Gravity and Strings

110 A perturbative introduction to general relativity
Furthermore, in the proof of invariance of the action the presence of the cosmological
constant terms is also crucial. Had we tried to prove the invariance of the equation of motion
directly, we would have seen the necessity for these terms to cancel out curvature terms
coming from the commutators of covariant derivatives. We can conclude that the theory,
with those terms, is massless.
It is interesting to see what kind of gauge identity and conserved current we obtain from
this invariance. We proceed as usual. We first find the variation of the action under an
arbitrary infinitesimal transformation of δhµν:
δSFP = 1
χ 2
SFP
δhαβ
d d x √ |¯g|
=− 1
2 ¯D αβ (h),
SFP
δhαβ + ¯∇µ
δhαβ
µ(αβ) l δhαβ
,
l µαβ = 1
2 ¯∇ µ h αβ − ¯∇ α h βµ + 1
2 ¯gµα ¯∇ β h + 1
2 ¯gαβ ¯∇νh µν − 1
2 ¯gαβ ¯∇ µ h.
(3.309)
Using now the particular form of the gauge transformation δɛhµν in the above equation and
integrating by parts, we obtain
δɛ S =
d d x |¯g|
− 1
χ 2 ɛβ ¯∇α ¯D αβ
1
(h) + ¯∇µ
χ 2 ¯D µβ ɛβ − 2
χ 2 lµ(αβ)
¯∇αɛβ , (3.310)
and, on comparing this with the first form of the variation of the action that we found, we
arrive finally at the identity, which is valid for arbitrary ɛ µ s and without the use of any
equations of motion,
0 = dd x √ |¯g|
j µ µ 1
N2 (ɛ) = j N1 (ɛ) +
χ 2 ¯D µβ (h)ɛβ,
j µ 2
N1 (ɛ) =−
χ 2 lµ(αβ) ¯∇αɛβ − s µ (ɛ).
From this identity we derive the gauge identity,
− 1
χ 2 ɛβ ¯∇α ¯D αβ (h) + ¯∇µ j µ
N2 (ɛ)
,
(3.311)
¯∇α ¯D αβ (h) = 0, (3.312)
and the off-shell covariant conservation of the above Noether current,
¯∇µ j µ
N2 (ɛ) = 0
∂µj µ
N2 (ɛ) = 0 . (3.313)
We know that this Noether current can always be written as j µ
νµ
N2 (ɛ) = ∂νjN2 (ɛ) with
j νµ
N2 (ɛ) =−jµν N2 (ɛ). Finding this antisymmetric tensor in the general case is complicated and
we are going to do it only for the most interesting case, in which ɛ µ is a Killing vector of
the background metric ɛ µ ≡ ¯ξ µ with ¯∇(µ ¯ξν) = 0. In this case, s µ (ξ) has to vanish identi-
(ɛ) also
cally, because the variations of hµν also vanish identically, and the first term of j µ
N1