Gravity and Strings

12 Differential geometry
a metric-compatible connection of a Riemann–Cartan spacetime always has the form
Ɣµν ρ =
ρ
µν
+ Kµν ρ . (1.50)
Observe that the symmetric part of the contorsion tension does not vanish, but
Tµρν + Tνρµ = 0. (1.51)
K(µν) ρ = 1
2
This means that the presence of torsion implies not only that the connection has a nonvanishing
antisymmetric part, but also that the symmetric part is not fully determined by
the metric but
Ɣ(µν) ρ
ρ
= + K(µν)
µν
ρ
ρ
= . (1.52)
µν
The curvature, Ricci, and Einstein tensors of a metric-compatible connection satisfy further
identities. On contracting the γ and σ indices in the third Bianchi identity Eqs. (1.30)
and using the metric postulate, we find the so-called contracted Bianchi identity
∇αGµ α + 2Tµαβ R βα − Tαβγ Rµ γαβ = 0. (1.53)
Furthermore, by applying the Ricci identity to the metric and using the metric postulate,
one can prove a fourth Bianchi identity:
Rαβ(γ δ) = 0. (1.54)
If we modify the connection according to Eq. (1.31) and Ɣ is metric-compatible, the
Ricci scalar is
R( ˜Ɣ) = R(Ɣ) − Tµν ρ τρ µν + 2∇µτν µν + τµ µλ τν ν λ + τν µρ τµρ ν . (1.55)
If ˜Ɣ is not a metric-compatible connection, then τ contains all the contributions of
the non-metricity tensor and the above formula allows us to work in the framework of a
Riemann–Cartan spacetime with non-metric-compatible connections.
If both ˜Ɣ and Ɣ are metric-compatible connections and ˜Ɣ has torsion but Ɣ = Ɣ(g), then
τ = ˜K , the contorsion tensor of ˜Ɣ, and the above formula takes a simpler form:
R( ˜Ɣ) = R[Ɣ(g)] + 2∇µ ˜Kν µν + ( ˜Kµ µλ ) 2 + ˜Kν µρ ˜Kµρ ν . (1.56)
Now, this formula allows us to work with torsion in a Riemann spacetime. Particularly
interesting is the case in which the contorsion ˜Kµνρ is a completely antisymmetric tensor
(proportional to the Kalb–Ramond field strength Hµνρ, for instance). Then we have, if
d d x
|g| R( ˜Ɣ) =
˜Kµνρ = 1
√ 12 Hµνρ, (1.57)
d d x
|g| R[Ɣ(g)] + 1
2 · 3! Hµνρ H µνρ
. (1.58)