Nonrelativistic spinless particle in vicinity of Schwarzschild-like black hole

1. Introduction

The black holes were first speculated by John Michel in 1784. However, Michel called these imaginary structures dark stars. In such astronomical bodies, the escape velocity would exceed the speed of light. What began with theoretical stars so dense, that not even a single particle of light would be able to escape and as such would not be seen, evolved into the current concept of black holes. The name black hole, however, only emerged in 1963, when John Wheeler, a professor at Princeton, supported by the theory of general relativity, popularized these fabulous physical objects. Since then, black holes have become an important subject to study because black holes allow scientists to study what happens in the most extreme conditions we know of in our universe with regard to temperature, mass, density and time. Thus, studies on black holes have become abundant in the scientific literature. In particular, research that deals with the movement of particles closer to black holes is an interesting topic of investigation, as it can reveal the behavior of matter in the presence of intense gravitational fields. One way to describe the movement of particles around hypermassive bodies, probably black holes, is to consider quantum equations in curved space–time.

The analysis of wave propagation opens a perspective on additional phenomena such as interference effects and radiation scattering in a black hole. Within the scope of quantum mechanics, particles are described by so-called wave functions. In such a way, free particles are effectively described by wave solutions. This subject has been widely discussed in the literature, especially in the case of relativistic particles. In this sense, Damour et al. developed a typical work describing the particles in the vicinity of a black hole by solving the Klein–Gordon equation in the Schwarzschild space–time,¹ in which the solution of the incident and emitted waves was obtained. The ideas of Damour et al. were used by Sannan to construct the particle probability distribution emitted from a black hole.² In the works cited, Hawking radiation and the evaporation of black holes were treated.^3,4 In the context of particles moving at relativistic speeds, other approaches involving Klein–Gordon and Dirac equations in curved spaces are presented in the literature; for instance, the relativistic wave equations are solved for different metrics, such as Kerr, Bertotti–Robinson and AdS black hole.^5,6,7 In the same aspect, the Klein–Gordon equation is used in the works of Vieira and Bezerra in which resonance frequencies, Hawking radiation and scattering of scalar waves from various types of black holes are investigated, such as the Kerr–Newman–Kasuya (dyon black hole) and Reissner–Nordströn black hole.^8,9 The Klein–Gordon equation in curved space can also be used to study the stability of black holes and quasi-normal modes of black holes from various points of view.^{10,11,12,13,14,15,16}

There are similar works in the literature involving nonrelativistic particles; however, the Schrödinger equation in curved space is generally obtained by the nonrelativistic limit of the Dirac or Klein–Gordon equation. This approach allows the generalization of the Schrödinger equation to the curved space.¹⁷ On the other hand, such a generalization leads to a profound question: would it be possible to describe a nonrelativistic field through a covariant structure? The answer is positive and the structure that makes this possible is known as Galilean covariance. This approach was introduced by Takahashi based on a covariant version of the Galilei group, which is based on fifth-dimensional tensors.^18,19 This method was used to analyze nonlinearized field equations from which the symmetries of superfluids were discussed in connection with the Goldstone bosons.²⁰ Since then, the notion of Galilean covariance has been developed in several areas, such as cosmology, gravitation and high-spin particle theory in condensed matter.^{21,22,23,24,25,26,27} In this work, we intend to apply the Galilean covariance method to analyze nonrelativistic particles in Schwarzschild-type geometry. Particularly this Schwarzschild-type solution is obtained within the scope of Galilean gravity. That is, Newtonian gravitation is described in a five-dimensional manifold. A covariant structure allows the understanding of the properties of the physical field from a new perspective. It facilitates algebraic manipulation and puts relativistic and nonrelativistic regimes on an equal footing. The covariant description of the nonrelativistic regime of the gravitational field invites us to investigate the coupling of this field with others described in the Galilean manifold, such as the scalar field, which represents the Schrödinger equation after the appropriate embedding, reducing the five dimensions back to the usual 4.

The presentation of this work is based on the following topics: In Sec. 2, we provide a brief review of Galilean covariance. In Sec. 3, we construct the covariant version of the Schrödinger equation. In Sec. 4, the Schrödinger equation for Schwarzschild-type space–time is solved. Section 5 is dedicated to solution analysis. Finally, in Sec. 6, we present the conclusion and perspectives.

2. Galilean Covariance

In this section, we construct the mathematical framework to write the covariant Schrödinger equation. For this purpose, we address the Galilean covariance.

In the three-dimensional Euclidean space ${𝔼}_3$ we can define the inner product between two vectors $x=(x^1,x^2,x^3)$ and $y=(y^1,y^2,y^3)$ as

Then, the square of norm of vector $x$ in ${𝔼}_3$, denoted by $\parallel x{\parallel }^2$, is given by Note that $\parallel x\parallel >0$, i.e. $\parallel x\parallel$ is positive defined.

In order to construct a nonrelativistic space with dimension up to three we define the inner product between the vectors $x=(x^1,x^2,x^3,x^4,x^5)$ and $y=(y^1,y^2,y^3,y^4,y^5)$ by

Equation (3) can be written by where We notice that $x^{\mu }={\eta }^{\mu \nu }{x}_{\nu }$, $x_{\mu }={\eta }_{\mu \nu }{x}^{\nu }$ and ${\eta }^{\mu \nu }={({\eta }_{\mu \nu })}^{-1}$. A vector space ${𝔾}_5$ constituted by inner product given in Eq. (4) is a pseudo-Riemannian five-dimensional manifold. In this sense, ${\eta }_{\mu \nu }$ is the metric of ${𝔾}_5$. The square of norm of a vector x, denoted by $\parallel x{\parallel }^2$, defined in ${𝔾}_5$ is given by Similarly to what occurs in the Minkowski space, we can obtain vectors such as $\parallel x{\parallel }^2>0$, $\parallel x{\parallel }^2<0$ and $\parallel x{\parallel }^2=0$.

Note that if we choose $x^5=\frac{\parallel x{\parallel }^2}{2t}$, $y^5=\frac{\parallel y{\parallel }^2}{2t}$, $x^4=y^4=t$, Eq. (3) reduces to

This last result can be considered as an embedding of ${𝔼}_3$ in ${𝔾}_5$ given by where $x\in {𝔼}_3$ and $x\in {𝔾}_5$.

The physical content of this embedding is more clear considering the dispersion relation $\frac{\parallel p{\parallel }^2}{2m}=E$, i.e. $\parallel p{\parallel }^2-2mE=0$. In this case, it is interesting to to define the five-momentum vector $p^{\mu }=p=(p,m,E)$, with $\mu =1,\dots ,5$, $p$ stands for the usual three-dimensional momentum, m is the mass and E is the energy. Fixing the notation $p^4=E/v$ and $p^5=vm$, in which v is a constant with unity of velocity, then the general five-dimensional dispersion relation can be written as

where $\kappa$ is a constant. It’s worth noting that the Galilean manifold is suitable for describing coordinates and moments in unified structures, namely tensors under Galilean transformations. This implies that nonrelativistic physics takes on a covariant form in a five-dimensional Galilean manifold. In this work, we adopt $v=1$, obtaining $p^4=E$, $p^5=m$ and $\kappa =0$. It is worth noticing that the coordinates of the five-vector $x^{\mu }=x$ defined in Sec. 2 are the canonical coordinates associated with the five-momentum $p^{\mu }$. Then, $x$ is the canonical three-vector associated with three-momentum $p$, $p^4$ is the canonical coordinate correspondent to E and $p^5$ is the canonical coordinate associated with m. In this sense, in accordance with Eq. (9), we define the dispersion relation coordinates $x^{\mu }{x}_{\mu }={\eta }_{\mu \nu }{x}^{\mu }{x}^{\nu }=\parallel x{\parallel }^2-2x^4{x}^5=s^2$. In the special case $p^{\mu }{p}_{\mu }=0$, we obtain the relations $x^5=\frac{\parallel x{\parallel }^2}{2t}$, $x^5=\frac{\parallel x{\parallel }^2}{2t}$, $x^4=t$, i.e. $s=0$. Here, it is necessary to clarify that time has no dimension of distance. So, a constant with velocity dimension is used to adjust the dimensionality of the coordinate $x^4$, say $v_0$. This constant represents some characteristic speed of the nonrelativistic system, such as the speed of sound. It is worth remembering that in the covariant formalism of nonrelativistic fields, there is no universal limit speed, such as the speed of light, in principle. But the speed $v_0$ can be identified with c by experimental criteria. Whatever the case, it is possible to adopt a system of units in which $v_0=1$, which we will use unless explicitly stated.

The results presented in this section are useful to construct the covariant Schrödinger equation.

3. Covariant Schrödinger Equation

In this section, it is shown how we can obtain the Schrödinger equation from a representation of the Galilei group. From this representation, we construct the covariant version for Schrödinger equation.

Galilei group in the covariant notation is defined by the transformations ${𝒢}:(x,t)\to (\overline{x},\overline{t})$ given by

where $R_j^i$ stands for rotations, $v^i$ stands for boost, $a^i$ spatial translation and $a^4$ time translation. The generators of this group can be written by $l_i=\frac{1}{2}{ϵ}_{ijk}{M}_{jk}$, $k_i=M_{5i}$, $c_i=M_{4i}$, $d=M_{54}$, where $M_{ij}=x_i{p}_j-x_j{p}_i$.

Nonrelativistic physical theories, specifically quantum theory, are obtained from unitary and faithful representations of Galilei group. In this context, covariant Schrödinger equation can be obtained from a faithful unitary representation of the Galilei group constructed by the operators defined as ($\hslash =1$) ${\widehat{x}}_{\mu }=x^{\mu }$, ${\widehat{p}}_{\mu }=-i{\partial }_{\mu }$. The generators of this group satisfy the nonnull commutations relations

It is worth noting that the Heisenberg uncertainty relation is obtained from the operators’ position and momentum because at this moment we are dealing with a quantum theory. In this work, we consider only spinless representations. In this case, the invariant of the Galilei algebra is given by

$$I_1={\widehat{p}}^{\mu }{\widehat{p}}_{\mu },$$(10)

$$I_2=\widehat{p_5}I,$$(11)

where $I$ is the identity operator. Applying these invariants in wave function $\mathrm{\Psi }(x)$, we obtain

$${\widehat{p}}^{\mu }{\widehat{p}}_{\mu }\mathrm{\Psi }=k^2\mathrm{\Psi },$$(12)

$${\widehat{p}}_5\mathrm{\Psi }=-m\mathrm{\Psi }.$$(13)

Once ${\widehat{p}}_5=i{\partial }_5$, we can write $\mathrm{\Psi }(x)=\psi (x,x^4)\phi (x^5)$. Then, the solution of Eq. (13) is given by In this sense, $\mathrm{\Psi }(x)=e^{im{x}^5}\psi (x,x^4)$. Using this last expression in Eq. (12), we obtain which is the usual Schrödinger equation, since we recognize $x^4=t$. Finally, from Eq. (14), we can write the covariant Schrödinger equation as This equation will be generalized to curved space in the following section.

4. Covariant Schrödinger Equation in Schwarzschild-Like Space–Time

The Galilean covariance is a formalism that aims to describe nonrelativistic fields in a covariant way. For this purpose, a five-dimensional manifold is introduced. In Sec. 3, it was shown that the structure of the scalar field in the Galilean manifold reduces to the Schrödinger equation when a dimensional reduction is performed. It should be noted that the curvature of the metric presented in this section vanishes identically. Thus, it is natural to think that nonzero curvature describes a covariant Newtonian gravitation. In fact, a Schwarzschild-type metric is presented in Ref. 26, so the scalar field in five dimensions can couple with this geometric structure. That is, the result is the description of the Schrödinger equation coupled to Newtonian gravitation. The Galilean Schwarzschild-like line element is given by

where $f(r)=1-2M/r$ and M stands for the nonrelativistic black hole mass. In this way, the metric tensor can be written as In this situation, covariant Schrödinger equation can be written as with $\sqrt{-g}=r^{2}sin\theta$ and $g^{\mu \nu }$ is the inverse tensor of $g_{\mu \nu }$, which is given by

In this way, the following differential equation is obtained:

As $\mathrm{\Psi }(x)=e^{im{x}^5}\psi (x,t),$ from Eq. (14), we get

$$\begin{array}{c}f(r)\left[\frac{\partial }{\partial r}\left(f(r)\frac{\partial \psi }{\partial r}\right)+\frac{2}{r}f(r)\frac{\partial \psi }{\partial r}+\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial \theta }\left(sin\theta \frac{\partial \psi }{\partial \theta }\right)+\frac{1}{r^2{sin}^2\theta }\frac{{\partial }^2\psi }{\partial {\varphi }^2}\right]\hfill \\ +im\frac{\partial \psi }{\partial t}=-k^{2}f(r)\psi .\hfill \end{array}$$(22)

Now, using the ansatz $\psi (x,t)=e^{-iEt}\mathrm{\Phi }(r,\theta ,\varphi )$, Eq. (22) can be rewritten as

$$\begin{array}{c}f(r)\left[\frac{\partial }{\partial r}\left(f(r)\frac{\partial \mathrm{\Phi }}{\partial r}\right)+\frac{2}{r}f(r)\frac{\partial \mathrm{\Phi }}{\partial r}+\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial \theta }\left(sin\theta \frac{\partial \mathrm{\Phi }}{\partial \theta }\right)+\frac{1}{r^2{sin}^2\theta }\frac{{\partial }^2\mathrm{\Phi }}{\partial {\varphi }^2}\right]\hfill \\ +(mE+k^{2}f(r))\mathrm{\Phi }=0.\hfill \end{array}$$(23)

Taking we obtain where $Y_{l,m}$ is the usual spherical harmonics. Simplifying Eq. (23) we get the following equation for the radius r :

In order to solve Eq. (26), we write it in the Sturm–Liouville form as

Through the transformations $z=1-r/2M$ and it is possible to rewrite Eq. (27) as a confluent Heun differential equation^28,29,30 in the form where

$$\alpha =-4M\sqrt{-mE-k^2},$$(28)

$$\beta =-4M\sqrt{-mE},$$(29)

$$\gamma =0,$$(30)

$$\delta =-4M^2\left(2mE+k^2\right),$$(31)

$$\eta =4M^2(2mE+k^2)-l(l+1),$$(32)

$$\nu =\delta -\mu +\alpha \frac{\beta +\gamma +2}{2},$$(33)

$$\mu =\frac{\left(\alpha -\gamma \right)\left(\beta +1\right)-\beta }{2}-\eta .$$(34)

This transformation gives the solution in terms of confluent Heun functions, valid for $0<r<4M$, as

$$\begin{array}{ccc}{R}_l(r)\hfill & =\hfill & c_1{R}_{l_f}(r)+c_2{R}_{l_{\infty }}(r)\hfill \\ \hfill & =\hfill & c_1{(2M-r)}^{2\sqrt{-mE}M}{\mathrm{\text{e}}}^{\sqrt{-mE-k^2}r}\hfill \\ \hfill & \hfill & \times HeunC\left(-4M\sqrt{-mE-k^2},4\sqrt{-mE}M,0,-4M^2(2Em+k^2),\right.\hfill \\ \hfill & \hfill & 4M^2(2Em+4k^2)-l(l+1),1-\frac{r}{2M})\hfill \\ \hfill & \hfill & +c_2{(2M-r)}^{-2\sqrt{-mE}M}{\mathrm{\text{e}}}^{\sqrt{-mE-k^2}r}\hfill \\ \hfill & \hfill & \times HeunC\left(-4M\sqrt{-mE-k^2},-4\sqrt{-mE}M,0,-4M^2(2Em+k^2),\right.\hfill \\ \hfill & \hfill & 4M^2(2Em+k^2)-l(l+1),1-\frac{r}{2M}),\hfill \end{array}$$(35)

noting that $HeunC=HeunC(\alpha ,\beta ,\gamma ,\delta ,\eta ,w)$ is defined for $|w|<1$ and $\alpha ,\beta ,\gamma ,\delta ,\eta \in ℂ$.^28,29,30 Moreover, Heun’s confluent function is defined satisfying the initial conditions and This result will be used in further sections.

5. Analysis of Solution

In this section, we will analyze some properties of the solution obtained in Sec.4. We will analyze how energy can be quantized and how a particle can leave the event horizon through the tunnel effect. It is worth noting that the solution of the Schrödinger equation coupled to the nonrelativistic gravitational field remains restricted to the vicinity of the black hole, as the Heun function is not defined at spatial infinity.

5.1. Energy levels near to Schwarzschild-like radius

Closer to the nonrelativistic Schwarzschild radius, $r\to 2M$, the confluent Heun’s function series expansion for the finite part of the solution, $R_{l_f}(r)$, gives us

so, for sufficiently small $\epsilon >0,$ such that $r-2M=\epsilon ,$ we get the approximation which gives us Then Finally, solving Eq. (39) for E the following approximation for the energy is obtained : for $l=0,1,2,\dots$. It should be noted that energy level $E_l$ depends on quantum number l, in addition it is inversely proportional to square of black hole mass. It is valid on the event horizon.

5.2. Energy levels for nonrelativistic supermassives black holes

In this section, we analyze the solution of the Schrödinger equation for Schwarzschild-like space–time, considering supermassive black holes. In this case, when $r\to 4M$, it is necessary that the wave of the function be finite and well behaved in all points where it is defined. Then, as $r\to 4M,$ the confluent Heun function must be a polynomial and the following necessary condition will be satisfied in this case^28,29,30 :

where n is an integer number and also represents the degree of the confluent Heun polynomials. Plugging the constant values in Eqs. (28)–(31) we get The condition expressed in Eq. (42) establishes the quantization of energy of the particle in vicinity of a Schwarzschild-like black hole. In the particular case $k=0$, we obtain In this sense, we notice that the maximum energy state has energy equal to $E_0=-\frac{1}{32M^{2}m}$. The negative signal of energy is because the particle of mass m is bounded to the mass M. Since M is much larger, those energy levels are very small. It should be noted that energy is defined as being less than a constant. Thus, k can be made null without loss of generality.

5.3. The tunnel effect

In this section, we use the solution given in Eq. (29) to discuss how a particle can pass through the event horizon via quantum tunneling. First, it should be noted that we can consider Eq. (29) as a progressive traveling wave if $-mE<0$, which means a nonbound state. In this case, we have

for $r<2M$ and

$$\begin{array}{ccc}{R}_{l\mathrm{\text{out}}}(r)\hfill & =\hfill & {(2M-r)}^{-2i\sqrt{2mE}M}{\mathrm{\text{e}}}^{\sqrt{-2mE}r}\hfill \\ \hfill & \hfill & \times HeunC\left(-4M\sqrt{-2mE},4\sqrt{-2mE}M,0,-4M^2(4Em),\right.\hfill \\ \hfill & \hfill & \left.4M^2(4Em)-l(l+1),1-\frac{r}{2M}\right),\hfill \end{array}$$(45)

for $2M<r<4M$. It is here considered $k=0$. The function $R_{l\mathrm{\text{in}}}$ represents a progressive wave inside the event horizon, while $R_{l\mathrm{\text{out}}}$ is the wave outside the black hole. It should be noted that $HeunC(\alpha ,\beta ,\gamma ,\delta ,\eta ,0)=1$, which is reached at $r\to 2M$, thus

$$\begin{array}{ccc}{R}_{l\mathrm{\text{in}}}(r)\hfill & =\hfill & {(2M-r)}^{2i\sqrt{2mE}M}{\mathrm{\text{e}}}^{\sqrt{-2mE}r}\hfill \\ \hfill & =\hfill & e^{2i\sqrt{2mE}M[log|2M-r|+i(2p\pi )]}{e}^{\sqrt{-2mE}r},p=0,1,2,\dots \hfill \end{array}$$(46)

and

$$\begin{array}{ccc}{R}_{l\mathrm{\text{out}}}(r)\hfill & =\hfill & {(2M-r)}^{-2i\sqrt{2mE}M}{\mathrm{\text{e}}}^{\sqrt{-2mE}r}\hfill \\ \hfill & =\hfill & e^{-2i\sqrt{2mE}M[log|2M-r|+i(\pi +2q\pi )]}{e}^{\sqrt{-2mE}r},q=0,1,2,\dots .\hfill \end{array}$$(47)

Then, there is a transmission coefficient given by which is hence

$$\begin{array}{ccc}\tau \hfill & =\hfill & |e^{4i\sqrt{2mE}M[log|2M-r|+i(\pi +2(p+q)\pi )]}{|}^2\hfill \\ \hfill & =\hfill & e^{-8\pi (2s+1)\sqrt{2mE}M},s=0,1,2,\dots ,\hfill \end{array}$$(49)

the parameter s in expression (49) represents the various branches of the complex number mentioned above. The reflection coefficient can also be written immediately, it reads

$$\begin{array}{ccc}N\hfill & =\hfill & \frac{\tau }{1-\tau }\hfill \\ \hfill & =\hfill & \frac{1}{e^{8\pi (2s+1)\sqrt{2mE}M}-1},s=0,1,2,\dots ,\hfill \end{array}$$(50)

that is, the coefficient respects the restriction given by a black hole’s event horizon, even in nonrelativistic physics. However, the transmission coefficient denotes a different prediction from that of black hole physics. Thus, a particle described by the progressive wave function can pass through the event horizon by means of quantum tunneling. It is undeniable that such an effect in relativistic physics would be a mechanism capable of explaining Hawking radiation. On the other hand, in the nonrelativistic description, there is no such radiation, as the particles do not leave the black hole towards spatial infinity. What occurs, therefore, in the nonrelativistic description is an evaporation of the black hole and concentration of matter in the vicinity of the black hole.

6. Concluding Remarks

In this work, we constructed a version for covariant Schrödinger equation for Schwarzschild-like geometry. In sequence, we solved analytically the equation obtained. The solution is given in terms of confluent Heun’s function. Using the properties of Heun’s function we analyze the solution aimed to study the nonrelativistic particles in vicinity of a Schwarzschild-like black hole. As results we obtained that the energy of particle interacting with black hole is quantized in terms of angular quantum number in the case of ordinary black holes and in terms of a integer number in the case of supermassive black holes. Furthermore, we study the tunnel effect through the black hole’s event horizon. We intend to extend the ideas presented in this work to other geometries.

ORCID

R. G. G. Amorim  https://orcid.org/0000-0001-6532-3087

V. C. Rispoli  https://orcid.org/0000-0003-0945-786X

S. C. Ulhoa  https://orcid.org/0000-0001-9994-958X

K. V. S. Araújo  https://orcid.org/0009-0002-7780-4958