Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes

0.2 MB·English

by  Benrong Mu

CTP-SCU/2015003
Minimal Length Effects on Tunnelling from Spherically
Symmetric Black Holes
Benrong Mua,b, Peng Wangb, and Haitang Yangb,c
∗ † ‡
aSchool of Physical Electronics, University of Electronic
Science and Technology of China ,Chengdu, 610054, China
5 bCenter for Theoretical Physics, College of Physical Science and Technology,
1
0 Sichuan University, Chengdu, 610064, PR China and
2
cKavli Institute for Theoretical Physics China (KITPC),
n
a
J Chinese Academy of Sciences, Beijing 100080, P.R. China

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2 Abstract

]
c Inthispaper,weinvestigateeffectsoftheminimallengthonquantumtunnellingfromspherically
q
- symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We
r
g
[ firstderivethedeformedHamilton-Jacobiequationsforscalarsandfermions,bothofwhichhavethe

1
same expressions. The minimal length correction to the Hawking temperature is found to depend
v
5
on the black hole’s mass and the mass and angular momentum of emitted particles. Finally, we
2
0
6 calculate a Schwarzschild black hole’s luminosity and find the black hole evaporates to zero mass
0
. in infinite time.
1
0
5
1
:
v
i
X
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
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Contents
I. Introduction 2
II. Deformed Hamilton-Jacobi Equations 4
III. Quantum Tunnelling 7
IV. Thermodynamics of Black Holes 9
V. Conclusion 14
References 14
I. INTRODUCTION
The classical theory of black holes predicts that anything, including light, couldn’t escape
from the black holes. However, Stephen Hawking first showed that quantum effects could
allow black holes to emit particles. The formula of Hawking temperature was first given
in the frame of quantum field theory[1]. After that, various methods for deriving Hawking
radiation have been proposed. Among them is a semiclassical method of modeling Hawking
radiation as a tunneling effect proposed by Kraus and Wilczek[2, 3], which is known as the
null geodesic method. Later, the tunneling behaviors of particles were investigated using
the Hamilton-Jacobi method[4–6]. Using the null geodesic method and the Hamilton-Jacobi
method, much fruit has been achieved[7–18]. The key point of the Hamilton-Jacobi method
is using WKB approximation to calculate the imaginary part of the action for the tunneling
process.
On the other hand, various theories of quantum gravity, such as string theory, loop
quantum gravity and quantum geometry, predict the existence of a minimal length[19–21].
The generalized uncertainty principle(GUP)[22] is a simply way to realize this minimal
length.
An effective model of the GUP in one dimensional quantum mechanics is given by[23, 24]
p
L k(p) = tanh , (1)
f
M
(cid:18) f(cid:19)
2
E
L ω(E) = tanh , (2)
f
M
(cid:18) f(cid:19)
where the generators of the translations in space and time are the wave vector k and the
frequency ω, L is the minimal length, and L M = ~. The quantization in position
f f f
representation xˆ = x leads to
k = i∂ , ω = i∂ . (3)
x t
−
Therefore, the low energy limit p M including order of p3 gives
≪ f Mf3
~2
p = i~∂ 1 ∂2 , (4)
− x − M2 x
f !
~2
E = i~∂ 1 ∂2 , (5)
t − M2 t
f !
where we neglect the factor 1. From eqn. (1), it is noted that although one can increase
3
p arbitrarily, k has an upper bound which is 1 . The upper bound on k implies that that
Lf
particles could not possess arbitrarily small Compton wavelengths λ = 2π/k and that there
exists a minimal length L . Furthermore, the deformed Klein-Gordon/Dirac equations
f
∼
incorporating eqn. (4) and eqn. (5) have already be obtained in [23], which will be briefly
reviewed in section II. So [23] provides a way to incorporate the minimal length with special
relativity, a good starting point for studying Hawking radiation as tunnelling effect.
The black hole is a suitable venue to discuss the effects of quantum gravity. Incorporating
GUP into black holes has been discussed in a lot of papers[25–30]. The thermodynamics of
black holes has also been investigated in the framework of GUP[29, 30]. In [31], a new form
of GUP is introduced
p0 = k0, (6)
pi = ki 1 αk+2α2k2 , (7)
−
(cid:0) (cid:1)
where pa is the modified four momentum, ka is the usual four momentum and α is a small
parameter. The modified velocity of photons, two-dimensional Klein-Gordon equation the
emission spectrum due to the Unruh effect is obtained there. Recently, the GUP deformed
Hamilton-Jacobi equation for fermions in curved spacetime have been introduced and the
corrected Hawking temperatures have been derived[32–36]. The authors consider the GUP
3
of form
x = x (8)
i 0i,
p = p 1+βp2 , (9)
i 0i
(cid:0) (cid:1)
where x and p satisfy the canonical commutation relations. Fermions’ tunnelling, black
0i, 0i
hole thermodynamics and the remnants are discussed there.
In this paper, we investigate scalars and fermions tunneling across the horizons of black
holes using the deformed Hamilton-Jacobi method which incorporates the minimal length
via eqn. (4) and eqn. (5). Our calculation shows that the quantum gravity correction is
related not only to the black hole’s mass but also to the mass and angular momentum of
emitted particles.
The organization of this paper is as follows. In section II, from the modified fundamental
commutation relation, we generalize the Hamilton-Jacobi in curved spacetime. In section
III, incorporating GUP, we investigate the tunnelling of particles in the black holes. In
section IV, we investigate how a Schwarzschild black hole evaporates in our model. Section
V is devoted to our conclusion. In this paper, we take Geometrized units c = G = 1, where
the Planck constant ~ is square of the Planck Mass m . We also assume that the emitted
p
particles are neutral.
II. DEFORMED HAMILTON-JACOBI EQUATIONS
To be generic, we will consider a spherically symmetric background metric of the form
dr2
ds2 = f (r)dt2 r2 dθ2 +sin2θdφ2 , (10)
− f (r) −
(cid:0) (cid:1)
where where f (r) has a simple zero at r = r with f (r ) being finite and nonzero. The
h ′ h
vanishing of f (r) at point r = r indicates the presence of an event horizon. In this section,
h
we will first derive the deformed Klein-Gordon/Dirac equations in flat spacetime and then
generalize them to the curved spacetime with the metric (10).
In the (3+1) dimensional flat spacetime, the relations between (p ,E) and (k ,ω) can
i i
simply be generalized to
p
i
L k (p) = tanh , (11)
f i
M
(cid:18) f(cid:19)
E
L ω(E) = tanh , (12)
f
M
(cid:18) f(cid:19)
4
~
where, in the spherical coordinates, one has for k
ˆ ˆ
∂ θ ∂ φ ∂
~
k = i rˆ + + . (13)
− ∂r r∂θ rsinθ∂φ
!
Expanding eqn. (11) and eqn. (12) for small arguments to the third order gives the energy
and momentum operator in position representation
~2
E = i~∂ 1 ∂2 , (14)
t − M2 t
f !
~ ~2∂3 ∂ ~2∂3 ∂ ~2∂3
~p = rˆ ∂ r +θˆ θ θ +φˆ φ φ , (15)
i r − M2 r − r3M2 rsinθ − r3sin3θM2
" f ! f ! f !#
where we also omit the factor 1. Substituting the above energy and momentum operators
3
into the energy-momentum relation, the deformed Klein-Gordon equation satisfied by the
scalar field with the mass m is
E2φ = p2φ+m2φ, (16)
where p2 = p~ p~. Making the ansatz for φ
·
iI
φ = exp , (17)
~
(cid:18) (cid:19)
and substituting it into eqn. (16), one expands eqn. (16) in powers of ~ and then finds that
the lowest order gives the deformed scalar Hamilton-Jacobi equation in the flat spacetime
2(∂ I)2 2(∂ I)2 (∂ I)2 2(∂ I)2
(∂ I)2 1+ t (∂ I)2 1+ r θ 1+ θ
t M2 − r M2 − r2 r2M2
f ! f ! f !
(∂ I)2 2(∂ I)2
φ 1+ φ = m2, (18)
−r2sin2θ r2sin2θM2
!
f
which is truncated at 1 .
O Mf2
Similarly, the deforme(cid:16)d Di(cid:17)rac equation for a spin-1/2 fermion with the mass m takes the
form as
(γ E +~γ ~p m)ψ = 0, (19)
0
· −
where γ ,γ = 2, γ ,γ = 2δ , and γ ,γ = 0 with the Latin index a running
0 0 a b ab 0 a
{ } { } − { }
over r,θ,and φ. Multiplying (γ E +~γ ~p+m) by eqn. (19) and using the gamma matrices
0
·
anticommutation relations, the deformed Dirac equation can be written as
[γ ,γ ]
E2ψ = p2 +m2 ψ a b p p ψ. (20)
a b
− 2
(cid:0) (cid:1)
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To obtain the Hamilton-Jacobi equation for the fermion, the ansatz for ψ takes the form of
iI
ψ = exp v, (21)
~
(cid:18) (cid:19)
where v is a vector function of the spacetime. Substituting eqn. (21) into eqn. (20) and
noting that the second term on RHS of eqn. (20) does not contribute to the lowest order of
~, we find the deformed Hamilton-Jacobi equation for a fermion is the same as the deformed
one for a scalar with the same mass, namely eqn. (18). Note that one can use the deformed
Maxwell’s equations obtained in [23] to get the deformed Hamilton-Jacobi equation for a
vector boson. However, for simplicity we just stop here.
In order to generalize the deformed Hamilton-Jacobi equation, eqn. (18), to the curved
spacetime withthemetric(10), we first consider theHamilton-Jacobiequation withoutGUP
modifications. In ref. [37], we show that the unmodified Hamilton-Jacobi equation in curved
spacetime with ds2 = g dxµdxν is
µν
gµν∂ I∂ I m2 = 0. (22)
µ ν
−
Therefore, the unmodified Hamilton-Jacobi equation in the metric (10) becomes
(∂ I)2 (∂ I)2 (∂ I)2
t f (r)(∂ I)2 θ φ = m2. (23)
f (r) − r − r2 − r2sin2θ
On the other hand, the unmodified Hamilton-Jacobi equation in flat spacetime can be ob-
tained from eqn. (18) by taking M ,
f
→ ∞
(∂ I)2 (∂ I)2
(∂ I)2 (∂ I)2 θ φ = m2. (24)
t − r − r2 − r2sin2θ
Comparing eqn. (23) with eqn. (24), one finds that the Hamilton-Jacobi equation in the
metric (10) can be obtained from the one in flat spacetime by making replacements ∂ I
r
→
f (r)∂ I and ∂ I ∂tI in the no GUP modifications scenario. Therefore, by making
r t → √f(r)
preplacements ∂ I f (r)∂ I and ∂ I ∂tI , the deformed Hamilton-Jacobi equation
r → r t → √f(r)
in flat spacetime, eqnp. (18), leads to the deformed Hamilton-Jacobi equation in the metric
(10), which is to 1 ,
O Mf2
(cid:16) (cid:17)
(∂ I)2 2(∂ I)2 2f (r)(∂ I)2 (∂ I)2 2(∂ I)2
t 1+ t f (r)(∂ I)2 1+ r θ 1+ θ
f (r) f (r)M2 − r M2 − r2 r2M2
f ! f ! f !
(∂ I)2 2(∂ I)2
φ 1+ φ = m2. (25)
−r2sin2θ r2sin2θM2
!
f
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III. QUANTUM TUNNELLING
In this section, we investigate the particles’ tunneling at the event horizon r = r of
h
the metric (10) where GUP is taken into account. Since the metric (10) does not depend
on t and φ, ∂ and ∂ are killing vectors. Taking into account the Killing vectors of the
t φ
background spacetime, we can employ the following ansatz for the action
I = ωt+W (r,θ)+p φ, (26)
φ
−
where ω and p are constants and they are the energy and the z-component of angular
φ
momentum of emitted particles, respectively. Inserting eqn. (26) into eqn. (25), we find
that the deformed Hamilton-Jacobi equation becomes
2f (r)p2 1 2ω2
p2 1+ r = ω2 1+ f (r) m2 +λ , (27)
r M2 f2(r) f (r)M2 −
f ! " f ! #
(cid:0) (cid:1)
where p = ∂ W, p = ∂ W and
r r θ θ
p2 2p2 p2 2p2
λ = θ 1+ θ + φ 1+ φ .
r2 r2M2 r2sin2θ r2sin2θM2
f ! f !
Since the magnitude of the angular momentum of the particle L can be expressed in terms
of p and p ,
θ φ
p2
L2 = p2 + φ , (28)
θ sin2θ
one can rewrite λ as
L2 1
λ = + . (29)
r2 O M2
f !
Solving eqn. (27) for p to 1 gives
r O Mf2
(cid:16) (cid:17)
1 2ω2
∂ W = ω2 1+ f (r)(m2 +λ)
r ∓ ±f (r)vu f (r)Mf2!− ×
u
t
2 2ω2
1 ω2 1+ f (r)(m2 +λ) , (30)
v − M2f (r) f (r)M2 −
u f " f ! #
u
t
where +/ represent the outgoing/ingoing solutions. In order to get the imaginary part of
−
W , we need to find residue of the RHS of eqn. (30) at r = r by expanding the RHS in a
h
±
7
Laurent series with respect to r at r = r . We then rewrite eqn. (30) as
h
1 2ω2
∂ W = ω2 f (r)+ f2(r)(m2 +λ)
r ∓ ±f (r)25vu Mf2!− ×
u
t
2 2ω2
f2(r) ω2 f (r)+ f2(r)(m2 +λ) . (31)
v − M2 M2 −
u f " f ! #
u
t
Using f (r) = f (r )(r r )+ f′′(rh) (r r )2 + (r r )3 , one can single out the 1
′ h − h 2 − h O − h r−rh
term of the Laurent series (cid:0) (cid:1)
a
1
∂rW ± − , (32)
∓ ∼ r rh
−
where we have
ω 2 L2 1
a = 1+ m2 + + . (33)
−1 f′(rh) " Mf2 (cid:18) rh2(cid:19)# O Mf4!
Using the residue theory for semi circles, we obtain for the imaginary part of W to 1
± O Mf2
(cid:16) (cid:17)
πω 2 L2
ImW = 1+ m2 + . (34)
± ±f′(rh) " Mf2 (cid:18) rh2(cid:19)#
However, when one tries to calculate the tunneling rate Γ from ImW , there is so
±
called “factor-two problem”[38]. One way to solve the ”factor two problem” is introduc-
ing a “temporal contribution”[14–16, 39, 40]. To consider an invariance under canonical
transformations, we also follow the recent work[14–16, 39, 40] and adopt the expression
p dr = p+dr p dr for the spatial contribution to Γ. The spatial and temporal
r r − −r
I
contributioRns to Γ arRe given as follows.
Spatial Contribution: The spatial part contribution comes from the imaginary part of
W (r). Thus, the spatial part contribution is proportional to
1
exp Im p dr
~ r
−
(cid:20) I (cid:21)
1
= exp Im p+dr p dr
−~ r − −r
(cid:20) (cid:18)Z Z (cid:19)(cid:21)
2πω 2 L2
= exp 1+ m2 + (35)
−f (r ) M2 r2
( ′ h " f (cid:18) h(cid:19)#)
Temporal Contribution: As pointed in Ref. [15, 38–40], the temporal part contribu-
tion comes from the ”rotation” which connects the interior region and the exterior region of
the black hole. Thus, the imaginary contribution from the temporal part when crossing the
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horizon is Im(ω∆tout,in) = ω π , where κ = f′(rh) is the surface gravity at the event horizon.
2κ 2
Then the total temporal contribution for a round trip is
2πω
Im(ω∆t) = . (36)
f (r )
′ h
Therefore, the tunnelling rate of the particle crossing the horizon is
1
Γ exp Im(ω∆t)+Im p dr
~ r
∝ −
(cid:20) (cid:18) I (cid:19)(cid:21)
4πω 1 L2
= exp 1+ m2 + . (37)
−f (r ) M2 r2
( ′ h " f (cid:18) h(cid:19)#)
This is the expression of Boltzmann factor with an effective temperature
f (r ) ~
′ h
T = , (38)
4π 1+ 1 m2 + L2
M2 r2
f h
(cid:16) (cid:17)
where T = ~f′(rh) is the original Hawking temperature.
0 4π
For the standard Hawing radiation, all particles very close to the horizon are effectively
massless on account of infinite blueshift. Thus, the conformal invariance of the horizon make
Hawingtemperaturesofallparticlesthesame. Themass, angularmomentumandidentity of
the particles are only relevant when they escape the potential barrier. However, if quantum
gravity effects are considered, behaviors of particles near the horizon could be different. For
example, if we send a wave packet which is governed by a subluminal dispersion relation,
backwards in time toward the horizon, it reaches a minimum distance of approach, then
reverse direction and propagate back away from the horizon, instead of getting unlimited
blueshift toward the horizon [41, 42]. Thus, quantum gravity effects might make fermions
andscalarsexperience different (effective) Hawking temperatures. However, ourresult shows
that in our model, the tunnelling rates of fermions and scalars depend on their masses and
angular momentums, but independent of the identities of the particles, to 1 . In other
O Mf2
words, effective Hawking temperatures of fermions and scalars are the same(cid:16)to (cid:17) 1 in
O Mf2
our model as long as their masses, energies and angular momentums are the same.(cid:16) (cid:17)
IV. THERMODYNAMICS OF BLACK HOLES
For simplicity, we consider the Schwarzschild metric with f (r) = 1 2M with the black
− r
hole’s mass, M. The event horizon of the Schwarzschild black hole is r = 2M. In this
h
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section, we work with massless particles. Near the horizon of the the black hole, angular
momentum of the particle L pr ωr . Thus, one can rewrite T
h h
∼ ∼
T
0
T , (39)
∼ 1+ 2ω2
M2
f
~
where T = for the Schwarzschild black hole. As reported in [43], the authors obtained
0 8πM
~
the relation ω & between the energy of a particle and its position uncertainty in the
δx
framework of GUP. Near the horizon of the the Schwarzschild black hole, the position un-
certainty of a particle will be of the order of the Schwarzschild radius of the black hole [44]
δx r . Thus, one finds for T
h
∼
T
0
T , (40)
∼ 1+ m4p
2M2M2
f
where we use ~ = m2. Using the first law of the black hole thermodynamics, we find the
p
corrected black hole entropy is
dM
S =
T
Z
A 4πm2 A
p
+ ln , (41)
∼ 4m2 M2 16π
p f (cid:18) (cid:19)
where A = 4πr2 = 16πM2 is the area of the horizon. The logarithmic term in eqn. (41)
h
is the well known correction from quantum gravity to the classical Bekenstein-Hawking
entropy, which have appeared in different studies of GUP modified thermodynamics of black
holes[27–29, 45–51]. In general, the entropy for the Schwarzschild black hole of mass M in
four spacetime dimensions can be written in form of
A A M2
f
S = +σln + , (42)
4 16π O A
(cid:18) (cid:19) (cid:18) (cid:19)
2M2 M2
where σ = f in our paper. Neglecting the terms f in eqn. (42), there could be
M2 O A
three scenarios depending on the sign of σ. (cid:16) (cid:17)
1. σ = 0 : This case is just the standard Hawking radiation. The black holes evaporate
completely in finite time.
2. σ < 0 : The entropy S as function of mass develops a minimum at some value of
M . This predicts the existence of black hole remnants. Furthermore, the black
min
holes stop evaporating in finite time. This is what happened in [27–29, 45–51], which
is consistent with the existence of a minimal length.
10