[Paper Review] Tests of Classical Gravity Description for Microscopic Black Hole Production

Slava Rychkov|ArXiv.org|May 12, 2004

Cosmology and Gravitation Theories21 references3 citations

TL;DR

This paper challenges the validity of classical gravity descriptions for microscopic black hole production in TeV-scale gravity scenarios. By analyzing quantum fluctuations in Aichelburg-Sexl shock waves using linearized quantum gravity, it finds that graviton occupation numbers are too low to resolve classical shock widths, implying quantum effects dominate and undermining the geometric cross-section estimate central to classical black hole formation models.

ABSTRACT

The classical Einstein gravity description of black hole production in transplanckian collisions in TeV-scale gravity is tested for self-consistency. In addition to the "curvature must be small" test, which was shown to be violated in [hep-ph/0401116], it is proposed to estimate quantum fluctuations in the Aichelburg-Sexl shock waves corresponding to the colliding particles. Using linearized quantum gravity, it is found that the occupation numbers of gravitons with characteristic frequency are too small to resolve the classical width of the shocks. This raises further doubts in the classical gravity picture of black hole creation and the geometric cross section estimate based on it.

Motivation & Objective

• To assess the self-consistency of classical gravity descriptions for microscopic black hole production in transplanckian collisions within large extra dimensions scenarios.
• To test whether the classical description of gravitational shock waves from colliding particles remains valid under quantum corrections.
• To evaluate whether the number of gravitons involved in the collision is large enough to justify a classical treatment, as required for consistency of classical field theory.
• To challenge the widely used geometric cross-section estimate for black hole production, which relies on classical general relativity.

Proposed method

• Modeling colliding particles as Aichelburg-Sexl shock waves in D-dimensional spacetime with large Lorentz factors.
• Using linearized quantum gravity to compute the occupation number of gravitons in the shock wave field.
• Expanding the classical shock profile in plane waves to identify the coherent state of gravitons.
• Deriving the mean occupation number of gravitons as a function of frequency, using the relation $ n_k \sim C^2 / (Z^2 k^3) $.
• Estimating the total number of gravitons with energy $ \sim \omega $ via integration: $ N_\omega \sim R_S^{D-4} \omega^{-2} $.
• Determining the frequency range $ \omega \ll \omega_{\text{max}} \sim R_S^{(D-4)/2} $ where classical behavior holds, and comparing it to the shock width scale $ w^{-1} $.

Experimental results

Research questions

• RQ1 Is the classical gravity description of microscopic black hole production self-consistent when quantum fluctuations are considered?
• RQ2 Do the gravitational shock waves from colliding high-energy particles contain a sufficiently large number of gravitons to justify a classical treatment?
• RQ3 Does the geometric cross-section estimate for black hole production remain valid if the classical description breaks down due to quantum effects?
• RQ4 What is the relationship between the shock wave width $ w $ and the characteristic frequency $ \omega $ at which quantum fluctuations become significant?
• RQ5 How do the quantum properties of the shock wave field compare to those in macroscopic black hole collisions, where classical gravity is well-justified?

Key findings

• The occupation number of gravitons with frequency $ \omega \sim w^{-1} $ is $ N_{\omega \sim w^{-1}} \sim E^{-(D-2)/(D-3)} \ll 1 $, indicating quantum fluctuations dominate.
• Classical gravity is only valid at distances $ |u| \gg \omega_{\text{max}}^{-1} $, which is strictly smaller than the region $ |u| \gtrsim w $ needed to resolve the shock structure.
• The condition $ N_\omega \gg 1 $ fails for modes near $ \omega \sim w^{-1} $, invalidating the classical description in the region critical for black hole formation.
• The curvature in the collision region exceeds Planck scale, violating the first classicality condition previously identified in [7], and quantum effects further undermine classical validity.
• The geometric cross-section estimate $ \sigma \sim \pi b_{\text{max}}^2 $ is fundamentally questionable, as it relies on classical gravity that breaks down in the relevant regime.
• In contrast to macroscopic black hole collisions, where $ N_{\omega \sim w^{-1}} \gg 1 $, the microscopic case exhibits strong quantum suppression due to low graviton occupation numbers.