Gravity and Strings

3.4 The Fierz–Pauli theory in a curved background 105
The first thing we have to do is to expand this equation in powers of the perturbation hµν.
The perturbation can be treated as a tensor on the background manifold. Then, it is natural
to lower and raise its indices (and those of all tensors) with the background metric ¯gµν and
its inverse ¯g µν .Inparticular, h µν =¯g µρ ¯g νσ hρσ is not the inverse of hµν (which need not
exist) and we also define h =¯g µν hµν. All barred covariant derivatives are also taken with
respect to the background metric’s Levi-Cività connection ¯Ɣµν ρ .Wefind
g µν =¯g µν − h µν + O(h 2 ),
Ɣµν ρ = ¯Ɣµν ρ + γµν ρ + O(h 2 ),
Rµνρ σ = ¯Rµνρ σ + 2 ¯∇[µγν]ρ σ + O(h 2 ),
(3.281)
with
γµν ρ = 1
2 ¯gρσ
¯∇µhσν + ¯∇νhµσ − ¯∇σ hµν . (3.282)
This equation is essentially the equation that gives the variation of the Levi-Cività connection
δƔµν ρ (≡ γµν ρ ) under an arbitrary variation of the metric53 δgµν (≡ hµν),
δƔµν ρ = 1
2 gρσ
∇µδgσν +∇νδgσρ −∇σδgµν . (3.284)
Now we can find the expansion of Rµνρ σ to first order in hµν using the so-called Palatini
identity that gives the variation of the curvature tensor under an arbitrary variation of the
connection
δRµνρ σ =+2∇[µδƔν]ρ σ . (3.285)
The Palatini identity follows from Eqs. (1.31) and (1.36), on setting the torsion equal to
zero, identifying τµν ρ with δƔµν ρ , and keeping only the linear terms. We stress that, unlike
Ɣµν ρ , the variation δƔµν ρ is a true tensor and its covariant derivative is well defined. 54
For the variation of Ɣµν ρ that we have just found we obtain
Rµνρ σ = ¯Rµνρ σ +¯g σλ ¯∇[µ ¯∇ν]hλρ + ¯∇[µ| ¯∇ρh|ν]λ − ¯∇[µ| ¯∇λh|ν]ρ
2
+ O(h ), (3.287)
and, on contracting the indices σ and ν,wefind55 Rµρ = ¯Rµρ + 1
¯∇ 2
2 hµρ − 2 ¯∇ λ ¯∇(µhρλ + ¯∇µ ¯∇ρh + O(h 2 ). (3.288)
53 For further use we quote here the generalization of this equation when there is torsion present:
δƔαβ γ = 1 2 gγδ
∇αδgβδ +∇βδgαδ −∇δδgαβ
+ 1
2 g δγ gσβδTαδ σ + g δγ gσαδTβδ σ
− δTαβ
γ
. (3.283)
54 Also for further use, here we quote the formula valid for a general connection:
δRµρ =∇µδƔνρ ν −∇νδƔµρ ν − Tµν λ δƔλρ ν . (3.286)
55 Sometimes the subindex L is used to indicate that the object is the part linear in hµν of the corresponding
tensor with the indices in the same position. Observe that for any tensor TL µ =¯g µν TL ν and for this reason
we try to avoid this notation.