  In order to study quark and anti-quark interaction, one should consider all
effects of the medium in motion of the pair. Because the pair, is not produced
at rest in QGP. So the velocity of the pair, has some effects on its
interactions that should be taken into account. In this paper we apply some
conformal transformations for a rotat- ing string dual to a rotating heavy
quark in $AdS_3$ which construct an accelerating string dual to an accelerating
quark and anti-quark pair. So, we can have a comparison between when pair has
angular velocity or not. Then we can study effects of angular velocity on the
accelerating quark and anti-quark which are constructed by performing special
con- formal transformations, conformal SO(2,2) transformation and particular
$SL(2,R)_L$ and $SL(2,R)_R$ transformation. The accelerating quark and
anti-quark show different behavior with increasing in angular velocity. With
useful numerical solutions we show that quark and anti-quark can deccelerate to
achieve each other or accelerate to get away from each other. We will see
variable behaviors of the pair in different transformations. There are various
behavior like, permanent behavior, increase or decrese in acceleration or we
will consider uncertain behavior.
