  This paper aims to study the $W$-curvature tensor on relativistic
space-times. The energy-momentum tensor T of a space-time is semi-symmetric
given that the $W$-curvature tensor is semi-symmetric whereas energy-momentum
tensor T of a space-time having a divergence free $W$-curvature tensor is of
Codazzi type. A space-time having a traceless $W$-curvature tensor is Einstein.
A $W$-curvature flat space-time is Einstein. Perfect fluid space-times which
admits $W$-curvature tensor are considered.
