[Paper Review] Creation of fermions by rotating charged black-holes

Dietrich Häfner|arXiv (Cornell University)|Dec 18, 2006

Cosmology and Gravitation Theories38 references21 citations

TL;DR

This paper provides a rigorous mathematical proof of the Hawking effect for fermions in the context of a rotating, charged black hole formed by stellar collapse. Using the Kerr-Newman metric and a novel coordinate system based on simple null geodesics, the authors establish that an observer at rest far from the black hole asymptotically detects a thermal state of fermions, confirming the quantum creation of particles via Hawking radiation in a non-spherically symmetric, rotating spacetime setting.

ABSTRACT

This work is devoted to the mathematical study of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal state when his proper time goes to infinity. We first introduce a model of the collapse of the star. We suppose that the space-time outside the star is given by the Kerr-Newman metric. The assumptions on the asymptotic behavior of the surface of the star are inspired by the asymptotic behavior of certain timelike geodesics in the Kerr-Newman metric. The Dirac equation is then written using coordinates and a Newman-Penrose tetrad which are adapted to the collapse. This coordinate system and tetrad are based on the so called simple null geodesics. The quantization of Dirac fields in a globally hyperbolic space-time is described. We formulate and prove a theorem about the Hawking effect in this setting. The proof of the theorem contains a minimal velocity estimate for Dirac fields that is slightly stronger than the usual ones and an existence and uniqueness result for solutions of a characteristic Cauchy problem for Dirac fields in the Kerr-Newman space-time. In an appendix we construct explicitly a Penrose compactification of block I of the Kerr-Newman space-time based on simple null geodesics.

Motivation & Objective

• To rigorously establish the Hawking effect for Dirac fermions in the gravitational field of a rotating, charged black hole formed by stellar collapse.
• To extend scattering theory and quantum field quantization to the non-spherically symmetric Kerr-Newman spacetime.
• To construct a Penrose compactification of block I of the Kerr-Newman spacetime using simple null geodesics for a global geometric description.
• To prove existence, uniqueness, and velocity estimates for solutions of the characteristic Cauchy problem for Dirac fields in Kerr-Newman spacetime.
• To demonstrate asymptotic completeness and thermalization of the quantum state at future null infinity, confirming the Hawking effect for fermions.

Proposed method

• Model the collapse of a rotating, charged star using the Kerr-Newman metric outside the star, with boundary conditions derived from timelike geodesics with zero angular momentum and energy.
• Introduce a new Newman-Penrose tetrad and coordinate system (star-Kerr coordinates) adapted to simple null geodesics to simplify the Dirac equation in the Kerr-Newman background.
• Formulate the Dirac equation on the collapsed spacetime manifold and prove existence and uniqueness of solutions to the characteristic Cauchy problem with data on a Lipschitz spacelike hypersurface.
• Establish a minimal velocity estimate for Dirac fields that is slightly stronger than standard estimates, crucial for scattering theory and wave operator construction.
• Apply second quantization in globally hyperbolic spacetimes to define the quantum field theory framework and analyze the asymptotic particle content.
• Construct an explicit Penrose compactification of block I of the Kerr-Newman spacetime using star-Kerr coordinates, enabling a geometric description of future and past null infinity.

Experimental results

Research questions

• RQ1 Does the Hawking effect for fermions persist in the non-spherically symmetric, rotating, and charged Kerr-Newman spacetime?
• RQ2 Can a rigorous scattering theory for Dirac fields be developed in the Kerr-Newman geometry, given the absence of spherical symmetry and the lack of separation of variables?
• RQ3 What is the asymptotic behavior of Dirac fields at future null infinity, and does it lead to a thermal state as predicted by Hawking?
• RQ4 How can the characteristic Cauchy problem for Dirac fields be solved uniquely and smoothly in the Kerr-Newman background, especially near the horizon?
• RQ5 Can a Penrose compactification of block I of the Kerr-Newman spacetime be explicitly constructed using simple null geodesics to describe the global structure?

Key findings

• An observer at rest in Boyer-Lindquist coordinates far from the black hole detects a thermal state of fermions as proper time tends to infinity, confirming the Hawking effect for Dirac fields.
• A new coordinate system and Newman-Penrose tetrad based on simple null geodesics allow a well-posed formulation of the Dirac equation in the Kerr-Newman spacetime.
• The paper proves a minimal velocity estimate for Dirac fields that is stronger than standard estimates, enabling improved control in scattering theory.
• An existence and uniqueness result is established for solutions to the characteristic Cauchy problem for Dirac fields on the collapsed spacetime manifold.
• An explicit Penrose compactification of block I of the Kerr-Newman spacetime is constructed using star-Kerr coordinates, with future and past null infinity defined as smooth hypersurfaces.
• The asymptotic dynamics of the Dirac field are shown to be unitarily equivalent to a free field at future null infinity, confirming the thermal nature of the emitted radiation.