Existence of Spherical Initial Data with Unit Mass, Zero Energy, and Virial less than - 1/2 for the Relativistic Vlasov-Poisson Equation with Attractive Coupling

TL;DR

This paper constructs specific initial data for the relativistic Vlasov-Poisson equation with attractive coupling that have unit mass, zero energy, and virial less than -1/2, confirming their existence which was previously uncertain.

Contribution

The paper demonstrates the existence of spherically symmetric initial data with the specified properties, resolving an open question in the field.

Findings

Existence of initial data with the desired properties confirmed.

Constructed two classes of such initial data.

Addresses an open problem regarding initial data with specific virial conditions.

Abstract

In a recent paper, Kiessling and Tahvildar-Zadeh proved that any classical solution of the relativistic Vlasov-Poisson equation with attractive coupling launched by spherically symmetric initial data with zero total energy and virial less than or equal to -1/2 will blow up in finite time. They left open whether such data exist. Subsequently, the question was raised whether any such data exist at all. In fact, the simplest conceivable ansatz, a nearly uniform ball of material centered at the origin with momenta directed inward, must have virial strictly larger than -1/2! In this brief note, we settle this issue by constructing two classes of such initial data.