Gravity and Strings

1.4 Tangent space 17
The significance of the torsion in this formalism, from the point of view of the gauge
theory of GL(d, R), isunclear. We can provide an interpretation in the framework of the
gauge theory of the affine group IGL(d, R) butwewill do it in the more restricted context
of the Lorentz and Poincaré groups in Section 4.5.
The first Vielbein postulate has allowed us to recover the structure of affinely connected
spacetime (Ld, g) with only one (independent) connection, generalized to allow the use
of an arbitrary basis in tangent space. Furthermore, we can recover the different particular
structures that we defined in the previous section, also generalized to allow the use of
arbitrary basis in tangent space. First, if Ɣ is a completely general connection, it is given
by Eq. (1.43) and then ω (which is related to Ɣ by the first Vielbein postulate) is given by
ωab c = ωab c (e) + Kab c + Lab c , (1.87)
where ω(e) is the (Cartan or even Levi-Cività) connection related to the Levi-Cività connection
Ɣ(g) by Eq. (1.84). It is completely determined by the Vielbeins:
ωab c
c
(e) = +
ab
−ab c + b c a − c
ab , (1.88)
where
c
=
ab
1
2 gcd {∂agbd + ∂bgad − ∂dgab}. (1.89)
Kab c is nothing but the contorsion tensor expressed in a tangent-space basis, i.e. Kab c =
ea µ eb νec ρ Kµν ρ and, similarly, Lab c = ea µ eb νec ρ Lµν ρ .
Observe that
ωa(bc) = 1
2 (Qabc + ∂agbc). (1.90)
We can impose the metric-compatibility condition Eq. (1.49), which in this context is
known as the second Vielbein postulate, and we have a Riemann–Cartan spacetime Ud.
The result is that Ɣ is again given by Eqs. (1.50), (1.44), and (1.45) and ω (which is related
to Ɣ by the first Vielbein postulate) is given by
ωab c = ωab c (e) + Kab c . (1.91)
If we now impose the vanishing of torsion, we obtain the Levi-Cività and Cartan connections
Ɣ(g) and ω(e) and we recover a Riemann spacetime Vd.
The two most important cases to which we can apply this general formalism are the
following.
1. The case in which we use a coordinate basis ea µ = δa µ ,sogab = gµν, = 0, and the
connections Ɣ and ω are identical.
2. The case in which we use an orthonormal basis gab = ηab in which
and
c
= 0
ab
ωabc = ωabc(e) + Kabc + Labc, ωabc(e) =−abc + bca − cab. (1.92)