3.2 Space-Time Fields
27
ρ
μν =
1
2
g
ρσ
∂ μ g νσ + ∂ ν g μσ − ∂ σ g μν
.
(3.2.24)
In tetrad GR, one can proceed in an analogous fashion. We start by requiring that
the connections are tetrad-compatible, i.e. the fully covariant equation (also known
as “the tetrad postulate”)
∇ μ e
a
ν ≡ ∂ μ e
a
ν − e
a
ρ
ρ
νμ +
a
bμ e
b
ν = 0 .
(3.2.25)
In particular, this implies that we can switch index types through the covariant derivative, e.g.
∇ μ X
a
≡ ∇ μ
e
a
ν X
ν
= e
a
ν ∇ μ X
ν
.
(3.2.26)
By contracting Eq. (3.2.25) with tetrads we derive a relation between the two connections, i.e. the affine connection is
μ
νρ = e
μ
a ∇
ρ e
a
ν ,
(3.2.27)
while the spin connection is
a
bμ = e
a
ν ∇
μ e
ν
b .
(3.2.28)
Note that both sides of (3.2.27) are consistently covariant under LLTs, but not MDs,
while both sides of (3.2.28) are consistently covariant under MDs, but not LLTs. In
particular, the last equation shows that now the spin connection basically amounts
to the information of the parallel transport of the tetrad vectors along themselves.
Now since the equation of tetrad compatibility (3.2.25) relates uniquely the two
connections and , it also relates uniquely the corresponding torsion and curvature tensors defined in Eqs. (3.2.16) and (3.2.21), respectively. Indeed, taking the
antisymmetric part of Eq. (3.2.27) we find that the affine and spin torsions are the
same, but just expressed in different bases
T
ρ
μν = e
ρ
a
a
μν .
(3.2.29)
On the other hand, plugging Eq. (3.2.28) in Eq. (3.2.21) one finds
ab
μν = e
a
ρ e
bσ R
ρ
σμν .
(3.2.30)
In analogy with the metric case, further demanding zero torsion fully determines the
connections in terms of the tetrad. The affine connection
ρ
μν is again the Christoffel
symbols of the metric, because we have the same conditions
∇ ρ g μν ≡ e aμ ∇ ρ e
a
ν + e aν ∇ ρ e
a
μ = 0 ,
T
ρ
μν = 0 .
(3.2.31)
As for the spin connection, the condition
a
μν = 0 alone fully determines it in terms
of e
a
μ , because these are as many equations as the number of components in
ab
μ , i.e.
27
ρ
μν =
1
2
g
ρσ
∂ μ g νσ + ∂ ν g μσ − ∂ σ g μν
.
(3.2.24)
In tetrad GR, one can proceed in an analogous fashion. We start by requiring that
the connections are tetrad-compatible, i.e. the fully covariant equation (also known
as “the tetrad postulate”)
∇ μ e
a
ν ≡ ∂ μ e
a
ν − e
a
ρ
ρ
νμ +
a
bμ e
b
ν = 0 .
(3.2.25)
In particular, this implies that we can switch index types through the covariant derivative, e.g.
∇ μ X
a
≡ ∇ μ
e
a
ν X
ν
= e
a
ν ∇ μ X
ν
.
(3.2.26)
By contracting Eq. (3.2.25) with tetrads we derive a relation between the two connections, i.e. the affine connection is
μ
νρ = e
μ
a ∇
ρ e
a
ν ,
(3.2.27)
while the spin connection is
a
bμ = e
a
ν ∇
μ e
ν
b .
(3.2.28)
Note that both sides of (3.2.27) are consistently covariant under LLTs, but not MDs,
while both sides of (3.2.28) are consistently covariant under MDs, but not LLTs. In
particular, the last equation shows that now the spin connection basically amounts
to the information of the parallel transport of the tetrad vectors along themselves.
Now since the equation of tetrad compatibility (3.2.25) relates uniquely the two
connections and , it also relates uniquely the corresponding torsion and curvature tensors defined in Eqs. (3.2.16) and (3.2.21), respectively. Indeed, taking the
antisymmetric part of Eq. (3.2.27) we find that the affine and spin torsions are the
same, but just expressed in different bases
T
ρ
μν = e
ρ
a
a
μν .
(3.2.29)
On the other hand, plugging Eq. (3.2.28) in Eq. (3.2.21) one finds
ab
μν = e
a
ρ e
bσ R
ρ
σμν .
(3.2.30)
In analogy with the metric case, further demanding zero torsion fully determines the
connections in terms of the tetrad. The affine connection
ρ
μν is again the Christoffel
symbols of the metric, because we have the same conditions
∇ ρ g μν ≡ e aμ ∇ ρ e
a
ν + e aν ∇ ρ e
a
μ = 0 ,
T
ρ
μν = 0 .
(3.2.31)
As for the spin connection, the condition
a
μν = 0 alone fully determines it in terms
of e
a
μ , because these are as many equations as the number of components in
ab
μ , i.e.