Lense-Thirring Acoustic Black Hole

• LTABH is an analogue gravity system that models rotating black holes through acoustic media while capturing key features like event horizons and frame-dragging.
• Its metric structure incorporates an acoustic parameter and a rotation term, resulting in distinct optical and acoustic horizons along with novel trapping regions.
• Observable phenomena such as geodesic dynamics, shadow formation, and resonance modes offer experimental testbeds in Bose–Einstein condensates and classical fluids.

The Lense-Thirring Acoustic Black Hole (LTABH) constitutes a class of analogue gravity systems whose effective spacetime geometry derives from the interplay of rotation and acoustic flow properties in media such as Bose–Einstein condensates and barotropic fluids. LTABHs mimic salient features of rotating gravitational black holes—including event horizons, ergospheres, frame-dragging, and shadow formation—within a fluid-dynamical or quantum context. The metric structure, trapping surfaces, geodesic dynamics, and observable signatures encode both photonic (optical) and phononic (acoustic) physics, governed centrally by the acoustic parameter $\xi$ and the rotation parameter $a$ (Balali et al., 28 Dec 2025, Vieira et al., 2021, Chakraborty et al., 2015).

1. Metric Structure of LTABHs

LTABHs generalize the Lense-Thirring (slowly rotating Kerr) spacetime by incorporating a tunable acoustic parameter $\xi$, yielding a line element: $$ds^2 = -F(r)\,dt^2 + \frac{dr^2}{F(r)} + r^2(d\theta^2+\sin^2\theta\,d\phi^2) - 2\,[1-F(r)]\,a\,\sin^2\theta\,dt\,d\phi$$ with

$$f(r) = 1-\frac{2M}{r}, \qquad F(r) = f(r)[1 - f(r)(1 - f(r))\xi]$$

where $M$ is the background black hole mass, $\xi>0$ the acoustic parameter, and $a\ll M$ the dimensionless rotation parameter. For $\xi\to 0$ one recovers standard Lense-Thirring/Kerr geometry (Balali et al., 28 Dec 2025, Vieira et al., 2021). In the 2+1D "draining bathtub" analogue, the fluid velocity profile is $\vec{v} = -A/r\,\hat r + B/r\,\hat\phi$, and the induced metric exhibits similar frame-dragging properties (Chakraborty et al., 2015).

2. Horizon and Region Structure

The LTABH spacetime possesses up to three distinct horizons derived from the roots of $F(r) = 0$:

• Standard (optical) event horizon: $r_H = 2M$.
• Acoustic horizons: $r_{\rm ac}^{(\pm)} = M(\xi \pm \sqrt{\xi(\xi-4)})$ (for $\xi > 4$).

This structure partitions the spacetime into four regions: | Region | Range | Physical Escape | |-----------------|----------------------|----------------------| | I | $r < r_H$ | No escape | | II | $r_H<r<r_{\rm ac}^{(-)}$ | Light only | | III | $r_{\rm ac}^{(-)}<r<r_{\rm ac}^{(+)}$ | Both trapped | | IV | $r>r_{\rm ac}^{(+)}$ | Phonons, photons |

The existence of double acoustic horizons for $\xi>4$ introduces novel trapping domains absent in standard Kerr scenarios. The ergoregion coincides with the acoustic horizons at leading order in $a$, and distortion only emerges at higher orders (Balali et al., 28 Dec 2025, Vieira et al., 2021).

3. Geodesic Dynamics and Shadow Phenomena

Null geodesics in LTABH backgrounds admit effective potentials whose maxima determine critical radii for photon and acoustic spheres: $$\left(\frac{dr}{d\tau}\right)^2 = E^2 - \frac{2 a E L}{r^2}[1-F(r)] - \frac{F(r)}{r^2}(L^2 + \mathcal K)$$ Unstable circular orbits (photon sphere $r_{ps}$, acoustic sphere $r_{as}$) satisfy $V_{\rm eff}=0$ and $V_{\rm eff}^\prime=0$. The observable shadows,

$$R = \sqrt{\zeta^2 + \eta}, \qquad \zeta(r), \eta(r) \text{ as in impact parameter formulas}$$

result in:

• The acoustic shadow $R_{as}$ is circular, insensitive to $a$.
• The optical shadow $R_s$ shifts rightward with increasing $a$.
• Both radii scale with $\xi$, with higher $\xi$ yielding larger shadows.

Distortion $\delta = (x_{\rm left}-x_{\rm right})/R$ is negligible for $R_{as}$ but grows with $a$ for $R_s$ (Balali et al., 28 Dec 2025).

4. Frame-Dragging and Lense-Thirring Precession

LTABHs reproduce the inertial frame dragging effect, quantified by the Lense-Thirring precession frequency. In the equatorial plane, the frequency reads: $$\Omega(r) = \frac{a\,M\,\bigl(r^2+\xi(r-2M)^2\bigr)\, [\xi(r-6M)(r-2M)+r^2]^{1/2}}{r^2(r-2M)[r^2-2M\xi(r-2M)]^{3/2}}$$ Key features:

• $\Omega(r) \to \infty$ near acoustic horizons ($r_{\rm ac}^{(\pm)}$), reflecting strong local frame dragging.
• Null-drag radii ($\Omega=0$) exist for certain values, found by solving $\Omega(r)=0$; at these radii, test spins feel no Lense-Thirring precession.
• In simpler "draining bathtub" analogues, $\Omega_{LT}(r)$ diverges at the ergosphere ($r=r_E$) and decays as $1/r^4$ at infinity (Chakraborty et al., 2015).

5. Quasinormal Modes, Resonances, and Hawking-Unruh Radiation

Scalar (sound) perturbations in the LTABH background obey a massless wave equation, admitting mode expansions: $$\theta_1(t, r, \theta, \phi) = e^{-i\omega t} R(r) Y_{\ell m}(\theta, \phi)$$ The radial equation is of Heun type, with resonance frequencies

$$\omega_{mn}^{(\pm)} = \frac{-i(n+1)B + AB \pm \sqrt{A^2 - 2i(n+1)A - n(n+2) - B^2}}{B^2-1}$$

Decay rates (Im $\omega<0$) confirm modal stability. The Hawking-Unruh temperature for acoustic radiation is

$$T_H = \frac{\kappa}{2\pi}, \quad \kappa = \frac{(r_h - r_{ac-})(r_h - r_s)}{2 r_h^2}$$

where $r_h = r_{ac+}$ (Vieira et al., 2021).

6. Light Deflection and Analogue Observables

The deflection of light in the LTABH spacetime, computed via the Gauss-Bonnet method, reads

$$\widehat\alpha = \frac{4M(1+\xi)}{b} - \frac{4aM(1+\xi)}{b^2} + \mathcal O(M^2, a^2)$$

The prefactor $(1+\xi)$ amplifies light bending compared to non-acoustic backgrounds. Experiments might image both acoustic and optical shadows, as phonon and photon trajectories are differentially trapped by the LTABH structure (Balali et al., 28 Dec 2025).

7. Experimental Realizations and Prospective Observables

LTABHs have been proposed for laboratory realization in systems such as:

• Bose–Einstein condensates: engineered via rotating traps and atom sinks to reproduce vortex flow and horizon conditions. Frame-dragging frequencies near the ergosphere reach detectable ranges (10–$10^3$ Hz).
• Classical fluids: dye advection and tracer-particle visualization can qualitatively demonstrate analogue frame dragging, despite metric extraction challenges due to viscosity and non-relativistic effects.
• Spinor BECs: magnon-phonon coupling enables gyroscopic probing of $\Omega_{LT}$.
• Optical media: nonlinear propagation and phase shifts in metamaterial vortex arrays could offer routes to simulate and observe enhanced light deflection.

Detection strategies focus on phase-shift measurement, shadow imaging, and resonance phenomena (such as superresonance and discrete spectral shifts). The LTABH architecture provides a platform for simultaneous study of classical and quantum aspects of horizon physics, including analogue Hawking radiation and modal resonances (Chakraborty et al., 2015, Balali et al., 28 Dec 2025, Vieira et al., 2021).