Rigidity of joinings for time-changes of unipotent flows on quotients of Lorentz groups

Let $u_{X}^{t}$ be a unipotent flow on $X=SO(n,1)/\Gamma$, $u_{Y}^{t}$ be a unipotent flow on $Y=G/\Gamma^{\prime}$. Let $\tilde{u}_{X}^{t}$, $\tilde{u}_{Y}^{t}$ be time-changes of $u_{X}^{t}$, $u_{Y}^{t}$ respectively. We show the disjointness (in the sense of Furstenberg) between $u_{X}^{t}$ and $\tilde{u}_{Y}^{t}$ (or $\tilde{u}_{X}^{t}$ and $u_{Y}^{t}$) in certain situations. Our method refines the works of Ratner and extends a recent work of Dong, Kanigowski and Wei.