Gravity and Strings

T = 0
1.3 Metric spaces 11
Affinely Connected Metric Spacetime
Riemann–Cartan Spacetime
U d
Riemann Spacetime Weitzenbock Spacetime
V d
R = 0
(L ,g)
d
Minkowski Spacetime
M d
Fig. 1.1. Particular structures in an affinely connected spacetime equipped with a metric
(Ld, g).
There is another way of reducing the number of independent fields: by imposing
the vanishing of the curvature tensor. In this case, both the metric and the connection
are completely determined by the Vielbein (to be defined latter). The connection
is called Weitzenböck connection [944, 945] and has torsion (also determined by the
Vielbein). A Riemann–Cartan spacetime with Weitzenböck connection is a Weitzenböck
spacetime Ad.
If both torsion and curvature vanish, the space has to be Minkowski spacetime Md since
the Minkowski metric gµν = ηµν is the only one that makes the full Riemann tensor vanish
in the absence of torsion.
The diagram in Figure 1.1 summarizes the different particular structures that we can have
on an affinely connected manifold equipped with a metric [522, 523].
In the rest of this section we are going to study the particular properties of some of these
spacetimes. The Weitzenböck spacetime will be studied after the introduction of Vielbeins
in Section 1.4.
1.3.1 Riemann–Cartan spacetime Ud
As has been said, this is an affinely connected metric spacetime with a metric-compatible
connection, so the non-metricity tensor vanishes, Qµνρ = 0. According to the general result,
Q = 0
R = 0
T = 0
A d