  By boosting the vertex operators of Witten's $SL(2,R)/U(1)$ black hole, we
show that in the region V they lead to the primary fields of $c=1$ matter
coupled to gravity at nonzero cosmological constant, while there is no such
correspondence in the region I, showing that Witten's black hole corresponds to
$2d$ gravity only in a certain region and in a specific limit. By using the
free field representation , we will show that the stress tensor of $SL(2,R)$
gauged by its nilpotent subgroup is equivalent to that of the Liouville theory
with zero cosmological constant.
