Consequently, since Levi-Civita (Γ
g
ð Þ σ
μν ) is a solution, then Γ
g
ð Þ σ
μν þ A μ δ
σ
ν is also a
solution. Indeed, the whole set of solutions can be separated into equivalence classes
of projectively related connections. So if a new solution Γ
sol σ
μν of a Lovelock theory
that has not the form Γ
g
ð Þ σ
μν þ A μ δ
σ
ν (for some A μ ) is found, then we could build a
family of new solutions just adding a term ℬ μ δ
σ
ν where ℬ μ is arbitrary. This property
also holds for any other lagrangians with projective invariance, such as f(R) gravity.
Other theories we have tested are those with quadratic torsion corrections to the
Einstein–Hilbert lagrangian. The torsion corrections we consider are only those with
even parity:
S g, Γ, ψ
½
¼
1
2κ
Z
g
μν R μν Γ
ð Þ
ffiffiffiffiffi ffi
g
j j
p
d
D x þ S matter g, ψ
½
þ
1
2κ
Â
Z
b 1 T
1
ð Þ
μνρ T
1
ð Þμνρ
þ b 2 T
2
ð Þ
μνρ T
2
ð Þμνρ
þ b 3 T
3
ð Þ
μνρ T
3
ð Þμνρ
h
i ffiffiffiffiffi ffi
g
j j
p
d
D x,
where b i are arbitrary dimensionless real constants and T
i
ð Þ
μνρ are the irreducible
components of the torsion (see (McCrea 1992)). For these extensions, the equivalence between metric and Palatini formalism holds.
5.6 Conclusions
To conclude we summarize our results. We have seen that Einstein–Hilbert gravity
in the Palatini formalism has some interesting features. If we couple this theory with
a matter action through the metric (and not the connection), the result is physically
indistinguishable from the dynamics obtained assuming Levi-Civita as the fundamental affine structure from the beginning (metric formalism).
The general solution of the equation of the connection is Levi-Civita plus the term
A μ δ
σ
ν where A μ is an undetermined field. However, the equations of motion are the
same as in metric formalism. Therefore, we get to different mathematical descriptions (related through the projective symmetry) that describe the same physics. In
other words, it is not necessary to set the connection to be Levi-Civita by hand. The
dynamics fixes the affine structure.
Another additional property of the Palatini connections is that they are the only
affine structures whose parallel transport is homothetic with respect to the LeviCivita transport. So the directions obtained in both cases are coincident.
We have also proved that an autoparallel of a given Palatini connection is a
trajectory with critical length (autoparallel of Levi-Civita). The undetermined field
A μ for a free falling observer can be absorbed in a reparametrization of its worldline,
so it has no physical meaning since a particular choice of the parameter is
meaningless.
58
B. Janssen et al.
g
ð Þ σ
μν ) is a solution, then Γ
g
ð Þ σ
μν þ A μ δ
σ
ν is also a
solution. Indeed, the whole set of solutions can be separated into equivalence classes
of projectively related connections. So if a new solution Γ
sol σ
μν of a Lovelock theory
that has not the form Γ
g
ð Þ σ
μν þ A μ δ
σ
ν (for some A μ ) is found, then we could build a
family of new solutions just adding a term ℬ μ δ
σ
ν where ℬ μ is arbitrary. This property
also holds for any other lagrangians with projective invariance, such as f(R) gravity.
Other theories we have tested are those with quadratic torsion corrections to the
Einstein–Hilbert lagrangian. The torsion corrections we consider are only those with
even parity:
S g, Γ, ψ
½
¼
1
2κ
Z
g
μν R μν Γ
ð Þ
ffiffiffiffiffi ffi
g
j j
p
d
D x þ S matter g, ψ
½
þ
1
2κ
Â
Z
b 1 T
1
ð Þ
μνρ T
1
ð Þμνρ
þ b 2 T
2
ð Þ
μνρ T
2
ð Þμνρ
þ b 3 T
3
ð Þ
μνρ T
3
ð Þμνρ
h
i ffiffiffiffiffi ffi
g
j j
p
d
D x,
where b i are arbitrary dimensionless real constants and T
i
ð Þ
μνρ are the irreducible
components of the torsion (see (McCrea 1992)). For these extensions, the equivalence between metric and Palatini formalism holds.
5.6 Conclusions
To conclude we summarize our results. We have seen that Einstein–Hilbert gravity
in the Palatini formalism has some interesting features. If we couple this theory with
a matter action through the metric (and not the connection), the result is physically
indistinguishable from the dynamics obtained assuming Levi-Civita as the fundamental affine structure from the beginning (metric formalism).
The general solution of the equation of the connection is Levi-Civita plus the term
A μ δ
σ
ν where A μ is an undetermined field. However, the equations of motion are the
same as in metric formalism. Therefore, we get to different mathematical descriptions (related through the projective symmetry) that describe the same physics. In
other words, it is not necessary to set the connection to be Levi-Civita by hand. The
dynamics fixes the affine structure.
Another additional property of the Palatini connections is that they are the only
affine structures whose parallel transport is homothetic with respect to the LeviCivita transport. So the directions obtained in both cases are coincident.
We have also proved that an autoparallel of a given Palatini connection is a
trajectory with critical length (autoparallel of Levi-Civita). The undetermined field
A μ for a free falling observer can be absorbed in a reparametrization of its worldline,
so it has no physical meaning since a particular choice of the parameter is
meaningless.
58
B. Janssen et al.