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We discuss the possibility of a dimensional reduction of the Einsteinequations in S3 black-hole lattices. It was reported in previous literaturethat the evolution of spaces containing curves of local, discrete rotationaland reflection Symmetry (LDRRS) can be carried out via a system of ODEs alongthese curves. However, 3+1 Numerical Relativity computations demonstrate thatthis is not the case, and we show analytically that this is due to the presenceof a tensorial quantity which is not suppressed by the symmetry. We calculatethe term analytically, and verify numerically for an 8-black-hole lattice thatit fully accounts for the anomalous results, and thus quantify its magnitude inthis specific case. The presence of this term prevents the exact evolution ofthese spaces via previously-reported methods which do not involve a full 3+1integration of Einstein's equation.

On the vacuum Einstein equations along curves with a discrete local rotational and reflection symmetry