Noncommutative QFT and Relative Entropy on Axisymmetric Bifurcate Killing Horizons

Abstract

We construct a deformed algebraic quantum field theory on bifurcate Killing horizons in stationary axisymmetric spacetimes. The deformation is generated by the commuting actions of affine dilations along the null generators of the horizon and rotations about the axis of symmetry, analogously to the Moyal-Rieffel deformation. Physically, this effectively implements a noncommutative geometric structure of the horizon. Moreover, we compute the relative entropy between coherent states in the deformed horizon theory, which remains strictly positive and exhibits a novel second-order correction in the deformation parameter, which becomes particularly significant for black holes whose horizon area is sufficiently small for Planck-scale effects to become non-negligible.

Figures (3)

• Figure 1: Structure of a bifurcate Killing horizon, consisting of the null hypersurfaces $\mathcal{H}_A, \mathcal{H}_B$, intersecting at the horizon cross-section $\mathcal{S}$, where $\mathcal{H}_A$ is affinely parametrized by $V$, and similarly $\mathcal{H}_B$ by $U$. The spacetime is separated into the wedge-shaped regions $\mathcal{L}, \mathcal{R}, \mathcal{F},\mathcal{P}$, in particular, dividing and the Killing horizons $\mathcal{H}_A, \mathcal{H}_B$ into the parts $\mathcal{H}_A^\mathcal{R},\mathcal{H}_A^\mathcal{L}$ and $\mathcal{H}_B^\mathcal{L},\mathcal{H}_B^\mathcal{R}$, respectively. The green line depicts the Killing flow $\left. \phi_t \right\vert_\mathcal{H}$ generated by $\xi^a$, projected to $\mathcal{H}^\mathcal{R}$.
• Figure 2: Schematic illustration of the future Killing horizon $\mathcal{H}_A$ (green cone), and a congruence of null geodesics along the horizon (green rays), affinely parametrized by the lightlike coordinate $V$. Fixing the coordinate $\vartheta$, the corresponding circular cross section $\mathcal{S}_\vartheta$ serves as a transverse section of the horizon, while the azimuthal coordinate $\varphi\in(-\pi,\pi)$ parametrizes the individual null rays within the congruence. In analogy with the null-plane analysis of MTW:2022npa, the horizon theory on $\mathcal{H}_A$ admits a natural interpretation in terms of a decomposition into identical chiral CFTs propagating along the affine parameter $V$, labelled by $\varphi$.
• Figure 3: Qualitative sketch illustrating the decomposition $f\star_\Theta g =: F_\Theta := F_N + F_\infty$. The functions $f,g\in\mathscr{D}_{\mathcal{H}_A}$ (drawn in blue and red) have compact $\varphi$-support contained within $(-\pi,\pi)$. The finite-order contribution $F_N$, defined by \ref{['FiniteExpansionProduct']}, is supported in $\mathrm{supp}(F_N) = \mathrm{supp}(f)\cap\mathrm{supp}(g)$, as depicted by the continuous purple curve. The higher-order remainder $F_\infty$ is shown as a dashed curve, indicating that its support generally extends beyond $(-\pi,\pi)$. This illustrates that, while finite truncations preserve the localization within $\mathscr{D}_{\mathcal{H}_A}$, the full deformed product generally belongs to an extended algebra. A similar picture holds for the $V$-support of $f$ and $g$.

Theorems & Definitions (17)

• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Proposition 3.4
• Lemma 3.5
• Corollary 3.6
• Corollary 3.7
• Theorem 3.8
• Theorem 3.9
• Theorem 3.10