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