### 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 [117, 105, 145]:

- Each line contains three points.
- Each point is on lines.
- 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.