de broglie–bohm interpretation for wave function of a toroidal black hole

Gen Relativ Gravit (2009) 41:1181–1193DOI 10.1007/s10714-008-0698-1

RESEARCH ARTICLE

de Broglie–Bohm interpretation for wave functionof a toroidal black hole

Bo-Bo Wang

Received: 22 February 2008 / Accepted: 19 September 2008 / Published online: 8 October 2008© Springer Science+Business Media, LLC 2008

Abstract The mass function of a toroidal black hole as a dynamical variable is givenby using Kuchar’s approach. Then the toroidal black hole is investigated by using thede Broglie–Bohm approach, and its quantum potential and quantum trajectories areobtained. In our process the vector potential of the electromagnetic field in the toroidalblack hole is treated as a canonical variable.

Keywords Quantum black hole · de Broglie–Bohm interpretation · Mass function

1 Introduction

The canonical formalism of gravity, as a theory of quantum gravities, has been for-mulated by Arnowitt et al. [1] and Dirac [2]. In this approach a gravitational sys-tem reduces to a totally constrained one, and the Hamiltonian and the momentumconstraints [3,4] can be obtained. In order to quantize the system, one imposes theconstraints as operators acting on the wave function of a black hole system. TheHamiltonian constraint corresponds to the well-known Wheeler–DeWitt (WD) equa-tion [5,6]. However it is very difficult to solve the WD equation when the spacetime isinhomogeneous. But when the spacetime has some symmetries, the WD equation canreduce to a simple and solvable equation with finitely many degrees of freedom. Thisis called a minisuperspace approximation. Nakamura et al. [7] studied the quantumblack hole with spherical symmetry and obtained the wave function of Schwarzschildblack hole. Then using the WKB approximation they found a static black with theapparent horizon separating from the event horizon due to the quantum fluctuationsof the metric evolution. Following this approach, Kenmoku, Kubotani, Takasugi and

B.-B. Wang (B)Department of Physics, School of Science, Beijing Jiaotong University, 100044 Beijing, Chinae-mail: [email protected]

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1182 B.-B. Wang

Yamazaki employed the canonical quantum formalism for spherically symmetric spa-cetimes to re-obtain the analytically exact solution of a wave function with newvariables [8–11]. In their approach they also employed a mass function as a dyna-mical variable, which is originally introduced by Kuchar in studying the Schwarschildblack hole [12]. For investigating quantum behaviors of black holes they further usedthe de Broglie–Bohm (dBB) interpretation of a wave function [13,14] instead of theWKB approximation. They found the location of the apparent horizon coincides withthat of the event horizon according to the dBB interpretation. In this paper, followingtheir approach, we will investigate the quantum geometry of a toroidally symmetricalblack hole. The two-dimensional surface of the black hole horizon with the constanttime coordinate and the constant radial space coordinate is a torus, which has topologyS1 × S1. Some classical properties of a toroidal black hole have been studied in [15].Its thermodynamically behaviours have been investigated by using the path-integralapproach in York’s formalism [16] and by using the thin film brick-wall model [17].

In the paper we first give the classical properties of a toroidal black, and then indetail calculate the mass function as canonical date, which is a dynamical variablein Kuchar’s approach. Based on the minisuperspace model, we give the Lagrangianand the Hamiltonian of the black hole. From the Lagrangian we recover the clas-sical solution of the spacetime by solving the Euler-Lagrange equations. From theHamiltonian the canonical quantum formulism is presented. Notice that, in our treat-ment, the vector potential of the electromagnetic field is regarded as a canonicalvariable. Next, by solving the WD equation and the mass operator equation simulta-neously, the wave function of the quantum black hole is obtained. Applying the dBBinterpretation to the wave function yields the quantum trajectories and the quantumpotential. In fact, Gao and Shen have studied the dBB approach to the toroidal blackhole [18]. But they did not treat the vector potential as a canonical variable. And infact they dealt with it as a constant. So they cannot give the quantum trajectory ofvector potential. For this reason our treatment seems to be more complete.

This paper is organized as follows. In Sect. 2, Some classical properties of a toroidalblack hole are given. In Sect. 3 the mass function is derived by using Kuchar’s approach.In Sect. 4, the canonically quantum formalism of the toroidal black hole is presented.Meanwhile the analytic solution of the wave function of the black hole is obtained.In Sect. 5, the dBB interpretation is employed to the wave function, and the quantumpotential and the quantum trajectories are obtained. we also discuss the quantumapparent horizon of the black hole. Summary and discussion are given in Sect. 6.

2 Some classical properties of a toroidal black hole

A generally statically plane-symmetric solution of the Einstein–Maxwell equationswith a cosmological constant λ can be expressed as [19,20]

ds2 = − f (r)dt2 + f (r)−1dr2 + r2(

dθ2 + dφ2), (1)

where

f (r) = −1

3λr2 − a

r+ b2

r2 , (2)

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de Broglie–Bohm interpretation for wave function of a toroidal black hole 1183

a and b are two integration constants. By identifying θ = 0 with θ = 2√π and

φ = 0 with φ = 2√π in Eq. (1), one can construct a toroidal solution [15]. Here, for

convenience, we set the identification in order to choose the solid angle (the area of thetorus divided by square radius) in the toroidal spacetime to be 4π instead of 4π2 as in[15]. The advantage is that some physical laws in toroidal black hole, such as Gausslaw of electromagnetic field, will have the same forms in the Euclidian spacetime,although the difference between the two choices can be regarded as adopting differentunits.

Substituting Eq. (1) into Einstein field equation with cosmological constant, we get

the non-vanishing components of the energy-momentum stress, T00 = b2 f8πr2 , T11 =

− b2

8π f r2 , T22 = T33 = b2

8πr2 . However, in the other hand, the electromagnetic stress also

can be obtained from Tab = 14π

(Fac F c

b − 14 gab Fcd Fcd

), which gives T00 = f (F01)

2

8π ,

T11 = − f −1(F01)2

8π , T22 = T33 = r2(F01)2

8π , where the non-vanishing components ofelectromagnetic tensors are F01 = −F10. So we can get F01 = −b/r2, which furthergives the only non-vanishing component of electric field intensity E1 = b/r2. Andthen using the Gauss theorem we obtain b = Q, where Q denotes the electric chargeof the black hole. For ADM mass, we can follow the method of [15,21], which givesthe mass of the black hole, m = a/2. The metric (1) becomes

ds2 = − f (r)dt2 + f (r)−1dr2 + r2(

dθ2 + dφ2), (3)

where

f (r) = −1

3λr2 − 2m

r+ Q2

r2 , (4)

The metric (3) becomes singular when f (r) = 0, which has two positive solutionsr+ and r− as long as 0 ≤ Q2 ≤ 3

4 (12m4/|λ|)1/3 for a negative λ < 0. So the metric(1) describes a black hole with a Cauchy horizon r− and a event horizon r+ in thecase of λ < 0. When 0 ≤ Q2 = 3

4 (12m4/|λ|)1/3, the event horizon and the Cauchyhorizon coincide. For simplicity we only consider the case of r+ > r−.

Inside the black hole, i.e. r− < r < r+, f (r) < 0, t represents the radial spacecoordinate while r denotes time coordinate. By using the exchange,

t → r, r → t, f (r) → −U (t),

we can rewrite the inside metric as

ds2 = −α(t)2

U (t)dt2 + U (t)dr2 + V (t)(dθ2 + dφ2), (5)

where U and V are two metric variables, and α(t) is a lapse function. In metric (5),the corresponding classical solution can be expressed as

α(t) = 1, U = −(

−1

3λt2 − 2m

t+ Q2

t2

), V (t) = t2. (6)

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1184 B.-B. Wang

3 The mass function of the toroidal black hole

In the section, by following Kuchar’s calculation [12], we give the mass function ofthe toroidal black hole. In Kuchar’s approach the mass of a black hole can be regardedas a dynamical variable, which can be expressed as a function of canonical data.

For convenience of applying Kuchar’s approach, we rewrite the element of thetoroidal spacetime as

ds2 = −F(R)dT 2 + F−1(R)d R2 + R2(dθ2 + dφ2), (7)

where

F(R) = −1

3λR2 − 2M

R+ Q2

R2 , (8)

M and Q are the mass and the charge of the black hole respectively. Using the Gausslaw, one can get

Q = R2 F01. (9)

According to the Kuchar’s method, a three-dimension space,

dσ 2 = �2(r)dr2 + R2(r)(

dθ2 + dφ2)

(10)

is embed into a four spacetime

ds2 = −(

N 2 −�2(Nr )2)

dt2 + 2�2 Nr dtdr

+�2dr2 + R2(

dθ2 + dφ2), (11)

where N is the lapse function and Nr the shift vector and both of them depend solelyon t and r variables. Here�(t, r) and R(t, r) are also assumed to depend not only onr but t .

The Einstein–Maxwell action with a cosmological constant can be written as

S = 1

16π

∫d4x

√−(4)g

((4)R − 2λ− FµνFµν

), (12)

where (4)R is the four-dimensional Riemann curvature scalar and Fµν = ∂µAν−∂ν Aµ the electromagnetic field strength, where Aµ is the vector potential which is thefunction of t and r . Here the only non-vanishing component of the strength is

F01 = A − A′0, (13)

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where A ≡ A1, and a dot and a prime denote the derivative with respect to t and rrespectively. Putting the metric (11) into action (12) we get the ADM action

S[R,�; N , Nr ] =

∫dt

∞∫

−∞dr

{− N−1

[R

(−�+ (

�Nr )′) (−R + R′Nr )

+ 1

2�

(−R + R′Nr )2]

+ N

(−�−1 R R′′ +�−2 R R′�′

− 1

2�−1 R′2

)− 1

2λN�R2+ 1

2N−1�−1 R2 (

A1− A′0

)2}.

(14)

By differentiating the ADM action with respect to the velocities �, R and A1, weobtain the momenta

P� = −N−1 R(R − R′Nr ) , (15)

PR = −N−1(

R(�− (

�Nr )′) +�(R − R′Nr )) , (16)

PA = N−1�−1 R2 (A1 − A′

0

) = N−1�−1 R2 F01. (17)

By transforming coordinates t and r into coordinates T and R as

T = T (t, r), R = R(t, r), (18)

and then substituting them into the toroidal line element (7) we can get

ds2 = −(

FT 2 − F−1 R2)

dt2 + 2(−FT ′T + F−1 R′ R

)dtdr

+(−FT ′2 + F−1 R′2) dr2 + R2

(dθ2 + dφ2

). (19)

Comparing this expression with the ADM line element (11) yields

�2 = −FT ′2 + F−1 R′2, (20)

�2 Nr = −FT ′T + F−1 R′ R, (21)

N 2 −�2 (Nr )2 = FT 2 − F−1 R2. (22)

Using the three equations and Eq. (15), one can obtain F as a function of the canonicaldeta:

F =(

R′

�

)2

−(

P�R

)2

. (23)

Substituting the Eq. (23) for F and the Eq. (9) for Q into the Eq. (8), we candetermine the mass of the toroidal black hole

M = 1

2R−1 P2

� − 1

2�−2 R R′2 − 1

6λR3 + 1

2N 2�2 R−1 P2

A, (24)

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1186 B.-B. Wang

which is a function of canonical date. Here Eq. (17) have been used. In the followingthe dynamical variable M will lead to a operator acting on a wave function when thesystem of the black hole is quantized.

4 Canonical quantum formalism

In this section we start with metric (5). By straightforward calculation from Eq. (5) we

have√−(4)g

(4)R = − 1

2αV U V 2− 1α

U V + ∂∂t (

1α

V U )+2 ∂∂t (

1α

U V ) and√−(4)g(−2λ−

FµνFµν) = αV (−2λ+ 2α2 F2

01). Inverting them into Eq. (12) and performing integra-tion by parts for t and integrations from 0 to 2

√π for θ and φ, we obtain the ADM

hypersurface action

S =∫

dt∫

dr L , (25)

where the Lagrangian is

L = 1

4

(−U V

α− U V 2

2Vα

)− λ

2αV + 1

2

V

αF 2

01 (26)

Substituting the Lagrangian (26) into the Euler–Lagrange equation ∂L∂qi − d

dt (∂Lqi ) =

0, where the general coordinates qi denote U , V , α and A respectively, we get fourequations

2V V α + α(

V 2 − 2V V)

= 0, (27)

UαV 2+2V 2(

2λα3+U α−α(

2F 201 +U

))−2V (−U V α+α(U V +U V )) = 0,

(28)

−4V 2(λα2 + F 2

01

)+ 2V U V + U V 2 = 0, (29)

∂

∂t

(V F01

α

)= 0. (30)

The lapse function can be fixed at any point. For convenience, we choose the lapsefunction as α = 1. In this case, these equations become

V 2 − 2V V = 0, (31)

U V 2 + 2V 2(

2λ− 2F 201 − U

)− 2V

(U V + U V

) = 0, (32)

−4V 2(λ+ F 2

01

)+ 2V U V + U V 2 = 0, (33)

∂

∂t(V F01) = 0. (34)

The general solution of Eq. (31) is V = (B1t + B2)2, where B1 and B2 are two

integration constants. But without loss of generality, we can set B1 = 1 and B2 = 0.

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So we get V = t2. The solution of Eq. (34) is V F01 = Q, where Q is a constant.

Inserting the two solutions into Eqs. (32) and (33), we obtain U = λ3 V − Q2

V + 2m√V

,where m is a constant. Finally we obtain the classical solutions,

α = 1, (35)

U = −(

− 2m√V

− λV

3+ Q2

V

), (36)

√V = t, (37)

V F01 = Q. (38)

This results are in a coincidence of Eqs. (35)–(37) with Eq. (3). And Eq. (38) is nothingbut the result of the Gauss law.

Next we will investigate the quantum solution of the toroidal black hole by followingthe method of [8–11]. Following them we also introduce two new variables z+ andz−,

z+ = U√

V , z− = √V . (39)

And the Lagrangian (26) changes into a simpler form

L = − 1

2αz+ z− + z2−

2αF2

01 − λ

2αz2−. (40)

Three canonical momenta conjugated to z+, z− and A respectively are:

�+ ≡ ∂L

∂ z+= − 1

2αz−, (41)

�− ≡ ∂L

∂ z−= − 1

2αz+, (42)

�A ≡ ∂L

∂ A= z2−

α

(A − A′

0

). (43)

The momenta conjugated to α and A0 are zeros and then yield the primary constraintsbecause the Lagrangian (40) does not include terms α and A0. Using Legendre trans-formation the action (25) changes

S =∫

dt∫

dr(z+�+ + z−�− + A�A − αH − A0 HA

), (44)

where H and HA are respectively,

H = −2�+�− + 1

2z2−�2

A + λ

2z2−, (45)

HA = −(�A)′. (46)

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1188 B.-B. Wang

In the following we quantize the black hole system. In the Schrödinger representa-tion, the canonical momenta are expressed by operators,

�+ = −i h∂

∂z+, �− = −i h

∂

∂z−, �A = −i h

∂

∂A. (47)

According to Dirac’s canonical quantization procedure, the Hamiltonian constraint(H ≈ 0) becomes an operator equation for the physical wave function �,

H� =(

−2�+�− + 1

2z2−�2

A + λ

2z2−

)� = 0, (48)

which is nothing but the WD equation. Here the wave function � = �(z+, z−, A)is the function of z+, z−, and A. Here the Weyl ordering has been chosen because itcan reduce to exactly classical results [8–11]. The Gaussian constraint (HA ≈ 0) alsogives a operator’s equation

�A� = Q�, (49)

where Q is the electric charge of the black hole.Now we use the mass function (24). Comparing Eq. (5) with the ADM metric (11),

we obtain

N 2 = α(t)2

U (t), �2 = U (t), R2 = V (t), Nr = 0. (50)

From the equation, we can see that R is independent of the variable r , so R′ = 0.Further by using the varialves z+ and z− of Eq. (39), we can express the mass function(24) as

M = 2z+�2+ − 1

6λz3− + 1

2z−�2

A, (51)

where �+ ≡ Pz+ and �A ≡ PA.In the process of the canonical quantization, the mass function (51) leads to a

operator M , which acts on physical states with a mass eigenvalue m,

M� =(

2�+z+�+ + α2

2z−�2

A − λ

6z3−

)� = m�. (52)

One can easily find, that the mass operator M weakly commutes with the Hamiltonianconstraint operator [

H , M]� = −2i h�+ H� = 0. (53)

By separating variables, �(z+, z−, A) = ψ(z+, z−)�(A), Eq. (49) yields

� = ψ(z+, z−)ei Q A/h . (54)

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Following the method in spherically symmetric black holes [8–11], we also intro-duce a new operator L ,

L� ≡[�+z+ H + �−

(M − m

)]� = 0. (55)

And by introducing a new variable,

z− = 1

z−

(−2mz− + λ

3z4− + Q2

), (56)

Equation (55) becomes

z−∂

∂ z−ψ − z+

∂

∂z+ψ = 0. (57)

Transforming variables z+ and z− to variables z and y,

y = 1

h

√−z+/z−, z = 1

h

√−z+ z−, (58)

we can express Eqs.(57) and (48), respectively, as

y∂ψ(y, z)

∂y= 0, (59)

1

z

∂

∂z

(z∂ψ(y, z)

∂z

)+ ψ(y, z) = 0. (60)

The two equations have the following solution

�(z, A) =(

c1 H (1)0 (z)+ c2 H (2)

0 (z))

ei Q A/h, (61)

z = 1

h

√−z+ z−, (62)

where c1 and c2 are two constants, and H (1)0 and H (2)

0 are Hankel functions.

5 de Broglie–Bohm interpretation

According to dBB interpretation the wave function should be expressed as�(z+, z−, A) = R(z+, z−, A) exp

[i S(z+, z−, A)/h

], where the amplitude R and the

phase S are two real functions. By substituting this expression into the WD equation(48), we obtain two equations,

2∂S

∂ z−∂S

∂z++ 1

2+ VQ = 0, (63)

∂

∂z

(z R2 ∂S

∂z

)= 0, (64)

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1190 B.-B. Wang

where VQ is a quantum potential:

VQ = −2h2

R

∂2 R

∂z+∂ z−. (65)

Notice that using the identity of the Hankel function, H (2) ∂∂z H (1)−H (1) ∂

∂z H (2) = 4iπ z ,

we can get

z R2 ∂S

∂z= 2h(c2

1 − c22)

π. (66)

Quantum trajectories Z+(T ), Z−(T ) and A(T ) are respectively introduced by

�+ = −1

2Z− = ∂S

∂z+

∣∣∣∣z+=Z+,z−=Z−,A=A

, (67)

�− = −1

2Z+ = ∂S

∂z−

∣∣∣∣z+=Z+,z−=Z−,A=A

, (68)

�A = Z2− A = ∂S

∂A

∣∣∣∣z+=Z+,z−=Z−,A=A

, (69)

where a dot denotes the derivative with respect to T . From Eqs. (67) and (68) we get

d Z+/d Z− = ∂z∂z− /

∂z∂z+

∣∣∣z+=Z+,z−=Z−

. Putting Eq. (58) into it yields d Z+/d Z− =Z+/Z−, and then gives a quantum trajectory Z+ = cZ−, where c is a integrationconstant. We further obtain,

U = −(

− 2m√V

− λ

3V + Q2

V

), (70)

where we have set c = −1 without loss of generality. From Eq. (67) we obtain thequantum trajectory for T-V relation,

T = − π

2(c2

1 − c22

)∫

Z∣∣∣c1 H (1)

0 (Z)+ c2 H (2)0 (Z)

∣∣∣2

d√

V , (71)

where Z = 1h |√V U | and Eq. (66) has been used. From Eq. (69) we easily get the

quantum trajectory for A,

A = Q

V. (72)

We can see that the quantum trajectory (72) is the same as the classical one (38).In dBB point of view, the quantum trajectories should be to correspond to those of

the classical black hole [11]. The quantum trajectory (72) corresponds to the classicalone (38). And the quantum trajectory (70) also go back to its calssical one (36).As for quantum trajectory (71) we should consider its asymptotical behaviour. The

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asymptotical form of the Hankel functions are

H (1)0 (Z) →

√2

π Zei(Z−π/4), (73)

H (2)0 (Z) →

√2

π Ze−i(Z−π/4), (74)

for Z � 1. So we get from Eq. (71)

T = −c21 + c2

2

c21 − c2

2

V 1/2 − 2c21c2

2

c21 − c2

2

∫cos 2

(Z − π

4

)dV 1/2, (75)

for Z � 1. We can see that we have to set c1 = 0 for reducing the quantum T-Vrelation (71) to the classical one (6). Otherwise we have T = −V 1/2 when c2 = 0,which do not correspond to Eq. (6). So the quantum function should be

�(z, A) = c2 H (2)0 (z)ei Q A/h . (76)

And the quantum potential (65) can be expressed as

VQ = −1

2

⎛⎜⎝1 − 4

π2 Z2∣∣∣H (2)

0 (Z)∣∣∣4

⎞⎟⎠ . (77)

Note that the quantum potential VQ ≈ 0 when Z � 1 and diverges in the vicinity ofthe horizon(Z = 0). The asymptotic behaviour is

VQ ≈ π2

8

1

Z2(ln Z)4, (78)

for Z → 0.Now, following [7,11], we consider the horizon of the quantum black hole. It is

easy to see that, for the toroidal black hole, the classical and quantum event horizonis located at U = 0. Let θ− and θ+ express the expansions for ingoing and outgoingnull rays respectively. Classically the apparent horizon can be defined by the equationθ+ = 0, which is equal to the equation θ−θ+ = 0 . Further one can express it as

θ−θ+ = U V −1(

.√V )2 = 1

V (2M√

V+ λ

3 V − �2A

V ), where M is the mass function, �A

the electromagnetic momentum, and the Eqs. (39), (41) and (51) have been used.The apparent horizon in quantum level is defined by �∗θ−θ+� = 0, which leads to2m√

V+ λ

3 V − Q2

V = 0. And then the equation can become U = 0 by noting Eq. (70).So the quantum apparent horizon and event horizon coincide.

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1192 B.-B. Wang

6 Conclusion and discussion

In this paper the mass function of a toroidal black hole is calculated by using Kuchar’sapproach. Its expression is similar to but differs from that of the spherically symmetricspacetimes. In the minisuperspace model we reobtain the classical solution of thetoroidal black hole from its Lagrangian. By using the dBB approach we obtain quantumtrajectories (70)–(72), and quantum potential (77) of the toroidal black hole. The dBBquantum behaviors of the toroidal black hole are very similar to those of sphericallysymmetric black holes [8–11]. For example, the quantum horizons of the two kinds ofblack holes are located at the null surface U = 0. And their quantum potentials havethe same expression as Eq. (77) although they have different expressions of Z . So thequantum potentials for those black holes are diverge in the vicinity of their horizons.

Now compare with the work of Gao and Shen [18]. We regard the vector potentialA as a canonical variable while Gao and Shen treat the vector potential (or electricalcharge) as a constant at starting. As a result, the wave function (76) is different fromthat in Gao’s work [17]. The difference is a phase factor, which has an essentiallyphysical meaning in quantum mechanics. For example, the phase factor of a wavefunction plays an important role in the well-known AB effect. In our work, the massfunction (51) is the function of canonical data (including the canonical momentum�A conjugated to the vector potential A). So we can give the quantum trajectory (72)although it is as the same as its classical counterpart. It is obvious that our treatment ismore complete. Of course our results about the quantum trajectories (70) and (71) arethe same as those of Gao and Shen. The reason seems to be that the quantum state ofthe black hole is exactly a eigenstate of the charge (and the mass, of course) with onlysingle eigenvalue (respectively) instead of a superposition state with some eigenstates.The difference of the two treatments should be found if some superposition state canbe obtained. Also in this reason, we can not explain the entropy of a black hole byusing the dBB approach up to now.

Acknowledgments This work is supported by the Science Paper Foundation of Beijing Jiaotong Univer-sity of China.

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de broglie–bohm interpretation for wave function of a toroidal black hole

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