Gravity and Strings

3.4 The Fierz–Pauli theory in a curved background 107
before. In fact, in the previous sections we have found the lowest-order term (quadratic in
h)oft µν in the case µν ¯ = ηµν ( = 0) which we denoted by t (0)µν
GR .
This equation, with the r.h.s. set to zero, is the equation of motion of a massless spin-2
field moving on a background spacetime ¯gµν, which we are going to study in the next section.
We can already see that this equation does not look like the typical wave equation for
amassless field because it has mass-like terms proportional to the cosmological constant.
However, we are going to argue that precisely those terms are necessary in order to describe
massless fields in a spacetime with ¯Rµν = ¯gµν.
Observe that, since ζ µν = h µν + O(h2 ) and h µν = ζ µν + O(ζ 2 ),wecould have arrived
at the same linear-order results by expanding around the inverse metric
g µν =¯g µν − ζ µν . (3.297)
We would like to have an action from which to derive the above equation of motion with
vanishing r.h.s. Instead of guessing, we simply expand the integrand of the Einstein–Hilbert
action to second order in hµν. Using the matrix identity
1
|M|=exp( tr ln M), (3.298)
2
and the expansions
(1 + x) −1 = 1 − x + x 2 − x 3 + ···,
ln (1 + x) = x − 1
2 x 2 + 1
3 x 3 − 1
4 x 4 + ···, (3.299)
exp y = 1 + y + 1
2! y2 + 1
3! y3 + ···,
we can easily calculate second- and higher-order terms:
gµν =¯gµν + hµν,
g µν =¯g µν − h µν + h µ σ h σν − h µ σ h σρ hρ ν + O(h 4 ), (3.300)
|g| = |¯g| 1 + 1 1
h +
2 8 h2 − 1
4 hµνh µν + 1
6 hµ ν hν ρ hρ µ
− 1
8 hhµνh µν + 1
48 h3
+ O(h 4 ).
For the Levi-Cività connection we can write the exact expression
Ɣµν ρ = ¯Ɣµν ρ + g ρσ γµνσ , γµνσ = 1
¯∇µhνσ + ¯∇νhµσ − ¯∇σ hµν , 2
(3.301)
and just have to substitute the above expansion of gρσ to the desired order. For the Riemann
curvature tensor and the Ricci tensor we can write also write exact expressions,
Rµνρ σ = ¯Rµνρ σ
+ 2 ¯∇[µ
σλ σδ λɛ g γν]ρλ + 2g g γ[µ|λδγ|ν]ρɛ,
Rµρ = ¯Rµρ + ¯∇µ
σλ g γσρλ − ¯∇σ
σλ σδ λɛ g γµρλ + g g (3.302)
γµλδγσρɛ − γσλδγµρɛ ,
on which, again, we simply have to expand the inverse metric. A similar expression can