[Paper Review] The Universal Area Product: An Heuristic Argument

Don N. Page, Andrey A. Shoom|arXiv (Cornell University)|Apr 21, 2015

Conflict of Laws and Jurisdiction1 references3 citations

TL;DR

This paper presents a heuristic argument for the universal area product $A_+A_- = (8\pi J)^2 + (4\pi Q^2)^2$ in four-dimensional, stationary, axisymmetric, electrically charged black holes distorted by external matter and fields. The argument uses adiabatic perturbations preserving symmetry and angular momentum, leading to a conjecture that this product remains invariant under such distortions and extends to higher-dimensional multi-horizon black objects with quantized charges, angular momenta, and cosmological constant.

ABSTRACT

We present an heuristic argument for the universal area product: A_{+}A_{-}=(8πJ)^{2}+(4πQ^{2})^{2} for a four-dimensional, stationary, axisymmetric, electrically charged black hole with an arbitrary stationary axisymmetric distribution of external matter (possibly charged), derived by Marcus Ansorg and Jorg Hennig. Here A_{+} and A_{-} are the areas of the event and Cauchy horizons, and J and Q are the angular momentum and electric charge. Based on this argument, we conjecture that a universal area product holds for higher-dimensional, stationary, multi-horizon black objects in the presence of an external stationary charged distribution of matter.

Motivation & Objective

• To investigate whether the universal area product of black hole horizons remains invariant under adiabatic distortion by external stationary, axisymmetric matter and fields.
• To extend the known area product relation for Kerr-Newman black holes to more general configurations with arbitrary external matter distributions.
• To conjecture that the area product depends only on quantized charges, angular momenta, and cosmological constant in higher-dimensional black holes.
• To explore the robustness of the area product in the presence of inner horizons and gravitational perturbations, including mass inflation effects.
• To provide a foundation for probing black hole microstates via area products in higher-dimensional and multi-horizon systems.

Proposed method

• Model a distorted black hole as a quasi-stationary transition from an undistorted Kerr-Newman solution via small, symmetric perturbations in metric and gauge potential.
• Use perturbation theory with small dimensionless parameters $\lambda$ and $\mu$ to describe gravitational and electromagnetic changes, preserving axial symmetry.
• Assume weak gravitational and electromagnetic waves during transition, decaying as inverse powers of advanced time, allowing the black hole to settle into a new stationary state.
• Apply the inverse scattering method and Ernst equations to justify the area product relation in the exact solution, then derive a heuristic argument from perturbative stability.
• Extend the argument to higher dimensions by assuming stability against weak gravitational waves and preserving angular momentum and charge during adiabatic distortion.
• Formulate a conjecture that the product of all horizon areas in d-dimensional black holes is a polynomial in $Q_j$, $J_k$, and $\Lambda^{-1/2}$, independent of mass and topology.

Experimental results

Research questions

• RQ1 Does the universal area product $A_+A_- = (8\pi J)^2 + (4\pi Q^2)^2$ remain valid for a Kerr-Newman black hole distorted by external stationary, axisymmetric, charged matter?
• RQ2 Can the area product relation be generalized to higher-dimensional, multi-horizon black objects under adiabatic perturbations?
• RQ3 How do inner horizons and potential mass inflation effects influence the validity of the area product in multi-horizon systems?
• RQ4 Is the area product independent of the black hole's mass and topology when external fields are present?
• RQ5 Can the area product serve as a probe of black hole microstates in higher-dimensional or asymptotically AdS spacetimes?

Key findings

• The area product $A_+A_-$ for a four-dimensional, stationary, axisymmetric, charged black hole with external matter is universally given by $(8\pi J)^2 + (4\pi Q^2)^2$, independent of the black hole's mass or the specific distribution of external matter.
• The heuristic argument relies on adiabatic perturbations that preserve axial symmetry and angular momentum, ensuring no net emission of angular momentum during the transition.
• The product remains invariant under weak gravitational and electromagnetic perturbations, as long as the final state is stationary and the perturbations decay over time.
• The conjecture extends to d-dimensional black holes ($d \geq 4$), where the product of all horizon areas is a polynomial function of charges $Q_j$, angular momenta $J_k$, and the inverse square root of the cosmological constant $\Lambda^{-1/2}$.
• For a five-dimensional, static, charged black hole distorted by neutral matter, the product of outer and inner horizon areas is proportional to the cube of the electric charge, supporting the conjecture.
• The validity of the conjecture may be affected by inner horizon singularities due to mass inflation, but the authors suggest treating eternal solutions formally to bypass this issue.