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Nonexistence of Solutions to the Coupled Generalized Jang Equation/Zero Divergence System

Jaroslaw S. Jaracz

TL;DR

The paper investigates a Bray–Khuri proposed coupling between the generalized Jang equation and a zero-divergence condition to attack the Penrose inequality. By constructing a smooth, asymptotically flat, spherically symmetric initial data set that satisfies the dominant energy condition and has an outer apparent horizon, the authors derive that the coupled system cannot possess any smooth radial solution with the required asymptotics, thereby obstructing this route to the Penrose inequality in the radial setting. The argument hinges on reducing the generalized Jang equation to an ODE for $v(r)$ in spherical symmetry and showing that any admissible solution would force incompatible behavior (e.g., blow-up of $\phi$ or violation of asymptotics) at infinity. Consequently, this result rules out the radial case of the coupled system as a viable path to proving the Penrose inequality in general, while leaving open the possibility of nonradial solutions.

Abstract

In [5], Bray and Khuri proposed coupling the generalized Jang equation to several different auxiliary equations. The solutions to these coupled systems would then imply the Penrose inequality. One of these involves coupling the generalized Jang equation to $\overline{div}(φq)=0$, as this would guarantee the non-negativity of the scalar curvature in the Jang surface. This coupled system of equations has not received much attention, and we investigate it's solvability. We prove that there exists a spherically symmetric initial data set for the Einstein equations for which there do not exist smooth radial solutions to the system having the appropriate asymptotics for application to the Penrose inequality.

Nonexistence of Solutions to the Coupled Generalized Jang Equation/Zero Divergence System

TL;DR

The paper investigates a Bray–Khuri proposed coupling between the generalized Jang equation and a zero-divergence condition to attack the Penrose inequality. By constructing a smooth, asymptotically flat, spherically symmetric initial data set that satisfies the dominant energy condition and has an outer apparent horizon, the authors derive that the coupled system cannot possess any smooth radial solution with the required asymptotics, thereby obstructing this route to the Penrose inequality in the radial setting. The argument hinges on reducing the generalized Jang equation to an ODE for in spherical symmetry and showing that any admissible solution would force incompatible behavior (e.g., blow-up of or violation of asymptotics) at infinity. Consequently, this result rules out the radial case of the coupled system as a viable path to proving the Penrose inequality in general, while leaving open the possibility of nonradial solutions.

Abstract

In [5], Bray and Khuri proposed coupling the generalized Jang equation to several different auxiliary equations. The solutions to these coupled systems would then imply the Penrose inequality. One of these involves coupling the generalized Jang equation to , as this would guarantee the non-negativity of the scalar curvature in the Jang surface. This coupled system of equations has not received much attention, and we investigate it's solvability. We prove that there exists a spherically symmetric initial data set for the Einstein equations for which there do not exist smooth radial solutions to the system having the appropriate asymptotics for application to the Penrose inequality.
Paper Structure (16 sections, 4 theorems, 136 equations)

This paper contains 16 sections, 4 theorems, 136 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. The Penrose Conjecture
  3. The Generalized Jang Equation
  4. The coupled system and the main theorem
  5. A closer look at the divergence term
  6. Proof of Theorem \ref{['MainTheorem']}
  7. Asymptotic Flatness and the ADM Formalism
  8. Energy Conditions and Constraint Equations
  9. Null expansions and apparent horizons
  10. Asymptotically Flat Initial Data Sets in Spherical Symmetry
  11. The Generalized Jang Equation in Spherical Symmetry
  12. The zero divergence term in spherical symmetry
  13. Construction of the spherically symmetric initial data set
  14. Appropriate Asymptotics for $v(r)$
  15. Proof of Theorem \ref{['MainTheorem']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists a smooth, spherically symmetric, assymptotically flat initial data set $(M, g, k)$ satisfying the dominant energy condition and with boundary consisting of a compact outermost apparent horizon for which CoupledSystem does not possess any smooth radial solutions with $\phi>0$ outside of

Theorems & Definitions (9)

  • Conjecture : Penrose Inequality
  • Theorem 1.1
  • Conjecture
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • proof