Quantum phenomena inside a black hole: quantization of the scalar field iniside horizon in Schwarzschild spacetime

TL;DR

This paper develops a unitary quantum framework for a scalar field inside a Schwarzschild black hole, revealing harmonic oscillator dynamics and analyzing particle creation, stability, and symmetry properties in the black hole interior.

Contribution

It introduces a novel unitary approach to quantize and describe the dynamics of a scalar field inside a black hole horizon, including a time-dependent Hamiltonian and Bogolyubov transformations.

Findings

The system behaves as coupled harmonic oscillators with symmetry-consistent interactions.

Heisenberg equations reduce to differential equations for Bogolyubov coefficients.

Gravitationally driven instability is shown to be absent in this model.

Abstract

We discuss the problem of the quantization and dynamic evolution of a scalar free field in the interior of a Schwarzschild black hole. A unitary approach to the dynamics of the quantized field is proposed: a time-dependent Hamiltonian governing the Heisenberg equations is derived. It is found that the system is represented by a set of harmonic oscillators coupled via terms corresponding to the creation and annihilation of pairs of particles and that the symmetry properties of the spacetime, homogeneity and isotropy are obeyed by the coupling terms in the Hamiltonian. It is shown that Heisenberg equations for annihilation and creation operators are transformed into ordinary differential equations for appropriate Bogolyubov coefficients. Such a formulation leads to a general question concerning the possibility of gravitationally driven instability, that is however excluded in this case.