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=π/2Λ, 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∈[0,π/2Λ], and the particles and antiparticles cannot be found elsewhere. Next, we provide a conditionally exact solution in the form of the general Heun function HG(a,q,α,β,γ,δ,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−β˜|E|)−1,h(u)=1; and f(u)=1,h(u)=1−β˜|E|υ, with υ=1,2, where u=|E|/Ep, β˜=β/Ep, and β 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 Ep 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.