Gravity and Strings

4.4 The Cartan–Sciama–Kibble theory 133
Now we couple the pure gravity Lagrangian to the matter Lagrangian and use the definition
of the Vielbein energy–momentum tensor Eq. (4.79) and the following definition of
the spin–energy potential, which generalizes Eq. (2.88),
matter µν a =− 2c
e
to obtain the equations of the CSK theory:
G (αβ) − ∗ ∗
∇µ
δSmatter
T µ(αβ) =
2 Tmatter αβ ,
∗
1
2 T γ αβ− ∗
T γ βα− ∗
T αβ
γ = χ 2
2 matter αβ γ .
, (4.109)
δTµν
a
χ 2
(4.110)
We have taken into account in the l.h.s. of the first equation that only the symmetric part
contributes to it, even though the r.h.s. (the Vielbein energy–momentum tensor) is not symmetric
in general (we have seen that the antisymmetric part vanishes on-shell).
These equations can be rewritten in a more suggestive form: taking the modified divergence
of the second equation, we find the equation
∗ ∗
∇µ
T µ(αβ) − 1
2
∗ ∗
∇µ
T αβµ =
χ 2
2
∗
∇µmatter µαβ , (4.111)
which, when subtracted from the first equation (4.110), gives a more elegant equation,
G αβ =
χ 2
2 Tcan αβ , (4.112)
where we have used Eq. (1.34) and have defined the canonical energy–momentum tensor
here by
Tcal βα = Tmatter αβ − ∗
∇µ matter µαβ . (4.113)
This identification is evidently based on the definition of the Belinfante tensor, but we will
prove that this tensor is indeed given by Eq. (4.84).
The second Eq. (4.110) can be simplified by raising the index γ and antisymmetrizing it
with β:
∗
T αβγ = χ 2 S γβα . (4.114)
Now we can use this equation to rewrite the Vielbein equation (the first of Eqs. (4.110))
in a general-relativistic form. First, we take the symmetric part of the equation that relates
the Einstein tensor of the torsionful connection Ɣ to the Einstein tensor of the Levi-Cività
connection Ɣ, which is
Gαβ(Ɣ) = Gαβ[Ɣ(g)] − 1
∗ ∗
2 ∇µ T α µ β + ∗
T β µ α − ∗
T αβ µ
− f (T 2 ), (4.115)
where f (T 2 ) is a complicated expression that is quadratic in the torsion whose explicit