CC BY
16
UDC 530.12, 539.12
Hawking Radiation in de Sitter Space: Calculation of the Reflection Coefficient for Quantum Particles
V. M. Red'kov*, E. M. Ovsiyukt, G. G. KryW
* Institute of Physics, NAS of Belarus 220072 Minsk, Republic of Belarus, Nezavisimosty avenue, 68
^ Mozyr State Pedagogical University, Belarus 28 Studencheskaya Street 2477660, Mozyr, Gomel region Belarus * Belarussian State University, Minsk, Belarus
Though the problem of Hawking radiation in de Sitter space-time, in particular details of penetration of a quantum mechanical particle through the de Sitter horizon, has been examined intensively there is still some vagueness in this subject. The present paper aims to clarify the situation. A known algorithm for calculation of the reflection coefficient REj on the background of the de Sitter space-time model is analyzed. It is shown that the determination of REj requires an additional constrain on quantum numbers sR/hc ^ j, where R is a curvature radius. When taking into account this condition, the value of REj turns out to be precisely zero. It is shown that the basic instructive definition for the calculation of the reflection coefficient in de Sitter model is grounded exclusively on the use of zero order approximation in the expansion of a particle wave function in a series on small parameter 1/R2, and it demonstrated that this recipe cannot be extended on accounting for contributions of higher order terms. So the result REj = 0 which has been obtained from examining zero-order term persists and cannot be improved.
It is claimed that the calculation of the reflection coefficient REj is not required at all because there is no barrier in the effective potential curve on the background of the de Sitter space-time.
Key words and phrases: Hawking radiation, de Sitter space-time, reflection coefficient.
1. Introduction
The problem of Hawking radiation [1], and in particular the radiation in de Sitter space-time [2] and details of penetration of quantum mechanical particles through the de Sitter horizon were examined in the literature [3-10]. Till now remains some vagueness in this point, and the present paper aims to clarify the situation.
In the paper, exact wave solutions for a particle with spin 0 in the static coordinates of the de Sitter space-time model are examined in detail, and the procedure for calculating of the reflection coefficient Rej is analyzed. First, for scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient Rej on the background of the de Sitter space-time model is analyzed. It is shown that the determination of Rej requires an additional constrain on quantum numbers eR/hc ^ j, where R is a curvature radius. When taking into account for this condition, the value of Rsj turns out to be precisely zero.
It is claimed that the calculation of the reflection coefficient Rej is not required at all because there is no barrier in the effective potential curve on the background of the de Sitter space-time.
Received 27th June, 2012.
Authors are grateful to participant of seminar of Laboratory of theoretical physics, Institute of Physics of National Academy of Sciences of Belarus for stimulating discussion.
We wish to thank the Organizers of the XLVIII All-Russia conference on problems in Particle Physics, Plasma Physics, Condensed Matter, and Optoelectronics Russia, Moscow, 15-18 May 2012, dedicated to the 100-th anniversary of Professor Ya.P. Terletsky, for opportunity to give a talk on the subject.
The same conclusion holds for arbitrary particles with higher spins, it was demonstrated explicitly with the help of the exact solutions for electromagnetic and Dirac fields in [11].
The structure of the paper is as follows. In Section 2 we state the problem; some more details concerning the approach used could be found in [11].
In Section 3 we demonstrate that the basic instructive definition for the calculation of the reflection coefficient in de Sitter model is grounded exclusively on the use of zero order approximation Ф(0)(г) in the expansion of a particle wave function in a series of the form
Ф(г) = ф(0)(г) + (Ф(1)(г) + (Ф(2)(г) + ... (1)
What is even more important and we will demonstrate it explicitly, this recipe cannot be extended on accounting for contributions of higher order terms. So the result Rej = 0 which will be obtained below from examining zero-order term Ф(0)(г) persists and cannot be improved.
2. Reflection Coefficient
Wave equation for a spin 0 particle (M is used instead of McR/h, R is the curvature radius) reads
dp + 2 + M 2) Ф(ж) = 0, (2)
and is considered in static coordinates
Ar2
dS2 = Фd^2 - — - r2(d$2 + sin2 $d(^2), 0 < r< 1, Ф = 1 - r2. (3) For spherical solutions Ф(ж) = e-l£tf (r)Yjm(*&,ф), s = ER/hc, the differential equa-
tion for ( ) is
dV + / 2 + Ф \ d/ + / e^ - M2 + 2 - j(J + 1) ^ f = 0.
dr2 \r Ф J dr \Ф2 Ф Фг2 J .
(4)
All solutions are constructed in terms of hypergeometric functions (let r2 = z): regular at = 0 standing waves are given as
f( z) = zj/2(1 - z)-i£/2F (a,b, c;z), к = j/2, a = -is/2, c = j + 3/2,
3/2 + j + iJM2 - 1/4 -is , 3/2 +j-i JM2 - 1/4 -is (5)
a =-, b=-;
2 ' 2 '
singular at = 0 standing waves are
g(z) = z-(i+1)/2(1 - z)-ie/2F(a, /3, 7; z), к = -(j + 1)/2, a = -i e/2, c= - j + 1/2,
1/2 -j + i^M2 - 1/4 -is 1/2 -j- i^M2 - 1/4 -is
With the use of the Kummer's relations, one can expand the standing waves into linear combinations of the running waves
_ г(с)г(с - a - b) T out( + r(c)r(a + b - c) f in 1 (2)_Г(С - а)Г(с - b) Iun (Z) + Г(а)Г(&) u ™n(z), (7)
Similarly for g(z)
Г(7)Г(7 - а - p)T, , Г(7)Г(а + ^ - .
U0Un(r - 0) - , ^(г - 1) - (1 - г2)—/2, i 1
и±Пп(г - 0) - , Щг - 1) - (1 - г2)+^/2,
f(r - 0) - г3, g(r - 0) - ,
f (г - 1) - 2Re д(г - 1) - 2Re
С) 2+iER/hcexp(-iEr* /he) Г(7)Г((а)+(^)-7) 2+^exp(-Er*/he)
(8)
U™£(z) _ ^'/2(1 - z)-i£/2F(a, b,a + b - с + 1; 1 - z), U^z) _ zj/2(1 - z)+i£/2F (с -а, с - b,c -a - 6 + 1; 1 - z), a* _ (c - a), b* _ (c - b), (a + b - c)* _ -(a + b - c),
Poun(*)]* _ urnn(z),
ti7) _ 2Re г( с)г(с -a - Ь) T ( _ 2 Re г(с)г(а + b - c) f ( (9)
f(z) _ 2Re г( c - а)Г(с - b)Uout(z) _ 2 Re Г(а)Г(&) Uin(z)- (9)
v(z) _ r( )Г(—ж^™ + г( )г(;) ^ГПп(^), (10) г(7 - а)г(1 - fv r(«)r(/3)
U?U£(z) _ ^'/2(1 - z)-i£/2F(a +1 -^,/3 + 1 + p + 1 - 7; 1 - z),
Uinn(z) _ zj/2(1 - z)+i£/2F(1 - a, 1 -P,^ + 1 - a - p; 1 - z), (11)
Ф) _ 2Re ЙТ>r<T):,a - % _ 2Re ^^ (12)
Г(7 - «)Г(7 - fv Г(а)Г(/3)
Asymptotic behavior of the running waves is given by the relations
(13)
or in new radial variable r* E [0, то):
* R 1 + r exp(2r*/R) - 1
r _ — ln-, r _----—--,
2 1 -r' exp(2r */ R) + 1,
U™£(r* -то) - (2-iER/h^ exp(+«Er*/he), (14)
^(r* - то) - (2+iER/h^ exp(-Er*/hс), e _ ER/hc). For the standing waves we have
1
(15)
On can perform the transition to the limit of the flat space-time in accordance with the following rules:
p +1 -isR + г JR2M2 - 1/4
a =-2-,
, p +1 -i sR -г JR2M2 - 1/4 , b=--, p = 3 + 1/2;
(16)
/т^2 n , \ ab z a(a +1)6(6+1) z2
дИт( Д2 ^ = Д2, F (a,b, c] z) = 1 + -1+ ( C(g)^(1) )^T + ..., (17)
and further (R ^ то)
a + n 1 p+1 L 1 \ n -is- iM
"й" = 2 £ + lM 2 - 4R2 i + R ,
b + n 1 p+1 . . / 2 1 \ n -ie + iM -R- = 2 ~R"-*£-iM 2 - 4r2J + л .
Let e2 - M2 = k2, then we arrive at
^ / k2R2/4)n
lim F(a,b, c;z) = T(1+p)Y, (г(1 + /+) ),
R^m ^ n!l(1 + n + p)
^ (-k2R2 /4)n
lim F (b - c+1,a - c+1, -c + 2;z) = Г(1 - p) V -/-Ц-.
R^m v 'о п!Г(1 + n - p)
Allowing for the know expansion for Bessel functions
(18)
x\p -m (ix/2)
<*> = (D' E n
2n
>2/ ^п!Г(1 + п + p)' we get (wave amplitude A will be determined below) lim A^OUnVz) = lim A -1 x
Д^то Д^то ^/r
_T(-isR + 1)Г(-р)Г(1 + p)(2/k)pR-p+1/2_
Г[ 1 (+гу'Л2М2 - 1/4 - ieR + p + 1)]Г[2(-i^R2M2 - 1/4 - ieR + p + 1)]
+_Г(-геД + 1)Г(+р)Г(1 - p)(2/fc)-^^+^+1/2_j '
Г[2(+i^R2M2 - 1/4 - ieR - p + 1)]Г[2(-i^R2M2 - 1/4 - ieR - p + 1)]
■(19)
Performing limiting procedure (see the detail in [11]) we derive the relation
Jp(kr) +
Ит ^outW ^ H^r),
(20)
R^ where
H<lll/2(^) = -np-) [e^Jp(x) -J-p(x)]
stands for Hankel spherical functions.
In connection with the limiting procedure, let us pose a question: when the relation (20) gives us with a good approximation provided the curvature radius R is finite. This
point is important, because when calculating the reflection coefficient in the de Sitter space just this approximation (20) was used [3-6].
To clarify this point, let us compare the radial equation in Minkowski model
d2 d 2
^ + + e2 -M2 - Щ1
2A d
/0 = о,
and the appropriate equation in de Sitter model
d2 2(1 - 2r2/R2) d_
+ /-i О / т^П \ 1 +
M2 + 2 j(j + 1)
d 2
r(1 - r2/R2) dr (1 - r2/R2)2 1 - r2/R2 r2
(21)
fej = 0. (22)
At the region far from the horizon r ^ R, the last equation reduces to
+ 1A + 2-M 2 - ^ + 1) + 2 i(i +1)
dR2 + RdR +c R2 R2
I
ej
0.
(23)
So, we immediately conclude that eq. (21) coincides with (23) only for solutions with quantum numbers obeying the following restriction
-M2 .
R2
(24)
In usual units, this inequality reads
h \ 2 \2 E = v— с2, X = —, R - 10-80, v2 - 1 » R Hd + 1) + 2]. (25)
Instead, in massless case we have r2(xi2
» [ ( + 1) + 2] or
R24TT2
X2
» Ш + 1) + 2].
(26)
So we can state that the above relation (20) is a good approximation at a finite R only for quantum numbers obeying (25)-(26).
One additional point should be emphasized, the radial equation in de Sitter space can be transformed to the form of the Schrodinger like equation with an effective barrierless potential. Indeed, in the variable r* eq. (22) reduces to
d
2
U(r*) =
d *2 1- r2
+ £ 2 -U(Г*)
G(r *) = 0,
R2
4(1 - r) + -^— + m2R2 +
1 + 2
(27)
It is easily verified that this potential corresponds to attractive force in all space
r dr *
d U 1 2
R2
+
2
) + m2R2 + Г
1+
+
+4(1 - )) + (1 - 2)
2, f2j(j + 1)
V"
+4-
(1 + )2
> 0.
At the horizon, г* ^ то, the potential function U(r*) tends to zero, so G(r*) exp(±« e r*).
2
2
1
3
r^j
The form of the effective Shcrodinger equation modeling a particle in the de Sitter space indicates that the problem of calculation of the reflection coefficient in the system should not be even stated. However, in a number of publications such a problem has been treated and solved. So we should reconsider these calculations and results obtained. Significant steps of our approach are given below:
The existing in literature calculations of non-zero reflection coefficients Rej were based on the usage of the approximate formula for ^*r°Un(R) for the region far from horizon (19). However, as noted above, the formula used is a good approximation only for solutions specified by (24), that is when j ^ e R.
It can be shown that the formula existing in the literature gives a trivial result when taking into account this restriction. The scheme of calculation (more detail see in [11]) is described below.
The first step consists in the use of the asymptotic formula for the Bessel functions: when x ^ v2 we have
Г(2г/ + 1)2-2^-1/2 J(X) ~ Г(и + 1)Г(и+1/2) V X
exp ( +i(x - Ж(v + + exp ^-i(x - Ж(v + 1))
2
2
2
2
so when j < j2 ^ eR we derive
o+ieR ( Ж 1 Ж 1 \
— I Ci exp(-2 (p + 2)) + °2 exp(-^^ + 2 V +
-,-isR
+
( Ж 1 Ж 1
f Ci exp(+i-(p + -)) + C2 exp(+i-(-p
R 1 2 2 2 2
where C1 and C2 are given by
Г( a + 6 + 1 - с)Г(1 - с) 2—-1Г(2р + 1)
(28)
Ci = Ci =
Г( b - c+ 1)Г( a - с + 1) (eR)iГ(р + 1/2)' Г( a + 6 + 1 - с)Г(с - 1) (eR)J'+1r(-2p + 1)
(29)
Г( а)Г( 6)
2-i Г(р+1/2)
The reflection coefficient Rsj is determined by the coefficients at e-teR/sR and e+teR/eR. It is the matter of simple calculation to verify that when sR ^ j, the coefficient Rej is precisely zero
eR > j, Rej = 0.
This conclusion is consistent with the analysis performed above.
x
3. Series Expansion on a Parameter R 2 of the Exact Solutions and Calculation of the Reflection Coefficient
In Section 3 we demonstrate that the basic instructive definition for the calculation of the reflection coefficient in de Sitter model, being grounded exclusively on the use of zero order approximation Ф(0)(г) in the expansion of a particle wave function in a series of the form (1), cannot be extended on accounting for contributions of higher order terms.
Let us start with the solution, the wave running to the horizon
Uout(z) = Г(а + b - c+ 1)[aF(z) + pG(z)], n = Г(1 - с) = Г(с - 1) (30)
a Г(& - с + 1)Г(а - c+1), P Г(а)Г(6),
where
F(z) = z(p-1/2)/2(1 - z)-ie/2F(a, b, c; z), с = j + 3/2 = 1 + p,
1+ p -i e + iJm2 - 1/4 1+p -is - iJm2 - 1/4 (31)
a =-, b=-,
2 ' 2 '
and
G(z) = z(-p-1/2)/2(1 - z)+is/2F(a - с + 1,b - c+1, 2 - c; z), 2 - с =1 -p,
1 -p-i e + i V—2 - 1/4 1 - p -ie-i ^Jm2 - 1/4 (32) a - с + 1 =-2-, v - с + 1 =-2-.
For the following we need the expressions for all quantities in usual units. It is convenient to change slightly the designation: now R stands for the curvature radius
r2 ER R ^2 McR R h
z = -, e = ^ = vx, E = vMc, m = — = X, X = M~c (33)
Now, the relations (30)-(32) read
rP f r2 \ r2
F(z) = R-p+1/2 ^ - F(а, b, C; R), с = 1 + p.
1 Л . R . R2 Л L 1 Л . R . [r2 Л (34)
a =2^ + p-+ *\IX2 - ij , --г\IX2 - y;;
and
G(Z) = RP+1/2^ - ^J F(a - C+ 1,6 - с + 1 2 - с; R),
^ - p - if!-- + i\J
(л • R • /
I 1 - p - «fy -гу
о 1 1 1 Л . R . —2 1\
2 с = 1 -p, a - c+1 = 2l1 -Р - ^X + г yX2 - 4/ , (35)
1 R R2 1
б-С+1 = 2 1 - Р - -г ^ - 4
The task consists in obtaining the approximate expressions for F( ) and G( ) in the region far from horizon, r ^ R; first let us preserve the leading and next to leading terms (we have a natural small parameter X/R).
First, let us consider the exponential factors
1 - -2 =exp
±2x ln(1 - -)
cos
fR r2
2л ln(1 - -)
sin
fR r2
2A ln(1 - -)
Using the expansion for the logarithmic function
, ж2 x3 . , r\ /г2 1 r4 1 r6 \
ln(1 -Ж) = -(Ж + T + T + ...), ln(1 - R2H R + 2R + 1 R + ...)
we get (assuming that r2 ^ XR)
2 \ ±i/J.R/2\
/ r^ ±1^п/2Л vR( r2 1 r4 1 r6 \
i1 - w) =cos~2X yR2 + 2R + 3R6 + ..J *
^R / r2 1 r4 1 r6 \ / p2r4 \ ^R / r2 1 r4 \
sin2A {& + 2 R4 + 3 R6 + ..j " ^ - 8X2R2) ** 2X\-R2 +2 rJ . (36)
Terms in eq. (36) can be written in descending order
j i^R
( v2 \ Г2 n2 y4 y4 X т2
I1 - R2) и 1 *n2XR- Y4XR4XR, X = R « 1, 2XR < 1 (37)
Now we turn to the hypergeometric function F(a,b, c;z) from (34). Because R ~ 1030, X 10 12, one can use the approximation of a leading and two next order terms in the expressions for the following parameters
1/ R R I XM 1+p n-1R X
a = 2 V+v - + l XV1 - №) = — -l~ X - lT6R,
, 1/ R R I XM 1+p n+1R X
b=«\ 1+P - г Пт -гТ V1 - = -JT" Т+г-
(38)
2 X X 4 R2 2 2 X 16 R
Then, the hypergeometric function is given as
2 1 a 2 1 1 a( a + 1) ( + 1) 2 2
F (a, r) = 1 + r ~г 2 + 2! R4 с(с+(1) > 2)2+
+ 12. a(a + 1)(a + 2)Ь(Ь + 1)(6 + 2) 2)з + + + 3! R6 с(с + 1)(с + 2) ) + ...+
+ 1^L a(a + 1)(a + 2)...(a + n - 1)Ь(Ь + 1)(Ь + 2)..(Ь + n - 1) ( 2)П + ( )
+ n\R2n с(с+1)(с + 2)...(с + п - 1) (Г ) + ... (39)
It is convenient to introduce a shortening notation for small quantity X = X/R), then a typical term is represented as
1 (a + n)(6 + n)_ 1 -г(ц - 1) Л .1+ p + 2n X2 ^ ^
1 + 1 Z X +
R2 (c + n) p +1 + n 2X V П - 1 8(n - 1) - (П +1) f1 + .1+ P + 2n x - X2
2 X n + 1 8( n + 1)
1 f n2 - , 2ifi_ n t ^ ^ (1 + p + 2n)2 - 1/4
n2 - 1
p + 1 + n
(-^)
1 + -^r (1 + p + 2 n)X -^J^pi^Zx2
n 2 - 1 n 2 - 1
Below we will use the notation k2 = —r-—, then
X2
1 ( a + n)(b + n) ^ 1 R2 (c + n) ~ p + 1 + n V 4
2i ff
(" t)
1+
f 2 - 1
d+r+^x - -1/4 x ^
f 2 - 1
Thus, we have the following approximate expressions for the first few terms in the series
2 a 1
R2 с p + 1
(-^)
1+
2 f
f2 - 1
(1+p)x -
(1+p)2 - 1/4 f2 - 1
x2
r2_ ( a + 1)( 6+ 1) _ 1 R2 ^T^ ~
(-^)
P + 2\ 4 2i f!
1+
f2 - 1
(1+P + 2 x 1)X -
(1+P + 2 x 1)2 - 1/4 f2 - 1
x2
r^ ( a + 2)(6 + 2) _ 1 ( k2r — (c + 2)
(-^)
P + 3\ 4 2i f
1+
f2 - 1
(1+„ + 2 x 2)X - (1 + f - 1/4X2
r2 ( a + 3)( 6 + 3) _ 1 ( k2r — (c + 3) ~ p + 4
(-^)
1 + J!^(1 + „ + 2 x 3)X - <1+', + 2г xf - 1/4 X2
f2 - 1 f2 - 1
r2 ( a + 4)(b + 4) ^ 1 f k2r — (c + 4) ~ p + 5
(-^)
1 + ^(1 + p + 2 x 4)X - <1+', + 2г x 4)2 - 1/4X2
f2 - 1 f2 - 1
r2 ( a + n)(b + n) — (c + n)
1
(-^)
p + 1 + n \ 4 2 f
1+
f2 - 1
(1+, + 2n)X - <1+''fз2_n>; - 1/4 X2
Now we are ready to write down the expressions for the coefficients of the hyper-geometric series preserving only leading and two next order terms
1 r2 ab 1
1! R2 с p+1
(-^ )
1 + 4^(1+ p)X - (1+ ^ - 1/4X2"
f2 - 1
f2 - 1
X
X
X
X
X
X
X
X
1 (r2)2 ab(a + 1)(6 + 1) _ 2!(R2 )2 c( c+1) -
- (-, -L , ол (1 + 7^[(1 + (1 + P + 2 X 1)]X-
4 ) 2!(p+1)(p + 2) \ n2 - 1
„ Г 4 n2
-X
V (1+ ,)(1+, + 2 X 1)+<1+ - 1/4 + <1+" + 2 x 1)2 - 1/4
( n - 1)2 n - 1 n - 1
1 (r2)3 a6(a + 1)(& + 1)(a + 2)(6 + 2) f k2r2\3 1
/ kVV V 4 ) 3\(
3\(R2)3 c(c+1)(с+ 2) V 4 у 3\(p+1)(p + 2)(p + 3)
n
4 n2
x (1 + X^^-[(1+ p) + (1 + p + 2 x 1) + (1 + p + 2 x 2)]-
n2 - 1
X2
L(n2 -1)2
[(1+ p) + (1+p + 2 x 1)](1 + p + 2 x 2)+
(1 +p)2 - 1/4 (1 + p + 2 x 1)2 - 1/4 (1 + p + 2 x 2)2 - 1/4
n 2 - 1 n 2 - 1 n 2 - 1
1 (r2)4 ab (a + !)(& + 1)(a + 2)(b + 2)(a + 3)(b + 3) _ 4! (R2)4 c(c+ 1)(c + 2)(c + 3) -
k2r2 \4 1
/_fcV\4_ V О 4!(
X
4 У 4!(p + 1)(p + 2)(p + 3)(p + 4)
2 n V-11
4 n2
x <| 1 + X-^T[(1+p) + (1+p + 2 x 1) + (1+ p + 2 x 2) + (1 + p + 2 x 3)]-
- [(1 + p) + (1 + p + 2 x 1) + (1 + p + 2 x 2)](1 + p + 2 x 3)+
kn2 -1)2
(1+ p)2 - 1/4 (1+p + 2 x 1)2 - 1/4 (1+p + 2 x 2)2 - 1/4
+ n2 - 1 + n2 - 1 + n2 - 1 +
(1 + p + 2 x 3)2 - 1/4
+ n2-!
x
x
1 (r2 )n ab (a +1)(b+1)....(a + n - 1)(6 + n - 1) n! (R2)" c(c+1)...(c + n)
k2r2\n 1
4 J n!(p+1)(p + 2)...(p + n)
x j 1 + X^^- [(1+ p) + (1+ p + 2 x 1) + (1 + p + 2 x 2) + ...
n2 - 1
+(1 + + 2 x ( n - 1))] - X
4 n
2
-[(1+p) + (1+ p + 2 x 1) + ..
.(n2 -1)2
+ (1 + p + 2 x ( n - 2))](1 + p + 2 x ( n - 1))+ (1 + p)2 - 1/4 (1 +p + 2 x 1)2 - 1/4 (1+p + 2 x (n - 1))2 - 1/4"
n 2 - 1 n 2 - 1 n 2 - 1
x
Thus, initial exact hypergeometric function can be approximated by the sum of three series
2
F = F(a, b, c; —) = Fo(r) + XF^r) + X^(r). (40)
The leading series F0(r), in fact, reduces to the Bessel function
- + (i kr/2)2 (ikr/2)4 + + (ikr/2)2n + =
0( ) + n!(p + 1) + 2!(p + 1)(p + 2) + ... + n!(p+1)(p + 2)...(p + n) + ...
= ^+ ч E n^h = Г(1(ki Ум* m
where
•w = (I)' E nT^ - = k„
The second series is given by
XF1 (r) = X^^- |(-k2r2/4^^Tr + (-k2r2/4)21^ ' ^J ,' 0V ^ +
^-k2r2/4) ^ + (— k2f2 /4)2 [(1+ P) + (1+P+2 x 1)] f2 - Ц( kr/4)p + 1+( kr/4) 2!(p+1)(p + 2)
+ (_k2r2 /4)3 [(1+p) + (1+ P + 2 x 1) + (1 + p + 2 x 2)] +
+ ( / ) 3!(p+1)(p + 2)(p + 3) +
+ (_k2 2 /4)4 [(1+p) + (1+p + 2 x 1) + (1+ p + 2 x 2) + (1 + p + 2 x 3)] +
+ (kr/) 4!(p+1)(p + 2)(p + 3)(p + 4) +..
(- k2r2/4)n [(1 + P) + (1 +P + 2 x 1) + ... + (1 +p + 2 x (n - 1))] j ;
that is
n!(p + 1)(p + 2)...(p + n) У (42)
YP(r) X 2if (-k2r2\ (+(-k2r2\ [(1+ p) + (1+p + 2 x 1)] +
XF-1 (0 = Xf2-~1{—) j1 4~J -2!(p+1)(p + 2)-+
/ - k2r2 \
V 4 ) 3!(p+1)(p + 2)(p + 3)
Hr)
- k2r2^2 [(1+p) + (1+ p + 2 x 1) + (1 + p + 2 x 2)] + '-k2r2\3 [(1 +p) + (1 + p + 2 x 1) + (1 + p + 2 x 2) + (1 +p + 2 x 3)] +
4 J 4!(p+1)(p + 2)(p + 3)(p + 4)
+
+ /-kV\ n [(1+ p) + (1+p + 2 x 1) + ... + (1 + p + 2 xn)] ... + V 4 ) ( n + 1)!(p+1)(p + 2)...(p + n + 1)
or shorter
XF1 (r) = Xf2f_Г(р+1)(x
- /- k2 r2\ n [(1+p) + ... + (1 + p + 2 xn)] -M 4 J (n + 1)!Г(р + 2 + n) . ( )
n=0
Using known sums
[(1+p) + ... + (1 + p + 2 x n) = (1 +p)(n +1) +2(1+ 2 + 3 + ...n) =
n) n
= (p + 1)(1 + n) + 2V^ 2 ' = (n + 1)(n + 1+p)
we derive
/k2r2\ У
XF1 (r) = X^^~Г(р+1)( У^(-k2r2/4) ,
1 () j2 - Л 4 1 ) (n + 1)!Г(р + 2 + n)
2ij / - k2r2 \ ^ (- k2r2/4)n
2in tV„ , n ^(-k2T2\ ,9 2/4)n_(n + 1)(n +1+ P)
r /4) (,
у Jdk. t=o n!r(P
/ - k2r2 \ 2ij w 1ЛДг
= X-^- r(p + 1)
(-k2r2\ ^ _±
V 4 / ^=0 n!r
n2 - 1 ^ y V 4 ) ^ nW(p + 1 + n)
\ / n=0 « '
^) = X(nPTrF0M.
Thus, the approximation (40) can be presented as follows F = F0(r) + XF^r) + X2F2(r) = FF0(r) + xj-2-^ (-j2^ i^0(r) + X2F2(r). (44)
Similar relations can be derived for the hypergeometric series G(r):
2
G = F(a - с+1,6 - c+1, 2 - c; r) = G0(f) + XG 1(f)+ X2G2(f). (45)
The leading term again reduces to the Bessel function
/kr \+p
G 0(г) = Г(1 - p)[ yJ J-P(kr). (46)
Next order term is given by
XG1(r) = X (^^(r). (47)
So, the approximation (45) is presented as
j2 - 1 V 4
Now let us consider the whole function
G = G0(r)+X(51(r)+X2^2(r) = ^0(r)+X (-"Т") ^0(r)+X2G2(r). (48)
, ^ / r2 \ / JN
F = +1/2 ^ ^ - -2J F[a,b, c; -2) =
= P-P+1/2 Г
R
2 2 4 4
r2 j2 r4 r4
1 + %iJn\ n —7Г л\2т>2 + %n-X-
2 X R 2 4 X2 R2 4 X2 R2
F0(r) + -*JL( -^)XF0(,) + X ^(r)
= R-+1/2^ F^0(r) + ^ (XF0(r) + X2F2(r)+
X
X
2 2 2 n k2 2 2 t2XRF0(r) + *j2xsj?-H") XF0(r) + *n2xrx
j2 r4 p (r) j2 r4 2ij (-k2r2\ XP(r)_ j2 r4 2 4 X2R2 0(') 2 4X2R2 J2 - 4 J 0(') 2 4X2R2
4 4 2 n k2 2 4 +Jx7t^2F0 (r) + Xp0(r) + ijX—-2^2X 2^2(r)
, rP
= R-p+1/2 —
2
Fp0(r) - ij+ X2F2(r)+ +
2 2 2 2 + >JJ 2XR^o(^•> + W 2XR F0W + yJ 2XRX
-т<Ixr+ ¥(2xr} F0W - £ (ш) X2F2M+
2 2 2 2 2
+2xr) + J2(2XR42xr) x3F'U
Preserving only first two terms we have
1 2 2 X2
, r+p
F (z) = R-p+1/2 —■
Similarly, for G( ) we obtain G(z) = R+p+1/2 —-
Fp0(0 + 8J2 X2R2F0(r) + R^to
1 2 2 X2 G0(r) + 8J2 X2R2^0(r) + R2^2(r)
(49)
(50)
One should emphasize one feature of the expansions (49) and (50): these approximations are real valued as we must expect remembering on relations (9) and (12).
In the known method of determining and calculating the reflection coefficients in de Sitter model [3-6], authors used the only leading terms in approximations (49) and (50), because only these terms allows to separate elementary solutions of the form e±ikr - see (28).
Let us perform some additional calculations to clarify the problem. The functions F( ) and F( ) enter the expression for (to horizon) running wave
U°ut(z) = r(a + b - с + 1)[aF(z) + $G(z)], n = Г(1 - с) = Г(с - 1) (51)
a Г(& - с + 1)r(a - c+1), P r(a)r(6),
where (introducing a very large parameter Y = R/2X)
Г(-Р)
a
Г[-(j - 1)Y + (1 - р)/2)]ГН(J + 1)Y + (1 - p)/2)]'
в «__ (52)
P Г[-i(j - 1)Y +(1+р)/2)]Г[-(J + 1)Y + (1+p)/2)]. ( )
For all physically reasonable values of quantum numbers j (not very high ones) and values of n (different from the critical value n = 1 and not too high ones - they
correspond in fact to energies of a particle in units of the rest energy) the argument of Г-functions in (52) are complex-valued with very large imaginary parts.
Let us multiply the given solution (51) by a special factor A which permits us to distinguish small and large parts in this expansion:
A = Г(д - р/2 + 1/4)Г(Ъ - p/2 + 1/4)
A = Г(а + b - c+1) • (53)
So, instead of (51), we get
AU out(z) = a'F (z) + /3'G(z), , = г(1_ )Г(о - р/2 + 1/4)Г(Ь - p/2 + 1/4) a 1(1 C) Г(Ь - с+1)Г(а - c+1) , (54)
0 _г(„ 1)Г(а - р/2 + 1/4)Г(Ь - р/ + 1/4)
0 =Г(С -1) тЩь) •
Instead of (52), the expressions for a' and 0' are
Г[-(p - 1)Y + 1/4] Г[-(p + 1)Y + 1/4]
a * Г(-p)
0'« r(+p)
Г[-(p - 1) Y + (1 - p)/2)] Г[-(p + 1) Y + (1 - p)/2)] ' Г[-(p - 1)Y +1/4] Г[-(p + 1)Y +1/4]
a
0' « +
Пг+I = ^ (1 + 1(A-B>(A + B + 1> +•) .ia^n.HH»; (58)
then (remembering that Y = R/2X) Г[-i(p - 1) Y +1/4] ^ Г[-i(p - 1)Y +(1 - p)/2)] ~
1 , 2 Л (2p - 1)(7 - 2p) + -i(p - 1)R 32
(55)
Г[-(p - 1) Y + (1 + p)/2)\ Г[-(p + 1) Y + (1 + p)/2)]' Allowing for identities
Г(-Р) = - -г^- , Г(Р) = +-тП- —^, (56)
sin pn 1(1 + p) sin pn 1(1 - p)
a and 0' (55) are transformed into
n 1 Г[-(p - 1)Y + 1/4] Г[-(p + 1)Y +1/4]
sin pn Г(1 + p) Г[-(p - 1)Y + (1 - p)/2)] Г[-(p + 1)Y + (1 - p)/2)] n 1 Г[-(p - 1)Y + 1/4] Г[-(p + 1)Y + 1/4]
(57)
sin pn Г(1 - p) Г[- (p - 1)Y + (1 + p)/2)] Г[- (p + 1)Y + (1 + p)/2)]' Now we have to take into account the asymptotic formula for Г-function
^ - 1)r1 P/2-1/4
L 2 Л J
Г[-(p + 1)Y +1/4] Г[-i(p + 1)Y +(1 - p)/2)\ *
,(p + 1)
2 Л
R
V/2-1/4
1+
2 Л (2p - 1)(7 - 2p)
-i (p + 1)R
32
(59)
and
Г[-(p - 1) Y +1/4] Г[-(p - 1)Y +(1+ p)/2)]
- (p - '4 -p/2-1/4
2 Л
1+
2 Л (-2p - 1)(7 + 2p)
-i(p - 1)R
32
Г[-(p + 1)Y +1/4] Г[-i (p + 1)Y +(1+ p)/2)]
- (p + 1>r1 -p/2-1/4
. 2 Л
1+
2 Л (-2p - 1)(7 + 2p)
-i (p + 1)R
Substituting (59) and (60) into (57), we obtain
32
(60)
a i=s —
R2)
p/2-1/4 2 \ (-1)P/2-1/4_
K
1
sinpn Г(1 + p)
1+
2 Л (2p - 1)(7 - 2p)
P'
+(^ R2)
-i(p - 1)R
-p/2-1/4
32
1+
2 Л (2p - 1)(7 - 2p)
1+
4 Л2 2 Л
(-1)-p/2-1/4
(-2p - 1)(7 + 2p)
■
- ( p + 1) R
1
32
-(p - 1)R
32
1+
sin pn Г(1 - p)
2 Л (-2p - 1)(7 + 2p)
- ( p + 1) R
32
(61)
Preserving the only terms of first two orders we have
к
1
+P-1/2
a'
sin pn Г(1 + p) \2
(2)
X R+P-1/2(-1)P/2-1/4 ^ 1 +
,(2p - 1)(7 - 2p)
г 8 p2 - 1 R,
P' «
n
1
(2)
-P-1/2
X
sin pn Г(1 - p) \2/
X R-P-1/2(-1)-P/2-1/4 f 1 - ,(-2p - 1)(7 + 2p) 2p *) . (62)
8 p2 - 1 R
X
X
X
Substituting expressions (62) into the following expansion
AU out(z) = a'F (z) + P'G(z),
k\ i ' 'p(kr), (63)
F(r) = R-P+1/2r(1 + p) (^ P _L Jp(kr) G(r) « Rp+1/2Г(1 - p)(kjP -1=J-P(kr),
we arrive at the following zero-order approximation
п 1 / к
sinрп Г(1 + p) \2
п 1 П\+р-1/2 AUout(z) = ^ut(r) = - . ^ fkj x
X R+P-l/2(-1)P/2-l/4R-P+l/2r(1 + p) ^^ P -L.Jp(kr) +
m-o (2 p'V*-^-1)-"2-^^^) (2)' -jи,
that is AU out(z) = ^°ut(r) =
п ^ [-(-1)p/2-l/4jp(kr) + (-1)-p/2-1/4J-p(kr)
sin п
= -(-1)-р/2-1/^/2к^ к-1)"^) - J-P(kr)]; (64)
which coincides with a spherical wave propagating in Minkowski space from the origin, expressed through the Hankel functions of the first kind
Hp ) = ^ [(-1)PJ^) - J-рк)] . (65)
4. Conclusion
The last but not the least mathematical remark should be given. All known quantum mechanical problems with potentials containing one barrier reduce to a second order differential equation with four singular points, the equation of Heun class. In particular, the most popular cosmological problem of that type is a particle in the Schwarzschild space-time background and it reduces to the Heun differential equation. Quantum mechanical problems of tunneling type are never linked to differential equation of hypergeometric type, equation with three singular points; but in the case of de Sitter model the wave equations for different fields, of spin 0, 1/2, and 1, after separation of variables are reduced to the second order differential equation with three singular points, and there exists no ground to search in these systems problems of tunneling class.
References
1. Hawking S. W. Particle Creation by Black Holes // Commun. Math. Phys. — 1975. — Vol. 43. — Pp. 199-220.
2. Hawking S. W., Gibbons G. W. Cosmological Event Horizons, Thermodynamics, and Particle Creation // Phys. Rev. D. — 1977. — Vol. 15. — Pp. 2738 - 2751.
3. Lohiya D., Panchapakesan N. Massless Scalar Field in a de Sitter Universe and its Thermal Flux // J. Phys. A. — 1978. — Vol. 11. — Pp. 1963-1968.
4. Lohiya D., Panchapakesan N. Particle Emission in the de Sitter Universe for Mass-less Fields with Spin // J. Phys. A. — 1979. — Vol. 12. — Pp. 533-539.
5. Khanal U., Panchapakesan N. Perturbation of the de Sitter-Schwarzchild Universe with Massless Fields // Phys. Rev. D. — 1981. — Vol. 24. — Pp. 829-834.
6. Khanal U., Panchapakesan N. Production of Massless Particles in the de Sitter-Schwarzschild Universe // Phys. Rev. D. — 1981. — Vol. 24. — Pp. 835-838.
7. Otchik V. S. On the Hawking Radiation of Spin 1/2 Particles in the de Sitter Space-Time // Class. Quantum Crav. — 1985. — Vol. 2. — Pp. 539-543.
8. Bogush A. A., Otchik V. S., Red'kov V. M. Vector Field in de Sitter Space // Vesti AN NSSR. — 1986. — Vol. 1. — Pp. 58-62.
9. Mishima T., Nakayama A. Particle Production in de Sitter Spacetime // Progr. Theor. Phys. — 1987. — Vol. 77. — Pp. 218-222.
10. Suzuki H., Takasugi E. Absorption Probability of De Sitter Horizon for Massless Fields with Spin Mod // Phys. Lett. A. — 1996. — Vol. 11. — Pp. 431-436.
11. Red'kov V. M., Ovsiyuk E. M. On Exact Solutions for Quantum Particles with Spin S = 0,1/2,1 and de Sitter Event Horizon // Ricerche di Matematica. — 2011. — Vol. 60, No 1. — Pp. 57-88.
УДК 530.12, 539.12
Хокинговское излучение в пространстве де Ситтера: вычисление коэффициента отражения для квантовых
частиц
В. М. Редьков*, Е. М. Овсиюк^ Г. Г. Крылов*
* ГНУ «Институт физики им,. Б.И Степанова Национальной академии наук
Беларуси»
пр.Независимости, 68, г.Минск, 220072 Беларусь ^ Мозырский государственный педагогический университет им. И.П. Шамякина ул. Студенческая, 28, г. Мозырь, Гомельская обл., 247760, Республика Беларусь * Белорусский государственный университет пр. Независимости 4, г. Минск, 220030 Республика Беларусь
Несмотря на то, что проблема хокинговского излучения в пространстве—времени де Ситтера и в частности детали прохождения квантовомеханических частиц через де сит-теровский горизонт уже многократно исследовались в литературе, до настоящего времени остаются некоторые неясности в исследовании этого явления. Цель настоящей работы — прояснить эти неясности. Для этого анализируется известный алгоритм, применяющийся для вычисления коэффициента отражения R£j на фоне пространства—времени де Ситтера. Показывается, что определение для величины R£J предполагает использование дополнительного ограничения на квантовые числа eR/hc ^ j, где R — радиус кривизны пространства. Если учесть это ограничение, то величина коэффициента Rsj оказывается равной точно нулю. Показывается, что процедура вычисления коэффициента отражения в де ситтеровской модели базируется на использовании исключительно только приближения нулевого порядка в разложении точной волновой функции частицы в ряд по малому параметру 1/ R2, при этом показывается также, что этот рецепт вычисления не может быть обобщен с учетом поправок более высокого порядка. Следовательно, результат Rsj = 0, который был найден из исследования приближения нулевого порядка, остается неизменным не может быть улучшен. Также утверждается, что задача о вычислении коэффициента отражения R£J не должна ставиться вообще, поскольку в случае пространства де Ситтера эффективная потенциальная кривая в радиальном уравнении является кривой безбарьерного типа.
Ключевые слова: излучение Хокинга, пространство де Ситтера, коэффициент отражения.
