2.2 Advantages of the Tetrad Formulation of Differential Geometry
15
• The local symmetry of observer transformations.
From Eq. (2.2.2) we see that the metric is invariant under a local Lorentz transformation (LLT) of the a index,
˜
e
a
μ (x) =
a
b (x) e
b
μ (x) ,
η cd
c
a
d
b ≡ η ab ,
(2.2.4)
which therefore corresponds to another observer family, i.e. with different 4velocities ˜
e
μ
0 and spatial frames ˜
e
μ
i . The corresponding tensor components transform accordingly
˜
T
b 1 ...b n
a 1 ...a m
(x) =
c 1
a 1
(x) . . . .
c m
a m
(x) )
b 1
d 1
(x) . . . .
b n
d n
(x) T
d 1 ...d n
b 1 ...b m
(x) ,
(2.2.5)
so they are diffeomorphism scalars, but they are Lorentz tensors. They correspond
to the components that are measured/emitted by the new observer family ˜
e
μ
a (x).
The LLTs are therefore “observer transformations”. Since the metric is invariant,
this is a symmetry of the equations when everything is expressed in terms of e
a
μ (x).
We can understand this by noting that a 6-dimensional gauge symmetry is required
in order to render the extra six components of the tetrad non-physical. These extra
components therefore contain the information of the tetrad orientation, i.e. the part
that is not captured by the internal scalar product in Eq. (2.2.2). We thus see that the
tetrad formulation provides an elegant unification of the gravitational and observer
information, i.e. the tetrad internal product and the tetrad orientation, respectively,
in a single mathematical object. Note also that the Lorentz group allows us to reach
all possible observer families, because the local boosts probe the full interior of
each light-cone, while the local rotations probe all possible spatial frames.
Finally, at the conceptual level, the LLT symmetry provides a coordinateindependent manifestation of the notion of relativity. Indeed, the fact that LLTs
are a symmetry of the action means that the physics is observer-independent, e.g.
whether two particles scattered or not is independent of the choice of e
a
μ (x), given
some g μν (x). However, their recorded initial and final momenta do depend on
the observer and change under an LLT. Consequently, the physics is observerindependent, but measurement is observer-dependent. Note that this has nothing
to do with the choice of coordinates, so the relativity of measurement is related
to the LLTs, not the coordinate transformations.
3 Rather, the symmetry under
coordinate transformations reflects the fact that the physics is independent of the
parametrization of space-time, which is independent of the notion of observer.
• Full covariance: freedom of gauge choice and control.
As already stressed in the case of the tetrad at the observer point, the fact that we
do not have to privilege a particular coordinate system means that we are free to
choose whichever diffeomorphism gauge we wish in cosmological perturbation
theory, say the Newtonian one. On the other hand, in the case of LLTs, a choice
3 Of course one can always associate an observer family 4-velocity to some coordinate system,
i.e. the family with u μ = (1, 0, 0, 0) in that system, but it cannot also have a trivial spatial frame,
because this would lead to a trivial tetrad field e
μ
a (x) = δ
μ
a and thus no curvature.
15
• The local symmetry of observer transformations.
From Eq. (2.2.2) we see that the metric is invariant under a local Lorentz transformation (LLT) of the a index,
˜
e
a
μ (x) =
a
b (x) e
b
μ (x) ,
η cd
c
a
d
b ≡ η ab ,
(2.2.4)
which therefore corresponds to another observer family, i.e. with different 4velocities ˜
e
μ
0 and spatial frames ˜
e
μ
i . The corresponding tensor components transform accordingly
˜
T
b 1 ...b n
a 1 ...a m
(x) =
c 1
a 1
(x) . . . .
c m
a m
(x) )
b 1
d 1
(x) . . . .
b n
d n
(x) T
d 1 ...d n
b 1 ...b m
(x) ,
(2.2.5)
so they are diffeomorphism scalars, but they are Lorentz tensors. They correspond
to the components that are measured/emitted by the new observer family ˜
e
μ
a (x).
The LLTs are therefore “observer transformations”. Since the metric is invariant,
this is a symmetry of the equations when everything is expressed in terms of e
a
μ (x).
We can understand this by noting that a 6-dimensional gauge symmetry is required
in order to render the extra six components of the tetrad non-physical. These extra
components therefore contain the information of the tetrad orientation, i.e. the part
that is not captured by the internal scalar product in Eq. (2.2.2). We thus see that the
tetrad formulation provides an elegant unification of the gravitational and observer
information, i.e. the tetrad internal product and the tetrad orientation, respectively,
in a single mathematical object. Note also that the Lorentz group allows us to reach
all possible observer families, because the local boosts probe the full interior of
each light-cone, while the local rotations probe all possible spatial frames.
Finally, at the conceptual level, the LLT symmetry provides a coordinateindependent manifestation of the notion of relativity. Indeed, the fact that LLTs
are a symmetry of the action means that the physics is observer-independent, e.g.
whether two particles scattered or not is independent of the choice of e
a
μ (x), given
some g μν (x). However, their recorded initial and final momenta do depend on
the observer and change under an LLT. Consequently, the physics is observerindependent, but measurement is observer-dependent. Note that this has nothing
to do with the choice of coordinates, so the relativity of measurement is related
to the LLTs, not the coordinate transformations.
3 Rather, the symmetry under
coordinate transformations reflects the fact that the physics is independent of the
parametrization of space-time, which is independent of the notion of observer.
• Full covariance: freedom of gauge choice and control.
As already stressed in the case of the tetrad at the observer point, the fact that we
do not have to privilege a particular coordinate system means that we are free to
choose whichever diffeomorphism gauge we wish in cosmological perturbation
theory, say the Newtonian one. On the other hand, in the case of LLTs, a choice
3 Of course one can always associate an observer family 4-velocity to some coordinate system,
i.e. the family with u μ = (1, 0, 0, 0) in that system, but it cannot also have a trivial spatial frame,
because this would lead to a trivial tetrad field e
μ
a (x) = δ
μ
a and thus no curvature.