[Paper Review] Consequences of a minimal length in a pseudo-complex extension of General Relativity

Leila Maghlaoui, Peter O. Hess|arXiv (Cornell University)|Apr 7, 2022

Black Holes and Theoretical Physics37 references1 citations

TL;DR

This paper investigates the consequences of a minimal length scale in a pseudo-complex extension of General Relativity (pcGR), showing that for small black hole masses (on the order of the minimal length), effective potential barriers emerge near the event horizon due to a conformal factor derived from maximal acceleration. These barriers inhibit particle infall, suggesting a quantum-gravitational cutoff that halts accretion and implies a lower bound on black hole mass.

ABSTRACT

The effects of a minimal length are investigated within an algebraically extended theory of General Relativity (GR). Former attempts, to include a minimal length in GR are first resumed, with a conformal factor of the metric as a consequence. Effective potentials for various black hole masses (as ratios to the minimal length) are deduced. It is found that the existence of a minimal length has, for a small mass black hole, important effects on the effective potential near the event horizon, creating barriers which inhibit that particles can pass the event horizon. Further, a new limit for the minimal mass of a black hole is derive

Motivation & Objective

• To examine the physical consequences of a minimal length scale within a pseudo-complex extension of General Relativity (pcGR), which incorporates maximal acceleration.
• To determine under what conditions the minimal length significantly alters black hole dynamics, particularly near the event horizon.
• To derive a new lower limit for black hole mass based on the interplay between minimal length and maximal acceleration.
• To analyze how the effective potential changes with black hole mass relative to the minimal length, especially in the Schwarzschild geometry.
• To assess whether quantum gravity effects—via minimal length—could be observable in macroscopic or microscopic black holes.

Proposed method

• Adopts a pseudo-complex extension of GR (pcGR) based on E.R. Caianiello’s quantum geometry framework, where spacetime is extended to a tangent bundle with eight dimensions.
• Introduces a conformal factor σ²(r) in the effective 4D metric, derived from the four-acceleration and maximal acceleration Am, leading to a modified line element: dw² = σ²(r) gμν dxμ dxν.
• Uses an iterative procedure to compute σ²(r), starting from geodesic solutions and refining the acceleration components at each step.
• Analyzes the effective potential for radial motion in Schwarzschild geometry, incorporating σ²(r) as a denominator, which leads to singularities when σ²(r) → 0.
• Considers three cases for the parameter B₄ (related to mass and minimal length) to explore different mass regimes: ϵ = l/m ≈ 1, ϵ ≪ 1, and B₄ > 81m⁴/8.
• Evaluates the behavior of the effective potential for varying particle energy E, identifying critical energy thresholds where the potential changes sign or develops barriers.

Experimental results

Research questions

• RQ1 How does the inclusion of a minimal length scale affect the effective potential near a black hole horizon in pcGR?
• RQ2 At what black hole mass scale do minimal length effects become non-negligible in the context of maximal acceleration?
• RQ3 Can the minimal length lead to a potential barrier that prevents particle infall into a black hole, and under what conditions?
• RQ4 Does the minimal length imply a lower bound on black hole mass, and if so, what is its magnitude?
• RQ5 How does the structure of the effective potential change when σ²(r) vanishes, and what physical consequences does this have?

Key findings

• For small black hole masses (m ≈ l, i.e., ϵ ≈ 1), the effective potential develops singularities and barriers near the event horizon due to σ²(r) → 0, which inhibit particle infall.
• The potential barrier at r = 3m/2 (event horizon) remains significant for black holes up to ~10¹⁸l (≈10⁸ kg), suggesting that accretion is suppressed even for macroscopic black holes.
• When ϵ ≪ 1 (m ≫ l), σ²(r) → 1 and the potential barrier vanishes, indicating that minimal length effects are negligible for large black holes.
• A critical energy threshold exists below which the effective potential becomes positive, and for λ ≠ 0, σ²(r) = 0 leads to divergences in the potential, halting particle motion.
• The positions r±₁ and r±₂, where maximal acceleration occurs, coincide with potential barriers that prevent particles from crossing, even in the absence of a classical horizon.
• The analysis suggests a new lower bound on black hole mass—on the order of 10⁸ kg (for l ~ 10⁻³⁵ m)—beyond which minimal length effects become negligible, implying a quantum-gravitational cutoff.