 The Einstein equations in vacuum in synchronous reference frame are
where formula_18 is the 3-dimensional tensor formula_19, and "P"αβ is the 3-dimensional [[Ricci tensor]], which is expressed by the 3-dimensional [[Metric tensor (general relativity)|metric tensor]] γαβ in the same way as "Rik" is expressed by "gik"; "P"αβ contains only the space (but not the time) derivatives of γαβ.Using triads, for one has simply

The components of "P"("a")("b") can be expressed in terms of the quantities η"ab" and the structure constants of the group by using the tetrad representation of the Ricci tensor in terms of quantities formula_21

After replacing the three-index symbols formula_23 by two-index symbols "C""ab" and the transformations