[Paper Review] Liouville Black Hole In A Noncommutative Space

K. Bilal, A. El Boukili+2 more|arXiv (Cornell University)|Sep 14, 2011

Noncommutative and Quantum Gravity Theories5 references3 citations

TL;DR

This paper investigates the effects of space noncommutativity on the two-dimensional Liouville black hole by deriving noncommutative solutions to the equations of motion. It computes the ADM mass, horizon, and scalar Ricci curvature in noncommutative spacetime, showing that the noncommutative corrections modify classical expressions through a deformation parameter $ heta$, with the commutative limit recovered when $ heta \to 0$. The key result is a modified horizon structure dependent on $ heta$ and momentum-dependent coordinates.

ABSTRACT

The space-noncommutativity adapted to the Liouville black hole theory is studied in the present work. Among our contributions, we present the solutions of noncommutative Liouville Black hole equations of motion and find their classical properties such as the ADM mass, the horizon and the scalar Ricci curvature.

Motivation & Objective

• To study the impact of space noncommutativity on the two-dimensional Liouville black hole model.
• To derive noncommutative solutions to the equations of motion in a 2D gravity–Liouville field system.
• To compute classical black hole properties—ADM mass, horizon, and scalar curvature—within the noncommutative framework.
• To establish a consistent link between noncommutative and commutative solutions by recovering the classical limit when the noncommutative parameter $θ \to 0$.

Proposed method

• Introduce noncommutative coordinates $̂{x}_{\mu}$ satisfying $[\u0302{x}_{\mu}, \u0302{x}_{\nu}] = i\theta_{\mu\nu}$, with $\theta_{\mu\nu}$ as a constant antisymmetric tensor.
• Apply the $\star$-product formalism to the metric and field equations, replacing ordinary products with star products to encode noncommutative geometry.
• Use the Weyl–Wigner correspondence to map noncommutative operators to functions, enabling the derivation of deformed classical quantities.
• Derive noncommutative versions of the metric $\alpha(\u0302{x})$ by replacing $x \to \u0302{x} = x - \frac{1}{2}\theta_{ij}p_j$ in classical expressions.
• Compute the ADM mass, horizon, and Ricci scalar in the noncommutative regime using the deformed metric and field configurations.
• Verify consistency by showing that all noncommutative expressions reduce to their commutative counterparts in the limit $\theta \to 0$.

Experimental results

Research questions

• RQ1 How does space noncommutativity modify the classical solutions of the Liouville black hole in two-dimensional gravity?
• RQ2 What is the form of the noncommutative ADM mass, and how does it differ from the commutative case?
• RQ3 How are the horizon location and scalar curvature deformed by the noncommutative parameter $\theta$?
• RQ4 Can the noncommutative black hole solutions be consistently reduced to the classical commutative limit when $\theta \to 0$?
• RQ5 What is the role of the noncommutative parameter $\theta$ in altering the geometric and physical properties of the black hole?

Key findings

• The noncommutative ADM mass remains equal to the commutative mass $\mathcal{M}$ when $\theta = 0$, indicating no quantum correction to the mass in the classical limit.
• The scalar Ricci curvature in the $\xi \neq 1/4, A \neq 0$ case is given by $R = -2M / \left(x - \frac{1}{2}\frac{L\theta}{x}\right)$, showing a $\theta$-dependent deformation of the classical $R = 2M/x$.
• For the $\xi = 1/4, A \neq 0$ case, the scalar curvature becomes $R = \frac{8\pi G a \Lambda}{a + 4\pi G \gamma} \exp\left(-2M\left(x - \frac{1}{2}\frac{L\theta}{x}\right)\right)$, with exponential dependence on the noncommutative correction.
• The horizon in the $\xi = 1/4, A = 0$ case is given by $x_H = \frac{M}{4D} \exp(2a\phi_H) \left(1 + \sqrt{1 + \frac{2\eta\theta}{(M/4D)^2 \exp(4a\phi_H)}}\right)$, exhibiting a nontrivial $\theta$-dependent shift.
• In the $\xi \neq 1/4, A \neq 0$ case, the horizon satisfies $x_H = \frac{x_0^{2-p}}{2} \exp(2a\phi) \left(1 + \sqrt{1 + \frac{2(2-p)L\theta}{x_0^{4-2p} \exp(4a\phi)}}\right)$, showing a $\theta$-induced correction to the classical horizon.
• All noncommutative solutions reduce to their classical counterparts from Ref. [2] when $\theta \to 0$, confirming consistency with the commutative limit.