Gravity and Strings

92 A perturbative introduction to general relativity
We already see here that the expression for Ɣµν ρ in terms of ϕ µν involves an infinite
series of terms. This is the reason why one iteration will be enough even though we had
expected an infinite series of corrections.
On substituting the last two results into the equation for Ɣµν ρ ,wefind
Ɣρµ σ gσν|g| 1
d−2 + Ɣρν σ gσµ|g| 1
d−2 = ∂ρ |g| 1
d−2 gµν . (3.241)
We see that, again, it is convenient to make the following definition:
gµν ≡|g| 1
d−2 gµν, ⇒ g µν = |g| g µν . (3.242)
In terms of the variable gµν, the above equation can be solved using the same procedure
as before. The result is that Ɣµν ρ is given by the Christoffel symbols associated with the
metric gµν (1.44). The two equations of motion can now be combined into one:
Rµν(g) = 0, (3.243)
where Rµν(g) is the Ricci tensor associated with the Levi-Cività connection of the metric
gµν. This is the vacuum Einstein equation, the equation of motion of GR, as we will see.
So far we have not shown that the corrected action has the required self-consistency
property. We are now going to do this, and this will allow us to claim that the vacuum
Einstein equation is the self-consistent extension of the Fierz–Pauli theory we were looking
for, written in terms of the new variable gµν, which turns out to have a geometrical meaning
that is really unexpected, given our starting point of view.
We turn back to the equation for Ɣµν ρ and try to solve it without the use of g µν and its
inverse, by raising and lowering indices with the Minkowski metric again. First, we contract
it with ηµ ρ ,giving
(η ρσ − χϕ ρσ )Ɣρσ ν = χ∂σ ϕ σν . (3.244)
Contracting now with ηµν and substituting into it the last result, we obtain
Ɣρδ δ =− 1
d − 2 χ −∂ρϕ + 2ϕ δ µƔρδ µ − ϕƔρδ δ , (3.245)
and, on plugging these results into the full equation, we arrive at
Ɣρµν + Ɣνρµ = χ∂ρhµν + fρµν,
fρµν = 2χϕδ (µ|Ɣρδ|ν) − χϕµνƔρδ δ − 1
d − 2 χηµν
δ 2ϕ λƔρδ λ − ϕƔρδ δ ,
which can be “solved” in exactly the same way, giving
(3.246)
Ɣρµν = 1
2χ{∂ρhµν + ∂µhνρ − ∂νhρµ}+ 1
2 { fρµν + fµνρ − fνρµ}. (3.247)
There are Ɣs onthe r.h.s. of this equation, but we do not need anything better (neither
can we obtain it without inverting the matrix ϕ µν ). On substituting into the equation for