  We propose a covariant holographic construction based on the extremal
entanglement wedge cross section, for the entanglement negativity of bipartite
states in $CFT_{1+1}$s dual to non static bulk $AdS_3$ geometries. Utilizing
our proposal we obtain the holographic entanglement negativity for bipartite
mixed states in $CFT_{1+1}$s dual to bulk extremal and non extremal rotating
BTZ black holes and the time dependent Vaidya-$AdS_3$ geometries. Our results
correctly reproduce the replica technique results modulo certain constants for
the corresponding dual $CFT_{1+1}$s in the large central charge limit for the
bulk black hole geometries. Furthermore for all the cases our results also
match upto such constants with the corresponding results from an earlier
alternate covariant holographic construction in the literature. This
demonstrates the equivalence modulo constants of the two constructions for the
$AdS_3/CFT_2$ scenario.
