R Γ
ð Þ ¼ R μν g
ð Þ,
respectively, where F μν ¼ ∂ μ A ν À ∂ ν A μ . As a consequence of these expressions,
the equation of motion of the metric (5.2) becomes the Einstein’s Eq. (5.1), see
(Dadhich and Pons 2012). We now present some properties of these solutions.
5.3.1 Projective Relation Between Solutions
Any two Palatini connections, for example
Γ
ρ
μν ¼ Γ
g
ð Þ σ
μν þ A μ δ
σ
ν , Γ
0 ρ
μν ¼ Γ
g
ð Þ σ
μν þ A
0
μ δ
σ
ν ,
are related by a transformation:
Γ
ρ
μν ! Γ
0 ρ
μν ¼ Γ
ρ
μν þ k μ δ
σ
ν ,
for certain covector k μ . This transformation is a projective transformation which
means that preserves autoparallels. This can be proved easily. First, consider an
autoparallel trajectory for the connection Γ
0 ρ
μν ,
dv
ρ
dβ
þ Γ
0 ρ
μν v
μ v
ν
¼ 0:
Then, imposing the projective relation between both connections and defining
Àk μ v
μ
f(β) we get to the expression:
dv
ρ
dβ
þ Γ
ρ
μν v
μ v
ν
¼ f β
ð Þv
ρ
:
And this is the equation of an autoparallel for the connection Γ
ρ
μν with a non-affine
parametrization. If we parametrize the path affinely (β ! α and v
ρ
! u
ρ ) we obtain:
du
ρ
dα
þ Γ
ρ
μν u
μ u
ν
¼ 0:
Q.E.D.
Consequently, the whole set of Palatini connections shares the same
autoparallels, up to reparametrizations, which have no physical meaning. As a
matter of fact, since Levi-Civita is a particular Palatini connection (the case with
A μ ¼ 0 ) we conclude: the autoparallels of any Palatini connection are critical
trajectories of the metric.
5 (Non-)Uniqueness of Einstein–Palatini Gravity
55