On the Stability of Tolman Oppenheimer Volkoff Fireholes

We investigate the radial stability of singular Tolman--Oppenheimer--Volkoff fireholes with a radiation equation of state. These configurations contain a naked, negative point-mass singularity surrounded by a positive-density fluid, whose mass profile develops a Schwarzschild-like plateau before approaching an extended, self-similar atmosphere. Using the Prüfer phase method to identify the fundamental radial mode, we consider two outer boundary conditions: vanishing Lagrangian pressure perturbation (open firehole) and vanishing radial displacement (boxed firehole). For both, the fundamental squared frequency changes from positive to negative as the outer boundary exceeds a critical radius R c. We find that R c follows approximately the same scaling with the central mass parameter as the equilibrium firehole radius R. However, the stability thresholds differ significantly: R c O <R for the open firehole, whereas R c B R for the boxed one. The regular zero mode is the derivative of the equilibrium family with respect to the inner lapse normalization at fixed central mass. This gives the boxed threshold a turning-point interpretation and, together with a scaling symmetry of the TOV equations, implies R c B to R in the limit of large central mass. Variational bounds show that the rigid wall is the most stabilizing separated, self-adjoint outer boundary condition and that the unbounded configuration is unstable. Tolman--Oppenheimer--Volkoff fireholes are therefore radially unstable unless confined within approximately their firehole radius.