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.
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
]:
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.
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