image of a curve in the manifold is what we will call trajectory or path. So a
trajectory is a set of spacetime points joined in a continuous and differentiable
way, while the curve is the function that generates this set.
Let M be a spacetime equipped with a connection Γ
ρ
μν . An autoparallel of this
affine structure is the image of a curve whose velocity is parallel to itself (with
respect to Γ
ρ
μν ). In other words, given a curve γ(α) with velocity v
μ (α), its image is an
autoparallel if the following equation holds:
v
μ
∇ μ v
ρ
dv
ρ
dα
þ Γ
ρ
μν v
μ v
ν
¼ f α
ð Þv
ρ
:
for some function f(α). If we reparametrize the trajectory, by doing α ! β(α), we
change the velocity as well as the function f. This function can always be set to zero
(identically) with an appropriate choice of the parameter. Those are called affine
parameters for the trajectory.
Consider that the manifold also has a metric structure. The autoparallels of the
associated Levi-Civita connection are special because they can be derived from a
completely metric approach, i.e. without using the Levi-Civita parallelism. If the
velocity is timelike or spacelike, they correspond to trajectories that are critical
points of the length functional:
s γ
½ α
ð Þ ¼
Z α
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
g μν v μ v ν
q
dα
0
:
The lightlike case should be treated separately, because the length functional
cannot be varied smoothly. However, they can be seen as critical points of other
functionals that, again, only involve the metric structure. For these reasons, in
general, we will call the autoparallels of Levi-Civita critical trajectories. The
associated affine parameters have a special meaning, because their changes are
proportional to the length between the considered points. In the timelike case, it is
essentially the proper time, so these parameters represent the rhythm of a proper
(i.e. freely falling) clock along them.
5.2 Einstein’s Equations and Variational Principles
General relativity is a geometric theory of the spacetime whose dynamics is
described by the Einstein’s equation:
R μν g
ð Þ À
1
2
g μν R g
ð Þ ¼ ÀκT μν ,
ð5:1Þ
52
B. Janssen et al.
trajectory is a set of spacetime points joined in a continuous and differentiable
way, while the curve is the function that generates this set.
Let M be a spacetime equipped with a connection Γ
ρ
μν . An autoparallel of this
affine structure is the image of a curve whose velocity is parallel to itself (with
respect to Γ
ρ
μν ). In other words, given a curve γ(α) with velocity v
μ (α), its image is an
autoparallel if the following equation holds:
v
μ
∇ μ v
ρ
dv
ρ
dα
þ Γ
ρ
μν v
μ v
ν
¼ f α
ð Þv
ρ
:
for some function f(α). If we reparametrize the trajectory, by doing α ! β(α), we
change the velocity as well as the function f. This function can always be set to zero
(identically) with an appropriate choice of the parameter. Those are called affine
parameters for the trajectory.
Consider that the manifold also has a metric structure. The autoparallels of the
associated Levi-Civita connection are special because they can be derived from a
completely metric approach, i.e. without using the Levi-Civita parallelism. If the
velocity is timelike or spacelike, they correspond to trajectories that are critical
points of the length functional:
s γ
½ α
ð Þ ¼
Z α
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
g μν v μ v ν
q
dα
0
:
The lightlike case should be treated separately, because the length functional
cannot be varied smoothly. However, they can be seen as critical points of other
functionals that, again, only involve the metric structure. For these reasons, in
general, we will call the autoparallels of Levi-Civita critical trajectories. The
associated affine parameters have a special meaning, because their changes are
proportional to the length between the considered points. In the timelike case, it is
essentially the proper time, so these parameters represent the rhythm of a proper
(i.e. freely falling) clock along them.
5.2 Einstein’s Equations and Variational Principles
General relativity is a geometric theory of the spacetime whose dynamics is
described by the Einstein’s equation:
R μν g
ð Þ À
1
2
g μν R g
ð Þ ¼ ÀκT μν ,
ð5:1Þ
52
B. Janssen et al.