Quasibound states of massless spin particles in Schwarzschild equivalent mediums

TL;DR

This work analyzes quasibound states of massless spin particles in Schwarzschild equivalent mediums—optical analogues that mimic Schwarzschild spacetime via the Gordon metric in isotropic coordinates. It derives a unified master equation valid for spin $s\le 2$, showing identical angular structure through spin-weighted harmonics and a radial envelope governed by a common equation, enabling a consistent treatment across spins. Under quasibound-state boundary conditions, the authors obtain simple closed-form frequencies that depend on the angular quantum number $l$ and a principal index $N$, with bosons and fermions following distinct spectra but sharing identical frequencies among particles with the same statistics. The results imply that electromagnetic waves in Schwarzschild-like metamaterials can simulate gravitational-wave dynamics, offering a realistic experimental pathway for analogue gravity and transformation-optics tests.

Abstract

We show that, in Schwarzschild equivalent mediums, the massless spin particles obey the same dynamical equation, from which we obtain remarkably simple formulae for the frequencies of the quasibound states. We find that the quasibound frequencies of different bosons can be identical at the same quantum number $l$, and the same is true of different fermions, but a quasibound frequency for bosons can never equal a quasibound frequency for fermions. These results mean that, in Schwarzschild equivalent mediums with the quasibound-state boundary conditions, characteristics of electromagnetic waves are the same as those for all the massless bosonic waves, thereby allowing electromagnetic waves to simulate gravitational waves. Our predictions can be tested in future experiments, building upon the successful preparation of Schwarzschild equivalent mediums.

Figures (2)

• Figure 1: Quasibound frequencies in units of $1/(3.5M)$ for $2l+1>\sqrt{15/7}N$. The absolute value of $Im(3.5M\omega)$ is an odd integer for bosons (B) and an even integer for fermions (F). All frequencies on the same curve have the same quantum number $l$. The curves towards to the top left of the graph correspond to larger $l$ valus, which are $0, 1, 2, 3, 4, 5, 6$, and $7$.
• Figure 2: Quasibound frequency spectrum with the smaller absolute value of $\omega$ in units of $1/(3.5M)$ for $N<2l+1<\sqrt{15/7}N$. $N$ is an odd integer for bosons (B) and an even integer for fermions (F). All frequencies on the same curve have the same quantum number $l$. From left to right, $l=0, 1, 2, 3, 4, 5, 6$, and $7$.