Fermions Tunnelling from Black Holes

Abstract

We investigate the tunnelling of spin 1/2 particles through event horizons. We first apply the tunnelling method to Rindler spacetime and obtain the Unruh temperature. We then apply fermion tunnelling to a general non-rotating black hole metric and show that the Hawking temperature is recovered.

Full text

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arXiv:0710.0612v5 [hep-th] 17 Mar 2008
Fermions Tunnelling from Black Holes
Ryan Kerner∗and R.B. Mann†
Department of Physics & Astronomy, University of Waterloo
Waterloo, Ontario N2L 3G1, Canada
March 17, 2008
Abstract
We investigate the tunnelling of spin 1/2 particles through event hori-
zons. We first apply the tunnelling method to Rindler spacetime and
obtain the Unruh temperature.We then apply fermion tunnelling to a
general non-rotating black hole metric and show that the Hawking tem-
perature is recovered.
1 Introduction
In recent years, a semi-classical method of modeling Hawking radiation as a
tunneling effect has been developed and has garnered a lot of interest [1]-[20].
The earliest work with black hole tunnelling was done by Kraus and Wilczek
[1], an approach that was subsequently refined by various researchers [2, 3, 4].
From this emerged an alternative way of understanding black hole radiation. In
particular one can calculate the Hawking temperature in a manner independent
of traditional Wick Rotation methods or Hawking’s originalmethod of modelling
gravitational collapse [21]. Tunnelling provides not only a useful verification of
thermodynamic properties of black holes but also an alternate conceptual means
for understanding the underlying physical process of black hole radiation.
has been shown to be very robust, having been successfully applied to a wide
variety of exotic spacetimes such as Kerr and Kerr-Newmann cases [8, 9, 12],
black rings [10], the 3-dimensional BTZ black hole [5, 11], Vaidya [16], other
dynamical black holes [17], Taub-NUT spacetimes [12], and G¨ odel spacetimes
[20]. Tunnelling methods have even been applied to horizons that are not black
hole horizons, such as Rindler Spacetimes [4],[12] and it has been shown the
Unruh temperature [22] is in fact recovered.
In general the tunnelling methods involve calculating the imaginary part of
the action for the (classically forbidden) process of s-wave emission across the
horizon, which in turn is related to the Boltzmann factor for emission at the
It
∗rkerner@uwaterloo.ca
†rbmann@sciborg.uwaterloo.ca
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Hawking temperature.
calculate the imaginary part of the action for the emitted particle.
black hole tunnelling method developed was the Null Geodesic Method used
by Parikh and Wilczek [1] which followed from the work of Kraus and Wilczek
[1]. The other approach to black hole tunnelling is the Hamilton-Jacobi Ansatz
used by Agheben et al which is an extension of the complex path analysis of
Padmanabhan et al [4].Both of these approaches to tunnelling use the fact
that the WKB approximation of the tunneling probability for the classically
forbidden trajectory from inside to outside the horizon is given by:
There are two different approaches that are used to
The first
Γ ∝ exp(−2ImI) (1)
where I is the classical action of the trajectory to leading order in ℏ (here set
equal to unity). Where these two methods differ is in how the action is calcu-
lated. For the Null Geodesic method the only part of the action that contributes
an imaginary term is?rout
it is possible to calculate the imaginary part of the action. For the Hamilton-
Jacobi ansatz it is assumed that the action of the emitted scalar particle satisfies
the relativistic Hamilton-Jacobi equation. From the symmetries of the metric
one picks an appropriate ansatz for the form of the action and plugs it into the
Relativistic Hamilton-Jacobi Equation to solve. (For a detailed comparison of
the Hamilton-Jacobi Ansatz and Null-Geodesic methods see [12]).
Since a black hole has a well defined temperature it should radiate all types
of particles like a black body at that temperature (ignoring grey body effects).
The emission spectrum therefore is expected to contain particles of all spins;
the implications of this expectation were studied 30 years ago [23].
application of tunnelling methods themselves to date have only involved scalar
particles. Specifically there is no other black hole tunnelling calculation (to the
best of our knowledge) that models fermions tunnelling from the black hole.
In fact comparatively little has been done for fermion radiation for black holes.
The Hawking temperature for fermion radiation has been calculated for 2d black
holes [24] using the Bogoliubov transformation and more recently was calculated
for evaporating black holes using a technique called the generalized tortoise
coordinate transformation (GTCT) [25]-[27]. The latter result [27] is interesting
because there is a contribution to the fermion emission probability due to a
coupling effect between the spin of the emitted fermion and the acceleration of
the Kinnersley black hole. From this one may infer that when fermions are
emitted from rotating black holes that will be a coupling between the spin of
the fermion and angular momentum of the rotating black hole present in the
tunnelling probability.
In this paper we extend the tunnelling method to model spin 1/2 particle
emission from non-rotating black holes.
analogous approach to the original approach used by Padmanabhan et al [4].
The Hamilton Jacobi ansatz emerged from an application of the WKB approx-
imation to the Klein Gordon equation. We will start by reviewing this general
rin
prdr, where pris the momentum of the emitted null
s-wave. Then by using Hamilton’s equation and knowledge of the null geodesics
However
In order to do this we will follow an
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calculation, and then apply a WKB approximation to the Dirac Equation. We
consider Rindler spacetime first and confirm that the Unruh temperature is re-
covered. Insofar as fermionic vacua are distinct from bosonic vacua and can
lead to distinct physical results [28], this result is non-trivial. We then extend
this technique to general 4-D black hole metric and show the Hawking temper-
ature is recovered.We illustrate this result in several coordinate systems –
Schwarzschild, Painlev´ e, and Kruskal – to demonstrate that the result is inde-
pendent of this choice. This last system is particularly interesting since it has no
coordinate singularities at the horizon. That we obtain the expected Hawking
temperature indicates that tunnelling can be understood as a bona-fide physical
phenomenon.
One of the assumptions of our semi-classical calculation is to neglect any
change of angular momentum of the black hole due to the spin of the emitted
particle. For zero-angular momentum black holes with mass much larger than
the Planck mass this is a good approximation. Furthermore, statistically parti-
cles of opposite spin will be emitted in equal numbers, yielding no net change
in the angular momentum of the black hole (although second-order statistical
fluctuations will be present). We confirm that spin 1/2 fermions are also emitted
at the Hawking Temperature. This final result, while not surprising, furnishes
an important confirmation of the robustness of the tunnelling approach.
2Review of the Hamilton-Jacobi Ansatz
We will consider a general (non-extremal) black hole metric of the form:
ds2= −f(r)dt2+dr2
g(r)+ C(r)hijdxidxj
(2)
The Klein Gordon equation for a scalar field φ is:
gµν∂µ∂νφ −m2
?2φ = 0
Applying the WKB approximation by assuming an ansatz of the form
φ(t,r,xi) = exp[i
?I(t,r,xi) + I1(t,r,xi) + O(?)]
and then inserting this back into the Klein Gordon equation we get the usual
result of the Hamilton-Jacobi equation to the lowest order in ?:
−?gµν∂µI∂νI + m2?+ O(?) = 0
(obtained after dividing by the exponential term and multiplying by h2).
For our metric the Hamilton-Jacobi equation is explicitly
−(∂tI)2
f(r)
+ g(r)(∂rI)2+
hij
C(r)∂iI∂jI + m2= 0(3)
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for the black hole metric (2) where we neglect the effects of the self-gravitation
of the particle. There exists a solution of the form
I = −Et + W(r) + J(xi) + K (4)
where
∂tI = −E,∂rI = W′(r),∂iI = Ji
and K and the Ji’s are constant (K can be complex). Since ∂tis the timelike
killing vector for this coordinate system, E is the energy. Solving for W(r) yields
W±(r) = ±
?
dr
?f(r)g(r)
?
E2− f(r)(m2+hijJiJj
C(r)
) (5)
since the equation was quadratic in terms of W(r). One solution corresponds to
scalar particles moving away from the black hole (i.e. + outgoing) and the other
solution corresponds to particles moving toward the black hole (i.e. - incoming).
Imaginary parts of the action can only come due the pole at the horizon or from
the imaginary part of K. The probabilities of crossing the horizon each way are
proportional to
Prob[out]∝ exp[−2
exp[−2
?ImI] = exp[−2
?ImI] = exp[−2
?(ImW++ ImK)] (6)
Prob[in]∝
?(ImW−+ ImK)](7)
To ensure that the probability is normalized so that any incoming parti-
cles crossing the horizon have a 100% chance of entering the black hole we set
ImK = −ImW−and since W+= −W−this implies that the probability of a
particle tunnelling from inside to outside the horizon is:
Γ ∝ exp[−4
?ImW+] (8)
Henceforth we set ? to unity and also drop the “+” subscript from W.
Integrating around the pole at the horizon leads to the result [12]
W =
πiE
?g′(r0)f′(r0)
(9)
where the imaginary part of W is now manifest. This leads to a tunnelling
probability of:
Γ = exp[−
4π
?f′(r0)g′(r0)E]
and implies the usual Hawking temperature of:
TH=
?f′(r0)g′(r0)
4π
(10)
It can be shown [19] that the proper Hawking temperature is recovered for
multiple choices of the form of the metric for the same black hole.
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3 Spin 1/2 particles and Rindler Space
We first consider the Rindler spacetime, for which the tunnelling calculation of
a scalar field has shown [4],[12] that the Unruh temperature [22] is recovered.
We will only show the calculation explicitly for spin up case; the final result
is also the same for the spin down case as can be easily shown using the methods
described below. Due to the statistical nature of the heat bath we assume that
no angular momentum is imparted to the accelerating detector (i.e. on average
there are as many spin up particles as spin down particles detected).
fermionic heat bath as seen by accelerated observers has many applications,
such as understanding the effects of acceleration on entanglement [?].
We will use the following metric for Rindler spacetime
The
ds2
=−f(z)dt2+ dx2+ dy2+dz2
a2z2− 1
a2z2− 1
a2z2
g(z)
f(z)=
g(z)=
so chosen for its convenience in extending the technique to normal black holes.
The Dirac equation is:
iγµDµψ +m
?ψ = 0(11)
where:
Dµ
=∂µ+ Ωµ
1
2iΓα β
1
4i[γα,γβ]
Ωµ
=
µΣαβ
Σαβ
=
The γµmatrices satisfy {γµ,γν} = 2gµν×1. There are many different ways
to choose the γµmatrices and we will use the following chiral form:
γt
=
1
?f(z)
?
?
?
?
σ1
0
σ2
0
01
0−1
?
?
0
σ3
?
γx
=
0
σ1
γy
=
0
σ2
γz
=
g(z)
?
σ3
0
?
5