- Email: [email protected]
Massive particles' black hole tunneling and de Sitter tunneling
Nuclear Physics B 725 (2005) 173–180
Massive particles’ black hole tunneling and de Sitter tunneling Jingyi Zhang, Zheng Zhao Department of Physics, Beijing Normal University, Beijing 100875, China Received 3 June 2005; accepted 21 July 2005 Available online 8 August 2005
Abstract Hawking radiation is viewed as a tunneling process. The behavior of the tunneling massive particles is investigated, and the emission rates at which massive particles tunnel across the event horizon and the de Sitter cosmological horizon of the Schwarzschild–de Sitter space–time are calculated. The results are consistent with an underlying unitary theory and take the same functional form as that of massless particles. 2005 Elsevier B.V. All rights reserved. PACS: 04.70.Dy Keywords: Black hole; Hawking radiation; Quantum theory
1. Introduction In 2000, Parikh and Wilczek proposed a method to calculate the emission rate at which particles tunnel across the event horizon [1–3]. They treat Hawking radiation as a tunneling process. The barrier is just created by the outgoing particle itself. Their key insight is to find a coordinate system well-behaved at the event horizon to calculate the emission rate. In this way they have calculated the corrected emission spectrum of the particles from the spherically symmetric black holes, such as Schwarzschild black holes and Reissner–Norström black holes. Their results are consistent with an underlying unitary theory. But in their treatE-mail addresses: [email protected] (J. Zhang), [email protected] (Z. Zhao). 0550-3213/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.07.024
174
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
ment, they only investigate the tunneling of the massless particles (massless shell). Since the massive particles are not lightlike particles, they do not follow the radial-lightlike geodesics when they tunnel across the horizon. In this paper, we extend Parikh’s method to the Schwarzschild–de Sitter space–time and calculate the massive particle’s emission rate. In the following discussion, we first investigate the behavior of a massive tunneling particle. For the sake of simplicity, we treat the particle as a massive shell (de Broglie s-wave). The phase velocity and group velocity of the de Broglie wave corresponding to the outgoing particle are obtained. Then, we apply Parikh’s approach to calculate the emission rates at which massive particles tunnel across the event horizon and de Sitter cosmological horizon of the Schwarzschild–de Sitter space–time. Throughout the paper, the geometrized units (G ≡ c ≡ h¯ ≡ 1) are used. 2. Behavior of the massive tunneling particles To describe tunneling, we need a coordinate system that is regular at the horizons; particularly convenient are Painlevé coordinates. The line element of the Schwarzschild– de Sitter space–time is [4] 2m λ 2 2m λ 2 −1 2 2 2 − r dts + 1 − − r dr + r 2 dΩ 2 , ds = − 1 − (1) r 3 r 3 where m is the mass of the black hole and λ is the cosmological constant. The event horizon rh and the cosmological horizon rc satisfy the equation 2m λ 2 − r = 0. r 3 The expressions of rh , rc and another negative root r− of above equation are ϕ 1 cos , rc = 2 λ 3 ϕ π 1 rh = −2 cos + , λ 3 3 ϕ π 1 r− = −2 cos − , λ 3 3 1−
with
√ cos ϕ = −3m λ.
(2)
(3) (4) (5)
(6)
To obtain the Painlevé–Schwarzschild–de Sitter coordinates, let us do the following transformation ts = t + f (r), which satisfies 1 1−
2m r
2m λ 2 2 − r f (r) = 1. − 1− r 3 − λ3 r 2
(7)
(8)
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
175
There is no need to integrate this; from dts = dt + f (r) dr, we can read off the Painlevé line element 2m λ 2 2m λ 2 2 2 − r dt ± 2 + r dt dr + dr 2 + r 2 dΩ 2 ds = − 1 − r 3 r 3 = g00 dt 2 + 2g01 dt dr + dr 2 + r 2 dΩ 2 .
(9)
Line element (9) displays the stationary, nonstatic, and nonsingular nature of the space– time. Note that we must choose the positive (negative) sign in Eq. (9) when we study the tunneling process taking place at the event horizon (cosmological horizon). Moreover, we find another important feature which is rarely stressed in the relevant literatures. We describe this as follows. According to Landau’s theory of the coordinate clock synchronization [5], in a space– time decomposed in (3 + 1), the coordinate time difference of two events, which take place simultaneously in different places, is g0i dx i (i = 1, 2, 3). T = − (10) g00 If the simultaneity of coordinate clocks can be transmitted from one place to another and has nothing to do with the integration path, components of the metric should satisfy [6] g0j g0i ∂ ∂ − = − (i, j = 1, 2, 3). (11) ∂x j g00 ∂x i g00 Obviously, the line element (9) satisfies condition (11), that is, the coordinate clock synchronization in the Painlevé coordinates can be transmitted from one place to another though the line element is not diagonal. In quantum mechanics, it is an instantaneous process that particle tunnels across a barrier. Thus, this feature is necessary for us to discuss the tunneling process. The radial null geodesics are given by [1] 2m λ 2 dr r˙ ≡ (12) = ±1 − + r , dt r 3 with the upper (lower) sign in Eq. (12) corresponding to outgoing (ingoing) geodesics, under the implicit assumption that t increases towards the future. But in this paper we investigate the tunneling of the massive particles. Since the worldline of a massive quanta is not lightlike, it does not follow radial-lightlike geodesics (12) when it tunnels across the horizon. For the sake of simplicity, we consider the outgoing massive particle as a massive shell (de Broglie s-wave). According to the WKB formula, the approximative wave equation is ψ(r, t) = Ce
i
r
ri −ε pr
dr−ωt
,
(13)
where ri − ε indicates the initial location of the particle. If let r pr dr − ωt = φ0, ri −ε
(14)
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J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
then, we have ω dr = r˙ = , dt k
(15)
where k is the de Broglie wave number. Comparing Eq. (15) with the definition of the phase velocity, we see that r˙ is the phase velocity of the de Broglie wave. Unlike the electromagnetic wave, the phase velocity vp of the de Broglie wave is not equal to the group velocity vg . The definitions and relationship between them are dr ω = r˙ = , dt k drc dω = , vg = dt dk 1 vp = vg . 2
vp =
(16) (17) (18)
To obtain the formula of the phase velocity r˙ , let us now investigate the behavior of a massive particle tunneling across the horizon. Since tunneling across the barrier is an instantaneous process, there are two simultaneous events during the process of emission. One event is particle tunneling into the barrier, another is particle tunneling out the barrier. In terms of Landau’s theory of the coordinate clock synchronization, the difference of coordinate times of these two simultaneous events is dt = −
g0i g01 dx i = − drc g00 g00
(dθ = dϕ = 0),
(19)
where rc is the location of the particle. The group velocity is vg =
g00 drc =− , dt g01
(20)
and the phase velocity is therefore 1 1 g00 . vp = r˙ = vg = − 2 2 g01
(21)
Substituting g00 and g01 into Eq. (21), we get the expression of r˙ 1 λr r 3 − λ3 r + 6m λ
r˙ = ∓ . 2r 3 r 3 + 6m
(22)
λ
Here, the upper sign corresponds to the motion equation of the outgoing particle near the event horizon, and the lower sign corresponds to that of the ingoing particle near the cosmological horizon. Moreover, if the self-gravitation is included, Eqs. (9), (20) and (21) should be replaced by m → m ∓ ω, where ω is the particle’s energy and with the upper (lower) sign corresponding to outgoing (ingoing) motion of the massive particles.
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
177
3. Black hole tunneling Let us first calculate the emission rate at which massive particles tunnel across the event horizon of the Schwarzschild–de Sitter. For a positive-energy s-wave, the imaginary part of the action has been found to have a conveniently simple form [1]: rf Im S =
rf pr pr dr =
ri
dpr dr,
(23)
ri 0
where Pr is the canonical momentum conjugate to r. We expect the initial radius, ri , to correspond roughly to the site of pair–creation, which should be slightly inside the event horizon rh . We expect the final radius, rf , to be slightly outside the final position of the horizon. To proceed with an explicit calculation, it is useful to apply Hamilton’s equation r˙ =
dH d(E − ω) dω = =− . dPr dPr dPr
(24)
Here, E represents the total energy of the black hole system, whereas E − ω can be regarded as the (varying) gravitational energy stored in the black hole. Let us emphasize that, by keeping E fixed, energy conservation will be enforced in a natural way. Substituting Eq. (24) into Eq. (23) yields rf ω Im S = − Im ri 0
dω dr. r˙
(25)
Before evaluating the above integral, we must necessarily obtain an expression for r˙ as a function of ω . If the self-gravitation is included, the form of this expression become 6(m−ω ) )(r − r )(r − r ) 1 λr (r − r− 1 λr r 3 − λ3 r + c h λ
=− , r˙ = − (26) 2r 3 2r 3 6(m−ω ) 6(m−ω ) 3 3 r + r + λ λ where
ϕ 1 cos , λ 3 ϕ π 1 cos + , rh = −2 λ 3 3 ϕ π 1 cos − , r− = −2 λ 3 3 rc
=2
(27) (28) (29)
with ϕ satisfies √ cos ϕ = −3(m − ω ) λ.
(30)
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J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
Substituting Eq. (26) into (25) yields
) rf ω 2r r 3 + 6(m−ω λ
Im S = Im dω dr. λr ri 0 3 (r − r− )(r − rh )(r − rc )
(31)
Switch the order of integration and do the r integral first. Since we
take into account the
tunneling of particle across the event horizon, we see that r = rh = 2 λ1 cos( ϕ3 + π3 ) is a pole. The integral can be evaluated by deforming the contour around the pole. In this way we obtain 6π Im S = − λ
ω 0
(rh
rh − r− )(rh
− rc )
dω .
(32)
From Eq. (30) we have dω =
− sin ϕ √ dϕ . 3 λ
(33)
(27)–(29) into (32), we have Substituting Eq. (33) and the expressions of rh , rc and r−
π Im S = − 3λ
ϕf
cos( ϕ3 + π3 ) sin ϕ
ϕi
[cos2 ( ϕ3 + π3 ) − 14 ]
dϕ .
(34)
Finishing the integral yields π 2 1 (35) r − ri2 = − SBH , 2 f 2 where SBH = SBH (m − ω) − SBH (m) is the difference of the entropies of the black hole before and after the emission. The tunneling rate is therefore Im S = −
Γ ∼ exp[−2 Im S] = eSBH ,
(36)
which is consistent with an underlying unitary theory.
4. de Sitter tunneling When the tunneling takes place at the de Sitter cosmological horizon, the particle is propagating from larger to smaller r, and the mass of the black hole increases from m to m + ω . If the self-gravitation is included, the form of r˙ become 6(m+ω ) )(r − r )(r − r ) 1 λr r 3 − λ3 r + 1 λr (r − r− c h λ
r˙ = (37) = , 2r 3 2r 3 ) 3 + 6(m+ω ) r 3 + 6(m+ω r λ λ
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
where
ϕ 1 cos , λ 3 ϕ π 1 rh = −2 cos + , λ 3 3 ϕ π 1 r− cos − , = −2 λ 3 3 rc
179
=2
(38) (39) (40)
with ϕ satisfies
√ cos ϕ = −3(m + ω ) λ.
(41)
For a positive-energy s-wave, the imaginary part of the action is r
r
f
f pr pr dr =
Im S = ri
ri
dpr dr.
(42)
0
Here, ri is slightly outside the cosmological horizon rc , and rf slightly inside the final position of the cosmological horizon. The total mass of the space–time is [4] M = −m.
(43)
So, the Hamilton’s equation is dM d(m + ω) dω dH = =− =− . (44) dPr dPr dPr dPr By keeping M fixed, energy conservation will be enforced in a natural way. Substituting Eq. (44) into (42) yields r˙ =
r
f ω Im S = Im ri
0
dω dr. r˙
(45)
Substituting Eq. (37) into (45), we have
r ) f ω 2r r 3 + 6(m+ω λ
Im S = Im dω dr. λr 3 (r − r− )(r − rh )(r − rc ) r 0
(46)
i
Since
the tunneling is taking place at the cosmological horizon, we see that r = rc = ϕ 2 λ1 cos 3 is a pole. The integral can be evaluated by deforming the contour around the pole. In this way we obtain
6π Im S = λ
ω 0
(rc
rc − r− )(rc
− rh )
dω .
(47)
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J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
From Eq. (41) we have dω =
sin ϕ √ dϕ . 3 λ
(48)
(38)–(40) into (47), we have Substituting Eq. (48) and the expressions of rh , rc and r−
π Im S = 3λ
ϕf ϕi
cos ϕ3 sin ϕ [cos2
ϕ 3
− 14 ]
dϕ .
(49)
Finally, finishing the integral we obtain 1 π 2 r − ri 2 = − SCH , 2 f 2 where SCH = SCH (m + ω) − SCH (m) is the change in entropy during the process of emission. The emission rate from the cosmological horizon is therefore Im S = −
Γ ∼ exp[−2 Im S] = eSCH .
(50)
It is also consistent with an underlying unitary theory.
5. Conclusion We have calculated the emission rates of massive particles tunneling across the event horizon and the cosmological horizon. The results are consistent with an underlying unitary theory and take the same functional form as that of massless particles [7].
Acknowledgements This research is supported by National Natural Science Foundation of China Grant Nos. 10373003, 10375051, 10475013 and National Basic Research Program of China Grant No. 2003CB716300.
References [1] [2] [3] [4] [5] [6] [7]
M.K. Parikh, F. Wilczek, Hawking radiation as tunneling, Phys. Rev. Lett. 85 (2000) 5042, hep-th/9907001. M.K. Parikh, Energy conservation and Hawking radiation, hep-th/0402166. M.K. Parikh, A secret tunnel through the horizon, hep-th/0405160. V. Balasubramanian, J. de Boer, D. Minic, Phys. Rev. D 65 (2002) 123508. L.D. Landau, E.M. Lifshitz, The Classical Theory of Field, Pergamon Press, London, 1975. H. Zhang, Z. Zhao, J. Beijing Normal Univ. (Natural Science) 37 (2001) 471 (in Chinese). A.J.M. Medved, Phys. Rev. D 66 (2002) 1240091.
Massive particles’ black hole tunneling and de Sitter tunneling Jingyi Zhang, Zheng Zhao Department of Physics, Beijing Normal University, Beijing 100875, China Received 3 June 2005; accepted 21 July 2005 Available online 8 August 2005
Abstract Hawking radiation is viewed as a tunneling process. The behavior of the tunneling massive particles is investigated, and the emission rates at which massive particles tunnel across the event horizon and the de Sitter cosmological horizon of the Schwarzschild–de Sitter space–time are calculated. The results are consistent with an underlying unitary theory and take the same functional form as that of massless particles. 2005 Elsevier B.V. All rights reserved. PACS: 04.70.Dy Keywords: Black hole; Hawking radiation; Quantum theory
1. Introduction In 2000, Parikh and Wilczek proposed a method to calculate the emission rate at which particles tunnel across the event horizon [1–3]. They treat Hawking radiation as a tunneling process. The barrier is just created by the outgoing particle itself. Their key insight is to find a coordinate system well-behaved at the event horizon to calculate the emission rate. In this way they have calculated the corrected emission spectrum of the particles from the spherically symmetric black holes, such as Schwarzschild black holes and Reissner–Norström black holes. Their results are consistent with an underlying unitary theory. But in their treatE-mail addresses: [email protected] (J. Zhang), [email protected] (Z. Zhao). 0550-3213/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.07.024
174
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
ment, they only investigate the tunneling of the massless particles (massless shell). Since the massive particles are not lightlike particles, they do not follow the radial-lightlike geodesics when they tunnel across the horizon. In this paper, we extend Parikh’s method to the Schwarzschild–de Sitter space–time and calculate the massive particle’s emission rate. In the following discussion, we first investigate the behavior of a massive tunneling particle. For the sake of simplicity, we treat the particle as a massive shell (de Broglie s-wave). The phase velocity and group velocity of the de Broglie wave corresponding to the outgoing particle are obtained. Then, we apply Parikh’s approach to calculate the emission rates at which massive particles tunnel across the event horizon and de Sitter cosmological horizon of the Schwarzschild–de Sitter space–time. Throughout the paper, the geometrized units (G ≡ c ≡ h¯ ≡ 1) are used. 2. Behavior of the massive tunneling particles To describe tunneling, we need a coordinate system that is regular at the horizons; particularly convenient are Painlevé coordinates. The line element of the Schwarzschild– de Sitter space–time is [4] 2m λ 2 2m λ 2 −1 2 2 2 − r dts + 1 − − r dr + r 2 dΩ 2 , ds = − 1 − (1) r 3 r 3 where m is the mass of the black hole and λ is the cosmological constant. The event horizon rh and the cosmological horizon rc satisfy the equation 2m λ 2 − r = 0. r 3 The expressions of rh , rc and another negative root r− of above equation are ϕ 1 cos , rc = 2 λ 3 ϕ π 1 rh = −2 cos + , λ 3 3 ϕ π 1 r− = −2 cos − , λ 3 3 1−
with
√ cos ϕ = −3m λ.
(2)
(3) (4) (5)
(6)
To obtain the Painlevé–Schwarzschild–de Sitter coordinates, let us do the following transformation ts = t + f (r), which satisfies 1 1−
2m r
2m λ 2 2 − r f (r) = 1. − 1− r 3 − λ3 r 2
(7)
(8)
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
175
There is no need to integrate this; from dts = dt + f (r) dr, we can read off the Painlevé line element 2m λ 2 2m λ 2 2 2 − r dt ± 2 + r dt dr + dr 2 + r 2 dΩ 2 ds = − 1 − r 3 r 3 = g00 dt 2 + 2g01 dt dr + dr 2 + r 2 dΩ 2 .
(9)
Line element (9) displays the stationary, nonstatic, and nonsingular nature of the space– time. Note that we must choose the positive (negative) sign in Eq. (9) when we study the tunneling process taking place at the event horizon (cosmological horizon). Moreover, we find another important feature which is rarely stressed in the relevant literatures. We describe this as follows. According to Landau’s theory of the coordinate clock synchronization [5], in a space– time decomposed in (3 + 1), the coordinate time difference of two events, which take place simultaneously in different places, is g0i dx i (i = 1, 2, 3). T = − (10) g00 If the simultaneity of coordinate clocks can be transmitted from one place to another and has nothing to do with the integration path, components of the metric should satisfy [6] g0j g0i ∂ ∂ − = − (i, j = 1, 2, 3). (11) ∂x j g00 ∂x i g00 Obviously, the line element (9) satisfies condition (11), that is, the coordinate clock synchronization in the Painlevé coordinates can be transmitted from one place to another though the line element is not diagonal. In quantum mechanics, it is an instantaneous process that particle tunnels across a barrier. Thus, this feature is necessary for us to discuss the tunneling process. The radial null geodesics are given by [1] 2m λ 2 dr r˙ ≡ (12) = ±1 − + r , dt r 3 with the upper (lower) sign in Eq. (12) corresponding to outgoing (ingoing) geodesics, under the implicit assumption that t increases towards the future. But in this paper we investigate the tunneling of the massive particles. Since the worldline of a massive quanta is not lightlike, it does not follow radial-lightlike geodesics (12) when it tunnels across the horizon. For the sake of simplicity, we consider the outgoing massive particle as a massive shell (de Broglie s-wave). According to the WKB formula, the approximative wave equation is ψ(r, t) = Ce
i
r
ri −ε pr
dr−ωt
,
(13)
where ri − ε indicates the initial location of the particle. If let r pr dr − ωt = φ0, ri −ε
(14)
176
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
then, we have ω dr = r˙ = , dt k
(15)
where k is the de Broglie wave number. Comparing Eq. (15) with the definition of the phase velocity, we see that r˙ is the phase velocity of the de Broglie wave. Unlike the electromagnetic wave, the phase velocity vp of the de Broglie wave is not equal to the group velocity vg . The definitions and relationship between them are dr ω = r˙ = , dt k drc dω = , vg = dt dk 1 vp = vg . 2
vp =
(16) (17) (18)
To obtain the formula of the phase velocity r˙ , let us now investigate the behavior of a massive particle tunneling across the horizon. Since tunneling across the barrier is an instantaneous process, there are two simultaneous events during the process of emission. One event is particle tunneling into the barrier, another is particle tunneling out the barrier. In terms of Landau’s theory of the coordinate clock synchronization, the difference of coordinate times of these two simultaneous events is dt = −
g0i g01 dx i = − drc g00 g00
(dθ = dϕ = 0),
(19)
where rc is the location of the particle. The group velocity is vg =
g00 drc =− , dt g01
(20)
and the phase velocity is therefore 1 1 g00 . vp = r˙ = vg = − 2 2 g01
(21)
Substituting g00 and g01 into Eq. (21), we get the expression of r˙ 1 λr r 3 − λ3 r + 6m λ
r˙ = ∓ . 2r 3 r 3 + 6m
(22)
λ
Here, the upper sign corresponds to the motion equation of the outgoing particle near the event horizon, and the lower sign corresponds to that of the ingoing particle near the cosmological horizon. Moreover, if the self-gravitation is included, Eqs. (9), (20) and (21) should be replaced by m → m ∓ ω, where ω is the particle’s energy and with the upper (lower) sign corresponding to outgoing (ingoing) motion of the massive particles.
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
177
3. Black hole tunneling Let us first calculate the emission rate at which massive particles tunnel across the event horizon of the Schwarzschild–de Sitter. For a positive-energy s-wave, the imaginary part of the action has been found to have a conveniently simple form [1]: rf Im S =
rf pr pr dr =
ri
dpr dr,
(23)
ri 0
where Pr is the canonical momentum conjugate to r. We expect the initial radius, ri , to correspond roughly to the site of pair–creation, which should be slightly inside the event horizon rh . We expect the final radius, rf , to be slightly outside the final position of the horizon. To proceed with an explicit calculation, it is useful to apply Hamilton’s equation r˙ =
dH d(E − ω) dω = =− . dPr dPr dPr
(24)
Here, E represents the total energy of the black hole system, whereas E − ω can be regarded as the (varying) gravitational energy stored in the black hole. Let us emphasize that, by keeping E fixed, energy conservation will be enforced in a natural way. Substituting Eq. (24) into Eq. (23) yields rf ω Im S = − Im ri 0
dω dr. r˙
(25)
Before evaluating the above integral, we must necessarily obtain an expression for r˙ as a function of ω . If the self-gravitation is included, the form of this expression become 6(m−ω ) )(r − r )(r − r ) 1 λr (r − r− 1 λr r 3 − λ3 r + c h λ
=− , r˙ = − (26) 2r 3 2r 3 6(m−ω ) 6(m−ω ) 3 3 r + r + λ λ where
ϕ 1 cos , λ 3 ϕ π 1 cos + , rh = −2 λ 3 3 ϕ π 1 cos − , r− = −2 λ 3 3 rc
=2
(27) (28) (29)
with ϕ satisfies √ cos ϕ = −3(m − ω ) λ.
(30)
178
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
Substituting Eq. (26) into (25) yields
) rf ω 2r r 3 + 6(m−ω λ
Im S = Im dω dr. λr ri 0 3 (r − r− )(r − rh )(r − rc )
(31)
Switch the order of integration and do the r integral first. Since we
take into account the
tunneling of particle across the event horizon, we see that r = rh = 2 λ1 cos( ϕ3 + π3 ) is a pole. The integral can be evaluated by deforming the contour around the pole. In this way we obtain 6π Im S = − λ
ω 0
(rh
rh − r− )(rh
− rc )
dω .
(32)
From Eq. (30) we have dω =
− sin ϕ √ dϕ . 3 λ
(33)
(27)–(29) into (32), we have Substituting Eq. (33) and the expressions of rh , rc and r−
π Im S = − 3λ
ϕf
cos( ϕ3 + π3 ) sin ϕ
ϕi
[cos2 ( ϕ3 + π3 ) − 14 ]
dϕ .
(34)
Finishing the integral yields π 2 1 (35) r − ri2 = − SBH , 2 f 2 where SBH = SBH (m − ω) − SBH (m) is the difference of the entropies of the black hole before and after the emission. The tunneling rate is therefore Im S = −
Γ ∼ exp[−2 Im S] = eSBH ,
(36)
which is consistent with an underlying unitary theory.
4. de Sitter tunneling When the tunneling takes place at the de Sitter cosmological horizon, the particle is propagating from larger to smaller r, and the mass of the black hole increases from m to m + ω . If the self-gravitation is included, the form of r˙ become 6(m+ω ) )(r − r )(r − r ) 1 λr r 3 − λ3 r + 1 λr (r − r− c h λ
r˙ = (37) = , 2r 3 2r 3 ) 3 + 6(m+ω ) r 3 + 6(m+ω r λ λ
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
where
ϕ 1 cos , λ 3 ϕ π 1 rh = −2 cos + , λ 3 3 ϕ π 1 r− cos − , = −2 λ 3 3 rc
179
=2
(38) (39) (40)
with ϕ satisfies
√ cos ϕ = −3(m + ω ) λ.
(41)
For a positive-energy s-wave, the imaginary part of the action is r
r
f
f pr pr dr =
Im S = ri
ri
dpr dr.
(42)
0
Here, ri is slightly outside the cosmological horizon rc , and rf slightly inside the final position of the cosmological horizon. The total mass of the space–time is [4] M = −m.
(43)
So, the Hamilton’s equation is dM d(m + ω) dω dH = =− =− . (44) dPr dPr dPr dPr By keeping M fixed, energy conservation will be enforced in a natural way. Substituting Eq. (44) into (42) yields r˙ =
r
f ω Im S = Im ri
0
dω dr. r˙
(45)
Substituting Eq. (37) into (45), we have
r ) f ω 2r r 3 + 6(m+ω λ
Im S = Im dω dr. λr 3 (r − r− )(r − rh )(r − rc ) r 0
(46)
i
Since
the tunneling is taking place at the cosmological horizon, we see that r = rc = ϕ 2 λ1 cos 3 is a pole. The integral can be evaluated by deforming the contour around the pole. In this way we obtain
6π Im S = λ
ω 0
(rc
rc − r− )(rc
− rh )
dω .
(47)
180
J. Zhang, Z. Zhao / Nuclear Physics B 725 (2005) 173–180
From Eq. (41) we have dω =
sin ϕ √ dϕ . 3 λ
(48)
(38)–(40) into (47), we have Substituting Eq. (48) and the expressions of rh , rc and r−
π Im S = 3λ
ϕf ϕi
cos ϕ3 sin ϕ [cos2
ϕ 3
− 14 ]
dϕ .
(49)
Finally, finishing the integral we obtain 1 π 2 r − ri 2 = − SCH , 2 f 2 where SCH = SCH (m + ω) − SCH (m) is the change in entropy during the process of emission. The emission rate from the cosmological horizon is therefore Im S = −
Γ ∼ exp[−2 Im S] = eSCH .
(50)
It is also consistent with an underlying unitary theory.
5. Conclusion We have calculated the emission rates of massive particles tunneling across the event horizon and the cosmological horizon. The results are consistent with an underlying unitary theory and take the same functional form as that of massless particles [7].
Acknowledgements This research is supported by National Natural Science Foundation of China Grant Nos. 10373003, 10375051, 10475013 and National Basic Research Program of China Grant No. 2003CB716300.
References [1] [2] [3] [4] [5] [6] [7]
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