On the Evolution of Relativistic Dust in Schwarzschild–de Sitter and Schwarzschild–Anti‐de Sitter Spacetimes. Part I: The Vanishing Mass Case

ABSTRACT Given that relativistic density and velocity propagate at the same speed, the Cauchy problem with smooth and compactly supported initial data for relativistic dust in Schwarzschild–de Sitter (SdS) and Schwarzschild–anti‐de Sitter (SAdS) spacetimes without black hole has been investigated using the characteristic method. Based on the explicit solution formulas under the spherically symmetric assumption, a precise classification of the initial data is provided, elucidating whether the classical solution for relativistic dust will persist globally or encounter a finite‐time blowup. Moreover, the paper offers a further analysis of the exact blowup phenomenon, including detailed insights into the blowup rate related to the blowup time.