3.2 Space-Time Fields
25
condition in field space, which therefore seems somewhat inelegant. In contrast, in the
tetrad formalism the metric (3.2.6) has Lorentzian signature whatever the eigenvalues
of e
a
μ , as long as they are all non-zero, so we now only need the condition e = 0 in
field space. The reason for this is that the signature information is now “hardwired”
through the choice of internal metric, or equivalently, of the internal group. Indeed,
it is the fact that we choose to contract the a indices with η ab , or equivalently, to act
on them with SO(1,3), which makes g μν Lorentzian. Had we chosen SO(4) as our
local group, the corresponding invariant metric would have been δ ab and would have
thus led to a g μν metric with Euclidean signature.
3.2.2 Connections and Curvatures
Given the local symmetries of the theory, i.e. MDs and LLTs, we introduce a gauge
field (or “connection”) for each one of them in order to form covariant derivatives.
For MDs we have the “affine connection”
ρ
μν . It is a scalar under LLTs, but under
a PMD it transforms as
˜
ρ
μν ( ˜
x) =
∂ ˜
x
ρ
∂x γ (x)
∂x
α
∂ ˜
x μ ( ˜
x(x))
∂x
β
∂ ˜
x ν ( ˜
x(x)) )
γ
αβ (x) +
∂ ˜
x
ρ
∂x γ (x)
∂
2 x
γ
∂ ˜
x μ ∂ ˜
x ν ( ˜
x(x)) ,
(3.2.12)
and under an AMD it varies by
δ ξ
ρ
μν = −∂ μ ∂ ν ξ
ρ
− L ξ
ρ
μν + O(ξ
2
)
(3.2.13)
≡ −∇ (μ
∇ ν) ξ
ρ
− T
ρ
ν)σ ξ
σ
+ R
ρ
(μν)σ ξ
σ
+
1
2
L ξ T
ρ
μν + O(ξ
2
), (3.2.14)
where ∇ is the covariant derivative with respect to MDs, e.g.
∇ μ X
ν
:= ∂ μ X
ν
+
ν
ρμ X
ρ
,
(3.2.15)
while
T
ρ
μν :=
ρ
νμ −
ρ
μν ,
R
ρ
σμν := ∂ μ
ρ
σν − ∂ ν
ρ
σμ +
ρ
αμ α
σν −
ρ
αν α
σμ ,
(3.2.16)
are the corresponding torsion and the curvature tensors. In Eq. (3.2.13), by “L ξ
ρ
μν ”
we mean the action of the Lie derivative L ξ on
ρ
μν as if it were a tensor of rank three
L ξ
ρ
μν := ξ
σ
∂ σ
ρ
μν −
σ
μν ∂ σ ξ
ρ
+
ρ
σν ∂ μ ξ
σ
+
ρ
μσ ∂ ν ξ
σ
.
(3.2.17)
Eq. (3.2.13) shows that the non-tensorial part of δ ξ
ρ
μν is simply −∂ μ ∂ ν ξ
ρ , i.e. the
linearization of the rightmost term in Eq. (3.2.12). On the other hand, Eq. (3.2.14)
has the advantage of being explicitly covariant to lowest order, as it should, since
25
condition in field space, which therefore seems somewhat inelegant. In contrast, in the
tetrad formalism the metric (3.2.6) has Lorentzian signature whatever the eigenvalues
of e
a
μ , as long as they are all non-zero, so we now only need the condition e = 0 in
field space. The reason for this is that the signature information is now “hardwired”
through the choice of internal metric, or equivalently, of the internal group. Indeed,
it is the fact that we choose to contract the a indices with η ab , or equivalently, to act
on them with SO(1,3), which makes g μν Lorentzian. Had we chosen SO(4) as our
local group, the corresponding invariant metric would have been δ ab and would have
thus led to a g μν metric with Euclidean signature.
3.2.2 Connections and Curvatures
Given the local symmetries of the theory, i.e. MDs and LLTs, we introduce a gauge
field (or “connection”) for each one of them in order to form covariant derivatives.
For MDs we have the “affine connection”
ρ
μν . It is a scalar under LLTs, but under
a PMD it transforms as
˜
ρ
μν ( ˜
x) =
∂ ˜
x
ρ
∂x γ (x)
∂x
α
∂ ˜
x μ ( ˜
x(x))
∂x
β
∂ ˜
x ν ( ˜
x(x)) )
γ
αβ (x) +
∂ ˜
x
ρ
∂x γ (x)
∂
2 x
γ
∂ ˜
x μ ∂ ˜
x ν ( ˜
x(x)) ,
(3.2.12)
and under an AMD it varies by
δ ξ
ρ
μν = −∂ μ ∂ ν ξ
ρ
− L ξ
ρ
μν + O(ξ
2
)
(3.2.13)
≡ −∇ (μ
∇ ν) ξ
ρ
− T
ρ
ν)σ ξ
σ
+ R
ρ
(μν)σ ξ
σ
+
1
2
L ξ T
ρ
μν + O(ξ
2
), (3.2.14)
where ∇ is the covariant derivative with respect to MDs, e.g.
∇ μ X
ν
:= ∂ μ X
ν
+
ν
ρμ X
ρ
,
(3.2.15)
while
T
ρ
μν :=
ρ
νμ −
ρ
μν ,
R
ρ
σμν := ∂ μ
ρ
σν − ∂ ν
ρ
σμ +
ρ
αμ α
σν −
ρ
αν α
σμ ,
(3.2.16)
are the corresponding torsion and the curvature tensors. In Eq. (3.2.13), by “L ξ
ρ
μν ”
we mean the action of the Lie derivative L ξ on
ρ
μν as if it were a tensor of rank three
L ξ
ρ
μν := ξ
σ
∂ σ
ρ
μν −
σ
μν ∂ σ ξ
ρ
+
ρ
σν ∂ μ ξ
σ
+
ρ
μσ ∂ ν ξ
σ
.
(3.2.17)
Eq. (3.2.13) shows that the non-tensorial part of δ ξ
ρ
μν is simply −∂ μ ∂ ν ξ
ρ , i.e. the
linearization of the rightmost term in Eq. (3.2.12). On the other hand, Eq. (3.2.14)
has the advantage of being explicitly covariant to lowest order, as it should, since