONCEPT OF A GRAVITON 29
Exploiting the Poincaré invariance of the flat background, one can calculate
from the Fierz–Pauli Lagrangian (2.20) without the T µν -term the canonical
energy–momentum tensor of the linearized gravitational field,
t µν =
∂L
∂f
ν
f αβ,µ − η µν L . (2.23)
αβ,
The resulting expression is lengthy, but can be considerably simplified in the TT
gauge where f µν assumes the form
⎛
⎞
00 0 0
f µν = ⎜ 00 0 0
⎟
⎝ 00f 22 f 23
⎠ .
00f 23 −f 22
Then,
TT 1
(
t 00 = f ˙
22 2 + f
16πG
˙
)
23
2 = −t 01 , (2.24)
which can be written covariantly as
TT 1
t µν =
32πG f αβ,µf αβ ,ν . (2.25)
It is sometimes appropriate to average this expression over a region of space–time
much larger than ω −1 ,whereω is the frequency of the weak gravitational wave,
so that terms such as exp(−2iω(t − x)) drop out. In the TT gauge, this leads to
¯t µν = k µk ν
16πG eαβ∗ e αβ . (2.26)
In the general harmonic gauge, without necessarily specifying to the TT gauge,
the term e αβ∗ e αβ is replaced by e αβ∗ e αβ − 1 2 |eα α |2 .Thisexpressionremainsinvariant
under the gauge transformations (2.9), as it should.
Regarding the Fierz–Pauli Lagrangian (2.20), the question arises whether it
could serve as a candidate for a fundamental helicity-2 theory of the gravitational
field in a flat background. As it stands, this is certainly not possible because, as
already mentioned, one has ∂ ν T µν = 0 and there is therefore no back reaction
of the gravitational field onto matter. One might thus wish to add the canonical
energy–momentum tensor t µν , eqn (2.23), to the right-hand side of the linearized
Einstein equations,
✷ ¯f µν = −16πG(T µν + t µν ) .
This modified equation would, however, lead to a Lagrangian cubic in the fields
which in turn would give a new contribution to t µν , and so on. Deser (1970, 1987)
was able to show that this infinite process can actually be performed in one single
step. The result is that the original metric η µν is unobservable and that all matter
couples to the metric g µν = η µν +f µν ; the resulting action is the Einstein–Hilbert