Classical and Quantum Descriptions of the Geometrodynamics of Black Holes

**Classical Geometrodynamics of Charged BH**

For non-rotating spherically symmetric configurations, we consider the space-time metric (4) and the electromagnetic field of the type [1]

$$ds^2 = \frac{R}{\xi} \left( N dx^0 \right)^2 - \frac{\xi}{R} \left( dr + N^r dx^0 \right)^2 - R^2 d\sigma^2$$

Eqn-(2)

$$A_p = \left\{ A_p = \phi, A = \phi, 0, 0 \right\}, F_{00} = F_{00} = E = \phi_0 - \phi_r, R_0 = \partial R / \partial x^0, R_r = \partial R / \partial r, R_{00} = \partial^2 R / \partial r^2$$

Eqn-(3)

where Field configuration variables

$$q^A = \left\{ q^1 = q^R = R, q^2 = q^ξ = \xi, q^3 = q^φ = \phi \right\}$$

Eqn-(4)

are generalized coordinates that depend on space-time coordinates $x^0, r$, besides $A, B = \{1, 2, 3\}$.

The action $S$ tot after dimensional reduction, can be written as [4]:

$$S_{tot} = \int \Lambda dx^0 dr, \Lambda = \frac{V^2}{2N} + NU$$

Eqn-(5)

where

$$V^2 = \Gamma_{AB} V^A V^B = -\frac{c^3}{\kappa} V^R V^\xi + \frac{R^2}{c} \left( V^\phi \right)^2$$

Eqn-(6)

is the velocity square of the kinetic part of the Lagrangian. Here $\Gamma_{AB}$ are the covariant components of the CS metric and $\Gamma = \det \left| \Gamma_{AB} \right| = -\frac{c^5}{4\kappa^2} R^2, V^A = q_{0}^A + K^A$ are the generalized velocity components

$$V^R = R_{0,0} + K^R, V^\xi = \xi_{0,0} + K^\xi, V^\phi = \phi_{0,0} + K^\phi$$

Eqn-(7)

where

$$K^R = -N^r R_{r,0}, K^\xi = -\xi_{r,0} N^r - 2\xi N_{r,0}^r, K^\phi = -\phi_{r,0}$$

Eqn-(8)

and

$$U = \frac{c^3}{2\kappa} \left( 1 + \frac{R^2}{\xi^2} R_r \xi_r - \frac{2R}{\xi} R_r^2 - \frac{2R^2}{\xi} R_{rr}^2 \right)$$

Eqn-(9)

is the potential part of the Lagrangian $\Lambda$.

Let us introduce the CS metric of the by the formula

$$d\Omega^2 = \Gamma_{AB} V^A V^B \left( dx^0 \right)^2 = \Gamma_{AB} Dq^A Dq^B$$

Eqn-(10)

Here $Dq^A = dq^A + K^A dx^0$ are the Lie differentials:

$$DR = dR + K^R dx^0, D\xi = d\xi + K^\xi dx^0, D\phi = d\phi + K^\phi dx^0$$

At that, the $\Gamma_{AB}$ components are defined in (6). Then

$$D\Omega^2 = -\frac{c^3}{\kappa} DR D\xi + \frac{R^2}{c} D\phi^2$$

Eqn-(11)

The canonical momenta conjugated to the configuration variables $\xi, R$ and $\phi$ are

$$P_\xi = \frac{\partial \Lambda}{\partial \xi_0} = -\frac{c^3}{2\kappa N} \left( R_{0,0} + K^R \right)$$

Eqn-(12)

$$P_R = \frac{\partial \Lambda}{\partial R_0} = -\frac{c^3}{2\kappa N} \left( \xi_{0,0} + K^\xi \right)$$

Eqn-(13)

$$P_\phi = \frac{\partial \Lambda}{\partial \phi_0} = -\frac{R^2}{cN} \left( \phi_{0,0} + K^\phi \right)$$

Eqn-(14)

The Legendre transformation of the system leads to the Hamiltonian action [2]

$$S = \int dx^0 \int_0^r dr \left\{ P_\xi \xi_{0,0} + P_R R_{0,0} + P_\phi \phi_{0,0} - NH - N^r H_r - \varphi H_\theta \right\}$$

Eqn-(15)

where

$$H = \frac{4\kappa}{c^4} P_R P_\xi + \frac{1}{R^2} P_\phi^2 - U \sim 0$$

Eqn-(16)

$$H_r = R_{r,0} P_r - \xi_{r,0} P_\phi - 2\xi P_{\xi,r} \sim 0$$

Eqn-(17)

$$H_\phi = P_\phi,r \sim 0$$

Eqn-(18)

so that 'H' is Hamiltonian, $H_r$ is momentum and $H_\phi$ is electromagnetic constraints expressed in terms of momenta. For convenience, we represent the Hamiltonian constraint in the form

$$H = \frac{1}{2} P^2 - U \sim 0$$

Eqn-(19)

where

Gladush VD, Classical and Quantum Descriptions of the Geometrodynamics of Black Holes, Open J of Astro 2024, 2(1): 000106.

$$ P ^ {2} = \Gamma^ {A B} P _ {A} P _ {B} = - \frac {4 \kappa}{c ^ {3}} P _ {R} P _ {\xi} + \frac {c}{R ^ {2}} P _ {\phi} ^ {2} \tag {20} $$ is the momentum square. Note that

AB Γ

are the contravariant

components of the metric in the CS introduced earlier in

(6) so that

$$ \Gamma_ {A B} \Gamma_ {A D} = \delta_ {D} ^ {B} $$

. Electromagnetic constraint (18)

determines the electric field E generated by the charge Q

according to the formula

2 $$ P _ {\phi} = \frac {R ^ {2}}{c N} E = \frac {q}{c} = c o n s t \Rightarrow E = N \frac {q}{R ^ {2}} \tag {21} $$ The system admits the motion integrals: the total mass Mtot and the charge q cPφ = of configuration [3]. The mass is determined by the mass function, which in terms of momenta has the form (5)

2 2 2 2 2 2 , 6 4 1 2 2 tot r c R M R P R P m const R c $$ = \frac {c ^ {2}}{2 \kappa} \left(R + \frac {4 \kappa^ {2}}{c ^ {6}} \xi P _ {\xi} ^ {2} - \frac {R ^ {2}}{\xi} R _ {, r} ^ {2}\right) + \frac {1}{2 R} P _ {\phi} ^ {2} = m = c $$ Eqn- (22) Using the Hamiltonian constraint and conservation laws, one can find analytical expressions for momenta as functions of configuration variables and parameters m and q. Indeed, using the relations (21), (21) and (16), we obtain

3 $$ P _ {\xi} = \frac {c ^ {3}}{2 \kappa} \sqrt {\frac {R}{\xi} F _ {t o t}} \tag {23} $$

2 $$ = \sqrt {\frac {\xi}{R F _ {t o t}}} \left(\frac {q ^ {2}}{2 c R ^ {2}} - U\right) $$ q P U RF cR Eqn- (24) where

2 2 R tot

2 2 , 2 4 2 2 1 tot r R m q F R Rc c R

$$ = \frac {R}{\xi} R _ {, r} ^ {2} - 1 + \frac {2 \kappa m}{R c ^ {2}} - \frac {\kappa q ^ {2}}{c ^ {4} R ^ {2}} \quad \mathrm {E q n -} (2 5) $$ The momenta obtained in this way identically satisfy the invariance condition of the action functional, i.e. momentum constraint (17).

Using implicitly the integrability conditions of functional equations $$ P _ {R} = \frac {\delta S}{\delta R}, P _ {\xi} = \frac {\delta S}{\delta \xi}, P _ {\phi} = \frac {\delta S}{\delta \phi} = \frac {q}{c} \tag {26} $$ , , R we find the action functional S as a solution of the Einstein- Hamilton-Jacobi equation in functional derivatives depending on the variables R, ξ and parameters m and q [4]:

3

( ) Eqn- (27)

To verify, we show that variations of S with respect to mass m and charge q lead to motion trajectories in the CS. Indeed, we have

0 tot RF S g c m FR m

$$ \frac {\delta S}{\delta m} = - c \frac {\sqrt {\xi R F _ {t o t}}}{F R} + \frac {\partial g}{\partial m} = 0 \tag {28} $$

0 tot RF S q g c m cFR R c q

$$ \frac {\delta S}{\delta m} = - c \frac {\sqrt {\xi R F _ {t o t}}}{c F R} \frac {q}{R} + \frac {\phi}{c} + \frac {\partial g}{\partial q} = 0 \quad \mathrm {E q n -} (2 9) $$ From this, follows the expressions for R ξ and electric potential ( ) ( ) 2 2 , 0 2 , r R f r f r q F R F c R c $$ \frac {\xi}{R} = \frac {f ^ {2} (r)}{c ^ {2}} F - \frac {R _ {, r} ^ {2}}{F}, \phi = \phi_ {0} - \frac {f (r)}{c} \frac {q}{R} \quad \mathrm {E q n -} [ 3 0 ] $$ where the designations are introduced ( ) ( ) 2

0 2 4 2 , ; , ; 2 1 , , g m q r g m q r m q F f c c m q c R c R $$ T = - 1 + \frac {2 \kappa m}{c ^ {2} R} - \frac {\kappa q ^ {2}}{c ^ {4} R ^ {2}}, f = - \frac {\partial g (m , q ; r)}{c \partial m}, \phi_ {0} = - c \frac {\partial g (m , \partial q)}{\partial q} $$ Eqn- (31) The resulting solution leads to the metric M(4):

1 2 2 2 2 2 2 , , 2 0 0 2 2 2 2 r r r R R f r f r ds F N dx F dr N dx R d F F c c σ $$ F ^ {2} = \left(\frac {f ^ {2} (r)}{c ^ {2}} F - \frac {R _ {, r} ^ {2}}{F}\right) ^ {- 1} \left(\tilde {N} d x ^ {0}\right) ^ {2} - \left(\frac {f ^ {2} (r)}{c ^ {2}} F - \frac {R _ {, r} ^ {2}}{F}\right) \left(d r + N ^ {r} d x ^ {0}\right) ^ {2} - F $$ ( ) ( ) ( ) ( )  Eqn- (32) $$ A = \frac {Q}{R} c d T = c \frac {Q}{R} \left(T _ {, 0} d x ^ {0} + T _ {, r} d r\right) \tag {33} $$ We determined ( ) $$ f (r) = c ^ {2} T _ {, r}, \mathrm {w h i l e} T _ {, 0} $$

T is found from the integrability of the form

$$ d T = T _ {0} d x ^ {0} + T _ {, r} d r. $$

. The value N

follows from the time recovery procedure.

Since time is nowhere explicitly included in system, we can transform action (5) from the spatiotemporal representation to the configuration one by writing it in a form analogous to the Jacobi action [5]. Eventually from Lagrangian (5) we obtain

2 V U N N $$ \frac {\partial \Lambda}{\partial N} = - \frac {V ^ {2}}{2 N ^ {2}} + U = 0 \quad \mathrm {E q n -} (3 4) $$

2 0 2 From here we find the multiplier

$$ N = \sqrt {V ^ {2} / 2 U} $$

. Excluding N

from the action (5), we rewrite it as follows [6, 7, 8]

2 2 2 2 0 2 2 2 tot tot tot V S d x NU d x dr UV dx dr D N   = Λ = + = = Ω       ∫ ∫ ∫ ∫ ∫ ∫ Eqn-(35) where

3 2 2 2 1 2 2 2 2

$$ D \Omega_ {t o t} ^ {2} = 2 U D \Omega^ {2} = 2 U T _ {A B} D q ^ {A} D q ^ {B} = U \left(- \frac {c ^ {3}}{2 \kappa} D \xi D R + \frac {1}{2 c} D \phi^ {2}\right) $$

is the CS supermetric, conformal to the original metric

2 DΩ,

which is built on the kinetic part of (5) (10)-(11).

A B tot AB c D UD UT Dq Dq U D DR D c ξ φ κ Note that the transformation of field variables to the new variables [6]

$$ \xi = c \tau - x - \frac {y ^ {2}}{R}, \phi = \frac {c ^ {2}}{\sqrt {\kappa}} \frac {y}{R}, R = c \tau + x \quad \mathrm {E q n -} (3 7) $$ reduces the metric to the quasi-Lorentzian form $$ D \Omega^ {2} = - \frac {c ^ {3}}{\kappa} D R D \xi + \frac {R ^ {2}}{c} D \phi^ {2} = - c ^ {2} D \tau^ {2} + D x ^ {2} + D y ^ {2} $$ Eqn- (38) Here we have introduced new the Lie differentials $$ \begin{array}{l} D \tau = d \tau + K ^ {\tau} d x ^ {0}, D x = d x + K ^ {x} d x ^ {0}, D y = d y + K ^ {y} d x ^ {0} \\ \mathrm {E q n -} \tag {39} \\ \mathrm {w h e r e} \\ \end{array} $$ $$ K ^ {\tau} = \frac {\partial \tau}{\partial R} K ^ {R} + \frac {\partial \tau}{\partial \xi} K ^ {\xi} + \frac {\partial \tau}{\partial \phi} K ^ {\phi} \tag {40} $$ R K K K K R The component Kx and Ky are found by similar formulas.

Thus, the metric 2 DΩ in CS can be considered as the metric of a flat nonholonomic section of a 4-dimensional space. So the structure of the CS is similar to the family of flat nonholonomic sections M(4) M. It can be shown that the squared momenta (20) under the transformation (37) also take the Lorentian form

2 2 2 2 2 3 2 2 4 1 AB

$$ \begin{array}{l} P ^ {2} = \Gamma^ {A B} P _ {A} P _ {B} = - \frac {4 \kappa}{c ^ {3}} P _ {R} P _ {\xi} + \frac {c}{R ^ {2}} P _ {\phi} ^ {2} = - \frac {1}{c ^ {2}} P _ {\tau} ^ {2} + P _ {x} ^ {2} + P _ {y} ^ {2} \\ \mathrm {E q n -} (4 1) \\ \end{array} $$

On the Quantum Geometrodynamics of CBH

The quantum states of the field configuration are determined by the wave functional ( ) , , R Ψ ξ ϕ in the CS.

At the same time, the momenta P A are associated with the momentum operators A P), which in the coordinate representation have the form of functional derivatives:

$$ \hat {P} _ {\tau} = - i \hbar \frac {\delta}{\delta \tau}, \hat {P} _ {x} = - i \hbar \frac {\delta}{\delta x}, \hat {P} _ {y} = - i \hbar \frac {\delta}{\delta y} \tag {42} $$ , , x y P i P i P i x y τ In the case of charge Q cP = ϕ from here we immediately obtain $$ \mathrm {Q} \rightarrow \hat {Q} = c \hat {P} _ {\phi} = - i c \hbar \frac {\delta}{\delta \phi} \mathrm {E q n -} [ 4 3 ] $$ Q Q cP ic φ Similarly, from (17) the quantum equation of the momentum constraint follows ) ) ) ) η Eqn- (44) and (18) implies the operator electromagnetic constraint equation $$ \hat {r} \Psi = \left(R, _ {r} \hat {P} _ {R} - \xi , _ {r} \hat {P} _ {\xi} - 2 \xi \hat {P} _ {\zeta}, _ {r}\right) \Psi = - i \hbar \left(R, _ {r} \frac {\delta \Psi}{\delta R} - \xi , _ {r} \frac {\delta \Psi}{\delta \xi} - 2 \xi \frac {\partial}{\partial r} \left(\frac {\delta \Psi}{\delta \xi}\right)\right) $$ δ δ δ ξ ξ ξ ξ δ δξ δξ ( ) , , , , , 2 2 r r R r r r r H R P P P i R R r ξ ζ ) ) η qn- (45) $$ \hat {H} _ {\varphi} \Psi = \hat {P} _ {\phi , r} \Psi = - i \hbar \frac {\partial}{\partial r} \left(\frac {\delta \Psi}{\delta \phi}\right) = 0 $$ With the mass function Mtot (22) is related to the problem of ordering momentum operators. For the hermiticity of the total mass operator, in the CS with the volume element

the following ordering is used

$$ \xi P _ {\xi} ^ {2} \rightarrow \hat {P} _ {\xi} \xi \hat {P} _ {\xi}. $$

. Therefore, the

mass function Mtot (22) corresponds with the operator,

2 2 2 2 2 2 2 , 6 2 2 4 2 r c R M R R c c R

$$ = \frac {c ^ {2}}{2 \kappa} \left(R - \frac {4 \kappa^ {2} \hbar^ {2}}{c ^ {6}} \frac {\delta}{\delta \xi} \xi \frac {\delta}{\delta \xi} - \frac {\kappa \hbar^ {2}}{c ^ {2} R} \frac {\delta^ {2}}{\delta \phi^ {2}} + \frac {R ^ {2}}{\xi} R _ {, r} ^ {2}\right) $$ κ δ δ κ δ ξ κ δξ δξ ξ δφ η η Eqn- (46) When the Hamiltonian constraint H = 0 (19) is quantized, it is associated with its quantum counterpart ˆ 0 HΨ = , the DeWitt equation. We note that the squared momentum in (20), as well as the CS metric can be reduced to the “quasi- Lorentzian” form using the transformation (37). Therefore, in the coordinates{ } , ,x y τ , when quantizing the constraint H = 0, you can use the usual quantization recipe in the form (42).

Thus, to construct a quantum Hermitian operator in the original curvilinear coordinates{ } R, , ξ ϕ , it is necessary to perform an inverse transformation of coordinates and operators, which is equivalent to passing to covariant derivatives A A P P i → = − ∇ ) η with respect to the metric AB Γ defined in (6). However, in the case under consideration, one should pass to covariant functional derivatives according to the formulas Here the covariant functional derivatives are defined as follows

$$\frac{D\Psi}{\delta q^A} = \frac{\delta\Psi}{\delta q^A}, \frac{D}{\delta q^A} \Upsilon_B = \frac{\delta}{\delta q^A} \Upsilon_B - \Gamma_{AB}^C \Upsilon_C$$

Then, for the momentum squared $P^2$ (20) in the Hamiltonian constraint (19) after the replacement (47) we have

$$P \rightarrow \bar{P}^2 = -\hbar^2 \Delta.$$

Here

$$\Delta = \gamma^{AB} \frac{D}{\delta q^A} \frac{D}{\delta q^B} = \frac{1}{\sqrt{-\gamma}} \frac{\delta}{\delta q^A} \left( \sqrt{\gamma^{AB}} \frac{\delta}{\delta q^B} \right) = \frac{2\kappa}{c^4} \frac{\delta^2}{\delta\xi\delta R} - \frac{2\kappa}{c^4} \frac{1}{R} \frac{\delta}{\delta R} R \frac{\delta}{\delta\xi} + \frac{1}{R^2} \frac{\delta^2}{\delta\phi^2}$$

is the Laplace-Beltrami operator, which is Hermitian in natural measure. Note that in the case of variable $\xi$ the formula $\nabla_\xi = \partial/\partial\xi$ takes place, so in the mass operator for the momentum operator it suffices to restrict ourselves to the functional derivative $\hat{P}_\xi = i\hbar\delta/\delta\xi$. As a result, the Hamiltonian constraint (19) leads to the following DeWitt operator

$$\hat{H} \rightarrow -\frac{c}{2} \hbar^2 \Delta - U$$

or to the DeWitt equation

$$\frac{c}{2} \hbar^2 = \left( \frac{2\kappa}{c^4} \frac{\delta^2}{\delta\xi\delta R} + \frac{2\kappa}{c^4} \frac{1}{R} \frac{\delta}{\delta R} R \frac{\delta}{\delta\xi} \Psi - \frac{1}{R^2} \frac{\delta^2}{\delta\phi^2} \right) - U\Psi = 0$$

where $\psi = \psi[R, \xi, \phi; m, q]$ is a functional. States with a certain charge $q$ and mass $m$ are found by solving problems on eigenvalues and Eigen functions of operators charge $Q$ and mass $M$

$$\bar{Q}\Psi = q\Psi, \bar{M}\Psi = m\Psi$$

These equations, taking into account (43) and (46), can be rewritten as follows

$$i\hbar \frac{\delta\Psi}{\delta\phi} = q\Psi$$

$$\left( R - \frac{4\kappa^2 \hbar^2}{c^6} \frac{\delta}{\delta\xi} \xi \frac{\delta}{\delta\xi} - \frac{\kappa\hbar^2}{c^2 R} \frac{\delta^2}{\delta\phi^2} + \frac{R^2}{\xi} R^2 \right) \Psi = \frac{2\kappa m}{c^2} \Psi$$

By virtue of the relation (54), it follows from the constraint (45) that $\psi$ does not depend on $r$. Moreover, (54) implies

$$\Psi(R, \xi, \phi; m, q) = \Psi[R, \xi; m, q] e^{i\int(q/\chi)\phi(r)}$$

Then, the DeWitt equation (49) becomes

$$\frac{\kappa}{c^4} \frac{\delta^2}{\delta\xi\delta R} + \frac{\kappa}{c^4} \frac{1}{R} \frac{\delta}{\delta R} R \frac{\delta}{\delta\xi} + \frac{1}{c^2} \left( \frac{q^2}{2cR^2} - U \right) \Psi = 0$$

The equation for the eigenvalues of the mass operator (52) can be rewritten as follows

$$\frac{\delta}{\delta\xi} \xi \frac{\delta\Psi}{\delta\xi} + \frac{c^6}{4\kappa^2 \hbar^2} RF_{tot}\Psi = 0$$

The joint solution of the (57) and (58) equations, together with the momentum constraint (44), describes the quantum state of the considered CBH model with fixed charge $q$ and mass $m$.

**Geometrodynamics of CBH in the T-Region**

The T-region $M^{(4)}$ CBH is region, where the vector Keeling $\xi^\mu$ is spatially similar and can be convert to form $\xi^\mu = \delta_\mu^\mu$ [7]. Then the metric (2) can be written as follows

$$ds^2 = \frac{R}{\xi} \left( N dx^0 \right)^2 - \frac{\xi}{R} dr^2 - R^2 d\sigma^2$$

In this case, $N^r = 0$ the coordinate system becomes orthogonal and all fields depend only on time, that $\xi_r = 0, R_r = 0, \varphi_r = 0$. Then the system of equations is greatly simplified, and we must also put $K^R = 0, K^\xi = 0, K^\varphi = 0$. Therefore integration over the coordinate $r$ is replaced by multiplication by the some constant: $\int dr \rightarrow 1 < \infty$. As a result, the action and the Lagrangian (5) take the form

$$S_{tot} \rightarrow S = \int L dx^0, \Lambda \rightarrow L = l \left( \frac{\bar{V}^2}{2\vec{N}} + \bar{N}\vec{U} \right)$$

While, $U \rightarrow \bar{U} = c^3/2\kappa$ is the potential part of the Lagrangian, is the square of the velocity:

3 2 2 2 ,0 ,0 ,0 A B AB

$$ V ^ {2} = \Gamma_ {A B} V ^ {A} V ^ {B} = - \frac {c ^ {3}}{\kappa} R _ {, 0} \xi_ {, 0} + \frac {R ^ {2}}{c} \phi_ {, 0} ^ {2} \quad \mathrm {E q n -} [ 6 1 ] $$ Here, { } ,0 ,0 ,0 , , A R V V R V V ξ φ ξ φ = = = = are generalized velocities. The CS metric is defined similarly to the general case (10), (11):

3 2 2 2 0 2 A B A B AB AB

$$ d \Omega^ {2} = \Gamma_ {A B} V ^ {A} V ^ {B} \left(d x ^ {0}\right) ^ {2} = \Gamma_ {A B} d q ^ {A} d q ^ {B} = - \frac {c ^ {3}}{\kappa} d R d \xi + \frac {R ^ {2}}{c} d \phi^ {2} $$ ( ) Eqn- (62) Legendre transformation of the system leads to the Hamiltonian action $$ S = \int d x ^ {0} \int_ {0} ^ {\infty} d r \left\{P _ {\xi} \xi_ {, 0} + P _ {R} R _ {, 0} + P _ {\phi} \phi_ {, 0} - N H - N ^ {r} H _ {r} - \varphi H _ {\theta} \right\} $$ Eqn- (63) where $$ H = \frac {1}{2 l} P ^ {2} - l \tilde {U} = \frac {c}{2 l} \left(- \frac {4 \kappa}{c ^ {4}} P _ {R} P _ {\xi} + \frac {1}{R ^ {2}} P _ {\phi} ^ {2} - \mu^ {2}\right) \sim 0 $$ Eqn- (64) is the Hamiltonian constraint, 2 , AB

2 2 2 2 4 2 1 4 1 2 tot c M R P P m const l c l R ξ φ κ ξ κ $$ = \frac {1}{2} \left(\frac {c ^ {2}}{\kappa} R + \frac {4 \kappa}{l ^ {2} c ^ {4}} \xi P _ {\xi} ^ {2} + \frac {1}{l ^ {2} R} P _ {\phi} ^ {2}\right) = m = c $$ Eqn- (65) Together with the Hamiltonian constraint, they lead to momenta

3 $$ P _ {\xi} = \frac {l c ^ {3}}{2 \kappa} \sqrt {\frac {R}{\xi} F} \tag {66} $$

3 2 $$ = \frac {l c ^ {3}}{2 \kappa} \sqrt {\frac {\xi}{R F}} \left(\frac {\kappa q ^ {2}}{c R ^ {2}} - 1\right) $$

2 1 2 R lc q P RF cR

ξ κ κ Eqn- (67)

where F is defined in (31).

To find the action

$$ S = S \left(R, \xi , \phi\right) $$

, as a function of field

variables , ,

R ξ φ and conserved quantities

tot M M = and Q q =

we can use the integrability condition of the relations

1 , , R S S S P P P q R c ξ φ ξ φ ∂ ∂ ∂ = = = = ∂ ∂ ∂ Eqn- (68) In fact, to find ( ) , , S R ξ φ it is enough to consider the differential $$ d S = P _ {R} d R + P _ {\xi} d \xi + P _ {\phi} d \phi = P _ {R} d R + P _ {\xi} d \xi + \frac {l q}{c} d \phi $$ R R Hence, using the integrability conditions $$ \partial P _ {\xi} / \partial R = \partial P _ {R} / \partial \xi $$

and (66-67), we find $$ S = S _ {q} + S _ {q} = 2 \xi P _ {\xi} + \frac {l q}{c} \phi = \frac {l c ^ {3}}{\kappa} \sqrt {\xi R F} + \frac {l q}{c} \phi \tag {69} $$ Further, from the formulas

$$ \text{Further, from the formulas} \frac {\partial S / \partial m = l a _ {m}}{\partial S / \partial q = l a _ {q}} \text{ where } a _ {q} \text{ and } a _ {m} \text{ are constants, we can find the trajectories of the system in the CS} $$

a are constants, we can find the trajectories

of the system in the CS.

As well as in the general case, we can transform action (60) from the space-time representation to the configuration representation of CS [8]. For this, from the Lagrangian L in (60), we obtain     Eqn- (70)

2 $$ \frac {\partial L}{\partial \tilde {N}} = l \left(\frac {\tilde {V} ^ {2}}{2 \tilde {N} ^ {2}} + \tilde {U}\right) = 0 $$ L V l U N N

2 0 2 which implies 2 2 N V U =    . Substituting into action N

(60) we get $$ S = \int L d x ^ {0} = \int l \sqrt {2 \tilde {U} \tilde {V} ^ {2}} d x ^ {0} = \mu \int \sqrt {c d \Omega^ {2}} \quad \mathrm {E q n -} (7 1) $$ where 2 DΩ received in (62). In a new representation of the action, we study an induced dynamical system that describes geodesics in a CS with the corresponding induced supermetric.

Using the transformation (37) of field variables, the metric

2 DΩ of CS is reduced to a flat form

4 2 2 2 2 2 2 4

$$ D \Omega^ {2} = \frac {c ^ {4}}{4 \kappa^ {2}} \left(- c ^ {2} d \tau^ {2} + d x ^ {2} + d y ^ {2}\right) \tag {72} $$ ( ) In this case the squared momentum also takes the Lorentzian form

2 2 2 2 2 3 2 2 4 1 R x y c P P P P P P P c R c ξ φ τ κ = + = − + + Eqn- (73) As we can see, the corresponding equations of geometrodynamics of the CBH in the T-region are greatly simplified. This is especially important in the case of quantization of the BH. In this case, the equations system of the quantum theory of BH for the wave functional $\Psi[R, \xi; m, q]e$ in functional derivatives is transformed over into the equations system in partial derivatives for the wave function. As a result, we have the DeWitt equation and equations for the eigenvalues of mass and charge. The equation for the charge eigenvalue leads to the wave function [9].

$$\Psi(R, \xi; \phi; m, q) = \Psi[R, \xi; m, q]e^{iql/a\hbar} \quad \text{Eqn- (74)}$$

where the function $\Psi(R, \xi; m, q)$ obeys equation on the mass eigenvalue and the reduced equation DeWitt

$$\frac{\partial}{\partial \xi} \xi \frac{\partial \Psi}{\partial \xi} + \frac{c^6}{4\kappa^2 \hbar^2} RF\Psi = 0 \quad \text{Eqn- (75)}$$

$$\frac{\partial^2 \Psi}{\partial \xi \partial R} + \frac{1}{R} \frac{\partial}{\partial R} R \frac{\partial}{\partial \xi} \Psi + \frac{c^6 l^2}{2\kappa^2 \hbar^2} \left(1 + \frac{\kappa q^2}{c^4 R^2}\right) \Psi = 0 \quad \text{Eqn- (76)}$$

The joint solution of the system of equations (75) and (76) leads to the following wave function [8]

$$\Psi_{m, q}(R, \xi, \phi) = \frac{C}{\sqrt{R}} J_0 \left( \frac{l c^3}{\hbar \kappa} \sqrt{\xi RF_T} \right) e^{iql/a\hbar} \quad \text{Eqn- (77)}$$

Where $J_0$ is the Bessel function of the first kind of order zero. We see that in this simplified formulation, the constructed model describes the CBH in the T-region with a continuous spectrum of mass $m$ and charge $q$. Note that $R=cT$.