In Einstein–Hilbert gravity, the conservation of the energy-momentum tensor
selects the critical paths. However, fortunately, the autoparallels set by the Palatini
dynamics coincide, as a consequence of the projective relation between Palatini
connections (a family that includes Levi-Civita). Indeed, we have seen that the field
A μ can be eliminated by the freely falling observer with an appropriate choice of the
parameter.
All of these ideas point in the same direction: the field A μ has no physical effects
(Bernal et al. 2017).
5.5 Equivalence in Other Theories
Finally, we add a few remarks about Palatini connections in other theories. One
example is Lovelock theory in Palatini formalism:
S g, Γ, ψ
½
¼S Lov g, Γ
½
þS matter g, ψ
½
, S Lov g, Γ
½
Z X K
n¼1
a n L n g, Γ
½
ffiffiffiffiffi ffi
g
j j
p
d
D x,
where a n are certain dimensionful parameters, ℕ 3 K ceiling(D/2 À 1) and the nthorder Lovelock lagrangian is defined by
L n ½g, Γ ¼ δ
½μ 1
ν 1
. . . δ
μ 2n
ν 2n
g
ρ 1 ν 1 . . . g
ρ n ν 2nÀ1 R μ 1 μ 2 ρ 1
ν 2 ðΓÞ . . . R μ 2nÀ1 μ 2n ρ n
ν 2n ðΓÞ:
It was shown in Borunda et al. (2008) that Levi-Civita is a solution for the
equation of the connection in any of these theories. As far as we know, the general
solution remains unknown, but we have found that the action presents the projective
symmetry, Γ
σ
μν ! Γ
σ
μν þ A μ δ
σ
ν . The proof is the following. Under the projective
transformation, the Riemann tensor is modified:
R μνρ
λ
! R μνρ
λ
þ F μν δ
λ
ρ :
Then, the lagrangian transforms:
L n ! δ
μ 1
½ν 1
. . . δ
μ 2n
ν 2n
ðR μ 1 μ 2
ν 1 ν 2 þ F μ 1 μ 2 g
ν 1 ν 2 Þ . . . ðR μ 2nÀ1 μ 2n
ν 2nÀ1 ν 2n þ F μ 2nÀ1 μ 2n g
ν 2nÀ1 ν 2n Þ
and the ν’s antisymmetrization cancels all the terms proportional to the metric, so
δ proj L n ¼ 0, 8n ) δ proj S Lov g, Γ
½
þS matter g, ψ
½
f
g ¼ 0:
Q.E.D.
5 (Non-)Uniqueness of Einstein–Palatini Gravity
57
selects the critical paths. However, fortunately, the autoparallels set by the Palatini
dynamics coincide, as a consequence of the projective relation between Palatini
connections (a family that includes Levi-Civita). Indeed, we have seen that the field
A μ can be eliminated by the freely falling observer with an appropriate choice of the
parameter.
All of these ideas point in the same direction: the field A μ has no physical effects
(Bernal et al. 2017).
5.5 Equivalence in Other Theories
Finally, we add a few remarks about Palatini connections in other theories. One
example is Lovelock theory in Palatini formalism:
S g, Γ, ψ
½
¼S Lov g, Γ
½
þS matter g, ψ
½
, S Lov g, Γ
½
Z X K
n¼1
a n L n g, Γ
½
ffiffiffiffiffi ffi
g
j j
p
d
D x,
where a n are certain dimensionful parameters, ℕ 3 K ceiling(D/2 À 1) and the nthorder Lovelock lagrangian is defined by
L n ½g, Γ ¼ δ
½μ 1
ν 1
. . . δ
μ 2n
ν 2n
g
ρ 1 ν 1 . . . g
ρ n ν 2nÀ1 R μ 1 μ 2 ρ 1
ν 2 ðΓÞ . . . R μ 2nÀ1 μ 2n ρ n
ν 2n ðΓÞ:
It was shown in Borunda et al. (2008) that Levi-Civita is a solution for the
equation of the connection in any of these theories. As far as we know, the general
solution remains unknown, but we have found that the action presents the projective
symmetry, Γ
σ
μν ! Γ
σ
μν þ A μ δ
σ
ν . The proof is the following. Under the projective
transformation, the Riemann tensor is modified:
R μνρ
λ
! R μνρ
λ
þ F μν δ
λ
ρ :
Then, the lagrangian transforms:
L n ! δ
μ 1
½ν 1
. . . δ
μ 2n
ν 2n
ðR μ 1 μ 2
ν 1 ν 2 þ F μ 1 μ 2 g
ν 1 ν 2 Þ . . . ðR μ 2nÀ1 μ 2n
ν 2nÀ1 ν 2n þ F μ 2nÀ1 μ 2n g
ν 2nÀ1 ν 2n Þ
and the ν’s antisymmetrization cancels all the terms proportional to the metric, so
δ proj L n ¼ 0, 8n ) δ proj S Lov g, Γ
½
þS matter g, ψ
½
f
g ¼ 0:
Q.E.D.
5 (Non-)Uniqueness of Einstein–Palatini Gravity
57