Quantum gravitational corrections at third-order curvature, acoustic analog black holes and their quasinormal modes

TL;DR

This work embeds quantum-gravity corrections at third order in curvature into an effective-field-theory framework with a single dimension-six operator $\mathcal{L}^{(3)}$ (coefficient $c_6$), yielding a corrected Schwarzschild metric with a horizon shift $r_h = 2 G_N M \left(1 - \frac{5 \pi c_6}{G_N^2 M^4}\right)$. It then establishes a quantitative gravity–acoustic analog by mapping perturbations on this background to sound waves in a de Laval nozzle, deriving the corresponding Schrödinger-type equation with a spin- and $c_6$-dependent $V_{\mathrm{eff}}$ and constructing nozzle profiles that reproduce the gravitational potential. Using the Mashhoon method with a Pöschl–Teller potential, the authors compute QNM frequencies $\omega_n$ for bosonic and fermionic perturbations across spins $s$, multipoles $\ell$, and higher overtones $n$, revealing sizable shifts controlled by $c_6$, most prominently for scalar and spin-1/2 channels. The results demonstrate that higher overtones extend the observable ringdown window and that laboratory analogs could provide empirical access to quantum-gravity signatures, with primordial BHs offering the most favorable regimes for sizable effects.

Abstract

Quasinormal modes for bosonic (scalar, electromagnetic, and axial gravitational) and fermionic field perturbations, radiated from black holes that carry quantum gravitational corrections at third order in the curvature to the Schwarzschild solution, are scrutinized from the propagation of analog transonic sound waves across a de Laval nozzle. The thermodynamic variables, the nozzle geometry, the Mach number, and the thrust coefficient are computed as functions of the parameter driving the effective action for quantum gravity containing a dimension-six local operator beyond general relativity. The quasinormal modes for quantum gravitational corrected analog black holes are also determined for higher overtones, yielding a more precise description of the quantum-corrected ringdown process and the gravitational waveform way before the fundamental mode sets in.

Figures (20)

• Figure 1: Saturation value for $c_6$ that maximises $V_\text{eff}$, versus its maximum effective potential value. The circle represents the global maximum value. The dotted black line considers $s=\ell = 0$, whereas the solid grey line considers $s=\ell=1/2$. Values calculated with natural units and $M=1$.
• Figure 2: Saturation value for $c_6$ that maximises $V_\text{eff}$, versus the orbital radius value relative to the Schwarzschild Radius $r_S = 2G_\text{N}M$. The circles represent the global minimum value for each combination of $s$ and $\ell$. The dotted black line considers $s=\ell = 0$, whereas the solid grey line considers $s=\ell=1/2$.
• Figure 3: Effective potential as a function of the longitudinal direction of the de Laval nozzle, varying the BH mass in the range $10^{27}$ kg $< M < 10^{36}$ kg and keeping $c_6$ fixed.
• Figure 4: Effective potential as a function of the longitudinal direction of the de Laval nozzle, varying $c_6$ for the quantum-corrected Schwarzschild metric \ref{['ck']}. The dashed black line represents the Schwarzschild solution. The values are calculated using $G_{{\hbox{N}}} = M = 1$.
• Figure 5: Nozzle shape geometry as a function of the longitudinal direction of the de Laval nozzle, varying $c_6$ for the quantum-corrected Schwarzschild metric \ref{['ck']}. The dashed black line represents the Schwarzschild solution. The values are calculated using $G_{{\hbox{N}}} = M = 1$.