Feynman Path Integral Formulation

4.2 First Order Formulation 105
ṗ = −∂H/∂q, then one needs to re-write the equations of motion in such a way that
only first derivatives of the metric appear.
In the first order (Palatini) formulation of general relativity one writes for the
Einstein-Hilbert pure gravity action
I = 1 ∫
16πG
d 4 x √ gg μν R μν (Γ ) , (4.8)
with R μν (Γ ) now considered only as a function of the affine connection Γ ,
R μν (Γ )=∂ λ Γ λ
μν − ∂ ν Γ λ
μλ +Γ λ
μνΓ σ
λσ −Γ λ
μσΓ σ
νλ . (4.9)
Variation of the gravitational action then requires that
∫
1
16πG
d 4 x δ[ √ gg μν R μν ]=0 . (4.10)
First by varying with respect to the metric g μν one obtains the Einstein field equations,
R μν − 1 2 g μν R = 0 . (4.11)
At this point the metric g μν and the connection Γ λ
μν are still thought of as independent
variables, and a relationship between the two needs to be established before
one can claim to have reproduced correctly the field equations.
The variation of R μν can be simplified by virtue of the Palatini identity
δ R λ μνσ
= δΓ λ
μσ;ν − δΓ λ
μν;σ . (4.12)
After integrating by parts one can then show that the term involving the variation of
the connection Γ implies
∂ λ g μν − g νσ Γ σ
μλ − g μσΓ σ
νλ = 0 , (4.13)
(normally one writes this as g μν;λ = 0). The last equation can then be solved to give
the usual relationship between the connection Γ and the metric g in Riemannian
geometry, namely
)
Γμν(g) λ = 1 2 gλσ( ∂ μ g νσ + ∂ ν g μσ − ∂ σ g μν . (4.14)
Instead of using the metric form, one can introduce local Lorentz frames and write
the gravitational action in terms of vierbeins (tetrads) e a μ(x) and spin connections
ω ab
μ (x). In this formalism (see for example Weinberg, 1972) the metric is written as
g μν (x) =η ab e a μ(x)e b ν(x) , (4.15)
where one can think of the four covariant vector fields e a μ as relating locally noninertial
coordinate system (described by g μν ) to locally inertial ones (described by