Klein-Gordon particles in a nonuniform external magnetic field in Bonnor-Melvin rainbow gravity background

We investigate the effect of rainbow gravity on Klein-Gordon (KG) bosons in a quantized nonuniform magnetic field in the background of Bonnor-Melvin (BM) spacetime with a cosmological constant. In the process, we show that the BM spacetime introduces domain walls (i.e., infinitely impenetrable hard walls) at $r = 0$ and $r = \pi/\sqrt{2\Lambda}$, as a consequence of the effective gravitational potential field generated by such a magnetized BM spacetime. As a result, the motion of KG particles/antiparticles is restricted indefinitely within the range $r \in [0, \pi/\sqrt{2\Lambda}]$, and the particles and antiparticles cannot be found elsewhere. Next, we provide a conditionally exact solution in the form of the general Heun function $H_G(a, q, \alpha, \beta, \gamma, \delta, z)$. Within the BM domain walls and under the condition of exact solvability, we study the effects of rainbow gravity on KG bosonic fields in a quantized nonuniform external magnetic field in the BM spacetime background. We use three pairs of rainbow functions: $f(u) = (1 - \tilde{\beta} |E|)^{-1}, \, h(u) = 1$; and $f(u) = 1, \, h(u) = \sqrt{1 - \tilde{\beta} |E|^\upsilon}$, with $\upsilon = 1,2$, where $u = |E| / E_p$, $\tilde{\beta} = \beta / E_p$, and $\beta$ is the rainbow parameter. We find that such pairs of rainbow functions, $(f(u), h(u))$, fully comply with the theory of rainbow gravity, ensuring that $E_p$ is the maximum possible energy for particles and antiparticles alike. Moreover, we show that the corresponding bosonic states form magnetized, rotating vortices, as intriguing consequences of such a magnetized BM spacetime background.