PDM KG-Coulombic particles in cosmic string rainbow gravity spacetime and a uniform magnetic field

Rainbow gravity (RG) has attracted research attention over the years R1 ; R2 ; R3 ; R4 ; R5 as a semi-classical extension of the deformed/doubly special relativity into general relativity (GR). It suggests that the energy of the probe particles affects the spacetime background, at the ultra-high energy regime, and the spacetime metric becomes energy-dependent R5 ; R6 ; R7 ; R8 ; R9 ; R10 ; R11 ; R12 ; R13 ; R14 ; R15 ; R16 ; R17 . That is, a cosmic string spacetime metric (in the natural units $c=\hbar=G=1$)

$$ds^{2}=-dt^{2}+dr^{2}+\alpha^{2}\,r^{2}d\varphi^{2}+dz^{2},$$

(1)

would, under RG, take an energy-dependent form

$$ds^{2}=-\frac{1}{g_{{}_{0}}\left(y\right)^{2}}dt^{2}+\frac{1}{g_{{}_{1}}\left(y\right)^{2}}\left(dr^{2}+\alpha^{2}\,r^{2}d\varphi^{2}+dz^{2}\right);\;y=E/E_{p},$$

(2)

where $\alpha$ is a constant related to the deficit angle of the conical spacetime and is defined as $\alpha=1-4G\mu$, $G$ is the Newton’s constant, $\mu$ is the linear mass density of the cosmic string so that $\alpha<1,E$ is the energy of the probe particle, and $E_{p}=\sqrt{\hbar c^{5}/G}$ is the Planck energy. Here, the signature of the line elements (1) and (2) is $\left(-,+,+,+\right)$. Moreover, the corresponding metric tensor $g_{\mu\nu}$ reads

$$g_{\mu\nu}=diag\left(-\frac{1}{g_{{}_{0}}\left(y\right)^{2}},\frac{1}{g_{{}_{1}}\left(y\right)^{2}},\frac{\alpha^{2}\,r^{2}}{g_{{}_{1}}\left(y\right)^{2}},\frac{1}{g_{{}_{1}}\left(y\right)^{2}}\right);\;\mu,\nu=t,r,\varphi,z,$$

(3)

and

$$\det(g_{\mu\nu})=-\frac{\alpha^{2}\,r^{2}}{g_{{}_{0}}(y)^{2}g_{{}_{1}}(y)^{6}}\Longrightarrow g^{\mu\nu}=diag(-g_{{}_{0}}(y)^{2},g_{{}_{1}}(y)^{2},\frac{g_{{}_{1}}(y)^{2}}{\alpha^{2}\,r^{2}},g_{{}_{1}}(y)^{2}).$$

(4)

where $g_{{}_{0}}\left(y\right)$, $g_{{}_{1}}\left(y\right)$ are the rainbow functions.

The Planck energy $E_{p}$, in the RG model, is considered to represent a threshold separating classical from quantum mechanical descriptions to introduce itself as yet another invariant energy scale alongside the speed of light. Consequently, rainbow gravity justifies the modification of the relativistic energy-momentum dispersion relation into

$$E^{2}g_{{}_{0}}\left(y\right)^{2}-p^{2}c^{2}g_{{}_{1}}\left(y\right)^{2}=m^{2}c^{4};\;0\leq\left(y=E/E_{p}\right)\leq 1,$$

(5)

where $mc^{2}$ is its rest mass energy. Such a modification is significant in the ultraviolet limit and is constrained to reproduce the standard GR dispersion relation in the infrared limit so that

$$\lim\limits_{y\rightarrow 0}g_{{}_{k}}\left(y\right)=1;\;k=0,1.$$

(6)

The effects of such modifications could be observed, for example, in the tests of thresholds for ultra high-energy cosmic rays R6 ; R13 ; R14 ; R15 , TeV photons R16 , gamma-ray bursts R6 , nuclear physics experiments R17 . Rainbow gravity settings have motivated interesting recent studies on the associated quantum gravity effects. Such studies include, for example, the thermodynamical properties of black holes R18 ; R19 ; R20 ; R21 ; R211 , the dynamical stability conditions of neutron stars R22 , thermodynamic stability of modified black holes R221 , charged black holes in massive RG R222 , on geometrical thermodynamics and heat engine of black holes in RG R223 , on RG and f(R) theories R224 , the initial singularity problem for closed rainbow cosmology R23 , the black hole entropy R24 , the removal of the singularity of the early universe R25 , the Casimir effect in the rainbow Einstein’s universe R8 , massive scalar field in RG Schwarzschild metric R26 , five-dimensional Yang–Mills black holes in massive RG R27 , etc.

On the other hand, the dynamics of Klein-Gordon (KG) particles (i.e., spin-0 mesons), Dirac particles (spin-1/2 fermionic particles), and Duffen-Kemmer-Peatiau (DKP) particles (spin-1 particles like bosons and photons) in different spacetime backgrounds in rainbow gravity are studied. For example, in a cosmic string spacetime background in rainbow gravity, Bezzerra et al. R8 ; R81 have studied Landau levels via Schrödinger and KG equations, Bakke and Mota R28 have studies the Dirac oscillator, they have also studied the Aharonov-Bohm effect R29 . Hosseinpour et al. R5 have studied the DKP-particles, Sogut et al. R11 have studied the quantum dynamics of photon, Kangal et al. R12 have studied KG-particles in a topologically trivial Gödel-type spacetime in rainbow gravity, and very recently KG-oscillators in cosmic string rainbow gravity spacetime in a non-uniform magnetic field are studied by Mustafa R291 (without and with the position-dependent mass (PDM) settings). In the current proposal, however, we extend such studies and consider PDM KG-Coulombic particles in cosmic string rainbow gravity spacetime and a uniform magnetic field.

One should be reminded, nevertheless, that PDM is a metaphoric notion that emerges as a manifestation of coordinate transformation/deformation that renders the mass to become effectively position-dependent R30 ; R31 ; R32 ; R33 ; R34 ; R35 ; R36 ; R37 . PDM concept has been introduced  in the study PDM KG-oscillators in cosmic string spacetime within Kaluza-Klein theory R38 , in (2+1)-dimensional Gürses spacetime backgrounds R39 , and in Minkowski spacetime with space-like dislocation R40 . Basically, for the PDM von Roos Schrödinger Hamiltonian R30 , it has been shown (c.f., e.g., R31 ; R32 ; R33 ) that the PDM momentum operator takes the form $\hat{p}_{j}(\mathbf{r})=-i[\partial_{j}-\partial_{j}f(\mathbf{r})/4f(\mathbf{r})]\,;\;j=1,2,3,$ where $f\left(\mathbf{r}\right)$ is a positive-valued dimensionless scalar multiplier. For more details on this issue the reader may refer to R31 ; R33 ; R38 ; R39 ; R40 ; R401 ; R402 . This assumption would, in turn, allow one to cast the PDM von Roos kinetic energy operator (using $\hbar=2m=1$ units in the von Roos Hamiltonian) as $\hat{T}(\mathbf{r})\psi(\mathbf{r})=-f(\mathbf{r})^{-1/4}(\mathbf{\nabla\,}f(\mathbf{r})^{-1/2})\cdot(\mathbf{\nabla\,}f(\mathbf{r})^{-1/4}\psi(\mathbf{r}))$ (known in the literature as Mustafa-Mazharimousavi’s PDM kinetic energy operator R32 ). Which suggests that the momentum operator for constant mass setting, $\hat{p}_{j}-i\,\partial_{j}$, should be replaced by the PDM operator $\hat{p}_{j}\left(\mathbf{r}\right)$ for PDM settings. We shall use such PDM recipe in the current study of PDM KG-Coulombic particles in cosmic string rainbow gravity spacetime and a uniform magnetic field. We shall be interested in three pairs of rainbow functions: (a) $g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}$, and $g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y}$, which belong to the set of rainbow functions $g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{n}}$ (where $\epsilon$ is a dimensionless constant of order unity) used to describe the geometry of spacetime in loop quantum gravity R41 ; R42 , (b) $g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=\left(1-\epsilon y\right)^{-1}$, a suitable set used to resolve the horizon problem R13 ; R43 , and (c) $g_{{}_{0}}(y)=(e^{\epsilon y}-1)/\epsilon y$ and $g_{{}_{1}}\left(y\right)=1$, which are obtained from the spectra of gamma-ray bursts at cosmological distances R6 .

The organization of our paper is in order. We discuss, in section 2, PDM KG-particles in the cosmic string rainbow gravity spacetime (2) and a uniform magnetic field. We show that the corresponding KG-equation collapses into the two-dimensional radial Schrödinger-Coulomb equation. In section 3, we discuss the RG effect (using the above mention rainbow functions sets) on the spectroscopic structure of KG-Coulombic constant mass particles. We discuss, in section 4, the effects of rainbow gravity as well as PDM on the energy levels of a PDM KG-Coulombic particle. Our concluding remarks are given in section 5.

In the cosmic string rainbow gravity spacetime background (2), a KG-particle of charge $e$ in a 4-vector potential $A_{\mu}$ is described (in $c=\hbar=G=1$ units) by the KG-equation

$$\frac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}g^{\mu\nu}D_{\nu}\Psi\right)=m^{2}\Psi,$$

(7)

where $D_{\mu}$ is the gauge-covariant derivative given by $D_{\mu}=\partial_{\mu}-ieA_{\mu}$, and $m$ is the rest mass energy of the KG-particle. At this point, we may also include position-dependent mass (PDM) settings (a metaphoric description of deformed coordinates and inherited from the von Roos Hamiltonian R30 ) using the PDM-momentum operator $\hat{p}_{j}(\mathbf{r})=-i[\partial_{j}-\partial_{j}\,f(\mathbf{r})/4\,f(\mathbf{r})]$ R31 ; R32 ; R33 ; R38 ; R39 ; R40 ; R401 ; R402 . In this case, $D_{\mu}\longrightarrow\tilde{D}_{\mu}=D_{\mu}+\mathcal{F}_{\mu}=\partial_{\mu}+\mathcal{F}_{\mu}-ieA_{\mu}$, where $\mathcal{F}_{\mu}=(0,\mathcal{F}_{r},0,0)$, $\mathcal{F}_{r}=f^{\prime}\left(r\right)/4\,f\left(r\right)$ and $f\left(\mathbf{r}\right)=f\left(r\right)$ is only radially dependent. One should notice that a KG-oscillator is obtained using $f(r)=\exp(2\beta r^{2})$, where $f\left(r\right)$ is a positive dimensionless scalar multiplier. Under such new structure our KG-equation (7) now describes PDM KG-particles in the cosmic string rainbow gravity spacetime and reads

$$\frac{1}{\sqrt{-g}}\tilde{D}_{\mu}\left(\sqrt{-g}g^{\mu\nu}\tilde{D}_{\nu}\right)\Psi=m^{2}\Psi\Longrightarrow\frac{1}{\sqrt{-g}}\left(D_{\mu}+\mathcal{F}_{\mu}\right)\sqrt{-g}g^{\mu\nu}\left(D_{\nu}-\mathcal{F}_{\nu}\right)\Psi=m^{2}\Psi.$$

(8)

Which, in a straightforward manner, yields

$$\left\{-g_{{}_{0}}\left(y\right)^{2}\partial_{t}^{2}+g_{{}_{1}}\left(y\right)^{2}\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}-M\left(r\right)+\frac{1}{\alpha^{2}\,r^{2}}\left(\partial_{\varphi}-ieA_{\varphi}\right)^{2}+\partial_{z}^{2}\right]\right\}\Psi\left(t,r,\varphi,z\right)=m^{2}\Psi\left(t,r,\varphi,z\right),$$

(9)

where

$$M\left(r\right)=\mathcal{F}_{r}^{\prime}+\frac{\mathcal{F}_{r}}{r}+\mathcal{F}_{r}^{2}=-\frac{3}{16}\left(\frac{f^{\prime}\left(r\right)}{f\left(r\right)}\right)^{2}+\frac{f^{\prime}\left(r\right)}{4rf\left(r\right)}+\frac{f^{\prime\prime}\left(r\right)}{4f\left(r\right)}$$

(10)

We now use the substitution

$$\Psi\left(t,r,\varphi,z\right)=\exp\left(i\left[\ell\varphi+k_{z}z-Et\right]\right)\psi\left(r\right),$$

(11)

in Eq. (9) to obtain

$$\left\{\tilde{E}^{2}+g_{{}_{1}}\left(y\right)^{2}\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}-M\left(r\right)-\frac{\left(\ell-eA_{\varphi}\right)^{2}}{\alpha^{2}\,r^{2}}\right]\right\}\psi\left(r\right)=0,$$

(12)

where

$$\tilde{E}^{2}=g_{{}_{0}}\left(y\right)^{2}E^{2}-g_{{}_{1}}\left(y\right)^{2}k_{z}^{2}-m^{2}$$

(13)

In what follows we shall consider $A_{\varphi}=\frac{1}{2}B_{\circ}r$, which in turn yields a non-uniform magnetic field $\mathbf{B}=\mathbf{\nabla}\times\mathbf{A}=B_{\circ}\,\hat{z}$. Consequently, Eq.(12) becomes

$$\left\{\lambda+\partial_{r}^{2}+\frac{1}{r}\partial_{r}-M\left(r\right)-\frac{\tilde{\ell}^{2}}{r^{2}}+\frac{\tilde{\ell}\,\tilde{B}}{r}\right\}\psi\left(r\right)=0,$$

(14)

where

$$\lambda=\frac{g_{{}_{0}}\left(y\right)^{2}E^{2}-g_{{}_{1}}\left(y\right)^{2}\left(k_{z}^{2}+\frac{\tilde{B}^{2}}{4}\right)-m^{2}}{g_{{}_{1}}\left(y\right)^{2}},\;\tilde{\ell}=\frac{\ell}{\alpha},\;\tilde{B}=\frac{eB_{\circ}}{\alpha}.$$

(15)

Moreover, with $\psi\left(r\right)=R\left(r\right)/\sqrt{r}$ we obtain

$$\left\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}-M\left(r\right)+\frac{\tilde{\ell}\,\tilde{B}}{r}+\lambda\right\}R\left(r\right)=0.$$

(16)

Under such spacetime and magnetic field structures, we shall consider two types of KG-particles: constant mass and PDM ones.

It is convenient to discuss the KG-particles with a standard constant mass, i.e., $f\left(r\right)=1\Longleftrightarrow$ $M\left(r\right)=0$, so that Eq.(16) reduces into the two-dimensional Schrödinger-oscillator form

$$\left\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}+\frac{\tilde{\ell}\,\tilde{B}}{r}+\lambda\right\}R\left(r\right)=0.$$

(17)

Which obviously admits exact solution in the form of hypergeometric function so that

$$R\left(r\right)\sim\,\left(2i\sqrt{\lambda}r\right)^{|\tilde{\ell}|+1/2}\exp\left(-i\sqrt{\lambda}r\right)\,_{1}F_{1}\left(\frac{1}{2}+\tilde{\ell}|-\frac{\tilde{\ell}\,\tilde{B}}{2i\sqrt{\lambda}},1+2|\tilde{\ell}|,2i\sqrt{\lambda}r\right).$$

(18)

However, to secure finiteness and square integrability we need to terminate the hypergeometric function into a polynomial of degree $n_{r}\geq 0$ so that the condition

$$\frac{1}{2}+\tilde{\ell}|-\frac{\tilde{\ell}\,\tilde{B}}{2i\sqrt{\lambda}}=-n_{r}.$$

(19)

is satisfied. This would in turn imply that

$$i\sqrt{\lambda}=\frac{\tilde{\ell}\,\tilde{B}}{2\tilde{n}};\;\tilde{n}=n_{r}+|\tilde{\ell}|+\frac{1}{2}\Rightarrow\lambda_{n_{r},\ell}=-\frac{\tilde{\ell}\,^{2}\tilde{B}^{2}}{4\tilde{n}^{2}},$$

(20)

and

$$\psi\left(r\right)=\frac{R\left(r\right)}{\sqrt{r}}=\mathcal{N}\,r^{|\tilde{\ell}|}\exp\left(-\frac{|\tilde{\ell}\,\tilde{B}|}{2\tilde{n}}\,r\right)\,_{1}F_{1}\left(-n_{r},1+2|\tilde{\ell}|,\frac{|\tilde{\ell}\,\tilde{B}|}{\tilde{n}}r\right).$$

(21)

Consequently, Eq.(15) would read

$$g_{{}_{0}}\left(y\right)^{2}E^{2}-m^{2}=g_{{}_{1}}\left(y\right)^{2}\,K_{n_{r},\ell};\;K_{n_{r},\ell}=\frac{\tilde{B}^{2}}{4}\left[1-\frac{\tilde{\ell}^{2}}{\left(n_{r}+|\tilde{\ell}|+\frac{1}{2}\right)^{2}}\right]+k_{z}^{2}.$$

(22)

At this point, it is convenient to mention that the choice of $i\sqrt{\lambda}=\tilde{\ell}\,\tilde{B}/2\tilde{n}\geq 0\Rightarrow i\sqrt{\lambda}=|\tilde{\ell}\,\tilde{B}|/2\tilde{n}$ is manifested by the requirement of finiteness and square integrability of $\psi\left(r\right)$ as $r\rightarrow\infty$. Interestingly, moreover, we notice that all $S$-states (i.e., $\ell=0$ states) are degenerate with each other (positive with positive and negative with negative states) and have the same $K_{n_{r},0}$ value

$$K_{0,0}=K_{1,0}=\cdots=K_{n_{r},0}=\tilde{B}^{2}/4+k_{z}^{2}$$

(23)

as suggested by Eq.(22). This is a consequence of the cosmic string spacetime (i.e., at $\lim\limits_{\epsilon\rightarrow 0}g_{{}_{0}}\left(y\right)=\lim\limits_{\epsilon\rightarrow 0}g_{{}_{1}}\left(y\right)=1$) and has nothings to do with rainbow gravity. Moreover, for a given $n_{r}$ and $\ell$ we have $K_{n_{r},|\ell|}=K_{n_{r},-|\ell|}$, which indicates that we shall have eminent degeneracies associated with the magnetic quantum number $\ell=\pm|\ell|\neq 0$ for each radial quantum number $n_{r}$. These effects are going to be reflected on the spectroscopic structure of constant mass KG-particles in cosmic string rainbow gravity spacetime, under different sets of rainbow functions.

To observe the rainbow gravity effects on such constant mass KG-particles we now consider different rainbow functions.

In this section, we consider a positive-valued dimensionless scalar multiplier in the form of $f\left(r\right)=\exp\left(4\eta r\right)$ in Eq.(10) so that $M(r)=\eta^{2}+\eta/r$. This would, in turn, imply that Eq.(16) reads

$$\left\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}+\frac{\left(-\eta+\tilde{\ell}\,\tilde{B}\right)}{r}+\tilde{\lambda}\right\}R\left(r\right)=0,\;$$

(28)

where

$$\tilde{\lambda}=\frac{g_{{}_{0}}\left(y\right)^{2}E^{2}-m^{2}}{g_{{}_{1}}\left(y\right)^{2}}-\left(k_{z}^{2}+\frac{\tilde{B}^{2}}{4}\right)-\eta^{2},$$

(29)

In this case, its exact solution is in the form of

$$R(r)=C\,r^{|\tilde{\ell}|+1/2}\exp(-i\sqrt{\tilde{\lambda}}r)\,_{1}F_{1}\left(\frac{1}{2}+|\tilde{\ell}|-\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{2i\sqrt{\tilde{\lambda}}},1+2|\tilde{\ell}|,2i\sqrt{\tilde{\lambda}}r\right),$$

(30)

which takes the form of a polynomial of degree $n_{r}\geq 0$ for

$$\frac{1}{2}+|\tilde{\ell}|-\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{2i\sqrt{\tilde{\lambda}}}=-n_{r}\Rightarrow\frac{1}{2}+|\tilde{\ell}|-\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{2i\sqrt{\tilde{\lambda}}}=-n_{r}\Rightarrow 2i\sqrt{\tilde{\lambda}}=\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{\tilde{n}};\;\tilde{n}=n_{r}+|\tilde{\ell}|+\frac{1}{2}.$$

(31)

Consequently, the eigen energies and wavefunctions, with $\tilde{\eta}=-\eta+\tilde{\ell}\,\tilde{B}$, are given, respectively, by

$$\tilde{\lambda}_{n_{r},\ell}=-\frac{\tilde{\eta}^{2}}{4\tilde{n}^{2}},$$

(32)

and

$$\psi\left(r\right)=\frac{R\left(r\right)}{\sqrt{r}}=C\,r^{|\tilde{\ell}|}\exp\left(-\frac{|\tilde{\eta}|}{2\tilde{n}}r\right)\,_{1}F_{1}\left(-n_{r},1+2|\tilde{\ell}|,\frac{|\tilde{\eta}|}{\tilde{n}}r\right).$$

(33)

Hence, Eq.s (29) and (32) along with (15) would result

$$g_{{}_{0}}\left(y\right)^{2}E^{2}-m^{2}=g_{{}_{1}}\left(y\right)^{2}\,\tilde{K}_{n_{r},\ell};\;\tilde{K}_{n_{r},\ell}=\left[k_{z}^{2}+\frac{\tilde{B}^{2}}{4}\left(1-\frac{\tilde{\ell}^{2}}{\tilde{n}^{2}}\right)+\eta^{2}\left(1-\frac{1}{4\tilde{n}^{2}}\right)+\frac{\eta\tilde{\ell}\,\tilde{B}}{2\tilde{n}^{2}}\right].$$

(34)

Evidently, the last term of $\tilde{K}_{n_{r},\ell}$ in (34) would lift the degeneracies associated with $\ell=\pm|\ell|\neq 0$ and states with both $\ell$ values would reappear in the spectrum, therefore. One should, moreover, notice that as $\tilde{B}\rightarrow 0$ our $\tilde{K}_{n_{r},\ell}\rightarrow k_{z}^{2}+\eta^{2}\left(1-1/4\tilde{n}^{2}\right)$ and consequently states with a specific $\ell=\pm|\ell|$ will emerge from the same $\tilde{B}=0$ and split as $\tilde{B}$ grows up from zero. We may now discuss the effects of rainbow gravity for different rainbow functions on the energy levels under the current metaphoric PDM KG-particles in cosmic string spacetime and uniform magnetic field of Eq.(34).

We have considered KG-particles in the cosmic string rainbow gravity spacetime (2) and uniform magnetic field (i.e., $\mathbf{B}=\mathbf{\nabla}\times\mathbf{A}=B_{\circ}\,\hat{z}$). We have shown that the corresponding KG-equation reduces to the two-dimensional radial Schrödinger Coulomb-like model (hence, the KG-Coulombic particle notion is used in the process). The exact textbook solution of which is used (along with the RG-modified energy-momentum dispersion relation (5)) and the effects of rainbow gravity on the spectra are discussed. We have also explored the effects of PDM (metaphorically speaking) on KG-Coulombic particles in cosmic string rainbow gravity and uniform magnetic field. In the process, we have studied the effects of four pairs of rainbow functions: (i) $g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}$, (ii) $g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y}$, (iii) $g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=\left(1-\epsilon y\right)^{-1}$, and (iv) $g_{{}_{0}}(y)=(e^{\epsilon y}-1)/\epsilon y$ and $g_{{}_{1}}\left(y\right)=1$.

Among the four pairs of rainbow functions, only $[g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}]$ provided energy levels that are symmetric about $E=0$ line. Yet, it is interesting to observe that for this particular pair of rainbow functions, the energy levels are destined to be within the range $\sqrt{(k_{z}^{2}+m^{2})/(1+\delta k_{z}^{2})}\leq|E_{n_{r},\ell}|\,\leq\sqrt{1/\delta}=E_{p}/\sqrt{\epsilon}$ as the value of $|eB_{\circ}|$ increases from zero (i.e., zero charge $e=0$ and/or zero magnetic field strength, $B_{\circ}=0$). This effect is also observed for PDM (using the same pair of rainbow functions). This effect is documented in Figures 1(b) and 5(c). Evidently, moreover, for $\epsilon=1$ we obtain the maximum possible value of the energy, $|E_{n_{r},\ell}|_{max}$ , of the probe KG-Coulombic particle (in cosmic string rainbow gravity spacetime and a uniform magnetic field) as the Planck energy $E_{p}$ so that $\sqrt{(k_{z}^{2}+m^{2})/(1+k_{z}^{2}/E_{p}^{2})}\leq|E_{n_{r},\ell}|\,\leq\sqrt{1/\delta}=E_{p}$.

We have, however, observed that the magnetic field did not remove the degeneracies of the energy levels associated with the magnetic quantum number $\ell=\pm|\ell|$, but the introduction of PDM-settings (through the PDM parameter $\eta$) has allowed the magnetic field to split $\ell=+|\ell|$ from $\ell=-|\ell|$. Which is, in fact, a common feature for the four pairs of rainbow functions we have considered. Yet, without the PDM parameter $\eta$, we have noticed that all $S$-states (i.e., $\ell=0$ states) are degenerate with each other (positive with positive and negative with negative states) and have the same $K_{n_{r},0}$ value as shown in Eq. (23). However, when $\eta$ is brought into action, such degeneracy is removed (documented in (34) and Figures 5, 6, 7, and 8).

The current study, in fact, supports and emphasises Bezerra’s et al. R81 statement that rainbow gravity is not just merely a mathematical re-scaling of both time and spatial coordinates. Rainbow gravity has deeply affected the spectroscopic structures for different rainbow function structures. The most interesting effect of which is observed for the pair $[g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}]$, which, in turn, implied that the energy of the probe KG-particle/antiparticle can not be more than the Planck’s energy $E_{p}$. This result clearly suggests that the Planck energy $E_{p}$, in the rainbow gravity model, is not only yet another invariant energy scale alongside the speed of light, but also a maximum possible particle/antiparticle (here, KG-particles) energy value (e.g., R44 ). More investigations should be carried out in this direction, we believe. Finally, to the best of our knowledge, the current methodical proposal did not appear elsewhere.