Gravity and Strings

3.2 Gravity as a self-consistent massless spin-2 SRFT 91
The correction to the action with the property (3.230) is precisely
S (2) = 1
χ 2
d d x −2χϕ αβ Ɣλ[α ρ Ɣρ]β λ . (3.234)
One could naively think that, with this correction, we can obtain only the first term (that
quadratic in Ɣµν ρ )inταβ.However, we have to take into account that the equation for Ɣµν ρ
changes and, hence, substituting its solution into the equation for ϕ µν will give us all the
terms we need. Observe also that this correction is cubic in fields whereas the action we
started from is quadratic. Finally, observe that this term will not contribute to the energy–
momentum tensor: there are no Minkowski metrics here to be replaced by the background
metric and there is no need to introduce √ |γ | because ϕ µν is, by hypothesis, a tensor
density. Thus, if this term really works, we will not need to introduce any more corrections.
For the total action
S (1) + S (2) = 1
χ 2
we find the following equations of motion:
χ δ(S(1) + S (2) )
δϕ µν
χ 2 (δS(1) + S (2) )
δƔµν ρ
d d x −χϕ µν 2∂[µƔρ]ν ρ + (η µν − χϕ µν )2Ɣλ[µ ρ Ɣρ]ν λ , (3.235)
=−Rµν(Ɣ) = 0,
= 2Ɣρ (µν) − η µν Ɣρλ λ −η λσ Ɣλσ (µ η ν) ρ − χ∂ρϕ µν + χηρ (µ ∂σ ϕ ν)σ
− 2χϕ δ(µ Ɣρδ ν) + χϕ µν Ɣρσ σ + χϕ λσ Ɣλσ (µ η ν) ρ = 0,
(3.236)
where Rµν(Ɣ) is nothing but the Ricci tensor associated with the connection Ɣµν ρ given in
Eq. (1.33). By defining
g µν = η µν − χϕ µν
(3.237)
and its inverse g µρ gρν = g µ ν = δ µ ν, which we are going to use as a metric to raise and
lower indices, we can write
χ 2 (δS(1) + S (2) )
δƔµν ρ
= 2g δ(µ Ɣρδ ν) − g µν Ɣρδ δ − g λσ Ɣλσ (µ g ν) ρ + ∂ρg µν − gρ (µ ∂σ g ν)σ = 0.
(3.238)
Now we proceed as before: we contract this equation of motion with gµ ρ ,giving
and then contract with gµν, using the last equation, giving
Ɣλ λν =−∂σ g σν , (3.239)
Ɣρλ λ = 1
d − 2 gµν∂ρg µν = 1
d − 2 ∂ρ ln |g|, |g|≡det g µν . (3.240)