Massless KG-oscillators in Som-Raychaudhuri cosmic string spacetime in a fine tuned rainbow gravity

Abstract

A fine tuned rainbow gravity describes both relativistic quantum particles and anti-particles alike. That is, the ratio $y=E/E_{P}$ in the rainbow functions $g_{_{0}}\left( y\right) $ and $% g_{_{1}}\left( y\right) $ should be fine tuned into $0\leq y=E/E_{P}\leq 1\Rightarrow y=\left\vert E\right\vert /E_{P}$, otherwise rainbow gravity will only secure Planck's energy scale $E_p$ invariance for relativistic particles and the anti-particles are left unfortunate (in the sense that their energies will be indefinitely unbounded). Using this fine tuning we discuss the rainbow gravity effect on Klein-Gordon (KG) oscillators in Som-Raychaudhuri cosmic string rainbow gravity spacetime background. We use the rainbow functions: (i) $g_{_{0}}\left( y\right) =1$, $% g_{_{1}}\left( y\right) =\sqrt{1-\epsilon y^{n}}, n=0,1$, loop quantum gravity motivated pairs, (ii) $% g_{_{0}}\left( y\right) =g_{_{1}}\left( y\right) =\left( 1-\epsilon y\right) ^{-1}$, a horizon problem motivated pair, and (iii) $g_{_{0}}\left( y\right) =\left( e^{\epsilon y}-1\right) /\epsilon y$, $g_{_{1}}\left( y\right) =1$, a gamma-ray bursts motivated pair. We show that the energies obtained using the first two rainbow functions in (i) completely comply with the rainbow gravity model (on the invariance of the Planck's energy scale $E_{p}$). The rainbow function pair in (ii) has no effect on massless KG-oscillators. Whereas, the one in (iii) does not show any eminent tendency towards the invariance of the Planck's energy scale. Yet, we suggest a new rainbow function pair $(g_0(y)=(1-\epsilon y)^{-1}, g_1 (y)=1)$, and show that it secures invariance of the Planck's energy scale $E_p$. Moreover, similar performance is observed when this new pair is used for KG-oscillators and KG-Coulombic particles in cosmic string rainbow gravity spacetime and magnetic fields.