Let $u_{X}^{t}$ be a unipotent flow on $X=\mathrm {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’s shearing argument. The method also extends a recent work of Dong, Kanigowski, and Wei [Rigidity of joinings for some measure preserving systems. Ergod. Th. & Dynam. Sys. 42 (2022), 665–690].

We introduce two properties: strong R-property and $C(q)$ -property, describing a special way of divergence of nearby trajectories for an abstract measure-preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$ -property. Moreover, we show that if $u_t$ is a unipotent flow on $G/\Gamma $ with $\Gamma $ irreducible, then $u_t$ satisfies the $C(q)$ -property provided that $u_t$ is not of the form $h_t\times \operatorname {id}$ , where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded-type Heisenberg nilflows.