On the Klein-Gordon bosonic fields in the Bonnor-Melvin spacetime with a cosmological constant in rainbow gravity: Bonnor-Melvin Domain Walls

We investigate the effect of rainbow gravity on Klein-Gordon (KG) bosons in the background of the magnetized Bonnor-Melvin (BM) spacetime with a cosmological constant. We first show that the very existence of the sinusoidal term \(\sin^2(\sqrt{2\Lambda}r)\), in the BM space-time metric, suggests that \(\sin^2(\sqrt{2\Lambda}r) \in [0,1],\) which consequently restricts the range of the radial coordinate \(r\) to \(r \in [0,\pi/\sqrt{2\Lambda}]\). Moreover, we show that at \(r = 0\) and \(r = \pi/\sqrt{2\Lambda}\), the magnetized BM-spacetime introduces domain walls (infinitely impenetrable hard walls) within which the KG bosonic fields are allowed to move. Interestingly, the magnetized BM-spacetime introduces not only two domain walls but a series of domain walls. However, we focus on the range \(r \in [0,\pi/\sqrt{2\Lambda}]\). A quantum particle remains indefinitely confined within this range and cannot be found elsewhere. Based on these findings, we report the effects of rainbow gravity on KG bosonic fields in BM-spacetime. We use three pairs of rainbow functions: \( f(\chi) = \frac{1}{1 - \tilde{\beta} |E|}, \, h(\chi) = 1 \); \( f(\chi) = (1 - \tilde{\beta} |E|)^{-1}, \, h(\chi) = 1 \); and \( f(\chi) = 1, \, h(\chi) = \sqrt{1 - \tilde{\beta} |E|^\upsilon} \), with \(\upsilon = 1,2\). Here, \(\chi = |E| / E_p\), \(\tilde{\beta} = \beta / E_p\), and \(\beta\) is the rainbow parameter. We found that while the pairs \((f,h)\) in the first and third cases fully comply with the theory of rainbow gravity and ensure that \(E_p\) is the maximum possible energy for particles and antiparticles, the second pair does not show any response to the effects of rainbow gravity. We show that the corresponding bosonic states can form magnetized, spinning vortices in monolayer materials, and these vortices can be driven by adjusting an out-of-plane aligned magnetic field.