Relativistic models of structure formation with stable end-state configuration

TL;DR

The paper develops an analytical reconstruction framework for relativistic collapse with tangential pressure arising from angular momentum, extending the LTB solution to the Datta-Bondi-Gair (DBG) family and targeting end-states that oscillate around an Einstein Cluster (EC) rather than forming black holes. By generalizing the Krasiński-Hellaby method to include the angular-momentum function $L(r)$, and introducing weak-field and small-$L$ approximations, the authors enable a (semi-)analytic reconstruction of the full spacetime from two initial-density profiles on distinct hypersurfaces, resolving three DBG functions $(M,E,L)$ together with the bang time $\tau_B$. The approach yields a piecewise analytic evolution: an early dust-like phase and a late-time oscillatory EC phase, connected via a controlled mapping that avoids intractable elliptic integrals until the final numerical step. The framework provides a tractable bridge between exact relativistic collapse models and realistic self-gravitating systems, with potential applications to galaxy clusters, halo dynamics, and gravitational lensing, while outlining future work to include nonzero radial pressure and cosmological backgrounds.

Abstract

The aim of this paper is to provide an analytical model for the formation of stable structures (cosmological or astrophysical), where stability is obtained through the tangential pressure countering the effect of gravity. We utilize the generalization of the Lemaitre-Tolman-Bondi (LTB) spacetime to matter with tangential pressure generated by the angular momentum of fluid particles. Extending the Krasiński-Hellaby (KH) LTB reconstruction method, we show how set of three functions defined on two arbitrary hypersurfaces can fully determine the spacetime geometry. We further restrict our attention to the bounded case and develop the weak-field and the small-angular-momentum approximations. We show how these can be applied to the exact solution on the initial hypersurface, together with the oscillatory solution on the final hypersurface, to considerably simplify the reconstruction scheme. The so obtained models exhibit explicit dust-like behaviour in the early and middle stages of the collapse, and reach the final state as oscillations around the static solution.