[Paper Review] A quantum mass-spectrum of Kerr black hole: superstrings

V. V. Kiselev|ArXiv.org|Dec 23, 2004

Black Holes and Theoretical Physics19 references3 citations

TL;DR

This paper proposes a quantum mass-spectrum for Kerr black holes using thermal quantization of geodesics confined within horizons, combined with a conjecture on massless modes and supersymmetry. By quantizing the ratio of horizon areas and extending boundary conditions to include Ramond-Neveu-Schwarz statistics, the resulting mass spectrum exhibits linear Regge trajectories identical to those of superstrings, with five distinct string tension parameters (α′) corresponding to different loop parameters l.

ABSTRACT

Using thermal quantization of geodesics confined under horizons and reasonable conjecture on massless modes, we evaluate quantum spectrum of Kerr black hole masses compatible with superstring symmetries.

Motivation & Objective

• To derive a quantum mass-spectrum for Kerr black holes using a quasi-classical thermal quantization approach.
• To extend the thermal quantization framework to include supersymmetry via Ramond-Neveu-Schwarz boundary conditions.
• To establish a connection between black hole mass spectra and the linear Regge trajectories of superstrings.
• To explore the role of area ratios and winding numbers in determining quantum mass levels.
• To investigate duality symmetries and entropy via the Cardy formula in the context of black hole microstates.

Proposed method

• Thermal quantization of radial geodesics confined under the inner and outer horizons of Kerr black holes, based on periodicity in imaginary time.
• Quantization of the ratio of horizon areas A₊/A₋ = l, where l ∈ ℕ, leading to discrete winding numbers n₊ and n₋.
• Derivation of the Christodoulou-Ruffini mass formula and its reduction to M² ∝ J for Kerr black holes (Q=0).
• Incorporation of supersymmetry by extending boundary conditions to include Ramond and Neveu-Schwarz sectors, conjecturing a link between massless modes and intercepts.
• Calculation of effective string tension α′ₗ = 2√l / (l+1) for l ∈ {1,2,3,∞}, yielding distinct Regge trajectories.
• Comparison with Cardy formula for entropy, identifying Virasoro algebra eigenvalues and central charge in terms of J and l.

Experimental results

Research questions

• RQ1 Can thermal quantization of confined geodesics reproduce the linear mass-spectrum of superstrings in the Kerr black hole context?
• RQ2 How do Ramond-Neveu-Schwarz boundary conditions affect the quantum spectrum and its relation to supersymmetry?
• RQ3 What is the role of the loop parameter l = A₊/A₋ in determining the effective string tension α′?
• RQ4 How does the duality l ↔ 1/l preserve the mass spectrum and string-like behavior?
• RQ5 Can the thermal quantization method be consistently linked to the Cardy formula for black hole entropy?

Key findings

• The quantum mass-spectrum of Kerr black holes exhibits linear Regge trajectories M² ∝ J, identical to those of superstrings, with slope α′ₗ = 2√l / (l+1).
• For l ∈ {1,2,3,∞}, the effective string tensions are α′ = {1, (2/3)√2, (1/2)√3, 0}, corresponding to five distinct superstring-like trajectories.
• The extremal black hole (l=1) reproduces the known result J = M², consistent with string theory at the leading Regge trajectory.
• The duality l ↔ 1/l preserves the mass spectrum and α′, indicating a symmetry between inner and outer horizon descriptions.
• Entropy S = 2π√(J²l) at large J matches the Cardy formula, suggesting a Virasoro algebra with central charge C = 6J² and L₀ = l.
• The method predicts a richer quantum spectrum than prior area-quantization approaches, which only apply to the extremal case (l=1).