Gauss law constraint in gauge theory ($D_i{E}_i=\rho$)

The time component of a gauge field's equation of motion constrains the spatial fields and their momenta. In a Yang-Mills theory it has the covariant form $D_i{E}_i=\rho$; its Abelian form is Gauss's law. Time-dependent collective coordinates of a gauge-theory soliton require a temporal gauge field, or an equivalent horizontal projection, that satisfies this constraint before the kinetic metric is read from the action.