### 10.1 Bianchi I models and eleven-dimensional supergravity

On the supergravity side, we will restrict the metric and the electromagnetic field to depend on time only,
Recall from Section 9.3 that with these ansätze the dynamical equations of motion of eleven-dimensional supergravity reduce to [61]
This corresponds to the truncation of the sigma model at level 2 which, as we have seen, completely matches the supergravity side. We also defined as in Section 9.3. Furthermore we have the following constraints,
which are, respectively, the Hamiltonian constraint, momentum constraint and Gauss’ law. Note that Greek indices correspond to the full eleven-dimensional spacetime, while Latin indices correspond to the ten-dimensional spatial part.

We will further take the metric to be purely time-dependent and diagonal,

This form of the metric has manifest invariance under the ten distinct spatial reflections
and in order to ensure compatibility with the Einstein equations, the energy-momentum tensor of the 4-form field strength must also be diagonal.

#### 10.1.1 Diagonal metrics and geometric configurations

Assuming zero magnetic field (this restriction will be lifted below), one way to achieve diagonality of the energy-momentum tensor is to assume that the non-vanishing components of the electric field are determined by geometric configurations with  [61].

A geometric configuration is a set of points and lines with the following incidence rules [117105145]:

1. Each line contains three points.
2. Each point is on lines.
3. Two points determine at most one line.

It follows that two lines have at most one point in common. It is an easy exercise to verify that . An interesting question is whether the lines can actually be realized as straight lines in the (real) plane, but, for our purposes, it is not necessary that it should be so; the lines can be bent.

Let be a geometric configuration with points. We number the points of the configuration . We associate to this geometric configuration a pattern of electric field components with the following property: can be non-zero only if the triple is a line of the geometric configuration. If it is not, we take . It is clear that this property is preserved in time by the equations of motion (in the absence of magnetic field). Furthermore, because of Rule 3 above, the products vanish when so that the energy-momentum tensor is diagonal.