On the Klein-Gordon oscillators in Eddington-inspired Born-Infeld gravity global monopole spacetime and a Wu-Yang magnetic monopole

We consider Klein-Gordon (KG) particles in a global monopole (GM) spacetime within Eddington-inspired Born-Infeld gravity (EiBI-gravity) and in a Wu-Yang magnetic monopole (WYMM). We discuss a set of KG-oscillators in such spacetime settings. We propose a textbook power series expansion for the KG radial wave function that allows us to retrieve the exact energy levels for KG-oscillators in a GM spacetime and a WYMM without EiBI-gravity. We, moreover, report some \textit{conditionally exact}, closed form, energy levels (through some parametric correlations) for KG-oscillators in a GM spacetime and a WYMM within EiBI-gravity, and for massless KG-oscillators in a GM spacetime and a WYMM within EiBI-gravity under the influence of a Coulomb plus linear Lorentz scalar potential. We study and discuss the effects of the Eddington parameter $\kappa$, GM-parameter $\alpha$, WYMM strength $\sigma$, KG-oscillators' frequency $\Omega$, and the coupling parameters of the Coulomb plus linear Lorentz scalar potential, on the spectroscopic structure of the KG-oscillators at hand. Such effects are studied over a vast range of the radial quantum number $n_r\geq 0$ and include energy levels clustering at $\kappa>>1$ (i.e., extreme EiBI-gravity), and at $|\sigma|>>1$ (i.e., extreme WYMM strength).

Global monopoles are characterized by spontaneous global symmetry breaking and behave like elementary particles, with their energy is mostly concentrated near the monopole core [9].Global monopoles are objects that modify the geometry of spacetime [5,8,9], and are spherically symmetric topological defects with the line element ds 2 = −B (r) dt 2 + A (r) dr 2 + r 2 dθ 2 + sin 2 θ dϕ 2 . (1) In their study of gravitational field of a GM, Barriola and Vilenkin (BV) [2] have shown that the monopole, effectively, exerts no gravitational force, and the space around and outside the monopole has a solid deficit angle that deflects all light.They have shown that where M is a constant of integration and in the flat space M ∼ M core ( M core is the mass of the monopole core).
Assuming very light GMs, one may neglect the mass term and rescale the variables r and t [9], so that the GM metric reads where 0 < α 2 = 1 − β ≤ 1, where β = 8πGη 2 is the deficit angle, α is a global monopole parameter that depends on the energy scale η, and G is the gravitational constant [9][10][11][12][13].Obviously, this metric collapses into the flat Minkowski one when α = 1.
Very recently, the incorporation of the nonlinear electrodynamics of Born-Infeld into Eddington theory of gravity [51][52][53] has introduced the so called Eddington-inspired Born-Infeld (EiBI) theory of gravity (EiBI-gravity, in short).
While the EiBI theory is known to be equivalent to the Einstein's General Relativity (GR) in vacuum, it provides distinctive features when matter is included, and it possesses internal consistency for being free of instabilities and ghosts [54].One of the most interesting feature of the EiBI-gravity is its ability to avoid cosmological singularity and yields entirely singularity-free states [51,55].It has been reported by [55][56][57][58] (to mention a recent few) that the GM spacetime in EiBI-gravity, generated by a source matter, is described, in spherical coordinates, by the metric where κ is the Eddington parameter that controls the nonlinearity of the EiBI-gravity, and It is obvious that this metric reduces into the GM one, Eq.( 3) above, when the Eddington parameter is switched off, ie., κ = 0, hence it is occasionally called modified GM or EiBI-GM.The study of the effects of the gravitational field, manifestly introduced by the GM-spacetime in EiBI-gravity, on the quantum mechanical spectroscopic structure were only very recently carried out by Pereira et al [57,59].Pereira and co-authors [57,59] were able to discuss one state (the lowest state with n = 1 for any value of the angular momentum quantum number ℓ).Whilst we emphasis that the the procedure and methodology they have used is just fine and good, their methodology does not work for n > 1.In the current methodical proposal, however, we report/provide an alternative approach to obtain a conditionally exact solution (or if you wish, quasi-exact solution) [60] for the Klein-Gordon (KG) oscillator in a GMspacetime within EiBI-gravity.Namely, we shall discuss KG-oscillators in a GM-spacetime and a Wu-Yang magnetic monopole (WYMM) [15][16][17] within EiBI-gravity for any n and ℓ (at a very specific correlation/constraint between the KG-oscillator's frequency and Eddington parameter).
In so doing, we are motivated by the fact that the influence of the topological defects, generated by gravitational fields, on the quantum mechanical systems, are interesting not only for quantum gravity but also for the geometrical theory of topological defects in condensed matter physics (e.g., [61][62][63]).To the best of our knowledge, the studies of Pereira and co-authors [57,59] are the only attempts made, in the literature, to study the KG-oscillators in a GM-spacetime within EiBI-gravity.In the current methodical proposal we shall add (in addition to Pereira et al.'s [57,59] approach) yet another degree of freedom that allows the reader to observe the effects of EiBI-GM gravitational field on the full spectrum of the KG-oscillators in a WYMM.Nevertheless, one should be reminded that although the WYMM is a theoretical concept that is yet to be observed, it remains an interesting model for modern theoretical particle physics and topological properties in gauge theories.
The organization of the current proposal is in order.In section 2, we consider KG-particles in GM spacetime within EiBI-gravity and in a WYMM, and recollect/recycle the corresponding angular and radial parts of the KGequations.In section 3, we discuss KG-oscillators in a GM spacetime within EiBI-gravity and in a WYMM.Therein, we argue that although the solution of the corresponding Schrödinger-like KG radial equation has a solution in the form of confluent Heun functions H C a, b, c, d, f, r 2 +κ κ , the power-series expansion should not be dictated by the argument of the confluent Heun functions (i.e., r 2 + κ /κ) but rather a textbook one that allows some freedom in the manipulations of a three recursion relations (usual product of such Heun equations) that in turn allow feasible reduction of such recursion relation into that of the KG-oscillator in no Eddington gravity (i.e.κ = 0).We show, in the same section, how to obtain the energy levels for the KG-oscillators in a GM spacetime in a Wu-Yang magnetic monopole in no EiBI-gravity, κ = 0, from the same recursion relations, in subsection 3-A.In subsection 3-B, we report a conditionally exact solution, through a parametric correlation, and obtain some conditionally exact energy levels for the KG-oscillators in a GM spacetime within EiBI-gravity, κ = 0, and a WYMM.Moreover, we discuss, in subsection 3-C, the conditionally exact energy levels for some massless KG-oscillators in a GM spacetime within EiBI-gravity and in a WYMM under the influence of a Coulomb plus linear Lorentz scalar potential.Our concluding remarks are given in section 4.

II. KG-PARTICLES IN GM SPACETIME WITHIN EiBI-GRAVITY AND IN A WYMM
A rescaling in the forms of (1 − β) dt → dt along with κ = κβ, would allow us to rewrite metric (4) as ( It should be noted that κ < 0 would describe a topologically charged wormhole [64][65][66], α = 1 and κ < 0 corresponds to Morris-Thorne-type wormhole spacetime [67,68], κ = 0 would describe a GM spacetime, and κ > 0 corresponds to a GM spacetime in EiBI-gravity.The corresponding metric tensor is to imply and Then, the KG-equation would reads where Dµ = D µ +F µ is in a non-minimal coupling form with that admits minimal coupling, A ν = (0, 0, 0, A ϕ ) is the 4-vector potential, S (r) is the the Lorentz scalar potential, and m • is the rest mass energy (i.e., m • ≡ m • c 2 , with = c = 1 units to be used through out this study).Moreover, we use F µ = (0, F r , 0, 0) to incorporate the KG-oscillators in the GM spacetime in EiBI-gravity.In particular, we are considering KG-oscillators [69], as it is a relativistic quantum model that has been extensively studied and applied in the literature, due to its analyticity and the possibility of recovering the Shrödinger oscillator in the non-relativistic limit [70].It is worth mentioning that the oscillator goes beyond a mere academic exercise, as any central potential, which has a minimum point, around this point, it behaves like an oscillator [71].Furthermore, oscillations are found At this point, one should notice that λ = ℓ (ℓ + 1) for A ϕ = 0 case, and for the WYMM [15][16][17] λ is given by where σ = eg (with g is the WYMM strength).The details of the result in Eq.( 13) are given in the Appendix (to keep this study self-contained).Consequently, Eq.( 11) would simply read With F r = Ωr, S (r) = 0, and σ = 0 (no WYMM), this equation would imply where This result is in exact accord with equations ( 6) and ( 7) reported by Pereira et al. [57] (where our Ω = mω of Pereira et al. [57]).In the following section, we shall consider the inclusion of a WYMM and describe a different solution to the problem at hand (i.e., a conditionally exact solution that deals not only with n = 1 states but also with n ≥ 1 states).

III. KG-OSCILLATORS IN A GM SPACETIME WITHIN EiBI-GRAVITY AND IN A WYMM
In what follows we shall consider the more general case that includes the WYMM and rewrite equation ( 14) with F r = Ωr, S (r) = 0, and σ = 0 so that This equation represents KG-oscillators in GM spacetime in EiBI-gravity and a WYMM, where and hence l is a new irrational angular momentum quantum number that collapses into the regular one ℓ = 0, 1, 2, • • • for α = 1 (i.e., flat Minkowski spacetime), σ = 0 (no Wu-Yang monopole), and κ = 0 (i.e., no EiBI-gravity).One should also notice that the case of κ = 0, α = 1, and σ = 0 corresponds to the GM spacetime background with a WYMM.This fact should provide a controling mechanism on the validity of the solution of the more general case that includes EiBI-gravity.
Apriori, one should notice that the solution of the KG-oscillators in ( 15) is known to be given by as reported by Pereira et al. [57] in their Eq.(11) along with the substitution used to obtain their Eq.( 8).In our opinion, nevertheless, the form of the confluent Heun function H C above should never dictate the form of the power series expansion (as the power series used by Pereira et al. [57] in their Eq.(12)).Instead, one should seek a power series solution that is reducible to the KG-oscillator's one at κ = 0 (no EiBI-gravity), σ = 0 (no Wu-yang monopole), and α = 1 (flar Minkowski spacetime).In the current methodical proposal, therefore, we follow a different route, but a textbook one, that allows not only reducible to the KG-oscillator's in the flat Minkowski spacetime but also allows the so called conditional exact solvability for the KG-oscillators in a GM spacetime in EiBI-gravity and a WYMM.
In so doing, we use in (17), to obtain where Let us use the change of variables y = r 2 to obtain We may now suggest a power series solution in the form of where ν is a parameter to be determined in the process, the usefulness of which is to be clear in the sequel.Under such settings, one obtains, using (24), Consequently, Under such settings, we obtain the set of relations and This set of recursion relations has to be dealt with diligently, and case-by-case, as shall be clarified in the sequel illustrative examples.Such illustrative examples are considered so that one obtains the exact solution for the KGoscillators in a GM spacetime and a WYMM in no EiBI-gravity (i.e., κ = 0), a conditionally exact solution for KG-oscillators in a GM spacetime within EiBI-gravity, κ = 0, and a WYMM, and a conditional exact solution for the massless KG-oscillators in a GM spacetime in WYMM and Coulomb plus linear Lorentz scalar potential within EiBI-gravity.
A. KG-oscillators in a GM spacetime in a Wu-Yang magnetic monopole in no EiBI-gravity, κ = 0.
At this point, we may test this procedure above and remove EiBI-gravity (i.e., κ = 0) to reduce the problem into KG-oscillators in GM spacetime with a WYMM.This would imply that Eq.( 28), with C 0 = 0 and P2 = P 2 /4 ( P 2 is given in (23)), now reads Therefore, we shall take ν = l/2 for it secures finiteness of the radial wavefunction at y = 0 (i.e., r = 0) in (25).
Consequently, Eq. ( 29) collapses into the two-terms recursion relation Next, we need to truncate the power series into a polynomial of order n r = 0, 1, 2, • • • (to secure square integrability of the wavefunction) by requiring that for ∀j = n r we have C nr +1 = 0 to imply This is the exact textbook result for the KG-oscillators in a GM spacetime with a WYMM in no EiBI-gravity (i.e., for κ = 0 in (17)).Moreover, with E 2 given in ( 18), we one obtains This result represents an exact solution for the KG-oscillators in a GM spacetime with a WYMM in no EiBI-gravity.
Moreover, it is in exact accord with that reported in Eq. ( 38) of Bragança et al. [19] and in (31) of Mustafa [73].The coefficients of the our polynomial in (31), on the other hand, are now given by Under such settings, for j = 0 we get, for j = 1 we get and so on.One should notice that we may consider C 0 = 1 hereinafter.Moreover, the radial part of the wave function is therefore given by B. KG-oscillators in a GM spacetime within EiBI-gravity, κ = 0, and in a WYMM .We now consider the more general case where EiBI-gravity is involved in the process.In this case, equation (27) suggests that in C 0 κν 2 = 0 ⇔ ν = 0 since C 0 = 0 = κ, and Moreover, equation ( 29) now reads We now wish to truncate the power series into a polynomial of order n r + 1 (or if you wish a polynomial of order n = n r + 1 ≥ 1) so that ∀j = n r we have C nr +2 = 0, C nr +1 = 0, and C nr = 0. We impose, moreover, the condition that This condition would facilitate conditional exact solvability of the problem at hand, as well as it provides a correlation, between the KG-oscillator frequency Ω and the Eddington gravity parameter κ, given by Consequently, since C nr = 0 and and all states' energies are given by where we have used Ω in (41).At this point, we have to emphasis that the condition used by Pereira et al. [57] (i.e., non-vanishing coefficients of C nr +1 = 0 in (39)) have constrained the validity of their reported solution to hold true for only n r = 0 states.Of course, their reported solution is one of a variety of conditionally exact solutions.Our condition in ( 40) is yet another feasible conditionally exact one, therefore.Moreover, our result in ( 42) is in exact accord with that reported in Eq. ( 14) of Ishkhanyan et al [74], with proper parametric mappings, of course.One should observe, however, that the correlation in (41) suggests that since Ω > 0 and κ > 0 (by definition), then the angular momentum quantum number should satisfy the relation where ℓ = 0, 1, 2, • • • .Hence, the maximum allowed values of ℓ are determined by the relation (44) above.For example: for α = 1, σ = 0, and n r = 0 we have ℓ < 2 (i.e., only states with ℓ = 0, 1 are allowed), for α = 1/2, σ = 0, and n r = 0 we have ℓ < 0.725 (i.e., only ℓ = 0 is allowed), and so on so forth.Moreover, one should also observe that our correlation (41) manifestly classifies our solution (43) as a conditionally exact solution (as known in the literature, e.g., [60]).This solution is not valid for Ω = 0 and/or κ = 0 (note that κ = 0 is a case discussed in the subsection above).Yet, using the left-hand-sides of Eq.s ( 41) and ( 43), our three terms recursion relation (39) now reads yields , for j = 0, (46) and so on.Therefore, our power series R (y) in ( 25) is now truncated into a polynomial of order n = n r + 1 ≥ 1 and is given by which indeed identifies a polynomial in even powers of the r.
In Fig. 1, we plot the energy levels in Eq.( 43) taking into account the restrictions on the orbital angular momentum quantum number Eq.( 44) and under the conditionally exact solvability correlation (41).In Fig. 1(a) one may observe that the quantum states tend to cluster for κ >> 1 values (i.e., at extreme Eddington gravity).In connection with the results discussed above, nevertheless, the following observation is unavoidably inviting in the process.Although the procedure discussed above is restricted with the condition in (41), it provides a significantly wider spectrum (for ∀n r ≥ 0, or equivalently ∀n = n r + 1 ≥ 1) for the KG-oscillators in a GM spacetime within EiBI-gravity, κ = 0, and a WYMM.The power series approach reported by Pereira et al. [57] cumbersome for ∀n = n r + 1 ≥ 2 states.Consequently, their solution, although a conditionally exact one, it falls short and fails to address ∀n = n r + 1 ≥ 2 quantum states.
C. Massless KG-oscillators in a GM spacetime within EiBI-gravity and in a WYMM under the influence of a Coulomb plus linear Lorentz scalar potential We consider massless KG-oscillators in a GM spacetime in EiBI-gravity and a WYMM subjected to a Coulomb plus linear Lorentz scalar potential S (r) = A/r + Br.In this case, one may show, in a straightforward manner, that Eq. ( 14) would yield where, Evidently, equation ( 49) is in the same form as that of Eq. ( 17).This would mandate that our equation ( 49) inherits the same forms of the reported solution in the preceding subsection.That is, with the substitution one obtains where Following the same steps as in the preceding subsection, one obtains and where the parametric correlation is obtained as which manifestly renders the solution to be classified as a conditionally exact solution.This relation would again suggest that since Ω > 0, and κ > 0 then the allowed values for angular momentum quantum number ℓ = 0, 1, 2  Moreover, our radial part of the wave function R (y) is similar to that in (25) and is again a polynomial in even powers of the r given by R (y) = In Fig. 2, we plot the energy levels in Eq.( 58) along with the restrictions on the orbital angular momentum quantum number Eq.( 60) and under the conditionally exact solvability correlation (59).In Fig. 2(a) we observe that the separation between the energy levels decreases with increasing GM-parameter α.In Fig. 2(d) we observe that the spacing between energy levels decreases with increasing values of A. Whereas, in Fig. 2(e) the separation between energy levels increase with increasing values of B. Finally, Fig. 2(f) shows that the effect of Eddington parameter κ remains the same as that in Eq.( 43) and shows the tendency to cluster the energy levels for κ >> 1.
This is exactly the result given in Eq. (13).Moreover, the so called Wu and Yang [10,15]  This Appendix section makes the current methodical proposal self-contained.

Fig. 1 (
b) shows that the separation between energy levels increase as the oscillator frequency Ω increases.In Fig.1(c) energy levels are observed to cluster at large values of the WYMM strength, i.e., |σ| >> 1. Hereby, one should notice that we have allowed σ to vary from negative values to positive ones depending on the charge involved in the WYMM structure (see Eq. (A3) in the Appendix below).Fig.1(d), documents that the separation between the energy levels increases with increasing GM-parameter α.

Fig. 2 (
b) shows that an obvious clustering of the energy levels for |σ| >> 1.The clustering of the energy levels is also observed for κ >> 1 in Fig.2(c).