28
3 Mathematical Framework
independently of whether tetrad compatibility is imposed or not. One thus finds the
torsion-free spin connection
ab
μ = e
aν
∂ [μ e
b
ν] − e
bν
∂ [μ e
a
ν] − e cμ e
aν e
bρ
∂ [ν e
c
ρ] ,
(3.2.32)
which can also be expressed as a linear combination of the structure coefficients of
Eq. (3.2.5)
abμ ≡
1
2
[C abc − C bca − C cab ] e
c
μ .
(3.2.33)
With the condition of zero torsion R μνρσ becomes the Riemann tensor of g μν and,
given Eq. (3.2.30),
ab
μν is nothing but the Riemann tensor partially expressed in
the tetrad basis, so from now on we write R
ab
μν instead. From this relation one sees
trivially why the first pair of indices of the Riemann tensor is antisymmetric as well,
because in the tetrad viewpoint these index pair parametrizes the Lorentz algebra.
One can then define the usual curvature tensors with only Lorentz indices
R abcd := R abμν e
μ
c e ν
d ,
R ab := R c
acb ≡ e
μ
a e ν
b R μν [g] ,
R := R a
a ≡ R[g] ,
(3.2.34)
and also the Weyl tensor
W abcd := R abcd − η a[c R d]b + η b[c R d]a +
1
3
η a[c η d]b R ≡ e
μ
a e
ν
b e
ρ
c e
σ
d W μνρσ [g] .
(3.2.35)
Because the latter is fully traceless and shares all the symmetries of the Riemann
tensor, all of its information lies in two symmetric traceless spatial tensors, the socalled electric and magnetic components associated with the observer e
μ
a
E i j := W 0i0 j ,
B i j := −
1
2
ε ikl W 0 jkl ,
(3.2.36)
since we then have
W 0i jk = −B il ε l jk ,
W i jkl = δ ik E jl − δ il E jk − δ jk E il + δ jl E ik . (3.2.37)
As for the electromagnetic field A μ , it is a MD covector and LLT scalar, but varies
under a U(1) gauge transformation (U(1)GT)
˜
A μ = A μ + ∂ μ θ ,
(3.2.38)
so that the invariant curvature is the Maxwell tensor
F μν := ∂ μ A ν − ∂ ν A μ ,
F ab := e
μ
a e
ν
b F μν .
(3.2.39)
The electric and magnetic fields measured by the observer family e
μ
a are then
3 Mathematical Framework
independently of whether tetrad compatibility is imposed or not. One thus finds the
torsion-free spin connection
ab
μ = e
aν
∂ [μ e
b
ν] − e
bν
∂ [μ e
a
ν] − e cμ e
aν e
bρ
∂ [ν e
c
ρ] ,
(3.2.32)
which can also be expressed as a linear combination of the structure coefficients of
Eq. (3.2.5)
abμ ≡
1
2
[C abc − C bca − C cab ] e
c
μ .
(3.2.33)
With the condition of zero torsion R μνρσ becomes the Riemann tensor of g μν and,
given Eq. (3.2.30),
ab
μν is nothing but the Riemann tensor partially expressed in
the tetrad basis, so from now on we write R
ab
μν instead. From this relation one sees
trivially why the first pair of indices of the Riemann tensor is antisymmetric as well,
because in the tetrad viewpoint these index pair parametrizes the Lorentz algebra.
One can then define the usual curvature tensors with only Lorentz indices
R abcd := R abμν e
μ
c e ν
d ,
R ab := R c
acb ≡ e
μ
a e ν
b R μν [g] ,
R := R a
a ≡ R[g] ,
(3.2.34)
and also the Weyl tensor
W abcd := R abcd − η a[c R d]b + η b[c R d]a +
1
3
η a[c η d]b R ≡ e
μ
a e
ν
b e
ρ
c e
σ
d W μνρσ [g] .
(3.2.35)
Because the latter is fully traceless and shares all the symmetries of the Riemann
tensor, all of its information lies in two symmetric traceless spatial tensors, the socalled electric and magnetic components associated with the observer e
μ
a
E i j := W 0i0 j ,
B i j := −
1
2
ε ikl W 0 jkl ,
(3.2.36)
since we then have
W 0i jk = −B il ε l jk ,
W i jkl = δ ik E jl − δ il E jk − δ jk E il + δ jl E ik . (3.2.37)
As for the electromagnetic field A μ , it is a MD covector and LLT scalar, but varies
under a U(1) gauge transformation (U(1)GT)
˜
A μ = A μ + ∂ μ θ ,
(3.2.38)
so that the invariant curvature is the Maxwell tensor
F μν := ∂ μ A ν − ∂ ν A μ ,
F ab := e
μ
a e
ν
b F μν .
(3.2.39)
The electric and magnetic fields measured by the observer family e
μ
a are then