Massive fields tend to form highly oscillating self-similarly expanding shells

TL;DR

The paper studies the evolution of massive scalar fields in Minkowski space, revealing the formation of highly oscillatory, self-similarly expanding shells that maintain their energy over time.

Contribution

It demonstrates that massive Klein-Gordon fields form self-similar expanding shells with high-frequency oscillations, a novel insight into their dynamic structure.

Findings

Shell structures are built by high-frequency oscillations.

Shell expansion is self-similar and modulated by a decaying factor.

Energy transported by each shell remains constant over time.

Abstract

The time evolution of self-interacting spherically symmetric scalar fields in Minkowski spacetime is investigated based on the use of Green's theorem. It is shown that a massive Klein-Gordon field can be characterized by the formation of certain expanding shell structures where all the shells are built up by very high frequency oscillations. This oscillation is found to be modulated by the product of a simple time decaying factor of the form $t^{-{3}/{2}}$ and of an essentially self-similar expansion. Apart from this self-similar expansion the developed shell structure is preserved by the evolution. In particular, the energy transported by each shell appears to be time independent.