QUANTUM PROBING OF SINGULARITIES AT EVENT HORIZONS OF BLACK HOLES

V. P<EMAIL_ADDRESS><EMAIL_ADDRESS>
Russian Federal Nuclear Center–All-Russian Research Institute of Experimental
Physics, Mira pr., 37, Sarov, 607188, Russia
National Research Nuclear University MEPhI, Moscow, 115409, Russia

###### Abstract

It is proved that coordinate transformations of the Schwarzschild metric to
new static and stationary metrics do not eliminate the mode of a particle
”fall” to the event horizon of a black hole. This mode is unacceptable for the
quantum mechanics of stationary states.

Keywords: Quantum mechanics in space-time of black holes; static, stationary
and non-stationary metrics of the general relativity; coordinate
transformations; unitary transformations; Lorentz transformations; particle
”fall” to the black hole horizon.

PACS numbers: 03.65.-w, 04.62.+v, 04.70.Dy

##  1 Introduction

In Refs. [1], [2], we considered the interaction of scalar particles
$\left({S=0}\right)$, photons $\left({S=1}\right)$, fermions
$\left({S=1/2}\right)$ with Schwarzschild, Reissner–Nordström, Kerr and
Kerr–Newman black holes with zero and non-zero cosmologic constants. For the
above metrics and particles with different spins, existence of the mode of a
particle ”fall” to event horizons has been found. The mode of a
quantum–mechanical ”fall” of particles to the singular center is, for instance
, examined in detail in Refs. [3]-[5]. This mode is unacceptable for quantum
mechanics.

Within the framework of the general relativity (GR), the coordinate
singularity of the Schwarzschild metric (S) can be eliminated by using the
appropriate coordinate transformations of an initial metric. Unlike the
Schwarzschild metric, in the space-time of transformed metrics, classical
particles can intersect the event horizon without emergence of singularities.
Many of the researchers apply this conclusion to the quantum theory as well.
However, it is not correct. We proved that the coordinate transformations of
the Schwarzschild metric in the quantum mechanics of stationary states do not
eliminate the mode of a particle ”fall” to the event horizon.

As an example, it is proved by using Eddington–Finkelstein (EF) [6, 7], and
Painlevè–Gullstrand (PG) [8, 9] stationary metrics. This conclusion is also
valid for any transformed static metrics, including the transition in the
Schwarzschild solution to the ”tortoise” coordinate [10].

The quantum mechanics in the space-time of non-stationary Lemaítre–Finkelstein
[7, 11], and Kruskal- Szekeres [12, 13] metrics leads to time-dependent
Hamiltonians [14]. Study of stationary states of particles, with
representation of wave functions in coordinates of these metrics in the form
of $\sim\Psi\left({{\rm{\bf R}}}\right)e^{-iET}$ for the Lemaítre-Finkelstein
metric and in the form of $\sim\Psi\left({{\rm{\bf u}}}\right)e^{-iE\nu}$ for
the Kruskal–Szekeres metric, is impossible in this case.

In this paper, our study will be considered for fermions. The obtained results
can be extended to the equations and wave functions of photons and spinless
particles.

The paper is arranged as follows. In Sec. 2, for coherence of the description,
we present basic features of the quantum-mechanical mode of a particle ”fall”
to the event horizon. In Secs. 3 and 4, we introduce the covariant Dirac
equation and provide the bases of the theory of coordinate transformations of
Hamiltonians for the Dirac equation in the space-time of (GR) metrics. These
results are presented in many textbooks, monographs and papers (see, for
instance, Refs. [15], [16] as well as our papers Refs. [14], [17]). In Sec. 5,
we analyze the solutions of the Dirac equation for a Schwarzschild metric. In
Secs. 6 – 8, we prove the presence of the quantum-mechanical mode of particle
”fall” by using stationary EF and PG metrics as well as by using the
”tortoise” coordinate in the Schwarzschild static metric. In Sec. 10, we
formulate basic results of the paper.

##  2 The mode of a particle ”fall” to the event horizon

For all black holes, the behavior of effective potentials of the radial
equation for Schrödinger-type fermions in the neighborhood of event horizons
has the form of an infinitely deep potential well [1, 2]

$\left.{U_{eff}\left(r\right)}\right|_{r\to
r_{\pm}}=-\frac{K_{1}}{\left({r-r_{\pm}}\right)^{2}},$ (1)

where $r_{\pm}$ are radii of external and internal event horizons and a
coefficient $K_{1}>1/8$. In this case, the mode of a particle ”fall” to the
event horizon is implemented (see Refs. [3] \- [5]).

The behavior of the real radial function of the Schrödinger-type equation is
given by††† In expression (2) and in what follows, the asymptotic behavior of
transformation operators and wave functions is examined for neighborhoods of
event horizons $\left({r>r_{+}}\right)$, $\left({r<r_{-}}\right)$.

$\left.{R\left(r\right)}\right|_{r\to
r_{\pm}}\sim\left|{r-r_{\pm}}\right|^{1/2}\sin\left({\sqrt{K_{2}}\ln\left|{\frac{r}{r_{\pm}}-1}\right|+\delta}\right),$
(2)

where $K_{2}=2\left({K_{1}-1/8}\right)$. At $r\to r_{\pm}$, the radial
functions of stationary states of discrete and continuous spectra have the
infinite number of zeros, the discrete energy levels emerge and ”dive” beyond
the allowed domains of functions $R\left(r\right)$. At $r=r_{\pm}$, the
functions of $R\left(r\right)$ do not have definite values.

In the Hamiltonian formulation, the mode of a particle ”fall” to the horizon
means that a Hamiltonian $H$ has non-zero deficiency indexes [18] \- [20].

To remove this singular mode, unacceptable for quantum mechanics, it is
necessary to choose additional boundary conditions on event horizons. The
self-conjugate extension of a Hermitian operator $H$ is determined by this
choice.

##  3 The Dirac equation

In the system of units of $\hslash=c=1$ and in the signature
$\left({+\,\,-\,\,-\,\,-}\right)$, the Dirac equation equals

$i\gamma^{\alpha}\left({\frac{\partial\psi}{\partial
x^{\alpha}}+\Phi_{\alpha}\psi}\right)-m\psi=0,$ (3)

where $m$ is a fermion mass, $\Phi_{\alpha}$ are bispinor connections, $\psi$
is a four-component bispinor, $\gamma^{\alpha}$ are 4x4 Dirac matrixes with
world indexes satisfying the relation of

$\gamma^{\alpha}\gamma^{\beta}+\gamma^{\beta}\gamma^{\alpha}=2g^{\alpha\beta}E,$
(4)

where $g^{\alpha\beta}$ is an inverse metric tensor, $E$ is a 4x4 unity
matrix.

In expressions (3), (4) and in what follows, the values designated by letters
of the Greek alphabet assume the values of 0, 1, 2, 3; those designated by the
letters of the Latin alphabet take the values of 1, 2, 3. When upper and lower
indices are the same, the summation of appropriate summands is implied.

Then, along with the Dirac matrixes of $\gamma^{\alpha}$ with world indices,
we use Dirac matrixes $\gamma^{{\underline{\alpha}}}$ with local indices
satisfying the relationship of

$\gamma^{{\underline{\alpha}}}\gamma^{{\underline{\beta}}}+\gamma^{{\underline{\beta}}}\gamma^{{\underline{\alpha}}}=2\eta^{{\underline{\alpha}}{\underline{\beta}}}E,$
(5)

where $\eta^{{\underline{\alpha}}{\underline{\beta}}}$ corresponds to the
inverse metric tensor of the plane Minkowski space with the signature of
$\eta_{{\underline{\alpha}}{\underline{\beta}}}=\mbox{diag}\left[{1,-1,-1,-1}\right].$

In Eq. (3), for determination of bispinor connections $\Phi_{\alpha}$, it is
necessary to choose a definite system of tetrad vectors
$H_{{\underline{\alpha}}}^{\mu}$, satisfying the relation of
$H_{{\underline{\alpha}}}^{\mu}H_{{\underline{\beta}}}^{\nu}g_{\mu\nu}=\eta_{{\underline{\alpha}}{\underline{\beta}}}$.
The bispinor connections are determined by using Christoffel derivatives of
tetrad vectors.

$\Phi_{\alpha}=-\frac{1}{4}H_{\mu}^{{\underline{\varepsilon}}}H_{\nu{\underline{\varepsilon}};\alpha}S^{\mu\nu}=\frac{1}{4}H_{{\underline{\mu}}}^{\varepsilon}H_{{\underline{\nu}}\,\varepsilon;\alpha}S^{{\underline{\mu}}{\underline{\nu}}},$
(6)

where

$\begin{array}[]{l}S^{\mu\nu}=\dfrac{1}{2}\left({\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}}\right),\\\\[10.0pt]
S^{{\underline{\mu}}{\underline{\nu}}}=\dfrac{1}{2}\left({\gamma^{{\underline{\mu}}}\gamma^{{\underline{\nu}}}-\gamma^{{\underline{\nu}}}\gamma^{{\underline{\mu}}}}\right).\\\
\end{array}$ (7)

The relation between$\gamma^{\alpha}$ and $\gamma^{{\underline{\alpha}}}$ is
specified by the equality of

$\gamma^{\alpha}=H_{{\underline{\beta}}}^{\alpha}\gamma^{{\underline{\beta}}}.$
(8)

For our analysis, it is convenient to use the Dirac equation in the
Hamiltonian form of

$i\frac{\partial\psi}{\partial t}=H\psi.$ (9)

Here $t=x^{0}$, $H$ is the Hamilton operator.

Taking into account the equation of $\gamma^{0}\gamma^{0}=g^{00}$ , we can
derive the following expression for the Hamiltonian from Eq. (3)

$H=\frac{m}{g^{00}}\gamma^{0}-\frac{1}{g^{00}}i\gamma^{0}\gamma^{k}\frac{\partial}{\partial
x^{k}}-i\Phi_{0}+\frac{1}{g^{00}}i\gamma^{0}\gamma^{k}\Phi_{k}.$ (10)

###  3.1 The formalism of pseudo-Hermitian quantum mechanics [21] \- [23]

Hamiltonians (10) describing the motion of Dirac particles in arbitrary
gravitational fields are pseudo-Hermitian [17].

The condition of pseudo-Hermitian character for Hamiltonians assumes the
existence of invertible operator $\rho_{{}_{P}}$, satisfying the relationship
of

$\rho_{{}_{P}}H\rho_{{}_{P}}^{-1}=H^{+}.$ (11)

If in this case, there exists an operator $\eta$ satisfying the condition of

$\rho_{{}_{P}}=\eta^{+}\eta,$ (12)

then for stationary case, we obtain the following Hamiltonian in
$\eta$-representation:

$H_{\eta}=\eta H\eta^{-1}=H_{\eta}^{+}.$ (13)

Hamiltonian $H_{\eta}$ is self-conjugate with the spectrum of eigenvalues
coinciding with the spectrum of an initial Hamiltonian $H$.

The wave-function in $\eta$-representation equals

$\psi_{\eta}=\eta\psi,$ (14)

where $\psi$ is a wave function in the Dirac equation (9).

The scalar product of wave functions for pseudo-Hermitian Hamiltonians is
written with a Parker weight operator $\rho_{{}_{P}}$ [14], [17], [25]. For a
wave function in $\eta$-representation, the scalar product has a form, typical
for the Hermitian quantum mechanics (a plane scalar product with
$\rho_{{}_{P}}=1$ )

$\left({\varphi_{\eta},\psi_{\eta}}\right)=\int{d{\rm{\bf
x}}\left({\varphi_{\eta}^{+}\psi_{\eta}}\right)}.$ (15)

Below, for the analysis, we will use Hamiltonians and wave functions in
$\eta$-representation.

###  3.2 The system of tetrad vectors in the Schwinger gauge

Many of the researchers use the convenient system of tetrad vectors of
$\left\\{{\tilde{{H}}_{{\underline{\alpha}}}^{\mu}}\right\\}$ in the Schwinger
gauge [25]. For this system,

$\tilde{{H}}_{{\underline{0}}}^{0}=\sqrt{g^{00}},\,\,\,\,\,\,\,\tilde{{H}}_{{\underline{0}}}^{k}=-{g^{k0}}/{g^{00}},\,\,\,\,\,\,\,\tilde{{H}}_{{\underline{k}}}^{0}=0.$
(16)

Any spatial tetrads, satisfying the following relations:

$\tilde{{H}}_{{\underline{k}}}^{m}\tilde{{H}}_{{\underline{k}}}^{n}=f^{mn},\,\,\,\,\,\,f^{mn}=g^{mn}+\frac{g^{0m}g^{0n}}{g^{00}},\,\,\,\,\,\,f^{mn}g_{nk}=\delta_{k}^{m}$
(17)

can be used as tetrad vectors of $\tilde{{H}}_{m}^{n}$.

Taking into account some freedom of choice for spatial tetrads, we can obtain
expressions for Hamiltonians non-coincident with each other. These
Hamiltonians are physically equivalent since they are connected by unitary
matrixes of spatial rotations [17].

##  4 Coordinate transformations

At coordinate transformations (at transition to a another space-time)

$\left\\{{x^{\alpha}}\right\\}\to\left\\{{{x}^{\prime\alpha}}\right\\}$ (18)

the following relationships are fulfilled:

${H^{\prime}}_{\underline{\beta}}^{\alpha}=\frac{\partial{x}^{\prime\alpha}}{\partial
x^{\mu}}H_{{\underline{\beta}}}^{\mu},\,\,\,\,\,\,{\gamma}^{\prime\alpha}=\frac{\partial{x}^{\prime\alpha}}{\partial
x^{\beta}}\gamma^{\beta},\,\,\,\,\,\,\,{\Phi}^{\prime\alpha}=\frac{\partial{x}^{\prime\beta}}{\partial
x^{\alpha}}\Phi_{\beta}.$ (19)

At transformations of (19), the form of wave functions of the Dirac equation
remains invariable except for an appropriate replacement of the variables.

In one and the same space-time, we can transfer from any system of tetrad
vectors of
$\left\\{{{H^{\prime}}_{{\underline{\alpha}}}^{\mu}\left(x\right)}\right\\}$
to another system of tetrad vectors of
$\left\\{{H_{{\underline{\alpha}}}^{\mu}\left(x\right)}\right\\}$ by using the
Lorentz transformation of $L\left(x\right)$. In this case,

$H_{{\underline{\alpha}}}^{\mu}\left(x\right)=\Lambda_{{\underline{\alpha}}}^{{\underline{\beta}}}\left(x\right){H^{\prime}}_{{\underline{\beta}}}^{\mu}\left(x\right).$
(20)

The values of $\Lambda_{{\underline{\alpha}}}^{{\underline{\beta}}}$ satisfy
the relations

$\Lambda_{{\underline{\alpha}}}^{{\underline{\mu}}}\left(x\right)\Lambda_{{\underline{\beta}}}^{{\underline{\nu}}}\left(x\right)\eta^{{\underline{\alpha}}{\underline{\beta}}}=\eta^{{\underline{\mu}}{\underline{\nu}}},$
(21)
$\Lambda_{{\underline{\alpha}}}^{{\underline{\mu}}}\left(x\right)\Lambda_{{\underline{\beta}}}^{{\underline{\nu}}}\left(x\right)\eta_{{\underline{\mu}}{\underline{\nu}}}=\eta_{{\underline{\alpha}}{\underline{\beta}}}.$
(22)

The matrixes of Lorentz transformation $L,L^{-1}$ are determined on the basis
of the invariability of Dirac matrixes with local indices of
$\gamma^{{\underline{\alpha}}}$ at the transformations of (20)

$L\left(x\right)\gamma^{{\underline{\alpha}}}L^{-1}\left(x\right)=\gamma^{{\underline{\beta}}}\Lambda_{{\underline{\beta}}}^{{\underline{\alpha}}}\left(x\right).$
(23)

At Lorentz transformations, Dirac currents of particles are preserved.

##  5 The Schwarzschild metric

The Schwarzschild metric in the coordinates of
$\left({t,r,\theta,\varphi}\right)$ is given by

$ds^{2}=f_{S}dt^{2}-\frac{dr^{2}}{f_{S}}-r^{2}\left({d\theta^{2}+\sin^{2}\theta
d\varphi^{2}}\right),$ (24)

where $f_{S}=1-{r_{0}/r}$, $r_{0}={2GM}/{c^{2}}$ is the event horizon
(gravitational radius).

###  5.1 The Dirac equation

The nonzero tetrads in the Schwinger gauge of
$\tilde{{H}}_{{\underline{\alpha}}}^{\mu}$ equal

$\tilde{{H}}_{{\underline{0}}}^{0}=1/{\sqrt{f_{S}}};\,\,\,\tilde{{H}}_{{\underline{1}}}^{1}=\sqrt{f_{S}};\,\,\,\,\,\tilde{{H}}_{{\underline{2}}}^{2}=1/r;\,\,\,\,\tilde{{H}}_{{\underline{3}}}^{3}=1/{\left({r\sin\theta}\right)}.$
(25)

In compliance with (8), the matrixes of $\tilde{{\gamma}}^{\alpha}$ are as
follows:

$\tilde{{\gamma}}^{0}={\gamma^{{\underline{0}}}}/{\sqrt{f_{S}}},\,\,\,\,\tilde{{\gamma}}^{1}=\sqrt{f_{S}}\gamma^{{\underline{1}}},\,\,\,\,\tilde{{\gamma}}^{2}={\gamma^{{\underline{2}}}}/r,\,\,\,\,\tilde{{\gamma}}^{3}={\gamma^{{\underline{3}}}}/{\left({r\sin\theta}\right)}.$
(26)

The bispinor connections of $\tilde{{\Phi}}_{\alpha}$ are as follows:

$\begin{array}[]{l}\tilde{{\Phi}}_{0}=\dfrac{r_{0}}{4r^{2}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}},\,\,\,\tilde{{\Phi}}_{1}=0,\,\,\,\tilde{{\Phi}}_{2}=-\dfrac{1}{2}\sqrt{f_{{}_{S}}}\gamma^{{\underline{1}}}\gamma^{{\underline{2}}},\\\\[5.0pt]
\tilde{{\Phi}}_{3}=-\dfrac{1}{2}\cos\theta\gamma^{{\underline{2}}}\gamma^{{\underline{3}}}+\dfrac{1}{2}\sqrt{f_{S}}\sin\theta\gamma^{{\underline{3}}}\gamma^{{\underline{1}}}.\end{array}$
(27)

Taking into account (25) - (27), the Dirac Hamiltonian of (10) is given by

$\begin{array}[]{l}\tilde{{H}}_{S}=\sqrt{f_{S}}m\gamma^{{\underline{0}}}-i\sqrt{f_{S}}\gamma^{{\underline{0}}}\left\\{{\gamma^{{\underline{1}}}\sqrt{f_{S}}\left({\dfrac{\partial}{\partial
r}+\dfrac{1}{r}}\right)+\gamma^{{\underline{2}}}\dfrac{1}{r}\left({\dfrac{\partial}{\partial\theta}+\dfrac{1}{2}\mbox{ctg}\theta}\right)}\right.+\\\\[10.0pt]
\left.{+\gamma^{{\underline{3}}}\dfrac{1}{r\sin\theta}\dfrac{\partial}{\partial\varphi}}\right\\}-i\dfrac{r_{0}}{4r^{2}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}.\\\
\end{array}$ (28)

Hamiltonian (28) is pseudo-Hermitian with the weight operator
$\rho_{{}_{P}}=f_{S}^{-1/2}$ [14], [17], [25].

The self-conjugate Hamiltonian with a plane scalar product of wave functions
has the following form (see Refs. [14] and [17]):

$\begin{array}[]{l}H_{\eta_{{}_{S}}}=H_{\eta_{{}_{S}}}^{+}=\eta_{{}_{S}}\tilde{{H}}_{S}\eta_{{}_{S}}^{-1}=\sqrt{f_{S}}m\gamma^{{\underline{0}}}-i\sqrt{f_{S}}\gamma^{{\underline{0}}}\left\\{{\gamma^{{\underline{1}}}\sqrt{f_{S}}\left({\dfrac{\partial}{\partial
r}+\dfrac{1}{r}}\right)}\right.+\\\\[10.0pt]
\left.{+\gamma^{{\underline{2}}}\dfrac{1}{r}\left({\dfrac{\partial}{\partial\theta}+\dfrac{1}{2}\mbox{ctg}\theta}\right)+\gamma^{{\underline{3}}}\dfrac{1}{r\sin\theta}\dfrac{\partial}{\partial\varphi}}\right\\}-i\dfrac{r_{0}}{2r^{2}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}},\\\
\end{array}$ (29)

where $\eta_{{}_{S}}=f_{S}^{-1/4}$.

For separation of variables, let us present a bispinor
$\psi_{\eta_{{}_{S}}}\left({t,r,\theta,\varphi}\right)$ as

$\psi_{\eta_{{}_{S}}}\left({t,r,\theta,\varphi}\right)=\left({\begin{array}[]{l}\,\,\,\,\,F\left(r\right)\xi\left(\theta\right)\\\
-iG\left(r\right)\sigma^{3}\xi\left(\theta\right)\\\
\end{array}}\right)e^{-iEt}e^{im_{\varphi}\varphi}.$ (30)

Hereinafter, $\sigma^{3}$ is a two-dimensional Pauli matrix.

Spinor $\xi\left(\theta\right)$ represents spherical harmonics of spin one-
half. $E,m$ are energy and mass of the Dirac particle,
$m_{\varphi}=-j,-j+1,...j-1$, $j$ is an azimuth component of a total momentum
$j$, $\kappa$ is a quantum number of the Dirac equation:

$\kappa=\mp 1,\mp
2...=\left\\{{\begin{array}[]{l}-\left({l+1}\right),\,\,\,j=l+\frac{1}{2}\\\
\,\,\,\,\,\,\,\,\,l,\,\,\,\,\,\,\,\,j=l-\frac{1}{2},\\\ \end{array}}\right.$

where $j,l$ are quantum numbers of the total and orbital momenta of a Dirac
particle.

For separation of variables , it is convenient to perform the following
equivalent substitution of matrixes:

$\gamma^{\underline{1}}\to\gamma^{\underline{3}},\,\,\,\,\,\gamma^{\underline{3}}\to\gamma^{\underline{2}},\,\,\,\,\,\gamma^{\underline{2}}\to\gamma^{\underline{1}}.$
(31)

After separation of the variables, the equations for radial functions have the
following form:

$\begin{array}[]{l}f_{S}\dfrac{dF}{dr}+\left({\dfrac{1+\kappa\sqrt{f_{S}}}{r}-\dfrac{r_{0}}{2r^{2}}}\right)F-\left({E+m\sqrt{f_{S}}}\right)G=0,\\\\[10.0pt]
f_{S}\dfrac{dG}{dr}+\left({\dfrac{1-\kappa\sqrt{f_{S}}}{r}-\dfrac{r_{0}}{2r^{2}}}\right)G+\left({E-\sqrt{f_{S}}}\right)F=0.\\\
\end{array}$ (32)

If at $r\to r_{0}$, we present the functions of
$F\left(\rho\right),G\left(\rho\right)$, as follows:

$\left.F\right|_{r\to
r_{0}}=\left({r-r_{0}}\right)^{p}\sum\limits_{k=0}^{\infty}{f_{k}\left({r-r_{0}}\right)^{k}},\,\,\,\left.G\right|_{r\to
r_{0}}=\left({r-r_{0}}\right)^{p}\sum\limits_{k=0}^{\infty}{g_{k}\left({r-r_{0}}\right)^{k}},$
(33)

then, the indicial equation for system (32) leads to the two solutions with
identical sense:

$\left(p\right)_{1,2}=-\frac{1}{2}\pm ir_{0}E.$ (34)
$\left.{F_{1}}\right|_{r\to r_{0}},\,\,\left.{G_{1}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/2}}\exp\left\\{{ir_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\},$
(35) $\left.{F_{2}}\right|_{r\to r_{0}},\,\,\left.{G_{2}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/2}}\exp\left\\{{-ir_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\}.$
(36)

In this case, solutions (32) can be represented by real functions [3, 5].
Taking into account (34), we can write down the asymptotics of the solutions
at $r\to r_{0}$ as

$\begin{array}[]{l}\left.F\right|_{r\to
r_{0}}=\dfrac{L}{\sqrt{r-r_{0}}}\sin\left({r_{0}E\ln\left({\dfrac{r}{r_{0}}-1}\right)+\delta}\right),\\\\[10.0pt]
\left.G\right|_{r\to
r_{0}}=\dfrac{L}{\sqrt{r-r_{0}}}\cos\left({r_{0}E\ln\left({\dfrac{r}{r_{0}}-1}\right)+\delta}\right),\\\
\end{array}$ (37)

where $L,\delta$ are integration constants. In some of the scattering
problems, one can use a complex phase $\delta$ [5].

In expressions (35) - (37), the functions of $F\left(r\right),G\left(r\right)$
are square-nonintegrable at $r\to r_{0}$. The oscillating form of the
functions of $F\left(r\right),G\left(r\right)$ for $E\neq 0$ testifies to
implementation of the mode of a particle ”fall” to event horizons (see Eq. (2)
in Sec. 2).

The solutions of the indicial equation for the system of equations for radial
functions of Hamiltonian (28) are as follows:

$\left({p_{S}}\right)_{1,2}=-\frac{1}{4}\pm ir_{0}E.$ (38)

The asymptotics of the radial functions at $r\to r_{0}$ are given by

$\left.{\left({F_{S}}\right)_{1}}\right|_{r\to
r_{0}},\,\,\left.{\left({G_{S}}\right)_{1}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/4}}\exp\left\\{{ir_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\},$
(39) $\left.{\left({F_{S}}\right)_{2}}\right|_{r\to
r_{0}},\,\,\left.{\left({G_{S}}\right)_{2}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/4}}\exp\left\\{{-ir_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\}.$
(40)

Functions $F_{S}\left(r\right),G_{S}\left(r\right)$ are square-nonintegrable
due to the presence of Parker weight operator $\rho_{{}_{P}}=f_{S}^{-1/2}$ in
the scalar product [14], [17], [25]. In this case, the mode of a particle
”fall” to the event horizon of $r_{0}$ is also present.

Below, the problem of the possibility to eliminate the mode of a particle
”fall” by means of the transformations of the coordinates of the
Schwarzschild’s metric is analyzed.

##  6 The stationary Eddington-Finkelstein metric

The coordinate transformation of the Schwarzschild metric

$\left({t,r,\theta,\varphi}\right)\to\left({T,r,\theta,\varphi}\right)$ (41)

has the form of

$dT=dt+\frac{r_{0}}{r}\frac{dr}{f_{S}}.$ (42)

Then, the EF metric is written as

$ds^{2}=f_{S}dT^{2}-2\frac{r_{0}}{r}dTdr-\left({1+\frac{r_{0}}{r}}\right)dr^{2}-r^{2}\left({d\theta^{2}+\sin^{2}\theta
d\varphi^{2}}\right).$ (43)

From Hamiltonian $\tilde{{H}}_{S}$ in the Schwarzschild space–time (28), we
will obtain Hamiltonian $H_{EF}$ in the EF space–time, by using relation (19)
and Eqs. (25) - (27).

As the result,

$\begin{array}[]{l}\left({H_{EF}}\right)_{{\underline{0}}}^{0}=\dfrac{1}{\sqrt{f_{S}}},\,\,\,\,\left({H_{EF}}\right)_{{\underline{1}}}^{0}=\dfrac{{r_{0}}/r}{\sqrt{f_{S}}},\,\,\,\,\left({H_{EF}}\right)_{{\underline{1}}}^{1}=\sqrt{f_{S}},\\\\[10.0pt]
\left({H_{EF}}\right)_{{\underline{2}}}^{2}=\dfrac{1}{r},\,\,\,\,\,\,\,\left({H_{EF}}\right)_{{\underline{3}}}^{3}=\dfrac{1}{r\sin\theta},\end{array}$
(44)
$\gamma_{EF}^{0}=\frac{1}{\sqrt{f_{S}}}\gamma^{{\underline{0}}}+\frac{r_{0}/r}{\sqrt{f_{S}}}\gamma^{{\underline{1}}},\,\,\,\,\gamma_{EF}^{1}=\sqrt{f_{S}}\gamma^{{\underline{1}}},\,\,\,\gamma_{EF}^{2}=\frac{1}{r}\gamma^{{\underline{2}}},\,\,\,\,\,\,\,\gamma_{EF}^{3}=\frac{1}{r\sin\theta}\gamma^{{\underline{3}}},$
(45)
$\begin{array}[]{l}\left({\Phi_{EF}}\right)_{0}=\dfrac{r_{0}}{4r^{2}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}},\,\,\,\,\,\,\,\left({\Phi_{EF}}\right)_{1}=\dfrac{r_{0}}{r}\dfrac{1}{f_{S}}\dfrac{r_{0}}{4r^{2}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}},\\\\[10.0pt]
\left({\Phi_{EF}}\right)_{2}=-\dfrac{1}{2}\sqrt{f_{S}}\gamma^{{\underline{1}}}\gamma^{{\underline{2}}},\,\,\,\,\,\,\,\left({\Phi_{EF}}\right)_{3}=-\dfrac{1}{2}\cos\theta\,\gamma^{{\underline{2}}}\gamma^{{\underline{3}}}+\dfrac{1}{2}\sin\theta\gamma^{{\underline{3}}}\gamma^{{\underline{1}}}.\\\
\end{array}$ (46)

For the EF metric (see Eq. (43)), the inverse metric tensor is

$g^{00}=1+\frac{r_{0}}{r}.$ (47)

By substituting (44) - (47) to Eq. (10), we obtain the Hamiltonian in the EF
space–time

$H_{EF}=\frac{1-\left({r_{0}/r}\right)\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}{1-{r_{0}^{2}}/{r^{2}}}\tilde{{H}}_{S}.$
(48)

Inverse equality is

$\tilde{{H}}_{S}=\left({1+\frac{r_{0}}{r}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right)H_{EF}.$
(49)

Let us demonstrate that Hamiltonians $H_{EF}$ and $\tilde{{H}}_{S}$ are
connected with each other through the unitary transformation.

At transformations (19), the form of wave functions does not vary except for
substitution of the variables. Let us denote

$\varphi_{{}_{EF}}\left(r\right)=\int{\frac{r_{0}}{r}\frac{dr}{f_{S}}}.$ (50)

It follows from (42), that
$t=T-\varphi_{{}_{EF}}\left(r\right)=T-\int{\dfrac{r_{0}}{r}\dfrac{dr}{f_{S}}}=T-\left({r_{0}\mbox{ln}\left({\frac{r}{r_{0}}-1}\right)+\mbox{const}}\right)$.
Since

$\tilde{{\psi}}_{S}\left({{\rm{\bf r}},t}\right)=\psi_{{}_{EF}}\left({{\rm{\bf
r}},T}\right),$ (51)

then

$\tilde{{\psi}}_{S}\left({{\rm{\bf
r}}}\right)e^{-iEt}=\tilde{{\psi}}_{S}\left({{\rm{\bf
r}}}\right)e^{iE\varphi_{{}_{EF}}\left(r\right)}e^{-iET}=\psi_{{}_{EF}}\left({{\rm{\bf
r}}}\right)e^{-iET}.$ (52)

The equation for stationary states in the Schwarzschild space–time equals

$\tilde{{H}}_{S}\tilde{{\psi}}_{S}\left({{\rm{\bf
r}}}\right)=E\tilde{{\psi}}_{S}\left({{\rm{\bf r}}}\right).$ (53)

It follows from (52), that $\tilde{{\psi}}_{S}\left({{\rm{\bf
r}}}\right)=e^{-iE\varphi_{EF}\left(r\right)}\psi_{EF}\left({{\rm{\bf
r}}}\right)$, and we obtain that

$e^{iE\varphi_{{}_{EF}}\left(r\right)}\tilde{{H}}_{S}e^{-iE\varphi_{{}_{EF}}\left(r\right)}=E\psi_{EF}\left({{\rm{\bf
r}}}\right).$ (54)

Using the explicit form of $\tilde{{H}}_{S}$ in (28) and equality (49), we
obtain that

$\left({1+\frac{r_{0}}{r}\gamma^{{0}}\gamma^{{1}}}\right)\left[{H_{EF}\psi_{EF}\left({{\rm{\bf
r}}}\right)-E\psi_{EF}\left({{\rm{\bf r}}}\right)}\right]=0.$ (55)

By multiplying (55) on the left by matrix
${\left({1-\frac{r_{0}}{r}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right)}\mathord{\left/{\vphantom{{\left({1-\frac{r_{0}}{r}\gamma^{{0}}\gamma^{{1}}}\right)}{\left({1-{r_{0}^{2}}/{r^{2}}}\right)}}}\right.\kern-1.2pt}{\left({1-{r_{0}^{2}}/{r^{2}}}\right)}$,
we will obtain an equation for stationary states in the EF space–time:

$H_{EF}\psi_{EF}\left({{\rm{\bf r}}}\right)=E\psi_{EF}\left({{\rm{\bf
r}}}\right).$ (56)

In this case, in compliance with (54) and (52), the following relations are
valid:

$H_{EF}=e^{iE\varphi_{{}_{EF}}}\tilde{{H}}_{S}e^{-iE\varphi_{{}_{EF}}},$ (57)
$\psi_{EF}\left({{\rm{\bf r}}}\right)=\tilde{{\psi}}_{S}\left({{\rm{\bf
r}}}\right)e^{iE\varphi_{{}_{EF}}}.$ (58)

Thus, it is demonstrated that Hamiltonians $H_{EF}$ and $\tilde{{H}}_{S}$ are
connected with each other through the unitary transformation:

$U_{EF}=e^{iE\varphi_{{}_{EF}}\left(r\right)}.$ (59)

It follows from (58) that the asymptotics of radial functions in the EF space-
time can be obtained by multiplication of asymptotics (39) and (40) by unitary
operator (59). Then,

$\left.{\left({F_{EF}}\right)_{1}}\right|_{r\to
r_{0}},\,\,\left.{\left({G_{EF}}\right)_{1}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/4}}\exp\left\\{{i2r_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\},$
(60) $\left.{\left({F_{EF}}\right)_{2}}\right|_{r\to
r_{0}},\,\,\left.{\left({G_{EF}}\right)_{2}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/4}}.$ (61)

The asymptotics (60) and (61) can be obtained by using another way.

We can separate variables in Eq. (56) by representing the wave function in the
form of (30). The equations for radial functions of
$F_{EF}\left(r\right),G_{EF}\left(r\right)$ are given by

$\begin{array}[]{l}f_{S}\dfrac{dF_{EF}}{dr}+\left({\dfrac{1+\kappa\sqrt{f_{S}}}{r}-\dfrac{3}{4}\dfrac{r_{0}}{r^{2}}}\right)F_{EF}-i\dfrac{r_{0}}{r}EF_{EF}-\left({E+\sqrt{f_{S}}m}\right)G_{EF}=0,\\\\[10.0pt]
f_{S}\dfrac{dG_{EF}}{dr}+\left({\dfrac{1-\kappa\sqrt{f_{S}}}{r}-\dfrac{3}{4}\dfrac{r_{0}}{r^{2}}}\right)G_{EF}-i\dfrac{r_{0}}{r}EG_{EF}+\left({E-\sqrt{f_{S}}m}\right)F_{EF}=0.\\\
\end{array}$ (62)

To determine the asymptotics of radial functions at $r\to r_{0}$, let us
present functions $\left.{\left({F_{EF}}\right)}\right|_{r\to
r_{0}},\left.{\left({G_{EF}}\right)}\right|_{r\to r_{0}}$ in the form of (33).
Let us write the indicial equation for system (62) as an equation for the
determinant:

$\begin{vmatrix}{\left({p+\dfrac{1}{4}}\right)\dfrac{1}{r_{0}}-iE}\hfill&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-E}\hfill\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E\hfill&{\left({p+\dfrac{1}{4}}\right)\dfrac{1}{r_{0}}-iE}\hfill\\\
\end{vmatrix}=0.$ (63)

The solutions for Eq. (63) are given by

$\begin{array}[]{l}p_{1}=-\dfrac{1}{4}+2ir_{0}E,\\\\[5.0pt]
p_{2}=-\dfrac{1}{4}.\\\ \end{array}$ (64)

Equalities (64) show that the required asymptotics coincide with asymptotics
(60), (61).

The Parker operator in the (EF) space-time with the set of tetrads (44) is

$\left({\rho_{{}_{P}}}\right)_{EF}=\gamma^{{\underline{0}}}\gamma_{EF}^{0}=\frac{1}{\sqrt{f_{S}}}\left({1+\frac{r_{0}}{r}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right).$
(65)

For both asymptotics (60), (61), the scalar product in the neighborhood of the
event horizon diverge lolgarithmically.

We can formally determine the form of the operator of transition to a self-
conjugate Hamiltonian in the EF space-time with a plane scalar product Eq.
(see (15))

$\begin{array}[]{l}\left({\rho_{{}_{P}}}\right)_{EF}=\eta_{{}_{EF}}^{+}\eta_{{}_{EF}},\\\
\eta_{{}_{EF}}=\dfrac{1}{f_{S}^{1/4}}\left({1+\dfrac{r_{0}}{r}\gamma^{\underline{0}}\gamma^{\underline{1}}}\right)^{1/2},\\\
\end{array}$ (66)

where $\eta_{{}_{EF}}^{+}=\eta_{{}_{EF}}$. However, the application of a
matrix operator $\eta_{{}_{EF}}$ is difficult in practice.

###  6.1 The Lorentz transformation

To obtain an operator $\eta_{{}_{EF}}$ in an acceptable matrix-free form, let
us transfer from the system of tetrad vectors of
$\left\\{{\left({H_{EF}}\right)_{{\underline{\alpha}}}^{\mu}}\right\\}$ to
tetrad vectors in the Schwinger gauge of
$\left\\{{\left({\tilde{{H}}_{EF}}\right)_{{\underline{\alpha}}}^{\mu}}\right\\}$
by using Lorentz transformation of (20) - (23). In this system, the tetrads of
$\tilde{{H}}_{{\underline{k}}}^{0}$ are equal to zero and an operator
$\eta_{{}_{EF}}$ is free from matrixes
$\gamma^{{\underline{0}}},\gamma^{{\underline{1}}}$. The nonzero tetrads of
$\left({\tilde{{H}}_{EF}}\right)_{{\underline{\alpha}}}^{\mu}$ in the
Schwinger gauge are determined by the expressions of

$\begin{array}[]{l}\left({\tilde{{H}}_{EF}}\right)_{{\underline{0}}}^{0}=\sqrt{1+\dfrac{r_{0}}{r}},\,\,\,\,\,\,\left({\tilde{{H}}_{EF}}\right)_{{\underline{0}}}^{1}=-\dfrac{{r_{0}}/r}{\sqrt{1+{r_{0}}/r}},\,\,\,\,\,\left({\tilde{{H}}_{EF}}\right)_{{\underline{1}}}^{1}=\dfrac{1}{\sqrt{1+{r_{0}}/r}},\\\\[10.0pt]
\,\left({\tilde{{H}}_{EF}}\right)_{{\underline{2}}}^{2}=\dfrac{1}{r},\,\,\,\,\,\,\,\left({H_{EF}}\right)_{{\underline{3}}}^{3}=\dfrac{1}{r\sin\theta}.\\\
\end{array}$ (67)

In considered case, the nonzero values of
$\Lambda_{{\underline{\alpha}}}^{{\underline{\beta}}}\left(r\right)$ in (20)
are equal to:

$\Lambda_{{\underline{0}}}^{{\underline{0}}}=\Lambda_{{\underline{1}}}^{{\underline{1}}}=\frac{1}{\sqrt{f_{S}}\sqrt{1+{r_{0}}/r}},\,\,\,\,\,\Lambda_{{\underline{0}}}^{{\underline{1}}}=\Lambda_{{\underline{1}}}^{{\underline{0}}}=-\frac{{r_{0}}/r}{\sqrt{f_{S}}\sqrt{1+{r_{0}}/r}}.$
(68)

Let us determine the form of the Lorentz transformation matrixes from Eqs.
(23) with the values of
$\Lambda_{{\underline{\alpha}}}^{{\underline{\beta}}}\left(x\right)$ from (68)

$L\gamma^{{\underline{0}}}L^{-1}=\frac{1}{\sqrt{1-{r_{0}}/r}\sqrt{1+{r_{0}}/r}}\left({\gamma^{{\underline{0}}}-\frac{r_{0}}{r}\gamma^{{\underline{1}}}}\right),\,$
(69)
$L\gamma^{{\underline{1}}}L^{-1}=\frac{1}{\sqrt{1-{r_{0}}/r}\sqrt{1+{r_{0}}/r}}\left({-\frac{r_{0}}{r}\gamma^{{\underline{0}}}+\gamma^{{\underline{1}}}}\right).$
(70)

It is well–known that Lorentz transformations can unambiguously be presented
as a product of either a boost (a Hermitian factor) by a spatial rotation
matrix $R$ (a unitary factor), or, vice versa, as a product of a matrix $R$ by
a boost. For our case, it is sufficient to use a unity matrix $R=1$.

The following forms of $L$ and $L^{-1}$ are reasonable:

$\begin{array}[]{l}L=\mbox{exp}\left({\dfrac{\theta}{2}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right)=\mbox{ch}\dfrac{\theta}{2}+\mbox{sh}\dfrac{\theta}{2}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}=\dfrac{1+\left({B/A}\right)\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}{\sqrt{1-{B^{2}}/{A^{2}}}},\\\\[10.0pt]
L^{-1}=\mbox{exp}\left({-\dfrac{\theta}{2}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right)=\mbox{ch}\dfrac{\theta}{2}-\mbox{sh}\dfrac{\theta}{2}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}=\dfrac{1-\left({B/A}\right)\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}{\sqrt{1-{B^{2}}/{A^{2}}}}.\\\
\end{array}$ (71)

In (71), the matrix $L$ represents a transformation of hyperbolic rotation
(i.e. boost) by an angle $\theta$.

Substituting (71) in Eqs. (69) and (70), we obtain that

$\frac{B}{A}=\frac{\sqrt{1+{r_{0}}/r}-\sqrt{1-{r_{0}}/r}}{\sqrt{1+{r_{0}}/r}+\sqrt{1-{r_{0}}/r}}.$
(72)

Then,

$L\left(r\right)=\frac{\sqrt{1+{r_{0}}/r}+\sqrt{1-{r_{0}}/r}+\left({\sqrt{1+{r_{0}}/r}-\sqrt{1-{r_{0}}/r}}\right)\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}{2\sqrt{\sqrt{1+{r_{0}}/r}\sqrt{1-{r_{0}}/r}}},$
(73)
$L^{-1}\left(r\right)=\frac{\sqrt{1+{r_{0}}/r}+\sqrt{1-{r_{0}}/r}+\left({\sqrt{1+{r_{0}}/r}-\sqrt{1-{r_{0}}/r}}\right)\gamma^{{\underline{1}}}\gamma^{{\underline{0}}}}{2\sqrt{\sqrt{1+{r_{0}}/r}\sqrt{1-{r_{0}}/r}}}.$
(74)

After Lorenz transformation of (73) and (74) , Hamiltonian (57) is transformed
to the Hamiltonian with tetrads in Schwinger gauge [14]:

$\begin{array}[]{l}\tilde{{H}}_{EF}=L\left(r\right)H_{EF}L^{-1}\left(r\right)=\\\
=\dfrac{m}{\sqrt{1+{r_{0}}/r}}\gamma^{{\underline{0}}}-i\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}\dfrac{1}{1+{r_{0}}/r}\left({\dfrac{\partial}{\partial
r}+\dfrac{1}{r}+\dfrac{r_{0}}{4r^{2}}\dfrac{1}{1+{r_{0}}/r}}\right)-\\\\[10.0pt]
-i\gamma^{{\underline{0}}}\gamma^{{\underline{2}}}\dfrac{1}{\sqrt{1+{r_{0}}/r}}\dfrac{1}{r}\left({\dfrac{\partial}{\partial\theta}+\dfrac{1}{2}\mbox{ctg}\theta}\right)-i\gamma^{{\underline{0}}}\gamma^{{\underline{3}}}\dfrac{1}{\sqrt{1+{r_{0}}/r}}\dfrac{1}{r\,\mbox{sin}\theta}\dfrac{\partial}{\partial\varphi}+\\\\[10.0pt]
+i\dfrac{r_{0}}{r}\dfrac{1}{1+{r_{0}}/r}\left({\dfrac{\partial}{\partial
r}+\dfrac{1}{r}-\dfrac{1}{4r\left({1+{r_{0}}/r}\right)}-\dfrac{1}{4r}}\right).\\\
\end{array}$ (75)

In this case,

$\tilde{{\psi}}_{EF}=L\left(r\right)\psi_{EF}.$ (76)

According to (73), the asymptotics of $L\left(r\right)$ at $r\to r_{0}$ are

$\left.L\right|_{r\to
r_{0}}=\frac{r_{0}^{1/4}\left({1+\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right)}{2^{3/4}\left({r-r_{0}}\right)^{1/4}}.$
(77)

In compliance with (76) and (77), after the similarity transformation
$L\left(r\right)$, the asymptotics of radial functions (60) and (61) are
transformed to the following form:

$\left.{\left({\tilde{{F}}_{EF}}\right)_{1}}\right|_{r\to
r_{0}},\,\,\left.{\left({\tilde{{G}}_{EF}}\right)_{1}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/2}}\exp\left\\{{i2r_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\},$
(78) $\left.{\left({\tilde{{F}}_{EF}}\right)_{2}}\right|_{r\to
r_{0}},\,\,\left.{\left({\tilde{{G}}_{EF}}\right)_{2}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/2}}.$ (79)

Now, let us obtain asymptotics of radial functions by using an another
technique. If we present the wave function of
$\tilde{{\psi}}_{EF}\left({{\rm{\bf r}},T}\right)$ in a form similar to Eq.
(30), then in the equation of

$\tilde{{H}}_{EF}\tilde{{\psi}}_{EF}=E\tilde{{\psi}}_{EF}$ (80)

we can perform separation of the variables. So,

$\tilde{{\psi}}_{EF}\left({r,\theta,\varphi,T}\right)=\left({\begin{array}[]{l}\,\,\,\,\,\tilde{{F}}_{EF}\left(r\right)\xi\left(\theta\right)\\\
-i\tilde{{G}}_{EF}\left(r\right)\sigma^{3}\xi\left(\theta\right)\\\
\end{array}}\right)e^{-iET+im_{\varphi}\varphi}.$ (81)

For convenience, while separating the variables, the equivalent substitution
of matrixes $\gamma^{{\underline{i}}}$ is carried out (see Eq. (31)). The
system of equations for radial wave functions of
$\tilde{{F}}_{EF}\left(r\right),\tilde{{G}}_{EF}\left(r\right)$ is given by

$\begin{array}[]{l}\left({1-\dfrac{r_{0}}{r}}\right)\dfrac{d\tilde{{F}}_{EF}}{dr}+\left({\dfrac{1-{r_{0}}/r+\kappa/{\sqrt{1+{r_{0}}/r}}}{r}+\dfrac{r_{0}}{4r^{2}}}\right)\tilde{{F}}_{EF}-\\\\[10.0pt]
-i\dfrac{r_{0}}{r}\left({E-\dfrac{m}{\sqrt{1+{r_{0}}/r}}}\right)\tilde{{F}}_{EF}+i\left({-\dfrac{1}{1+{r_{0}}/r}\dfrac{r_{0}}{2r^{2}}+\dfrac{r_{0}}{r^{2}}\dfrac{\kappa}{\sqrt{1+{r_{0}}/r}}}\right)\tilde{{G}}_{EF}-\\\\[10.0pt]
-\left({E+\dfrac{m}{\sqrt{1+{r_{0}}/r}}}\right)\tilde{{G}}_{EF}=0,\\\\[10.0pt]
\left({1-\dfrac{r_{0}}{r}}\right)\dfrac{d\tilde{{G}}_{EF}}{dr}+\left({\dfrac{1-{r_{0}}/r-\kappa/{\sqrt{1+{r_{0}}/r}}}{r}+\dfrac{r_{0}}{4r^{2}}}\right)\tilde{{G}}_{EF}-\\\\[10.0pt]
-i\dfrac{r_{0}}{r}\left({E+\dfrac{m}{\sqrt{1+{r_{0}}/r}}}\right)\tilde{{G}}_{EF}+i\left({\dfrac{1}{1+{r_{0}}/r}\dfrac{r_{0}}{2r^{2}}+\dfrac{r_{0}}{r^{2}}\dfrac{\kappa}{\sqrt{1+{r_{0}}/r}}}\right)\tilde{{F}}_{EF}+\\\\[10.0pt]
+\left({E-\dfrac{m}{\sqrt{1+{r_{0}}/r}}}\right)\tilde{{F}}_{EF}=0.\\\
\end{array}$ (82)

If in the neighborhood of the event horizon of $\left({r\to r_{0}}\right)$, we
determine

$\left.{\tilde{{F}}_{EF}}\right|_{r\to
r_{0}}=\left({r-r_{0}}\right)^{s}\sum\limits_{k=0}^{\infty}{\tilde{{f}}_{k}\left({r-r_{0}}\right)^{k}},\,\,\,\left.{\tilde{{G}}_{EF}}\right|_{r\to
r_{0}}=\left({r-r_{0}}\right)^{s}\sum\limits_{k=0}^{\infty}{\tilde{{g}}_{k}\left({r-r_{0}}\right)^{k}},$
then the indicial equation for system (82) is

$s\left({s+\frac{1}{2}-i2r_{0}E}\right)=0.$ (83)

The solutions of (83) are

$s_{1}=-\frac{1}{2}+i2r_{0}E,\,\,\,s_{2}=0.$ (84)

The solution of $s_{1}=-1/2+i2r_{0}E$ corresponds to asymptotics for square-
nonintegrable wave functions with the mode of a particle ”fall” to the event
horizon. The asymptotics with solution $s_{1}$ coincide with asymptotics (78).
These asymptotics agree with the transformations of wave function asymptotics
$F_{1},G_{1}$ (see Eq. (35)) at transition to the EF metric and to the
Hamiltonian with tetrads in Schwinger gauge.

In this case, the transition to the stationary EF metric does not eliminate
the mode of a particle ”fall” to the event horizon. The wave functions are
square-nonintegrable in the neighborhood of the event horizon.

Since for the (EF) metric, $\eta_{{}_{EF}}=\left({1+{r_{0}}/r}\right)^{1/4}$,
these conclusions are also valid for the wave functions of the self–conjugate
Hamiltonian of

$H_{\eta_{{}_{EF}}}=\eta_{{}_{EF}}\tilde{{H}}_{{}_{EF}}\eta_{{}_{EF}}^{-1},$
(85) $\psi_{\eta_{{}_{EF}}}=\eta_{{}_{EF}}\tilde{{\psi}}_{{}_{EF}}.$ (86)

The smooth square-integrable wave functions of
$\left.{\left({\tilde{{F}}_{EF}}\right)_{2}}\right|_{r\to
r_{0}}=\mbox{const}\,\,1$,
$\left.{\left({\tilde{{G}}_{EF}}\right)_{2}}\right|_{r\to
r_{0}}=\mbox{const}\,\,2$ without a mode of a particle ”fall” to event
horizons correspond to asymptotics with the solution of $s_{2}=0$ (see Eq.
(84)). Some of the researches use this solution to prove the possibility of
intersection of the event horizon by quantum mechanical particles and their
”sink” to singularity at $r\to 0$ (see, for instance, Ref. [26]). However, the
solution of $s_{2}=0$ does not correspond to the history of wave function
transformations (see (58), (76) and (86)) at the transition from the
self–conjugate Hamiltonian with the Schwarzschild metric (29) to the
self–conjugate Hamiltonian with the EF metric (85).

Of course, the wave functions with $s_{2}=0$ are accurate solutions of GR
equations in the EF space-time. However, these solutions are not associated
with the Schwarzschild metric. Such an unambiguous link exists only for wave
functions with the index of $s_{1}=-\frac{1}{2}+i2r_{0}E$ (see Eq. (84)).

In accordance with Birkhoff’s theorem, the geometry of any spherically
symmetric vacuum region of spacetime with $r>r_{0}$ is a piece of the
Schwarzschild geometry. Hence, at coordinate transformations of a static
Schwarzschild metric to a stationary EF metric, only wave functions with an
index $s_{1}$ should be used. This leads to conservation of the mode of a
particle ”fall” to the event horizon.

##  7 The Painlevè-Gullstrand stationary metric

For a PG metric, as well as in Sec. 6, we prove the impossibility to eliminate
the singular mode of a particle ”fall” by using coordinate transformations of
the Schwarzschild metric. Because of the logic closeness of Secs. 6 and 7, we
present a short form of the proofs.

The coordinate transformation is given by

$dT=dt+\sqrt{\frac{r_{0}}{r}}\frac{dr}{f_{S}}.$ (87)

The PG metric is written as

$ds^{2}=f_{S}dT^{2}-2\sqrt{\frac{r_{0}}{r}}dTdr-
dr^{2}-r^{2}\left({d\theta^{2}+\sin^{2}\theta d\varphi^{2}}\right).$ (88)

From a Hamiltonian $\tilde{{H}}_{S}$ in the Schwarzschild space-time (28), the
Hamiltonian $H_{PG}$ is obtained in the PG space–time by using relationship
(19) and equalities (25) - (27).

As the result, we obtain that

$\begin{array}[]{l}\left({H_{PG}}\right)_{{\underline{0}}}^{0}=\dfrac{1}{\sqrt{f_{S}}},\,\,\,\,\left({H_{PG}}\right)_{{\underline{1}}}^{0}=\dfrac{\sqrt{{r_{0}}/r}}{\sqrt{f_{S}}},\,\,\,\,\left({H_{PG}}\right)_{{\underline{1}}}^{1}=\sqrt{f_{S}},\\\\[10.0pt]
\left({H_{PG}}\right)_{{\underline{2}}}^{2}=\dfrac{1}{r},\,\,\,\,\,\,\,\left({H_{PG}}\right)_{{\underline{3}}}^{3}=\dfrac{1}{r\sin\theta},\end{array}$
(89)
$\gamma_{PG}^{0}=\frac{1}{\sqrt{f_{S}}}\gamma^{{\underline{0}}}+\frac{\sqrt{{r_{0}}/r}}{\sqrt{f_{S}}}\gamma^{{\underline{1}}},\,\,\,\,\gamma_{PG}^{1}=\sqrt{f_{S}}\gamma^{{\underline{1}}},\,\,\,\gamma_{PG}^{2}=\frac{1}{r}\gamma^{{\underline{2}}},\,\,\,\,\,\,\,\gamma_{PG}^{3}=\frac{1}{r\sin\theta}\gamma^{{\underline{3}}},$
(90)
$\begin{array}[]{l}\left({\Phi_{PG}}\right)_{0}=\dfrac{r_{0}}{4r^{2}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}},\,\,\,\,\,\,\,\left({\Phi_{PG}}\right)_{1}=\dfrac{\sqrt{{r_{0}}/r}}{f_{S}}\dfrac{r_{0}}{4r^{2}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}},\\\\[10.0pt]
\left({\Phi_{PG}}\right)_{2}=-\dfrac{1}{2}\sqrt{f_{S}}\gamma^{{\underline{1}}}\gamma^{{\underline{2}}},\,\,\,\,\,\,\,\left({\Phi_{EF}}\right)_{3}=-\dfrac{1}{2}\cos\theta\,\gamma^{{\underline{2}}}\gamma^{{\underline{3}}}+\dfrac{1}{2}\sqrt{f_{S}}\sin\theta\gamma^{{\underline{3}}}\gamma^{{\underline{1}}}.\\\
\end{array}$ (91)

For the (PG) metric

$g^{00}=1.$ (92)

By substituting (89) - (92) to Eq. (10), we obtain the Hamiltonian in the PG
space-time

$H_{PG}=\frac{1-\sqrt{{r_{0}}/r}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}{f_{S}}\tilde{{H}}_{S}.$
(93)

By analogy with the EF metric (see Eqs. (50) - (58)), it is possible to
demonstrate that Hamiltonians $H_{PG}$ and $\tilde{{H}}_{S}$ are connected
with each other through the unitary transformation:

$H_{{}_{PG}}=e^{iE\varphi_{{}_{PG}}}\tilde{{H}}_{S}e^{-iE\varphi_{{}_{PG}}},$
(94) $\psi_{{}_{PG}}\left({{\rm{\bf
r}}}\right)=\tilde{{\psi}}_{{}_{S}}\left({{\rm{\bf
r}}}\right)e^{iE\varphi_{{}_{PG}}},$ (95)

where

$\varphi_{{}_{PG}}\left(r\right)=\int{\sqrt{\frac{r_{0}}{r}}\frac{dr}{f_{S}}}.$
(96)

Multiplication of asymptotics (39) and (40) by the unitary factor of
$U_{PG}=e^{iE\varphi_{{}_{PG}}}$ leads to a form similar to the formulas (60)
and (61) for the EF metric:

$\left.{\left({F_{PG}}\right)_{1}}\right|_{r\to
r_{0}},\,\,\left.{\left({G_{PG}}\right)_{1}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/4}}\exp\left\\{{i2r_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\},$
(97) $\left.{\left({F_{PG}}\right)_{2}}\right|_{r\to
r_{0}},\,\,\left.{\left({G_{PG}}\right)_{2}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/4}}.$ (98)

The asymptotics (97) and (98) can be obtained in a different way. For this, in
the equation:

$H_{PG}\psi_{PG}=E\psi_{PG},$ (99)

it is necessary to separate the variables and obtain the system of equations
for the radial wave functions $F_{PG},G_{PG}$. Then, in the neighborhood of
the event horizon for this system of equations, we should obtain the solutions
of the indicial equation. As the result, by analogy with (62) - (64), we
obtain the asymptotics of the radial functions coinciding with (97) and (98).

The Parker operator in the PG space–time with the set of tetrads (89) is

$\left({\rho_{{}_{P}}}\right)_{PG}=\gamma^{{\underline{0}}}\gamma_{PG}^{0}=\frac{1}{\sqrt{f_{S}}}\left({1+\sqrt{\frac{r_{0}}{r}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right).$
(100)

The operator of transition to the self-conjugate Hamiltonian with a plane
scalar product is given by

$\eta_{{}_{PG}}=\frac{1}{f_{S}^{1/4}}\left({1+\sqrt{\frac{r_{0}}{r}}\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right)^{1/2}.$
(101)

The practical use of a matrix operator $\eta_{{}_{PG}}$ seems to be difficult.

###  7.1 The Lorentz transformation

In order to obtain $\eta_{{}_{PG}}$ in an acceptable matrix-free form, let us
transfer, as well as in Sec. 6, from the system of tetrad vectors of
$\left\\{{\left({H_{PG}}\right)_{{\underline{\alpha}}}^{\mu}}\right\\}$ to
tetrad vectors of
$\left\\{{\left({\tilde{{H}}_{PG}}\right)_{{\underline{\alpha}}}^{\mu}}\right\\}$
in the Schwinger gauge using the Lorentz transformation of (20) - (23).
Non–zero tetrads
$\left({\tilde{{H}}_{PG}}\right)_{{\underline{\alpha}}}^{\mu}$ in the
Schwinger gauge are determined by the following expressions:

$\begin{array}[]{l}({\tilde{H}_{PG}})_{{\underline{0}}}^{0}=1,\,\,\,\,\,\,({\tilde{{H}}_{PG}})_{{\underline{0}}}^{1}=-\sqrt{\dfrac{r_{0}}{r}},\,\,\,\,\,({\tilde{{H}}_{PG}})_{{\underline{1}}}^{1}=1,\\\\[10.0pt]
({\tilde{{H}}_{PG}})_{{\underline{2}}}^{2}=\dfrac{1}{r},\,\,\,\,\,\,\,({H_{PG}})_{{\underline{3}}}^{3}=\dfrac{1}{r\sin\theta}.\end{array}$
(102)

For the PG metric, the non–zero values of
$\Lambda_{{\underline{\alpha}}}^{{\underline{\beta}}}\left(r\right)$ in (20)
are

$\Lambda_{{\underline{0}}}^{{\underline{0}}}=\Lambda_{{\underline{1}}}^{{\underline{1}}}=\frac{1}{\sqrt{f_{S}}},\,\,\,\,\,\Lambda_{{\underline{0}}}^{{\underline{1}}}=\Lambda_{{\underline{1}}}^{{\underline{0}}}=-\frac{\sqrt{{r_{0}}/r}}{\sqrt{f_{S}}}.$
(103)

Similarly to Eqs. (69) - (74), we obtain the following form of a matrix
$L,L^{-1}$ for the PG metric:

$L\left(r\right)=\frac{1+\sqrt{{r_{0}}/r}+\sqrt{f_{S}}+\left({1+\sqrt{{r_{0}}/r}-\sqrt{f_{S}}}\right)\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}{2\sqrt{\sqrt{f_{S}}\left({1+\sqrt{{r_{0}}/r}}\right)}},\,$
(104)
$L^{-1}\left(r\right)=\frac{1+\sqrt{{r_{0}}/r}+\sqrt{f_{S}}+\left({1+\sqrt{{r_{0}}/r}-\sqrt{f_{S}}}\right)\gamma^{{\underline{1}}}\gamma^{{\underline{0}}}}{2\sqrt{\sqrt{f_{S}}\left({1+\sqrt{{r_{0}}/r}}\right)}}.$
(105)

After Lorentz transformation (104) and (105), Hamiltonian (94) is transformed
to a self–conjugate Hamiltonian with tetrads in the Schwinger gauge [14],
[26].

For PG metrics, $\eta_{{}_{PG}}=1$ and $\tilde{{H}}_{PG}=H_{\eta_{{}_{PG}}}$,
where

$\begin{array}[]{l}H_{\eta_{{}_{PG}}}=L\left(r\right)H_{PG}L^{-1}\left(r\right)=\\\\[10.0pt]
=m\gamma^{{\underline{0}}}-i\gamma^{{\underline{0}}}\left\\{{\gamma^{{\underline{1}}}\left({\dfrac{\partial}{\partial
r}+\dfrac{1}{r}}\right)+\gamma^{{\underline{2}}}\dfrac{1}{r}\left({\dfrac{\partial}{\partial\theta}+\dfrac{1}{2}\mbox{ctg}\theta}\right)+\gamma^{{\underline{3}}}\dfrac{1}{r\,\mbox{sin}\theta}}\right\\}+\\\\[10.0pt]
+i\sqrt{\dfrac{r_{0}}{r}}\left({\dfrac{\partial}{\partial
r}+\dfrac{3}{4}\dfrac{1}{r}}\right),\\\\[10.0pt]
\psi_{\eta_{{}_{PG}}}=L\left(r\right)\psi_{PG}.\\\ \end{array}$ (106)

According to (104), the asymptotics $L\left(r\right)$ at $r\to r_{0}$ equals

$\left.L\right|_{r\to
r_{0}}=\frac{r_{0}^{1/4}\left({1+\gamma^{{\underline{0}}}\gamma^{{\underline{1}}}}\right)}{2^{1/2}\left({r-r_{0}}\right)^{1/4}}.$
(107)

In compliance with (106) and (107), after the similarity transformation
$L\left(r\right)$, the asymptotics of radial functions (97), (98) become equal
to

$\left.{\left({\tilde{{F}}_{pG}}\right)_{1}}\right|_{r\to
r_{0}},\,\,\left.{\left({\tilde{{G}}_{PG}}\right)_{1}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/2}}\exp\left\\{{i2r_{0}E\ln\left({\frac{r}{r_{0}}-1}\right)}\right\\},$
(108) $\left.{\left({\tilde{{F}}_{PG}}\right)_{2}}\right|_{r\to
r_{0}},\,\,\left.{\left({\tilde{{G}}_{PG}}\right)_{2}}\right|_{r\to
r_{0}}\sim\frac{1}{\left({r-r_{0}}\right)^{1/2}}.$ (109)

Then, we will obtain asymptotics of radial functions in a different way.

While presenting a wave function $\psi_{\eta_{{}_{PG}}}\left({{\rm{\bf
r}},T}\right)$ in the form of (30) in the equation of

$H_{\eta_{{}_{PG}}}\psi_{\eta_{{}_{PG}}}=E\psi_{\eta_{{}_{PG}}}$ (110)

we can separate the variables. While separating the variables, as well as in
Secs. 5 and 6, let us perform an equivalent substitution of matrixes
$\gamma^{{\underline{i}}}$ (see Eq. (31)).

The system of equations for radial wave functions
$\tilde{{F}}_{PG}\left(r\right),\tilde{{G}}_{PG}\left(r\right)$ is given by

$\begin{array}[]{l}f_{S}\dfrac{d\tilde{{F}}_{PG}}{dr}+\left({\dfrac{1+\kappa}{r}-\dfrac{3}{4}\dfrac{r_{0}}{r^{2}}-i\left({E-m}\right)\sqrt{\dfrac{r_{0}}{r}}}\right)\tilde{{F}}_{PG}-\\\\[10.0pt]
-\left({E+m+i\sqrt{\dfrac{r_{0}}{r}}\dfrac{1/4-\kappa}{r}}\right)\tilde{{G}}_{PG}=0,\\\\[10.0pt]
f_{S}\dfrac{d\tilde{{G}}_{PG}}{dr}+\left({\dfrac{1-\kappa}{r}-\dfrac{3}{4}\dfrac{r_{0}}{r^{2}}-i\left({E+m}\right)\sqrt{\dfrac{r_{0}}{r}}}\right)\tilde{{G}}_{PG}+\\\\[10.0pt]
+\left({E-m+i\sqrt{\dfrac{r_{0}}{r}}\dfrac{1/4+\kappa}{r}}\right)\tilde{{F}}_{PG}=0\\\
\end{array}$ (111)

The indicial equation for system (111) coincides with equation (83) for the
(EF) metric.

The solution of $s_{1}=-1/2+i2r_{0}E$ corresponds to square-nonintegrable wave
functions with the mode of a particle ”fall” to the event horizon. This
solution corresponds to the wave function transformations at transition to the
PG metric.

The solution of $s_{2}=0$ does not correspond to the transformation history of
wave functions at transition from the Schwarzschild metric to the PG metric.
As well as for the EF metric, the solution of $s_{2}=0$ for the PG metric,
being an accurate solution of the GR equations, is not connected with the
Schwarzschild solution.

The connection with the original Schwarzschild metric is implemented only by
the solution with the index of $s_{1}=-1/2+i2r_{0}E$.

It follows therefrom that the transition to the (PG) stationary metric does
not eliminate the mode of a particle ”fall” to the event horizon. The wave
functions are square-nonintegrable in the neighborhood of the event horizon.

##  8 Static metrics and the Schwarzschild metric with ”tortoise” coordinate

At spatial coordinate transformations of the Schwarzschild metric, the radius
of the event horizon is changed. For instance, for the metric in isotropic
coordinates [27, 28], the gravitational radius is
$\left({R_{is}}\right)_{0}={r_{0}}/4$, for the metric in spherical harmonic
coordinates, $\left({R_{g}}\right)_{0}={r_{0}}/2$. In this case, the mode of a
particle ”fall” on the event horizons is preserved (see, in Ref. [14]
solutions (35) and (41), Hamiltonian (61) and the system of equations for
radial functions (62)).

The similar situation is preserved when using the transformation of Ref. [10]

$dr\ast=dr\left({1-\frac{r_{0}}{r}}\right)^{-1},$ (112)

where

$r\ast=r+r_{0}\mbox{ln}\left({\frac{r_{0}}{r}-1}\right)+\mbox{const.}$ (113)

A value $r$ is the function of $r\ast$.

The coordinate $r\ast\to-\infty$ at $r\left({r\ast}\right)\to r_{0}$. In the
system of equations (32), for the Schwarzschild metric at transformation
(112), the first summands become equal to ${dF}/{dr\ast},\,\,{dG}/{dr\ast}$.
In the rest of the summands, the function of $r\left({r\ast}\right)$ is used.
Asymptotically, at $r\ast\to-\infty$ and $r\left({r\ast}\right)\to r_{0}$, the
solutions of system (32) are given by

$\begin{array}[]{l}\left.F\right|_{r\ast\to-\infty}=L\ast\left({r\left({r\ast}\right)-r_{0}}\right)^{{-1}/2}\sin\left({Er\ast+\delta\ast}\right),\\\\[10.0pt]
\left.G\right|_{r\ast\to-\infty}=L\ast\left({r\left({r\ast}\right)-r_{0}}\right)^{{-1}/2}\cos\left({Er\ast+\delta\ast}\right),\\\
\end{array}$ (114)

where $L\ast,\delta\ast$ are integration constants.

The wave functions in (114) preserve the mode of a particle ”fall” and are
square-nonintegrable (see also Eq. (37)).

##  9 Nonstationary metrics

After transformations from the Schwarzschild metric to the space-time of
Lemaítre–Finkelstein and Kruskal-Szekeres nonstationary metrics, Dirac
Hamiltonians explicitly depend on time coordinate [14]. In these cases,
examination of stationary states of particles is impossible [4].

##  10 Conclusions

The coordinate transformations of the Schwarzschild metric do not eliminate
the singular mode of a particle ”fall” to the event horizon of a black hole.
This mode is unacceptable for quantum mechanics of stationary states.

This conclusion has been rigorously proved for EF and PG stationary metrics
and is also valid for any transformed static metric, including Schwarzschild
metric with a ”tortoise” coordinate $r\ast$.

## Acknowledgments

The author thanks M. A. Vronskiy and V .E. Shemarulin for the useful remarks
and discussions. The author also thanks L. P. Babich and A. L. Novoselova for
the essential technical assistance in preparation of the paper.

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