Bouncing Dust Collapse and Black-to-White Hole Transition on the Brane

Abstract

We study the spherical collapse of a homogeneous dust cloud on a three-brane in the Shtanov--Sahni braneworld, in which the extra dimension is timelike. In the high-energy regime the brane Friedmann equation acquires a negative quadratic term in the matter density, controlled by a critical density $\rhoc=2λ$, where $λ$ is the magnitude of the negative brane tension. For a spatially closed interior we integrate the dynamics exactly. Every turning point of the scale factor lies above $\astar=\sqrt{3/4πλ}$, a bounce exists if and only if a simple condition on the comoving density holds, and all curvature invariants stay below bounds fixed by $λ$ alone along the entire cycle. Comoving worldlines are geodesically complete. We then match the cloud across a comoving boundary to a spherically symmetric exterior with a trace-free effective source and solve the Israel conditions in closed form. The exterior is a charged Vaidya geometry whose mass function and tidal charge are fixed by the boundary radius. The tidal charge is positive, opposite in sign to the Randall--Sundrum case, and the exterior cannot be static. A trapped region forms on the boundary if and only if the mass exceeds $2/(3\sqrt{πλ})$, and in that case the bounce still takes place in an untrapped neighbourhood of the boundary. For trapped clouds we construct an exterior over the full contraction, bounce and re-expansion by joining an advanced and a retarded charged Vaidya patch through a static wedge. The matched metric is of class $C^{1,1}$, the effective energy conditions hold, and the cloud re-emerges through a white hole region into a second asymptotic region. In the general relativistic limit $λ\to\infty$ the Oppenheimer--Snyder solution is recovered.