From vacuum field equations on principal bundles to Einstein equations with fluids

In the present work we show that the Einstein equations on $M$ without cosmological constant and with perfect fluid as source, can be obtained from the field equations for vacuum with cosmological constant on the principal fibre bundle $P(\frac{1}{I}M,U(1))$, $M$ being the space-time and $I$ the radius of the internal space $U(1)$.