Bosonic and Fermionic love number of static acoustic black hole

TL;DR

This work computes static tidal responses (tilde Love numbers) for scalar and Dirac perturbations of static acoustic black holes in $(3+1)$ and $(2+1)$ dimensions, using horizon regularity and large-radius matching to relate decaying and growing modes. In $(3+1)$ dimensions, the scalar Love number is generically nonzero with a closed-form in terms of Gamma functions, while the Fermionic Love numbers obey a universal power law $\mathcal{F}_{\pm\tfrac{1}{2},\ell m}=\pm 4^{-(\ell+1/2)}$; in $(2+1)$ dimensions, the scalar LN exhibits a logarithmic structure, vanishing for even $m$ and remaining nonzero for odd $m$, whereas the Fermionic LN remains a simple nonzero power law $F_m=4^{-m}$. These results illuminate how analogue gravity systems emulate (or fail to mimic) black-hole tidal responses across spin and dimensionality, highlighting qualitative differences between integer- and half-integer-spin fields. The findings provide benchmarks for experimental analogue setups and motivate further exploration of rotation and full frequency-dependent responses. $${}$

Abstract

We compute static ($ω\to0$) tilde Love numbers for scalar ($s=0$) and Dirac ($s=1/2$) perturbations of static acoustic black holes (ABHs) in (3+1) and (2+1) dimensions respectively. By imposing horizon regularity condition and matching to the large-radius expansion, we extract the ratio between decaying and growing modes. It turns out that in (3+1) dimensions the scalar Love number is generically nonzero for ABHs, while the Fermionic Love numbers follow a universal power-law form $F^{\pm1/2}_{\ell m}=\pm 4^{-(\ell+1/2)}$. In (2+1) dimensions the scalar field exhibits a strange logarithmic structure, causing the Bosonic Love number to vanish for even $m$ but remain nontrivial for odd $m$; In contrast, the Fermionic Love number in this case retains a simple power-law form $F_m=4^{-m}$ and is generically nonzero. These results provide insights into tidal response in analogue gravity systems and highlight qualitative differences between integer- and half-integer-spin fields.