Scalar field as an intrinsic time measure in coupled dynamical matter-geometry systems. II. Electrically charged gravitational collapse

TL;DR

This study explores whether scalar fields or charge functions can serve as internal clocks during the gravitational collapse of charged scalar fields in Einstein and Brans-Dicke theories, especially near singularities relevant for quantum gravity.

Contribution

It demonstrates that certain scalar fields and charge measures can act as time indicators near singularities only when coupled and present in the system, advancing understanding of internal time in quantum gravity contexts.

Findings

Scalar fields and charge functions can quantify time near singularities when coupled.

Maxwell potential alone cannot serve as a time measure in this scenario.

None of the quantities studied are suitable for measuring time near the Cauchy horizon.

Abstract

Investigating the dynamics of gravitational systems, especially in the regime of quantum gravity, poses a problem of measuring time during the evolution. One of the approaches to this issue is using one of the internal degrees of freedom as a time variable. The objective of our research was to check whether a scalar field or any other dynamical quantity being a part of a coupled multi-component matter-geometry system can be treated as a `clock' during its evolution. We investigated a collapse of a self-gravitating electrically charged scalar field in the Einstein and Brans-Dicke theories using the 2+2 formalism. Our findings concentrated on the spacetime region of high curvature existing in the vicinity of the emerging singularity, which is essential for the quantum gravity applications. We investigated several values of the Brans-Dicke coupling constant and the coupling between the Brans-Dicke and the electrically charged scalar fields. It turned out that both evolving scalar fields and a function which measures the amount of electric charge within a sphere of a given radius can be used to quantify time nearby the singularity in the dynamical spacetime part, in which the apparent horizon surrounding the singularity is spacelike. Using them in this respect in the asymptotic spacetime region is possible only when both fields are present in the system and, moreover, they are coupled to each other. The only nonzero component of the Maxwell field four-potential cannot be used to quantify time during the considered process in the neighborhood of the whole central singularity. None of the investigated dynamical quantities is a good candidate for measuring time nearby the Cauchy horizon, which is also singular due to the mass inflation phenomenon.