State-space Manifold and Rotating Black Holes

TL;DR

This paper investigates the geometric structure of fluctuations in higher-dimensional black holes within string theory, revealing stability, attractive interactions, and links to phase transitions through state-space geometry analysis.

Contribution

It introduces a Riemannian geometric framework for analyzing black hole microstate fluctuations, uncovering stability and correlation properties across various black hole solutions in string theory.

Findings

State-space geometry is non-degenerate and negatively curved for studied black holes.

Principal components of the metric are positive definite, off-diagonals are not.

Brane-brane correlations show asymmetry and effective attraction, indicating stability.

Abstract

We study a class of fluctuating higher dimensional black hole configurations obtained in string theory/ $M$-theory compactifications. We explore the intrinsic Riemannian geometric nature of Gaussian fluctuations arising from the Hessian of the coarse graining entropy, defined over an ensemble of brane microstates. It has been shown that the state-space geometry spanned by the set of invariant parameters is non-degenerate, regular and has a negative scalar curvature for the rotating Myers-Perry black holes, Kaluza-Klein black holes, supersymmetric $AdS_5$ black holes, $D_1$-$D_5$ configurations and the associated BMPV black holes. Interestingly, these solutions demonstrate that the principal components of the state-space metric tensor admit a positive definite form, while the off diagonal components do not. Furthermore, the ratio of diagonal components weakens relatively faster than the off diagonal components, and thus they swiftly come into an equilibrium statistical configuration. Novel aspects of the scaling property suggest that the brane-brane statistical pair correlation functions divulge an asymmetric nature, in comparison with the others. This approach indicates that all above configurations are effectively attractive and stable, on an arbitrary hyper-surface of the state-space manifolds. It is nevertheless noticed that there exists an intriguing relationship between non-ideal inter-brane statistical interactions and phase transitions. The ramifications thus described are consistent with the existing picture of the microscopic CFTs. We conclude with an extended discussion of the implications of this work for the physics of black holes in string theory.