International Journal of Theoretical Physics …
Volume 17, Number 3, pp. 197-204
ISSN 1525-4674
© 2015 Nova Science Publishers, Inc.
ZITTERBEWEGUNG NEAR TOY MODEL
OF BLACK HOLE
Natalia N. Konobeeva1,*, Alexandra A. Polunina1,†
and Mikhail B. Belonenko1,2,‡
1
2
Volgograd State University, Volgograd, Russia
Volgograd Institute of Business, Volgograd, Russia
Abstract
This chapter studies Zitterbewegung effect (ZB) near toy model of new type of black hole. We
obtain the analytical expression for the density of current, which induced by the motion of a
wave packet of electrons. We analyse the dependences of the current on the wave packet
width and the black hole`s parameters.
Keywords: Zitterbewegung; black hole; Dirac equation
PACS numbers: 04.25.-g., 04.62.+v
1. Introduction
In the last few years there has been a growing interest in the study of gravity models with
different dimensions. Considering different tendencies, we have to mention that, apart of the
solution to the Einstein equations with a negative cosmological constant (known as the BTZ
black hole) [1], a considerable attention has been attributed to the so-called Topologically
Massive Gravity (TMG). Another trend is associated with the New Massive gravity,
introduced by Bergshoeff, Hohm, and Townsend (BHT), dealing with the action of the
standard Einstein-Hilbert term with a specific combination of scalar curvature term and Ricci
tensor square one.+2-7 The latter, being linearized, is equivalent to the Fierz-Pauli action for
a massive spin-2 field.
At the same time Zitterbewegung has attracted the attention of scientists [8,9]. Initially a
lot of physicists considered this effect as a draw-back of the Dirac theory. It was connected
with the fact that the projection of the velocity of an electron can take only two values ±c (с is
the light velocity). Further, with the discovery of graphene which is described by the Dirac
*
E-mail address: yana_nn@inbox.ru
E-mail address: polunina_a@mail.ru
‡
E-mail address: mbelonenko@yandex.ru
†
198
Natalia N. Konobeeva, Alexandra A. Polunina and Mikhail B. Belonenko
equation in the long-wave approximation, the possibility of observing ZB leads to practical
applications [10]. Now, we have a point of view that Zitterbewegung effect exists in any
system if the Hamiltonian does not commute with the velocity operator [11]. This paper
presents the theory of ZB in the curved surface, for example new type of black holes. We
propose to use the Shrödinger representation, which allows us to consider the “trembling
motion” in the case of the curved space. Thus, purpose of the paper is show the possibility of
applying the Shrödinger picture for describing ZB effect near black hole. This approach can
be useful for the solving of the problems of the electron radiation near the black holes and
other astrophysical problems. It should be noted that since the estimates of the mass black
hole is very real costs (from 6.9 Msun to 8∙106Msun) [12] therefore a quantitative analysis is
rather difficult. We can say that expected influence of curved space on ZB has a qualitative
nature, which manifests itself in additional radiation from the fermions in the field of the
black hole, which has not been previously considered. This approach may also be applied to
the problem of neutrino oscillations near massive objects.
2. Basic Equations
Metric for the black hole when cosmological constant λ has the following form:
ds 2  f  r  dt 2 
dr 2
 r 2 d 2 .
f r 
(1)
where f(r)=Ar2+Br+C.
The parameter А is proportional to λ and is determined form the solution of Einstein`s
equations Rij-0.5R/gij=λ∙gij (with metric (1)), Rij is the Ricci tensor, R is the scalar curvature,
B is a kind of "gravitational hair", the constant C is associated with the mass of the black hole.
It should be noted, depending of the values of the parameters А, В, С metric (1) may describes
a black hole or not.
The roots of f(r) can be written as:
r 

1
B 
2A

B 2  4 AC .
(2)
In the case when r+>0 the metric represents a new type of black holes with an event
horizon near r+.
Further we consider the case of massless fermionic perturbations (m=0) for asymptotically
de Sitter black holes (A<0). Properties of fermions near the black hole will be described on the
basis of the Dirac equation generalized for the case of a curved space-time [13]:
  (      )  0 .
(3)
199
Zitterbewegung near Toy Model of Black Hole
Here and below the repeated indices assume summation; ∂μ is the partial derivative with
respect to coordinate μ, Ωμ is the component of the spin connection,      is the wave
 
 
function (column vector).
As is known [13], if we are given the metric tensor
ds 2  g  dx  dx 
g  g    
,
(4)

(   -delta Kronecker symbol) then we can define the field frames (tetrads):
g   ea eb  ab
g   ea eb  ab ,
 ab
bc

(5)
c
a
Then:
1
 b
    a  b ea g  (  eb  
e )
4
1


 g  ( g  ,   g ,  g  , )
2

  ea  a
Note that all the Christoffel symbols are equal to zero for metric (1), except:
 001 
f
ff  1
f
1
2
, 100 
, 11  
, 12
 , 122   fr .
2f
2
2f
r
And spin connection:
0  
 0 1
4
f , 1  0,  2 
 1 2
2
f .
From the Dirac equation we have that Hamiltonian is described by Eq. (6):
0
Hˆ  if 
 r
 r   if  if f
 
0   4
2r 2
0 1 f  0
 
 2 
  1 0  r  
The Heisenberg equations can be written as [12]:
 
.
0
(6)
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Natalia N. Konobeeva, Alexandra A. Polunina and Mikhail B. Belonenko
rˆ  i[ Hˆ , rˆ ] .
(7)
А standard approach of ZB (namely to Heisenberg picture) gives us some problem. It's
easy to see that solving of the ZB problem in a curved space-time is rather difficult. This is
due to the non closed system of operator equations follow from Eq. (7) in the curved space.
So, we apply the Shrödinger picture:
rˆ  rˆ 0 ,    (t) .
The current can be calculated as:
j   (t) v  (t)   (t) r  (t) ,
here v is the electron`s velocity. Because i  H
representation:
and we have in the eigenvalues
 (t)   C ei t  (t)  ,
0

we can obtain:
j  i  (t)  r , H   (t) .
Next we suppose there is a solution:  (r) 
(8)
 1 (r)
 2 (r)
Using the change of variables: y=1-r+/r- [14]. We obtain the Dirac equation in the
following form:
y  y  1  Q  1 y   0.5  2 y  1  Q  1  y  
.
 k 2Q
 2Q 2
iQ 
1
1 
 2  2

   1  y   0

 Ar
Ar y  y  1  Q  2 Ar  y  1  Q y  


(9)
Here we have defined Q=r+/r-, where r- corresponds to the cosmological horizon (r-> r+).
Performing the changes of variables according the work of P.A. Gonzalez and Y. Vasquez [15].
 1 y   y  y  1  Q  F  y  .

The equation (6) is reduced to
201
Zitterbewegung near Toy Model of Black Hole
y  y  1  Q  F   y   0.5   Q  11  4   2 y 1  2  2    F   y  
 k 2Q
2
   2       F  y   0
  r

iQ
   
Ar 1  Q 
.
(10)
Then, making the changes of variables: (1-Q)z=y, we obtain the hypergeometric
equation:
z 1  z  F   z    c  z 1  a  b   F   z   abF  z   0
a    
ik Q
,b     
A r
ik Q
, c  0.5  2
.
(11)
A r
The general solution of Eq. (11) is:
F  z   С1  2 F1  a, b, c; z   С2 z1c  2 F1  a  c  1, b  c  1, 2  c; z  ,
(12)
where 2F1(a,b,c;z) is a hypergeometric function, С1, С2 are constants.
Thus, the solution for the radial function can be written as:
 1  z   С1 z 1  z   2 F1  a, b, c; z   С2 z1/ 22  2 F1  a  c  1, b  c  1, 2  c; z  . (13)

The wave function is:
  z  
 (r ,  )   1
    .
 2  z  
 (r , t)   d e  
0
2
(14)
 (  )  
/ 2
 1    it
 1     it .
 (  )  C1 
 e  C2 
e
 2   
 2    
(15)
We choose wave packet in a Gaussian form in the energy representation. It does not
matter because the developed method uses a specific form of the packet. In our case the
commutator is:
0
r r   r r 

 rˆ , Hˆ   i 
.


0
 r r   r r

202
Natalia N. Konobeeva, Alexandra A. Polunina and Mikhail B. Belonenko
The result for the current is given by:
j(r , t)   d  e
 C1 C 
*
2
*
1
  0  /  2
2

2cos t  f  r  C1*C2 1*   2     C1*C2 1    2*    .
   2    C1 C  1      
*
2
(16)
*
2
3. Results
Evolution of the average of the current operator for the different values r is shown in Fig. 1.
Figure 1. Dependence of the current density (in relativity units) on time (A=-1, B=-3, C=4, Δ=0.5): a)
r=2; b) r=5; c) r=10.
Figure 2. Dependence of the current density (in relativity units) on time (A=-1, B=-3, C=4, r=2): a)
Δ=0.1; b) Δ=0.7.
Zitterbewegung near Toy Model of Black Hole
203
From this dependence, it can be concluded that we can observe ZB-effect near black hole.
The influence of the parameter r is obvious. At the r=10 we can observe the greatest
“trembling motion”, i.e. when r increases. It is easy to explain, if we take into account that r
corresponds to the radial distribution of the electrons density and with increasing of parameter
r the electron density is distributed more delocalized.
An influence of the packet width on the current density can be found in Fig. 2.
Note that the “damping” value of the trembling motion is directly related with the packet
width and there is an agreement with all the results on the ZB-effect. There isn't damping if
the packet have a "zero" width, the trembling motion continues indefinitely and “damping” is
absent.
An influence of parameter Q (Q=r+/r-)on the density of current is presented in Fig. 3.
Figure 3. Dependence of the current density (in relativity units) on time (r=2, Δ=0.5): a) Q=-4 (A=-1,
B=-3, C=4); b) Q=-6 (A=-1, B=-5, C=6); c) Q=-9 (A=-1, B=-8, C=9).
As can be seen from Fig.3, ZB effect decreases when the value of |Q| decreases too.ь
Note, the parameter Q is calculated directly through the constants in the metric tensor and is
associated with space-time geometry. It can be concluded that the effect ZB depends on the
geometry of the space-time.
Based on the results it can be concluded that the proposed approach is based on the
Shrödinger representation for evolution, allows us to consider the “trembling motion” in the
case of the curved space. The above example demonstrates the possibility of applying this
approach to astrophysical problems, in particular to problems of the electron radiation near
the black holes. More precisely, we can say that in our case the result can be applied to the
problem of neutrino oscillations near massive objects in the anti-de Sitter universe. Thus, this
effect can be a reason of extra electron radiation near the black holes and of a change in the
emitted spectrum of the electromagnetic waves. This research is not aimed at a detailed
examination of such influence and to find possible evidence in its favor. But from general
considerations it can be assumed this effect may amend the discussed spectra. Also, it should
be noted that given consideration of the ZB effect can be readily used in the investigation of
the neutrino dynamics in the field of the black holes.
204
Natalia N. Konobeeva, Alexandra A. Polunina and Mikhail B. Belonenko
The most controversial is the question of a real application of the metric of the form (1)
to astrophysical objects. It should be noted that the essence of the method developed by us is
to use the Schrödinger representation for consideration ZB effect and only requires
knowledge of the system of eigenfunctions of the corresponding Dirac equation. We choose
this model because of the fact that the corresponding system functions are known in analytic
form. Obviously, to talk about the real-world applications we should to consider the
Schwarzschild black hole, which existence has the evidence. In this case, there are some
difficulties with the analytical expression of the corresponding system functions, but there is
no fundamental limitations on the proposed method.
Acknowledgments
The authors are grateful to Dr. Shura Hayryan (Academia Sinica) for the inspiration and
valuable discussions.
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