3.2 Space-Time Fields
29
E
i
:= F
0i
,
B
i
:=
1
2
ε
i jk F
jk
,
(3.2.40)
respectively. Finally, it is also useful to define
abc := abμ e
μ
c , A a := e
μ
a A μ , ∂ a := e
μ
a ∂ μ , ∇ a := e
μ
a ∇ μ ,
(3.2.41)
where the abc are also known as the “Ricci rotation coefficients”. Rearranging
(3.2.28) we find
∇
a e
μ
b ≡
c
ba e
μ
c ,
(3.2.42)
so these coefficients control the parallel transport of the tetrad vectors along themselves. In particular, since e
μ
0 is the 4-velocity, the 4-acceleration vector a
μ
:= ∇
0 e
μ
0
in the tetrad basis is nothing but
a
0
≡ 0 ,
a
i
≡
i
00 .
(3.2.43)
We can also compute the Fermi-Walker derivative of the spatial frame along the
4-velocity field
∇
FW
0 e
μ
i := ∇
0 e
μ
i − g (e i , a) e
μ
0 + g (e i , e 0 ) a
μ
≡ − i j0 e
μ
j − i00 e
μ
0 . (3.2.44)
We see that the time components of the spin connection i00 and i j0 control the
4-acceleration of the observer family and the precession of its spatial frame, respectively.
To conclude this subsection, we stress again that, although the tetrad has both
types of indices, it only requires the notion of spin connection to form fully covariant objects. This is because the condition of zero spin torsion
a
μν = 0 amounts to
as many equations as the number of components in
ab
μ and thus determines the
latter uniquely in terms of the tetrad (Eq. (3.2.32)). One can then form full scalars
through the curvature of
ab
μ . The tetrad-compatibility condition ∇ μ e
a
ν = 0 is therefore superfluous and can be alternatively understood as a way of constructing an
affine connection
ρ
μν out of the tetrad, i.e. Eq. (3.2.27), if one wants to introduce
one. As for the matter sector, note that there too
ρ
μν is not required because all
derivatives can be expressed as antisymmetric (“exterior”) derivatives of differential forms (at least in the Standard Model). On the other hand, a tetrad and a spin
connection are required in order to couple to spinors, as we will see in Sect. 3.2.4.
3.2.3 Tetrad General Relativity
The action we will consider is made of three terms
S = S EH + S EM + S m ,
(3.2.45)
29
E
i
:= F
0i
,
B
i
:=
1
2
ε
i jk F
jk
,
(3.2.40)
respectively. Finally, it is also useful to define
abc := abμ e
μ
c , A a := e
μ
a A μ , ∂ a := e
μ
a ∂ μ , ∇ a := e
μ
a ∇ μ ,
(3.2.41)
where the abc are also known as the “Ricci rotation coefficients”. Rearranging
(3.2.28) we find
∇
a e
μ
b ≡
c
ba e
μ
c ,
(3.2.42)
so these coefficients control the parallel transport of the tetrad vectors along themselves. In particular, since e
μ
0 is the 4-velocity, the 4-acceleration vector a
μ
:= ∇
0 e
μ
0
in the tetrad basis is nothing but
a
0
≡ 0 ,
a
i
≡
i
00 .
(3.2.43)
We can also compute the Fermi-Walker derivative of the spatial frame along the
4-velocity field
∇
FW
0 e
μ
i := ∇
0 e
μ
i − g (e i , a) e
μ
0 + g (e i , e 0 ) a
μ
≡ − i j0 e
μ
j − i00 e
μ
0 . (3.2.44)
We see that the time components of the spin connection i00 and i j0 control the
4-acceleration of the observer family and the precession of its spatial frame, respectively.
To conclude this subsection, we stress again that, although the tetrad has both
types of indices, it only requires the notion of spin connection to form fully covariant objects. This is because the condition of zero spin torsion
a
μν = 0 amounts to
as many equations as the number of components in
ab
μ and thus determines the
latter uniquely in terms of the tetrad (Eq. (3.2.32)). One can then form full scalars
through the curvature of
ab
μ . The tetrad-compatibility condition ∇ μ e
a
ν = 0 is therefore superfluous and can be alternatively understood as a way of constructing an
affine connection
ρ
μν out of the tetrad, i.e. Eq. (3.2.27), if one wants to introduce
one. As for the matter sector, note that there too
ρ
μν is not required because all
derivatives can be expressed as antisymmetric (“exterior”) derivatives of differential forms (at least in the Standard Model). On the other hand, a tetrad and a spin
connection are required in order to couple to spinors, as we will see in Sect. 3.2.4.
3.2.3 Tetrad General Relativity
The action we will consider is made of three terms
S = S EH + S EM + S m ,
(3.2.45)