18
2 Motivation
ˆ
e
a
μ ≡
∂ x
a
ˆ
P
∂ x μ ( ˆ
P) .
(2.3.2)
Going from that arbitrary system x
μ to some other arbitrary system ˜
x
μ then reproduces the tensorial transformation rule of the μ index
˜ ˆ
e
a
μ ≡
∂ x
a
ˆ
P
∂ ˜
x μ ( ˆ
P) =
∂ x
a
ˆ
P
∂ x ν ( ˆ
P)
∂ x
ν
∂ ˜
x μ ( ˆ
P) ≡ ˆ
e
a
ν
∂ x
ν
∂ ˜
x μ ( ˆ
P) ,
(2.3.3)
and so on for the tetrad vectors ˆ
e
μ
a . In this process, the observer coordinates x
a
ˆ
P
are
fixed, hence the consistency with their Lorentz index. Indeed, these coordinates can
only be transformed to the ones of some other observer at ˆ
P, by definition, since
we must maintain ˆ
g ab ≡ η ab . The allowed coordinate transformations are the ones
preserving (2.3.1), i.e. those that reduce to Poincaré around ˆ
P
˜
x
a
ˆ
P
= q
a
+
a
b x
b
ˆ
P
+ O((x ˆ
P − ˆ
x ˆ
P )
2
) .
(2.3.4)
The corresponding frame would then only be sensitive to the Lorentz transformation
(i.e. not the translation)
˜ ˆ
∂ a, ˆ
P =
b
a
ˆ
∂ b, ˆ
P ,
⇒
˜ ˆ
e a =
b
a ˆ
e b ,
(2.3.5)
and therefore reproduces the Lorentz transformation of the a index. Here we wish to
stress that the interpretation of the tetrad as a Jacobian of some coordinate transformation x
a
ˆ
P
(x) is relevant only in the case where it is used at a single point ˆ
P ∈ M.
This is because a tetrad at some other point ˆ
P
will correspond to the Jacobian of
some other transformation x
a
ˆ
P (x), i.e. the one trivializing the metric at ˆ
P
, not ˆ
P.
Indeed, if the tetrad were the Jacobian of a single coordinate transformation all over
M, then space-time would be flat, as one could perform that transformation to get
g μν → η μν everywhere.
Thus, in the presence of a tetrad field the underlying observer coordinates associated with each point lose their relevance and one only retains the necessary information for observations, i.e. the basis at each tangent space T P M. We therefore believe
that the “Jacobian matrix” interpretation of the tetrad loses its appeal in this context
and can even become misleading. We prefer the more gauge-theoretical viewpoint
where the Lorentz indices “a” simply correspond to some internal gauge symmetry,
just as in Yang-Mills theory, with no reference whatsoever to any particular coordinate system x
a
P . The e
μ
a (x) are therefore simply a set of four vector fields and we
privilege no coordinate system in describing their dynamics. The only aspects that
one can retain from the Jacobian matrix picture is that, for a given P ∈ M, there
always exists a coordinate system in which e
μ
a (P) = δ
μ
a .
2 Motivation
ˆ
e
a
μ ≡
∂ x
a
ˆ
P
∂ x μ ( ˆ
P) .
(2.3.2)
Going from that arbitrary system x
μ to some other arbitrary system ˜
x
μ then reproduces the tensorial transformation rule of the μ index
˜ ˆ
e
a
μ ≡
∂ x
a
ˆ
P
∂ ˜
x μ ( ˆ
P) =
∂ x
a
ˆ
P
∂ x ν ( ˆ
P)
∂ x
ν
∂ ˜
x μ ( ˆ
P) ≡ ˆ
e
a
ν
∂ x
ν
∂ ˜
x μ ( ˆ
P) ,
(2.3.3)
and so on for the tetrad vectors ˆ
e
μ
a . In this process, the observer coordinates x
a
ˆ
P
are
fixed, hence the consistency with their Lorentz index. Indeed, these coordinates can
only be transformed to the ones of some other observer at ˆ
P, by definition, since
we must maintain ˆ
g ab ≡ η ab . The allowed coordinate transformations are the ones
preserving (2.3.1), i.e. those that reduce to Poincaré around ˆ
P
˜
x
a
ˆ
P
= q
a
+
a
b x
b
ˆ
P
+ O((x ˆ
P − ˆ
x ˆ
P )
2
) .
(2.3.4)
The corresponding frame would then only be sensitive to the Lorentz transformation
(i.e. not the translation)
˜ ˆ
∂ a, ˆ
P =
b
a
ˆ
∂ b, ˆ
P ,
⇒
˜ ˆ
e a =
b
a ˆ
e b ,
(2.3.5)
and therefore reproduces the Lorentz transformation of the a index. Here we wish to
stress that the interpretation of the tetrad as a Jacobian of some coordinate transformation x
a
ˆ
P
(x) is relevant only in the case where it is used at a single point ˆ
P ∈ M.
This is because a tetrad at some other point ˆ
P
will correspond to the Jacobian of
some other transformation x
a
ˆ
P (x), i.e. the one trivializing the metric at ˆ
P
, not ˆ
P.
Indeed, if the tetrad were the Jacobian of a single coordinate transformation all over
M, then space-time would be flat, as one could perform that transformation to get
g μν → η μν everywhere.
Thus, in the presence of a tetrad field the underlying observer coordinates associated with each point lose their relevance and one only retains the necessary information for observations, i.e. the basis at each tangent space T P M. We therefore believe
that the “Jacobian matrix” interpretation of the tetrad loses its appeal in this context
and can even become misleading. We prefer the more gauge-theoretical viewpoint
where the Lorentz indices “a” simply correspond to some internal gauge symmetry,
just as in Yang-Mills theory, with no reference whatsoever to any particular coordinate system x
a
P . The e
μ
a (x) are therefore simply a set of four vector fields and we
privilege no coordinate system in describing their dynamics. The only aspects that
one can retain from the Jacobian matrix picture is that, for a given P ∈ M, there
always exists a coordinate system in which e
μ
a (P) = δ
μ
a .