Time-dependent compactification to de Sitter space: a no-go theorem

It is known that the Einstein gravitational field equations in D > 4 spacetime dimensions have no time-independent non-singular compactification solutions to de Sitter space if the D-dimensional stress tensor satisfies the Strong Energy Condition (SEC). Here we show, by example, that the SEC alone does not exclude time-dependent non-singular compactifications to de Sitter space, in Einstein conformal frame. However, this possibility is excluded by the combined SEC and Dominant Energy Condition (DEC) because the DEC forces a time-evolution towards a singular D-metric.