Fermion-antifermion pairs in magnetized spacetime generated by a point source

In this research, we study fermion-antifermion pairs in a magnetized spacetime induced by a point-like source and characterized by an angular deficit parameter, $\alpha$. In the rest frame, the relative motion ($\propto r$) of these pairs is analyzed using exact solutions of a two-body Dirac equation with a position-dependent mass expressed as $m(r) = m_0 + \mathcal{S}(r)$. We select the Lorentz scalar potential $\mathcal{S}(r) = -\alpha_c/r$, which modifies the rest mass in a manner analogous to an attractive Coulomb potential, and derive analytical solutions to the resulting radial wave equation. Our findings are applicable to pairs in flat spacetime when $\alpha = 1$ without loss of generality. We elucidate how the spectra of such pairs are influenced by the spacetime background. Additionally, we observe that even the well-known non-relativistic energy ($\propto \alpha_c^2$) reflects the influence of the parameter $\alpha$ in positronium-like fermion-antifermion systems. We propose that our results can also be extended to study charge carriers in magnetized monolayer materials. Furthermore, we demonstrate that the metric for a 2+1-dimensional spinning point source background can be transformed into the metric describing the near-horizon region of a rotating BTZ black hole, a result not previously reported in the literature. This metric holds potential for providing meaningful insights into topics such as holographic superconductivity and quantum critical phenomena in future research