Correspondence between multileaf topology of closed geodesics and spatiotemporal autocorrelations of hotspot images in Schwarzschild spacetime

Abstract: The classification of relativistic closed orbits by their multileaf structures provides a framework for studying strong-field dynamics. Identifying these structures in astronomical images remains challenging. Using ray tracing, we construct the spatiotemporal autocorrelations of the primary image of a pointlike hotspot moving along bound closed geodesics around a Schwarzschild black hole. These orbits are classified by three integers $(z,w,v)$, where $z$ counts the leaves, $w$ counts the additional whirls during each radial period, and $v$ specifies the order in which the orbital leaves are traced. For the orbit families examined, our numerical results establish a correspondence between the topological integers $(z,w,v)$ and the numbers of correlation bands $N_{\rm band}$ and recurrence points $N_{\rm rec}$. The integers are recovered as $z=N_{\rm rec}-1$, $$w=\left\lfloor (N_{\rm band}+1)/(N_{\rm rec}-1)\right\rfloor-1$$, and $v=(N_{\rm band}+1)\bmod(N_{\rm rec}-1)$. Here, $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$, and $\bmod$ denotes the remainder operation. These relations provide a quantitative method for recovering closed-orbit topology from hotspot image autocorrelations.