Conserved quantities and dual turbulent cascades in Anti-de Sitter spacetime

TL;DR

This paper uncovers two new conserved quantities in the dynamics of a scalar field in Anti-de Sitter spacetime, revealing dual turbulent cascades and constraining the system's phase space evolution.

Contribution

It introduces the conservation of particle number and Hamiltonian in the TTF framework, explaining dual cascades and limiting energy equipartition in AdS scalar field dynamics.

Findings

Existence of conserved particle number and Hamiltonian in TTF system.

Dual turbulent cascades of energy and particle number.

Constraints on phase space and energy distribution among modes.

Abstract

We consider the dynamics of a spherically symmetric massless scalar field coupled to general relativity in Anti--de Sitter spacetime in the small-amplitude limit. Within the context of our previously developed two time framework (TTF) to study the leading self-gravitating effects, we demonstrate the existence of two new conserved quantities in addition to the known total energy $E$ of the modes: The particle number $N$ and Hamiltonian $H$ of our TTF system. Simultaneous conservation of $E$ and $N$ implies that weak turbulent processes undergo dual cascades (direct cascade of $E$ and inverse cascade of $N$ or vice versa). This partially explains the observed dynamics of 2-mode initial data. In addition, conservation of $E$ and $N$ limits the region of phase space that can be explored within the TTF approximation and in particular rules out equipartion of energy among the modes for general initial data. Finally, we discuss possible effects of conservation of $N$ and $E$ on late time dynamics.