36
3 Mathematical Framework
which, according to Eq. (3.2.44), amounts to choosing non-precessing frames. Along
with Eq. (3.2.86), we thus derive
∇
0 e
μ
a = 0 ,
⇔ ab0 = 0 ,
(3.2.87)
and we can therefore refer to it as the “parallel-transported tetrad gauge” (PTT). The
fact that we can express this gauge in terms of the spin connection is convenient,
because we directly see that it corresponds to exactly as many conditions as the
number of dimensions of the Lorentz group (six). This, however, does not correspond
to a complete gauge fixing. Indeed, the condition (3.2.87) is similar to the Weyl gauge
of electrodynamics, where the time component of the vector potential A μ is set to
zero
A t = 0 .
(3.2.88)
The subtlety here is that we actually have the Lorentz time-component ab0 instead
of the coordinate one abt . In the Weyl gauge of electrodynamics one has a residual
gauge symmetry that are the time-independent transformations, i.e. Eq. (3.2.38) with
θ = θ( x). Here, using the transformation of e
a
μ and
ab
μ under a LLT, we find that
the condition (3.2.87) is maintained by a further LLT if the latter satisfies
c
0 ∂ c
ab
+
i
0
a
c
bc
i = 0 .
(3.2.89)
To get more insight into this equation we can express the Lorentz matrix in terms of
the generators = e
−θ and consider the transformation to linear order in θ ab
∂ 0 θ ab − θ 0i abi + O(θ
2
) = 0 .
(3.2.90)
Since e
μ
0 is time-like, ∂ 0 ≡ e
μ
0 ∂ μ takes the form of a convective derivative, up to
some multiplicative factor, so we know that this type of equation admits solutions,
at least locally. We thus have a residual gauge symmetry of the same kind as in
electrodynamics.
We can now understand this situation as follows. The fact that the PTT gauge
(3.2.87) involves only a time-like derivation of the tetrad means that it only determines
its evolution in time. The residual LLTs then correspond to the freedom to choose the
frame arbitrarily on some time-like hypersurface, say the initial data surface. Once
this is done, the full tetrad field is uniquely determined by Eq. (3.2.87). In the case
of boosts, the residual gauge freedom amounts to the freedom of choosing among all
possible free-falling observer 4-velocity fields, while for rotations, it amounts to the
freedom of choosing among all possible parallel-transported spatial frames along e
μ
0 .
Although the PTT gauge is physically suitable in many cases, here we will not fix
the LLT symmetry to maintain generality.
3 Mathematical Framework
which, according to Eq. (3.2.44), amounts to choosing non-precessing frames. Along
with Eq. (3.2.86), we thus derive
∇
0 e
μ
a = 0 ,
⇔ ab0 = 0 ,
(3.2.87)
and we can therefore refer to it as the “parallel-transported tetrad gauge” (PTT). The
fact that we can express this gauge in terms of the spin connection is convenient,
because we directly see that it corresponds to exactly as many conditions as the
number of dimensions of the Lorentz group (six). This, however, does not correspond
to a complete gauge fixing. Indeed, the condition (3.2.87) is similar to the Weyl gauge
of electrodynamics, where the time component of the vector potential A μ is set to
zero
A t = 0 .
(3.2.88)
The subtlety here is that we actually have the Lorentz time-component ab0 instead
of the coordinate one abt . In the Weyl gauge of electrodynamics one has a residual
gauge symmetry that are the time-independent transformations, i.e. Eq. (3.2.38) with
θ = θ( x). Here, using the transformation of e
a
μ and
ab
μ under a LLT, we find that
the condition (3.2.87) is maintained by a further LLT if the latter satisfies
c
0 ∂ c
ab
+
i
0
a
c
bc
i = 0 .
(3.2.89)
To get more insight into this equation we can express the Lorentz matrix in terms of
the generators = e
−θ and consider the transformation to linear order in θ ab
∂ 0 θ ab − θ 0i abi + O(θ
2
) = 0 .
(3.2.90)
Since e
μ
0 is time-like, ∂ 0 ≡ e
μ
0 ∂ μ takes the form of a convective derivative, up to
some multiplicative factor, so we know that this type of equation admits solutions,
at least locally. We thus have a residual gauge symmetry of the same kind as in
electrodynamics.
We can now understand this situation as follows. The fact that the PTT gauge
(3.2.87) involves only a time-like derivation of the tetrad means that it only determines
its evolution in time. The residual LLTs then correspond to the freedom to choose the
frame arbitrarily on some time-like hypersurface, say the initial data surface. Once
this is done, the full tetrad field is uniquely determined by Eq. (3.2.87). In the case
of boosts, the residual gauge freedom amounts to the freedom of choosing among all
possible free-falling observer 4-velocity fields, while for rotations, it amounts to the
freedom of choosing among all possible parallel-transported spatial frames along e
μ
0 .
Although the PTT gauge is physically suitable in many cases, here we will not fix
the LLT symmetry to maintain generality.