[Paper Review] The general Penrose inequality: lessons from numerical evidence

Janusz Karkowski, Edward Malec|ArXiv.org|Dec 21, 2004

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper investigates the general Penrose inequality through numerical simulations of vacuum and matter-filled initial data sets, testing multiple formulations of the inequality. It finds that the inequality holds for both past and future apparent horizons—regardless of whether they intersect—whenever they bifurcate from an outermost minimal surface, provided the dominant energy condition is satisfied, offering strong numerical support for the conjecture in physically relevant scenarios.

ABSTRACT

Formulation of the Penrose inequality becomes ambiguous when the past and future apparent horizons do cross. We test numerically several natural possibilities of stating the inequality in punctured and boosted single- and double- black holes, in a Dain-Friedrich class of initial data and in conformally flat spheroidal data.The Penrose inequality holds true in vacuum configurations for the outermost element amongst the set of disjoint future and past apparent horizons (as expected)and (unexpectedly) for each of the outermost past and future apparent horizons, whenever these two bifurcate from an outermost minimal surface, regardless of whether they intersect or remain disjoint. In systems with matter the conjecture breaks down only if matter does not obey the dominant energy condition.

Motivation & Objective

• To test various formulations of the general Penrose inequality in non-spherically symmetric spacetimes where past and future apparent horizons may intersect.
• To determine whether the Penrose inequality holds for individual outermost past and future apparent horizons, not just the globally outermost one.
• To investigate the role of the dominant energy condition in the validity of the inequality, especially in matter-filled spacetimes.
• To explore whether numerical evidence can support a local analytic proof of the inequality in initial data with small extrinsic curvature.
• To clarify the ambiguity in formulating the Penrose inequality when apparent horizons cross in generic foliations.

Proposed method

• Constructing initial data using the conformal method with conformally flat metrics and specified traceless extrinsic curvature.
• Solving the Lichnerowicz-York equation numerically for the conformal factor $\phi$ under boundary conditions: $\phi \to 1$ at spatial infinity and constant at inner boundaries.
• Defining initial data for punctured black holes (Bowen-York and Dain-Friedrich types) and spheroidal systems with adjustable matter density.
• Computing the ADM mass $m$ and the horizon area $S_H$ to evaluate $m \geq \sqrt{S_H / 16\pi}$, testing multiple versions of the Penrose inequality.
• Varying the parameter $C$ in the energy density $\rho$ to test cases with $C=1$ (dominant energy condition satisfied) and $C=0$ (violated).
• Using numerical relativity techniques to locate apparent horizons via the optical scalar condition $\theta_{\pm} = \nabla_i t^i \pm K_{ij}t^i t^j = 0$.

Experimental results

Research questions

• RQ1 Does the Penrose inequality hold for both the outermost past and future apparent horizons when they bifurcate from a common minimal surface, even if they intersect?
• RQ2 Can the Penrose inequality be violated in systems where the dominant energy condition is not satisfied?
• RQ3 How do different formulations of the Penrose inequality (e.g., PIS, PIM) perform in non-spherically symmetric, vacuum initial data?
• RQ4 Is there a consistent numerical pattern supporting a local analytic proof of the inequality in initial data with small extrinsic curvature?
• RQ5 What is the role of the minimal surface in determining the validity of the Penrose inequality in non-vacuum, matter-filled configurations?

Key findings

• The Penrose inequality holds numerically for both the outermost past and future apparent horizons in vacuum configurations, regardless of whether they intersect or remain disjoint.
• In all tested vacuum cases with bifurcating horizons from an outermost minimal surface, the inequality $m \geq \sqrt{S_H / 16\pi}$ is satisfied for each horizon individually.
• For $C=1$, the energy density satisfies the dominant energy condition, and the inequality holds: e.g., for $\phi(r=1)=2.5$, $m \approx 3.00096$ and $m_H \approx 3.00067$, so $m > m_H$.
• When $C=0$, the dominant energy condition is violated, and the inequality breaks down in the first two examples: e.g., $m \approx 2.80004$ and $m_H \approx 2.80011$, so $m < m_H$.
• The outermost minimal surface consistently serves as the bifurcation point for both past and future apparent horizons in the tested configurations.
• The numerical results suggest a plausible path toward a local analytic proof of the inequality for small extrinsic curvature, as sketched in Section 5.