5.3.2 Homothety Property
Let γ(τ) be a general curve in the spacetime and v
μ its velocity, and a vector W
μ
. If we
compare the change of the vector along γ(τ) under Palatini and Levi-Civita parallel
transport, we see that the difference between both transports is proportional to W
μ :
v
μ
∇ μ À v
μ
∇
g
ð Þ
μ
W
ρ
¼ ÀA μ v
μ W
ρ
λ τ
ð ÞW
ρ
:
The module is not conserved but the direction does. Due to this, we say the
Palatini parallel transport is homothetic with respect to the Levi-Civita transport. It
can be proved that the only connections with this property are the Palatini connections (Bernal et al. 2017). Any other connection would generate a perturbation in the
direction of W
μ .
5.4 Observability and Physical Implications
We introduced the Palatini formalism in order to see if the dynamics could fix LeviCivita as the fundamental connection of the theory, in contrast with the metric
formalism in which it is selected by hand. However, we have obtained a family of
connections that differ in a vector field, with Levi-Civita as a particular case. In this
section, we analyse the physical implications of this field. Indeed, we will see that it
is undetectable or, equivalently, that metric and Palatini formalism describe the same
physics.
The main point is that the gravitational dynamics is the same in both formalisms.
The equation of the matter is clearly the same, because the corresponding action does
not depend on the connection, so the difference between formalisms does not affect
this part of the total action. And, as we previously showed, Palatini connections
imply the reduction of the equation of the metric to the Einstein’s equation. The
resulting dynamics for the metric and the matter content is given by
R μν g
ð Þ À
1
2
g μν R g
ð Þ ¼ ÀκT μν ,
δS matter
δψ
¼ 0,
in both formalisms.
Furthermore, in Einstein–Hilbert gravity the distinction between critical and
autoparallel trajectories disappears due to the projective symmetry. In fact, defining
the trajectory of a test particle is often presented as a basic problem of metric-affine
theories. Those of critical length and those with covariantly constant velocity are
candidates because both of them infinitesimally reduce to straight lines. The critical
paths are the simplest approach, but there are authors who defend the description
with autoparallels (Kleinert and Pelster 1999) and others who state that only the
conserved currents determine the test paths (Hehl and Obukhov 2007).
56
B. Janssen et al.
Let γ(τ) be a general curve in the spacetime and v
μ its velocity, and a vector W
μ
. If we
compare the change of the vector along γ(τ) under Palatini and Levi-Civita parallel
transport, we see that the difference between both transports is proportional to W
μ :
v
μ
∇ μ À v
μ
∇
g
ð Þ
μ
W
ρ
¼ ÀA μ v
μ W
ρ
λ τ
ð ÞW
ρ
:
The module is not conserved but the direction does. Due to this, we say the
Palatini parallel transport is homothetic with respect to the Levi-Civita transport. It
can be proved that the only connections with this property are the Palatini connections (Bernal et al. 2017). Any other connection would generate a perturbation in the
direction of W
μ .
5.4 Observability and Physical Implications
We introduced the Palatini formalism in order to see if the dynamics could fix LeviCivita as the fundamental connection of the theory, in contrast with the metric
formalism in which it is selected by hand. However, we have obtained a family of
connections that differ in a vector field, with Levi-Civita as a particular case. In this
section, we analyse the physical implications of this field. Indeed, we will see that it
is undetectable or, equivalently, that metric and Palatini formalism describe the same
physics.
The main point is that the gravitational dynamics is the same in both formalisms.
The equation of the matter is clearly the same, because the corresponding action does
not depend on the connection, so the difference between formalisms does not affect
this part of the total action. And, as we previously showed, Palatini connections
imply the reduction of the equation of the metric to the Einstein’s equation. The
resulting dynamics for the metric and the matter content is given by
R μν g
ð Þ À
1
2
g μν R g
ð Þ ¼ ÀκT μν ,
δS matter
δψ
¼ 0,
in both formalisms.
Furthermore, in Einstein–Hilbert gravity the distinction between critical and
autoparallel trajectories disappears due to the projective symmetry. In fact, defining
the trajectory of a test particle is often presented as a basic problem of metric-affine
theories. Those of critical length and those with covariantly constant velocity are
candidates because both of them infinitesimally reduce to straight lines. The critical
paths are the simplest approach, but there are authors who defend the description
with autoparallels (Kleinert and Pelster 1999) and others who state that only the
conserved currents determine the test paths (Hehl and Obukhov 2007).
56
B. Janssen et al.