Gravity and Strings

1.4 Tangent space 19
For more general tensors one has to add a curvature (ω) term for each flat index and a curvature
(Ɣ) term for each world index. The curvatures have the same form as in Eqs. (1.27)
and (1.81) but now Rµν ab is antisymmetric in ab.
The following expression is sometimes used:
Rab =−∂aωc c b − ∂cωab c + ωcdaω dc b + ωabdωc cd . (1.95)
The Vielbein formalism allows us to study the Weitzenböck spacetime defined on
page 11.
1.4.1 Weitzenböck spacetime Ad
This spacetime is defined by a metric-compatible connection that we denote by Wµν ρ and
call the Weitzenböck connection [944, 945] whose Riemann curvature is identically zero,
Rµνρ σ (W ) = 0. Trying to solve this equation directly for W = 0isavery difficult task.
However, we can use the Vielbein formalism to find a solution. Let us denote by Ws µ ab the
tangent-space connection associated with W via the first Vielbein postulate
∇µe a ν = ∂µe a ν − Wµν a + Ws µν a = 0. (1.96)
The curvature of Ws is obviously zero on account of Eq. (1.85). Now, however, we can
use the trivial solution to the equation Rµν ab (Ws) = 0, namely Ws = 0, because, according
to the above relation, Ws = 0 does not imply W = 0but
Wµν ρ = ea ρ ∂µe a ν. (1.97)
This is the Weitzenböck connection whose curvature vanishes identically. It cannot be
rewritten in terms of the metric: it is necessary to use the Vielbein formalism. Observe
that, using this connection, we can write the relation between any two connections Ɣ and
ω satisfying the first Vielbein postulate in the form
Ɣµν ρ = Wµν ρ + ωµν ρ . (1.98)
ωµν ρ is a tensor, but Ɣµν ρ is not (it is an affine connection), and responsible for this is
the Weitzenböck connection Wµν ρ .Wecan also write
ωµ ab = Ɣµ ab − Wµ ab , Wµ ab = e aν ∂µe b ν. (1.99)
Now Ɣµ ab is a GL(d, R) tensor in the upper two indices whereas ωµ ab is not (because
it is a GL(d, R) connection). Again, the Weitzenböck connection Wµ ab is responsible for
this.
Even though we have to search explicitly for a metric-compatible connection to find W ,
it is easy to check that it is indeed metric-compatible. Then, it can be decomposed into the
sum of the Levi-Cività connection and the contorsion tensor. The torsion tensor is
and, therefore, the contorsion tensor is given by
Tµν ρ =−2µν ρ , (1.100)
Kµνρ(W ) = µνρ − νρµ + ρµν =−ωµνρ(e), (1.101)