On the use of the Kodama vector field in spherically symmetric dynamical problems

TL;DR

This paper demonstrates that using the Kodama vector field as a preferred time vector in spherically symmetric dynamical systems simplifies the evolution equations, separating true degrees of freedom and identifying hyperbolic subsystems, applicable to scalar fields coupled to gravity.

Contribution

It introduces the use of the Kodama vector field to simplify and clarify the structure of evolution equations in spherically symmetric dynamical systems, including scalar fields.

Findings

Separation of true degrees of freedom.

Identification of a symmetric hyperbolic subsystem.

Generalization to self-interacting scalar fields.

Abstract

It is shown that by making use of the Kodama vector field, as a preferred time evolution vector field, in spherically symmetric dynamical systems unexpected simplifications arise. In particular, the evolution equations relevant for the case of a massless scalar field minimally coupled to gravity are investigated. The simplest form of these equations in the 'canonical gauge' are known to possess the character of a mixed first order elliptic-hyperbolic system. The advantages related to the use of the Kodama vector field are two-folded although they show up simultaneously. First, it is found that the true degrees of freedom separate. Second, a subset of the field equations possessing the form of a first order symmetric hyperbolic system for these preferred degrees of freedom is singled out. It is also demonstrated, in the appendix, that the above results generalise straightforwardly to the case of a generic self-interacting scalar field.