Dynamical Collapse of Boson Stars

TL;DR

This paper investigates the dynamical collapse of boson stars by analyzing the mean field limit of many-body quantum systems with attractive Coulomb interactions, linking finite-time blow-up in nonlinear equations to gravitational collapse.

Contribution

It provides the first dynamical description of gravitational collapse at the many-body quantum level, connecting nonlinear Hartree blow-up with many-body Schrödinger dynamics.

Findings

Approximation of many-body dynamics by nonlinear Hartree equation in non-blow-up regime.

Demonstration that blow-up in Hartree solutions implies collapse in many-body Schrödinger dynamics.

First rigorous link between gravitational collapse phenomena and quantum many-body evolution.

Abstract

We study the time evolution in system of $N$ bosons with a relativistic dispersion law interacting through an attractive Coulomb potential with coupling constant $G$. We consider the mean field scaling where $N$ tends to infinity, $G$ tends to zero and $\lambda = G N$ remains fixed. We investigate the relation between the many body quantum dynamics governed by the Schr\"odinger equation and the effective evolution described by a (semi-relativistic) Hartree equation. In particular, we are interested in the super-critical regime of large $\lambda$ (the sub-critical case has been studied in \cite{ES,KP}), where the nonlinear Hartree equation is known to have solutions which blow up in finite time. To inspect this regime, we need to regularize the Coulomb interaction in the many body Hamiltonian with an $N$ dependent cutoff that vanishes in the limit $N\to \infty$. We show, first, that if the solution of the nonlinear equation does not blow up in the time interval $[-T,T]$, then the many body Schr\"odinger dynamics (on the level of the reduced density matrices) can be approximated by the nonlinear Hartree dynamics, just as in the sub-critical regime. Moreover, we prove that if the solution of the nonlinear Hartree equation blows up at time $T$ (in the sense that the $H^{1/2}$ norm of the solution diverges as time approaches $T$), then also the solution of the linear Schr\"odinger equation collapses (in the sense that the kinetic energy per particle diverges) if $t \to T$ and, simultaneously, $N \to \infty$ sufficiently fast. This gives the first dynamical description of the phenomenon of gravitational collapse as observed directly on the many body level.