Class Z Reference Frames and Rate Synchronizability near the Horizon of Kerr-(A)dS Black Holes

We investigate the behavior of clock synchronization and class Z (rate-synchronizable) reference frames near the horizon of Kerr-de Sitter (Kerr-dS) and Kerr-anti-de Sitter (Kerr-AdS) black holes. Landau's time-orthogonality criterion \(g_{0i}=0\) is required to establish global simultaneity surfaces; Zhao subsequently introduced the weaker condition of ``transitivity of clock rate synchronization,'' \(\partial (g_{0i}/g_{00}) / \partial t = 0\), which was later formalized by Gao et al. as the geometric framework of class Z reference frames. While this reduced criterion has been successfully applied to the Kerr and Kerr-Newman black holes in asymptotically flat backgrounds, its extension to spacetimes with a non-zero cosmological constant has remained unexplored. In this paper, we restrict the class Z criterion to a two-dimensional spherical surface and apply it to the Kerr-(A)dS metric in Boyer-Lindquist coordinates. Extracting the metric components \(g_{tt}\) and \(g_{t\phi}\), we substitute them into the integrability condition \(\partial_\theta (g_{t\phi}/g_{tt}) = 0\) on the thin shell \(r = r_+ + \varepsilon\), and take the limit \(\varepsilon \to 0\) approaching the event horizon. Using the horizon condition \(\Delta_r(r_+) = 0\) to eliminate the mass parameter \(M\), we rigorously demonstrate that the condition is fully satisfied in the limit. The result shows that the ratio \(g_{t\phi}/g_{tt}\) becomes precisely independent of the angular coordinate \(\theta\) on the horizon. This conclusion holds for both Kerr-dS (\(\Lambda > 0\)) and Kerr-AdS (\(\Lambda < 0\)), independent of the sign of the cosmological constant. Although a global three-dimensional simultaneity surface cannot be established, a well-defined two-dimensional rate-synchronizable surface exists on the infinitely thin layer approaching the event horizon. This work provides the necessary geometric support for applying thin-layer quantization schemes in these non-asymptotically flat rotating spacetimes.