---
abstract: 'We rewrite the Klein-Gordon (KG) equation in an arbitrary space-time transforming it into a generalized Schrödinger equation. Then, we take the weak field limit and show that this equation has certain differences with the traditional Schrödinger equation plus a gravitational field. Thus, this procedure shows that the Schrödinger equation derived in a covariant manner is different from the traditional one. We study the KG equation in a Newtonian space-time to describe the behavior of a scalar particle in an inertial system. This particle is immersed in a gravitational field with the new Schrödinger equation. We study particular physical systems given examples for which we find their energy levels, effective potential and the wave function of the systems. The results contain the gravitational effects due to the curvature of the space-time. Finally, we discuss the possibility of the experimental verification of these effects in a laboratory using non-inertial reference frames.'
author:
- 'Omar Gallegos[^1]'
- 'Tonatiuh Matos[^2]'
bibliography:
- 'ref.bib'
date: February 2019
title: Weak gravitational quantum effects in boson particles
---

Introduction
============

In the last century, General Relativity (GR) and Quantum Mechanics (QM), the two pillars of modern physics, have been developed and verified independently with great precision, while quantum physics describes successfully the behavior of tiny particles, GR is very accurate for forces at cosmic scales. However, in some cases, the two theories produce incompatible results which give rise to different definitions for the same concept. We think the inconsistency between GR and QM owes to the concept of interaction between particles. In QM two particles interact when they exchange a virtual particle, while in GR the interaction is just due to the geometry of space-time. In this work we adopt the geometrical GR concept, instead of the exchange of virtual particles to test if a boson gas follows the Klein- Gordon equation in a curved space-time, and to test GR in quantum regime. These results could be useful either for laboratory particles as well as for the study of the quantum character of boson particles proposed as dark matter (see for example [@Matos:1998vk], [@Hui:2016ltb]).

An important problem in theoretical and fundamental physics is to have a Theory of Everything where the principal theories in physics, GR and QM, can be compatible. In the last decades, some theories have been proposed [@Loop],[@String] to this unification. Nevertheless, the experimental verification of these candidates and their theoretical problems are so far too complex. Several physicists have tried to test if gravity has a quantum nature with different proposals for experiments and observations[@Casimir][@entaglement], but some proposals are not feasible with the technology available to this day. We do not pretend to propose a new Theory of Everything, instead we want to give a new different way to measure the gravitational effect due to the curvature of space-time on quantum systems, specially on scalar particles. These results do not definitely prove if gravity is or not a quantum interaction but they present a closer path to answer this fundamental question.

Einstein’s Equivalence Principle (EEP), one of the foremost ideas for developing General Relativity, where the concept of inertia takes a different role from the one used in Newtonian mechanics. EEP states that experiments in a sufficiently small freely falling laboratory, over a sufficiently short time, give results that are indistinguishable from those of the same experiments in an inertial frame in the presence of a gravitational field. Hence, we use this principle to develop this work, studying some different examples of QM on an inertial frame immersed in a gravitational field. To measure these effects in a laboratory, we will place a quantum system on a non-inertial frame, hoping to get the same results both in the theoretical part and in the experimental one. Furthermore, we expect to obtain results on quantization as in QM. So, one of the objectives of this paper is to test the EEP in quantum scales, thus a proof of GR in a quantum scale.\
We would expect that there exists a regime where the quantum aspects of gravity are detectable. With a simple dimensional analysis we may have an idea where the gravitational effects on a quantum system are important, i.e. the physical scale where these effects can be observed. It is possible to compare two well-known quantities in QM and GR, which are the Bohr radius $r_{bohr}$ and the Schwarzschild radius $r_{sch}$. Both radii are defined as $r_{bohr}=\hbar/(\mu c \alpha)$ and $r_{sch}=2GM/c^2$ where $c$, $G$, $\alpha$ are the speed of light, the gravitational constant and the fine-structure constant respectively. Also $\hbar=h/2\pi$ is the Planck’s constant, while $M$ is the mass that produces the gravitational field, $\mu$ is the reduced mass of a couple of particles. In a case where the comparison between masses is such that $M>>m$, it leads us to the approximation $\mu\approx m$, where $m$ is the mass of a boson particle, which is inside a gravitational field generated by a source of mass $M$. Thereby, the following expression is obtained when both radii $r_{bohr}\sim r_{sch}$ are comparable $$\label{dimensional analysis}
M\sim\dfrac{\hbar c}{2G\alpha m}=\dfrac{m_{pl}^2}{2\alpha m}\approx\dfrac{3.25\times10^{-14}}{m}\text{kg}^2,$$ where $m_{pl}=\sqrt{\hbar c/G}$ is the Planck’s mass. This result was inspired by the micro black holes described in[@microBH] and it enables us to find the limit for measuring quantum gravitational effects. If we assume $m=m_e$ is the mass of an electron, we obtain that $M\sim 3.57\times10^{16}$kg is the mass for which an electron should feel a gravitational effect. On the other hand, the mass of Earth is $\sim 5\times 10^{24}$kg, namely, we should measure in Earth the quantum gravitational effects on an electron. If we consider now the mass of an ultralight scalar particle $\sim 10^{-22}eV/c^2$, the mass proposed in models of Scalar Field Dark Matter (SFDM)[@Matos:1998vk], [@Hui:2016ltb], we calculate from Eq.(\[dimensional analysis\]) that $M\sim 10^{12}M_\odot$ (the galaxies have a mass of this order). Thus, we can say, if dark matter is a scalar field with mass $m\sim 10^{-22}eV/c^2$ [@Matos:1998vk], [@Hui:2016ltb], the galaxy should present a gravitational quantum behavior. In this work, we study well-known examples of QM for scalar particles with corrections due to the curvature of space-time.

Generalized Schrödinger Equation
================================

In order to analyze the gravitational effect due to curvature of space-time in a quantum system, we focus on a scalar field following reference [@Matos:2016uxo], in which the KG covariant equation with an external potential is described $$\label{KG covariant}
\Box\Phi-\dfrac{d\mathscr{V}}{d\Phi^*}=0,$$ here $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu=\dfrac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}g^{\mu\nu}\partial_\nu)$ is the D’Alembertian operator associated to an arbitrary metric $g_{\mu\nu}$, $\Phi=\Phi(t,\vec{x})$ is the scalar field, $\Phi^*=\Phi^*(t,\vec{x})$ is its conjugated complex and the scalar field potential is $\mathscr{V}=\mathscr{V}(\Phi,\Phi^*)$ endowed with an external potential $V$ just as it is shown in the following equation $$\label{potential}
    \mathscr{V}=\left(\dfrac{m^2c^2}{\hbar^2}+\dfrac{\lambda n_0}{2}+\dfrac{2m}{\hbar^2}V\right)\Phi\Phi^*,$$ where $m$ is the mass of the scalar field and $n_0=n_0(t,\vec{x})$ is defined as the scalar field density, namely $n_0=|\Phi|^2$. The space-time is expanded in a 3+1 slices, such that the coordinate $t$ here is the parameter of evolution, the 3+1 metric then reads $$\label{metric}
    \mathrm{d}s^2 = - N^2 c^2 
    \mathrm{d}t^2 + \gamma_{ij} \left(\mathrm{d}x^i + N^i c \, \mathrm{d}t \right) \left(
    \mathrm{d}x^j + N^j c \, \mathrm{d}t \right) ,$$ $N$ represents the lapse function which measures the proper time of the observers traveling along the world line, $N^i$ is the shift vector that measures the displacement of the observers between the spatial slices and $\gamma_{ij}$ is the spatial metric.

The KG equation is a covariant equation whose origin lies in quantum field theory. We can obtain a Schrödinger equation, starting from equation (\[KG covariant\]) with the potential in Eq.(\[potential\]), using the metric (\[metric\]) and transforming the scalar field by $\Phi(t,\vec{x})=\Psi(t,\vec{x}) e^{-i\omega t}$. Following [@Matos:2016uxo] we obtain $$\begin{aligned}
    i \nabla^0 \Psi - \frac{1}{2\omega} \Box_G \Psi 
    + \frac{1}{2\omega} \left( \widetilde{m}^2 + \lambda n_0+V \right) \Psi &&
     \nonumber\\
    +\quad \frac{1}{2} \left( - \frac{\omega}{N^2}
    + i \, \Box_G \, t \right) \Psi &=& 0 ,
\label{eq:GP}\end{aligned}$$ being $\lambda$ the coupling parameter. Here $\omega=mc^2/\hbar$ is the characteristic frequency of the scalar field. The D’Alembertian operator $\Box_G=\nabla^\mu\nabla_\mu$ is associated to the metric (\[metric\]), and $\widetilde{m}$ stands for the mass in units where $c=\hbar=1$, i.e, $\widetilde{m}=m^2c^2/\hbar^
2$ (see [@Matos:2016uxo]). We can interpret the function $\Psi$ as a wave function analogously as in QM.

Beginning with Eq.(\[KG covariant\]) but now using the $\Psi$ variable, we obtain Eq.(\[eq:GP\]) using the 3+1 metric (\[metric\]) and the potential (\[potential\]). Equation (\[eq:GP\]) can be interpreted as the covariant generalization of the Schrödinger equation for any curved space-time, (see[@Chavanis:2016shp] and [@Matos:2016uxo]) where this equation in the weak field limit reduces to the standard Schrödinger one. Hereafter, we use the Newtonian geometry given by the Newtonian metric $$\label{Newtonian metric}
ds^2=-\left(1-\dfrac{2GM}{rc^2}\right)c^2dt^2+\left(1+\dfrac{2GM}{rc^2}\right)dx_idx^i,$$ Newtonian gravity is known to be valid when the gravitational fields are weak, that is $GM/rc^2<<1$. Using the Newtonian metric in Eq.(\[eq:GP\]) yields $$\label{KG-N equation}
\dfrac{\hbar^2}{2m}\Box_N\Psi+\left(1+\dfrac{2U}{c^2}\right)^{-1}\left(i\hbar\dfrac{\partial \Psi}{\partial t}+\dfrac{mc^2}{2}\Psi\right)-\dfrac{\hbar^2}{2m}V\Psi=0,$$ where $U=-GM/r$ is the gravitational potential, hither the D’Alembertian operator $\Box_N$ is related to the metric given by Eq.(\[Newtonian metric\]). Furthermore, we do not consider self-interaction, here the contribution of the term $\lambda$ is negligible. We take into account that the evolution of this function is small, thus the contribution of $\partial_0^2\Phi$ can be taken as null because the Newtonian potential fulfills $U/c^2<<1$. Additionally, one can ignore terms of equal or greater order than $\left(\dfrac{2U}{c^2}\right)^2$. With all this in mind, the linearization of the generalized Schrödinger equation (\[KG-N equation\]) gives rise to the following equation $$\label{KG-N motion equation}
-\dfrac{\hbar^2}{2m}\nabla^2\Psi+V\Psi+mU\Psi +\left(\dfrac{2U}{c^2}V-\dfrac{2\hbar^2U}{mc^2}\nabla^2\right)\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$ Henceforth, we are going to work with the Laplacian operator $\nabla^2=\nabla\cdot\nabla$ in flat space, specially for the examples in the following chapters where we use spherical symmetry. In other words, equation (8) comes from the KG equation in a curved space-time immersed in a weak gravitational field, which is described by a Newtonian geometry.

Focusing on the comparison of the traditional Schrödinger equation plus a gravitational potential, we note that now there are two additional terms inside the parenthesis in Eq.(\[KG-N motion equation\]). These terms were not simply added in the Schrödinger equation, they appear from the covariant meaning of the equation. From a qualitative point of view, this means that the QM version of the interactions between particles fulfills the Schrödinger equation, while the GR version of these interactions fulfills the generalized Schrödinger Eq.(\[KG-N motion equation\]). This difference is the main idea of this work. In what follows, we calculate the quantum quantities for different external potentials of well known problems in QM and compare them with those corresponding to GR. For the subsequent chapters our equation of motion will be given by Eq.(\[KG-N motion equation\]).

With the goal of testing EEP in a quantum region, we are going to show in the subsequent chapters distinct calculations about well known examples in QM. In them, the correction terms due to the curvature of space-time which were introduced in the KG covariant equation derived in Eq.(\[KG-N motion equation\]) are going to appear. To compare the cases with and without a gravitational source, the extra terms in Eq.(\[KG-N motion equation\]) will be taken as perturbations in the Schrödinger equation, for this reason we decide to opt for the QM formalism, using perturbation theory to give a clearer comparison between the cases with and without gravitational field. The results of the gravitational field contribution have been well studied in the standard literature.

Free Particle
=============

Firstly, we consider a free scalar particle with mass $m$ under the influence of a gravitational field generated by a source of mass $M$. Using an external potential $V_{free}=0$ in the equation of motion (\[KG-N motion equation\]), we have that $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t},$$ we recall that the gravitational potential is defined by $U=-GM/r$ and that an extra term $-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi$ appears in the Schrödinger equation, although the treatment on these equations will be in the QM formalism. Using perturbation theory, from the previous equation we can write a principal Hamiltonian operator $\hat{H}_0$ and a perturbed Hamiltonian operator $\hat{H}_p$ in the following way $$\begin{aligned}
\label{H0 free particle}
&\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi-\dfrac{GMm}{r}\Psi,\\
\label{Hp free particle}
&\hat{H}_p\Psi=-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=\dfrac{4U}{c^2}(E_n^{(0)}-mU)\Psi.\end{aligned}$$ In general, to find the corrections to eigenvalues of energy due to a perturbation in QM, the following expression is used $$\label{energy correction}
E_n=E_n^{(0)}+\bra{\psi^{(0)}_n}\hat{H}_p\ket{\psi^{(0)}_n}+\sum_{m\neq n}\dfrac{|\bra{\psi^{(0)}_m}\hat{H}_p\ket{\psi^{(0)}_n}|^2}{E_n^{(0)}-E_m^{(0)}}+...,$$ we can associate the Eq.(\[energy correction\]) as a series of contributions of higher order correction of energies $E_n=E_n^{(0)}+E_n^{(1)}+E_n^{(2)}+...$, where $E_n^{(j)}$ is the $j$th-order correction of the eigenvalues in the $n$th-state of energy, hence we can say that the first-order correction is given by $E_n^{(1)}=\bra{\psi^{(0)}_n}\hat{H}_p\ket{\psi^{(0)}_n}$. The zero-order correction for eigenvalues of energy $E_n^{(0)}$ and the eigenfunctions are the well-known exact solutions of QM (without gravitational field). Equation (\[energy correction\]) gives the correction for a non-degenerated quantum system, though first-order correction is valid for both cases (degenerate and non degenerate systems). We do not present an expression for the second order correction in a degenerate system because the corrections we make are at most of first-order. Nevertheless, such a case can be found in standard QM textbooks.\
Going back to our example of a free scalar particle inside a gravitational field, the first order correction of energy using the perturbed Hamiltonian from the hydrogen atom-like problem in QM is $$\begin{aligned}
E_n&\approx E^{(0)}_n\left(1-\dfrac{4GM}{\rho_0c^2n^2}+...\right),
\\&=E^{(0)}_n\left(1+\dfrac{8E^{(0)}_n}{mc^2}+...\right)\notag,\end{aligned}$$ being $E^{(0)}_n$ the well known energy for the hydrogen atom, that is $$\label{energy atom gravitation}
E^{(0)}_n=-\dfrac{(GM)^2m^3}{2\hbar^2n^2}=-\dfrac{GMm}{2\rho_0n^2},$$ where $\rho_0=\hbar^2/(GMm^2)$ is the Bohr radius for the gravitational case. Thus, we expect that the gravitational field modifies the energy of a free particle in a quadratic level, suppressed by the rest energy of the boson particle. This result is surprising for an ultra-light dark matter boson [@Matos:1998vk][@Matos:2000ss], because this model postulates a boson particle with a mass of the order of $10^{-22}$eV$/c^2$. With such a mass the self-gravitation effects of the boson field are important in a system of particles, a feature that a heavy boson system does not present.

Isotropic Harmonic Oscillator
=============================

Another important example, not only in QM but in physics as a whole, is the study of the harmonic oscillator potential. With this in mind, we analyze the case of potential $V_{osc}$ for an isotropic harmonic oscillator. This analysis can be done in two ways. The first one is to start from Eq.(\[KG-N motion equation\]) with the principal potential given by the gravitational type $U=-GM/r$ and taking the isotropic harmonic oscillator $V_{osc}=\dfrac{1}{2}m\omega_0^2r^2$ as perturbation. The other way is to regard the principal Hamiltonian like as that of an isotropic harmonic oscillator, while the perturbation is taken from the gravitational potential. Therefore, equation (\[KG-N motion equation\]) with the potential $V_{osc}$, transforms into $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi +\dfrac{1}{2}m\omega_0^2r^2\Psi+\left[\dfrac{2U}{c^2}\left(\dfrac{1}{2}m\omega_0^2r^2\right)-\dfrac{2\hbar^2U}{mc^2}\nabla^2\right]\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$

Harmonic Oscillator inside a Gravitational Field
------------------------------------------------

We start with the first cases exposed previously, we suppose that one has an isotropic harmonic oscillator immersed in a gravitational field, that means, $V_{osc}<<U$. To solve this problem we can take the principal Hamiltonian operator $\hat{H}_0$ with the gravitational part and the perturbed Hamiltonian $\hat{H}_p$ in the following way $$\begin{aligned}
\label{principal H IHO in curved space}
\hat{H}_0\Psi&=-\dfrac{\hbar^2}{2m}\nabla^2\Psi-\dfrac{GMm}{r}\Psi,\\
\label{Hp IHO in curved space}
\hat{H}_p&=\dfrac{1}{2}m\omega_0^2r^2-\dfrac{GMm\omega_0^2r}{c^2}+\dfrac{4GME^{(0)}_n}{rc^2}.\end{aligned}$$ We are interested in the wave function with spherical symmetry $\Psi=\Psi(t,r,\theta,\phi)$. Thus, we apply the separation of variables method for $\Psi=R_{nl}(r)Y_{lj}(\theta,\phi)\exp(-\dfrac{iE_n}{\hbar}t)$, where $Y_{lj}$ are the spherical harmonics and $R_{nl}(r)$ is the radial function for the hydrogen atom problem. Here $n,l$, and $j$ play the role of quantum numbers as in QM. The $\hat{H}_0$ Hamiltonian contains the well-known solutions of the eigenvalues from the hydrogen atom problem in terms of the recurrence relation for powers of $r$. It can be shown that the correction for first order of energy from Eq.(\[energy correction\]) is given by $$\begin{aligned}
E_n=&E^{(0)}_n\left[1-\dfrac{8E^{(0)}_n}{mc^2}-\dfrac{\omega_0^2\rho_0^2}{c^2}(3n^2-l(l+1))\right]
\\ &+\dfrac{1}{4}m\omega_0^2\rho_0^2\left[n^2(5n^2+1-3l(l+1))\right],\notag\end{aligned}$$ where $E^{(0)}_n$ is given by Eq.(\[energy atom gravitation\]). Note that if $\omega_0=0$, we return to the case of a free particle on a gravitational field. Observe that as in the previous case, the modifications due to the gravitational field are of second order, but now, an additional term is added which is proportional to the mass and the frequency $\omega_0$. In this case, the second term of the contributions of the perturbations of the gravitational potential due to the harmonic oscillator are negligible for ultra-light masses. Nevertheless, if it is a massive boson then the quadratic contributions are not important. In any case, for any boson mass there is a contribution of the harmonic oscillator that must be taken into account.

Gravitational Field inside Harmonic Oscillator
----------------------------------------------

On the other hand, assuming the gravitational field as a perturbation in an isotropic harmonic oscillator with spherical symmetry, the principal Hamiltonian operator $\hat{H}_0$ is given by $$\begin{aligned}
\hat{H}_0\Psi&=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+\dfrac{1}{2}m\omega_0^2\left[\left(r-\dfrac{GM}{c^2}\right)^2-\left(\dfrac{GM}{c^2}\right)^2\right]\Psi,\notag
\\&\approx-\dfrac{\hbar^2}{2m}\nabla^2\Psi+\dfrac{1}{2}m\omega_0^2r^2-\Psi-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2\Psi.\end{aligned}$$ Since the quantum system is far from the source, it is possible to do the previous approximation. For the perturbed Hamiltonian $\hat{H}_p$ we have $$\hat{H}_p=-\dfrac{GMm}{r}+\dfrac{4GME^{(0)}_n}{rc^2}.$$ The principal Hamiltonian is that of an isotropic harmonic oscillator with spherical symmetry whose eigenfunctions are well known from QM. These solutions are obtained after applying the separation of variables method, just as in the previous case $\Psi_{nklj}(t,r,\theta,\phi)=R_{kl}(r)Y_{lj}(\theta,\phi)\exp(-iE_nt)$. Here $R_{nl}(r)$ is the radial function for an isotropic harmonic oscillator $R_{kl}(r)=r^le^{-\gamma r^2}L^{(l+1/2)}_k(2\gamma r^2)$, with $\gamma=\dfrac{m\omega}{2\hbar}$ and $L^{p}_q(x)$ being the generalized associated Laguerre polynomials. The set $(n,k,l,j)$ become the quantum numbers for the isotropic harmonic oscillator with spherical symmetry from QM.Thus, the first order correction for the energy from Eq.(\[energy correction\]) is given by $$E_n=\hbar\omega_0\left(n+\dfrac{3}{2}\right)-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2-GMm\left[1-\dfrac{4E^{(0)}_n}{mc^2}\right]\left\langle \dfrac{1}{r}\right\rangle,$$ where $E^{(0)}_n=\hbar\omega_0\left(n+\dfrac{3}{2}\right)$ is a well known result from QM. Also note that there is a degeneration of these numbers since $n=2k+l$. A more general solution of $\left\langle \dfrac{1}{r}\right\rangle$ was found in [@Amrmstrong], that for the case we are discussing reduces to the expression $$\begin{aligned}
\label{iho correction}
\left\langle \dfrac{1}{r}\right\rangle=&\sqrt{\dfrac{m\omega_0}{\hbar}}\Gamma(l+1)[\dfrac{1}{2}(n-l-1)]!\times
\\&\sum_t\dfrac{(-1)^t}{[1/2(n-l-1)-t]!\Gamma(1/2(2l+1)+t+1)}\binom{-1/2}{t}^2\notag,\end{aligned}$$ where $\Gamma(z)$ is the Gamma function $\Gamma(z)=\int^\infty_0t^{z-1}\exp(-t)dt$. When only the first term is taken from the sum of Eq.(\[iho correction\]), we obtain the energy value $$\begin{aligned}
\label{correction oscillator}
E_n=&\hbar\omega\left(n+\dfrac{3}{2}\right)-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2\notag\\
&-GMm\sqrt{\dfrac{m\omega_0}{\hbar}}\left[1-\dfrac{4E^{(0)}_n}{mc^2}\right]\Gamma(l+1)[\dfrac{1}{2}(n-l-1)]!.\end{aligned}$$

Note that if $G=0$, we return to the solution for an isotropic harmonic oscillator with spherical symmetry without gravitational field. It is peculiar that the Schwarzschild radius $r_{sch}$ is obtained in a very natural way.

The two cases we have analyzed in this section are extreme, and correspond to those where the gravitational field is much more intense than the harmonic oscillator and vice-versa. The case where the two potentials are comparable is much more complex and we leave it for a future work.

Infinite Spherical Well {#Infinite Spherical Potential G}
=======================

In this section we continue discussing well known examples from QM with the novelty of the presence of a gravitational field. We now deal with the problem of an infinite spherical well barrier $V_{inf}(r)$ that has two regions, this problem is commonly used in experimental verification or applications of QM. The barrier potential is given by $$V_{inf}(r)= \left\{
           \begin{array}{ll}
        0\ \ \ \text{if} \ \ \ 0<r<a, \\
         \infty \ \ \ \text{otherwise}. \\
           \end{array}
         \right.$$ We are only interested in studying the region where $0<r<a$, since outside of this region the wave function is zero, thus the probability to find a particle in the $r>a$ is null. The equation of motion (\[KG-N motion equation\]) in the region of interest is $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi -\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$ We can choose a principal Hamiltonian operator $\hat{H}_0$ from the motion equation, where $$\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi=E^{(0)}\Psi.$$ The zero-order correction of the energy is then $E^{(0)}_{ln}=\dfrac{\hbar^2}{2m}\dfrac{q_{ln}^2}{a^2}$. We can define the perturbed Hamiltonian $\hat{H}_p$ from the KG equation $$\begin{aligned}
\hat{H}_p=-\dfrac{GMm}{r}\left(1+\dfrac{4E^{(0)}}{mc^2}\right).\end{aligned}$$ The first-order correction of the energy $E^{(1)}$ can be calculated using the zero-order correction of the eigenfunction $\Psi_{nlk}^{(0)}(r,\theta,\phi)=A_{ln}j_l\left(\dfrac{q_{ln}}{a}r\right)Y_{lj}(\theta,\phi)$. Here $q_{ln}$ is n-th root of the Bessel spherical functions $j_l(x)$ and $A_{ln}$ is the constant of normalization $$A_{ln}^2=\dfrac{2}{a^3[j_{l+1}(q_{ln})]^2}.$$ Therefore $$\label{first order infinity sph energy}
E_{ln}^{(1)}=-GMm\left(1+\dfrac{4E^{(0)}}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle,
%$$ where, from perturbation theory we have $$\left\langle \dfrac{1}{r}\right\rangle=A_{ln}^2\int_0^{a}\abs{j_l\left(\dfrac{q_{ln}}{a}r\right)}^2rdr.$$ Unfortunately, it is not possible to solve the last integral with analytic methods, thus we will integrate it numerically. In general, there exists for each state $n=2l+1$ degeneration, thus for $l=0$, we have that $j_0(x)=\sin(x)/x$, whose n-th root is $q_{0n}=n\pi$ and the normalization’s constant is $A_{01}=\sqrt{2/a}$. Hence $$\left\langle \dfrac{1}{r}\right\rangle\approx\dfrac{2a}{\pi^2}(1.218),$$ Now, for $l=1$ the first excited state is three-fold degenerate, which means we need the first three roots of $j_1(x)=\dfrac{\sin(x)}{x^2}-\dfrac{\cos(x)}{x}$. The roots are $q_{11}\approx 4.49340$, $q_{12}\approx 7.72525$ and $q_{13}\approx 10.90412$. Therefore for $n=1$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{11}^2\left( \dfrac{a}{q_{11}}\right)^2 (0.4124),$$ for $n=2$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{12}^2\left( \dfrac{a}{q_{12}}\right)^2 (0.4590),$$ and for $n=3$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{13}^2\left( \dfrac{a}{q_{13}}\right)^2 (0.4778).$$ We can continue this process for the next excited states, as in the previous cases. If we make $G=0$, we return to solutions for QM without gravitational field.

Spherical Potential Barrier
===========================

The problem of square well potential with certain symmetry (spherical, cartesian or cylindrical) is important for some experiments of quantum systems. In this section, we study a square well barrier with spherical symmetry. Similarly as in the previous sections, we analyze the case of QM for a square well potential with space-time curvature. We use the Newtonian metric from (\[Newtonian metric\]), so the equation of motion (\[KG-N motion equation\]) transforms into $$\label{spherical KG motion eq}
-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi +V_B\left(1+\dfrac{2U}{c^2}\right)\Psi-\dfrac{4U}{c^2}\dfrac{\hbar^2}{2m}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}$$ where again $U=-GM/r$ is the gravitational potential. We take the Laplacian operator $\nabla^2$ in spherical coordinates and the potential $V_B$ as $$V_B(r)= \left\{
           \begin{array}{ll}
        -U_0\ \ \ \text{if} \ \ \ r<a, \\
         0 \ \ \ \ \ \ \ \text{if} \ \ \ r>a,
           \end{array}
         \right.$$ Thus, from Eq.(\[spherical KG motion eq\]) we can identify a principal Hamiltonian operator $\hat{H}_0$ $$\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+V_B\Psi=E^{(0)}\Psi.
\label{perturbed sph barrier}$$ The perturbed Hamiltonian $\hat{H}_p$ can be defined from Eq.(\[spherical KG motion eq\]) $$\hat{H}_p=-\dfrac{GMm}{r}\left(1+\dfrac{4E^{(0)}}{mc^2}-\dfrac{2V_B}{mc^2}\right)$$ In general, we can find the fist-order correction of energy from Eq.(\[energy correction\]) with the perturbed Hamiltonian operator in Eq.(\[perturbed sph barrier\]). This yields $$E^{(1)}=-GMm\left(1+\dfrac{4E^{(0)}}{mc^2}-\dfrac{2V_B}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle.$$ The case where $E<0$ shows the quantum nature of the system due to the fact that the energy spectrum is discrete. Potential $V_B$ defines naturally two regions, $Region\ I$ ($r<a$) and $Region\ II$ ($r>a$). Both regions without gravitational field are well-known. For $Region\ I$ the first-order correction of energy $E^{(1)}_{I}$ is given by $$E^{(1)}_I=-GMm\left(1+\dfrac{4E_{I}^{(0)}}{mc^2}+\dfrac{2U_0}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle_I,$$ where the zero-order correction of the eigenvalue is $E_{I}^{(0)}=\dfrac{\hbar^2}{2m}(\gamma_{ln})^2$ and the zero-order correction eigenfunction reads $\Psi_{lnj}(r,\theta,\phi)=R_{ln}(r)Y_{lp}(\theta,\phi)$. As usual, $Y_{lp}(\theta,\phi)$ are the spherical harmonic functions and $R_{ln}(r)=A_{ln}j_l(\gamma_{ln}r)$ is the radial solution, such that $j_l(x)$ are the spherical Bessel functions. Also $k_1=\gamma_{ln}$ is the n-th solution of the transcendental equation due to the boundary and continuity condition given by $$\label{trascendental eq}
\dfrac{1}{h_l^{(1)}(ik_2r)}\dfrac{dh_l^{(1)}(ik_2r)}{dr}|_{r=a}=\dfrac{1}{j_l(ik_1r)}\dfrac{dj_l(ik_1r)}{dr}|_{r=a}.$$ For $l=0$, it reduces to the transcendental equation $$-k_2=k_1\cot(k_1a).$$ In the limit $|E|<<U_0$, we return to the solution for the first root. When $l=0$ we recover the same result as in the last section, namely $k_1a\approx \pi/2$. In general for Region I, we need to integrate $$\left\langle \dfrac{1}{r}\right\rangle_I=A_{ln}^2\int^a_0\abs{j_l\left(\dfrac{\sigma_{ln}}{a}r\right)}^2rdr,$$ where $\sigma_{ln}=a\gamma_{ln}$. If we define the parameter $\eta=\sigma_{ln}/q_{ln}$ that compares the n-th solution of the transcendental equation with the n-th root, we see that when $\eta=1$ we return to case of Section \[Infinite Spherical Potential G\]. However, the solution of this integral should be calculated using numerical methods.

In $Region\ II$, the zero-order correction is given by $$E_{II}^{(0)}=\dfrac{\hbar^2}{2m}(\gamma^*_{ln})^2=\dfrac{\hbar^2}{2M}\left(\dfrac{\sigma^*_{ln}}{a}\right)^2,$$ where we have introduced the new parameters $\sigma^*_{ln}=a\gamma^*_{ln}$, and $\eta^*=\gamma^*_{ln}/q_{ln}$. The first-order correction is similar to that of $Region\ I$ $$E^{(1)}_{II}=-GMm\left(1+\dfrac{4E_{II}^{(0)}}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle_{II},$$ with the eigenfunction $\Psi_{lnj}(r,\theta,\phi)=R_{ln}(r)Y_{lj}(\theta,\phi)$. Nevertheless, the radial function has a new form $R_{ln}(r)=Bh_l^{(1)}(i\gamma_{ln}^*r)=Bj_l(i\gamma_{ln}^*r)+i Bn_l(i\gamma_{ln}^*r)$, where $B$ is a constant of normalization, $h_l^{(1)}(x)$ are the spherical Hankel functions of the first kind, $n_l(x)$ are the spherical Neumann function and $k_2=\gamma_{ln}^*$ is the solution of the transcendental equation (\[trascendental eq\]). Therefore $$\begin{aligned}
\left\langle \dfrac{1}{r}\right\rangle_{II}=&B^2\int_a^\infty\abs{h_l^{(1)}\left(i\gamma^*_{ln}\right)}^2rdr\notag\\
=&B^2\int_a^\infty\left(\abs{j_l(i\gamma_{ln}^*r)}^2- \abs{n_l(i\gamma_{ln}^*r)}^2\right) rdr,\end{aligned}$$ This is a general solution of this perturbation, and if we want to solve this integral, we need to set every parameter for a specific case. As expected, just like in the previous cases, if $G=0$ we return to the solutions without a gravitational field.

Conclusions
===========

Throughout the course of this work we studied the KG equation in a weak gravitational field for different external potentials with the objective of understanding the quantum effects of a boson gas on a gravitational field. Typical examples of QM were analyzed featuring the addition of space-time curvature (or gravitational effects). Starting from the most general equation for bosons in Quantum Field Theory in curved space-times, we found a generalized Schrödinger equation that is simply the KG covariant one. To solve the differential equation, we identified the principal and perturbed Hamiltonian operator in each case, and compared them with the well-known results in QM. Each example was worked out on an inertial frame. This is an important aspect to highlight due to the main idea of this work. We expect to obtain the same results when gravitational effects are measured on a quantum system in the laboratory. This quantum system (with scalar particles) will be on a non-inertial frame, where we predict that the Einstein’s equivalence principle gives us a correspondence between experimental and theoretical results.\
These results let us find the limit, where we could measure quantum gravitational effects, namely $2\alpha\, mM/m_{pl}^2\sim 1$. For this analysis we could be able to apply our results in micro black holes scale[@microBH]. For example, if we consider a mass of a scalar particle $m$ as the mass of the electron, we obtain that the gravitational mass $M$ that affects the electron by quantum gravitational effects is $\sim 10^{16}$kg. In the same way, a $M\sim 10^{13}$kg could affect a particle with a proton mass. If we think on the easiest non-inertial frame, we could think on a rotating system, for which there would not be a real mass $M$, but rather an effective mass given by angular frequency $\Omega$, where $\Omega^2=GM/r^3$. Using a detector to $r=1$ meter, the corresponding gravity can be reached with a rotational wheel spinning with an angular frequency of $\Omega\approx 1543$ rad/s=245 rev/s for $M\sim 10^{16}$kg, and $\Omega\approx 25$ rad/s=3.9 rev/s for $M\sim 10^{13}$kg. However, for a mass $M\sim M_\odot$, we would need an angular frequency of $\Omega\sim 10^{9}$rev/s, to obtain relativistic effects.\
Analyzing the typical experiments on a harmonic oscillator [@oscillator] and on an infinite well barrier [@pozo] for a particle with an electron mass, the correction due to the gravitational effects has, in both cases, a dominant term which is given by $GMm\left\langle \dfrac{1}{r}\right\rangle$. For the case of a harmonic oscillator with frequency $\omega_0\sim G$Hz and using the value for the mass $M\sim 10^{16}$kg. The dominant term of the first-order correction of the energy in this case is $\sim 10^{-5}E_{osc}^{(0)}$ from Eq.(\[correction oscillator\]), while other contributions in this equation are $<10^{-9}E^{(0)}_{osc}$. On the other hand, for a particle with an electron mass confined in an infinite well barrier of width $a\sim 10$nm, it is possible to obtain that $E_{inf}^{(0)}\sim 10$eV. The dominant term in Eq.(\[first order infinity sph energy\]) is $\sim 10^{-5}E_{inf}^{(0)}$ for the first-order correction of energy, and the other term in this correction is $ <10^{-20}E_{inf}^{(0)}$. Note that in either cases, the first-order corrections of energy have the same order if we compare them with their zero-order counterparts. With this analysis it is possible to conclude that we can measure in a laboratory the effects of the quantization of a weak gravitational field directly or using non-inertial systems.

Acknowledgments
===============

This work was partially supported by CONACyT México under grants CB-2011 No. 166212, CB-2014-01 No. 240512, Project No. 269652 and Fronteras Project 281; Xiuhcoatl and Abacus clusters at Cinvestav, IPN; I0101/131/07 C-234/07 of the Instituto Avanzado de Cosmología (IAC) collaboration (http:// www.iac.edu.mx). O.G. acknowledge financial support from CONACyT doctoral fellowship. Works of T.M. are partially supported by Conacyt through the Fondo Sectorial de Investigación para la Educación, grant CB-2014-1, No. 240512

[^1]: ogallegos@fis.cinvestav.mx

[^2]: tmatos@fis.cinvestav.mx
