QUANTUM TUNNELING
IN
BLACK HOLES

Thesis submitted for the degree of
Doctor of Philosophy (Science)
of
University of Calcutta, India

December, 2010

Bibhas Ranjan Majhi
Department of Theoretical Sciences

$\mathcal{TO}$
$\mathcal{MY\ NEPHEW}$

## Acknowledgements

This thesis is the culminative outcome of four years work, which has been made
possible by the blessings and support of many individuals. I take this
opportunity to express my sincere gratitude to all of them.

First and foremost, I would like to thank Prof. Rabin Banerjee, my thesis
supervisor. His uncanny ability to select a particular problem, a keen and
strategic analysis of it and deep involvement among the students makes him
something special. Thank you Sir for giving me all that could last my entire
life.

In addition to this, I like to express my sincere thanks to Prof. Claus
Kiefer, Universität zu Köln, Germany and Dr. Elias C. Vagenas, Academy of
Athens, Greece for their constant academic help and stimulating collaboration
throughout my research activity. I thank Prof. Subir Ghosh, ISI Kolkata. It
was a quite nice experience to work with them.

I would also like to express my gratitude Prof. Sandip K. Chakrabarti, Dr.
Debashis Gangopadhyay, Dr. Archan S. Majumdar, Dr. Biswajit Chakraborty and
Prof. Subhrangshu Sekhar Manna for helping me in my academics at Satyendra
Nath National Center for Basic Sciences (SNBNCBS).

Prof. Jayanta Kumar Bhattacharjee and Prof. Binayak Dutta Roy have always been
available to clarify very elementary but at same time subtle concepts in
physics. I sincerely thank them for giving me their valuable attention.

I thank Prof. Arup K. Raychaudhuri, Director of SNBNCBS, for providing an
excellent academic atmosphere.

I am also thankful to the Library staff of SNBNCBS for their support.

I am indebted to Prof. Manoranjan Saha, Prof. Amitava Raychaudhuri, Prof.
Anirban Kundu, Dr. Debnarayana Jana, University of Calcutta, for giving me
crucial suggestions and guidance, whenever I required. I would also like to
thank Prof. Tapan Kumar Das , University of Calcutta and my college teachers
Dr. Debabrata Das, Prof. Maynak Gupta.

My life during this voyage has been made cherishable and interesting by some
colourful and memorable personalities to whom I would like to express my
heartiest feelings. I am indebted to my seniors - Chandrashekhar da and Mitali
di. Things, which I gained from them, turn out to be most important in my
thesis work. I would like to thank all of them for their brotherly support.
Saurav Samanta and Shailesh Kulkarni, seniors and friends, deserve special
mention for those strategic discussions, on both the academic as well as non-
academic fronts. Then come my brilliant and helpful group mates, ranging from
enthusiastic Sujoy, Debraj, Sumit, Dibakar to Biswajit. I am grateful to
Himadri for his sometimes meaningless but still delightful gossips and also to
his group for sharing precious moments. I would also like to thank all the
cricket and football players of SNBNCBS for making my stay enjoyable.

I owe my deepest gratitude to each and every member of my of family.
Especially, I would like to express my heartiest love and respect to my
parents for their constant care, encouragement and blessings. Another two
pairs of my family, who stands in the same footing as my parents, are my uncle
(Jethu) and aunt (Jethima) and my would be father-in-law and mother-in-law. I
give paranams to all. I express my love to my elder brother Pinku and elder
sisters Tinku, Rinku and my brother-in-law, Subikash and sister-in-law, Dona
for understanding and helping me, on many occasions.

I am indebted to my better halves: my grandmothers (Didima and Thakurma) and
my beloved nephew, Oishik (Sona) for making my life fill of joy and greetings.
Finally, I express my heartiest love to Priyanka (Mousam, my would be wife)
with whom I shared all my moments (sweet or bitter) for making everything
possible from my post-graduate life.

## List of publications

  1. 1.

Gauge Theories on A(dS) space and Killing Vectors.
Rabin Banerjee and Bibhas Ranjan Majhi
Annals Phys. 323, 705 (2008) [arXiv:hep-th/0703207].

  2. 2.

Crypto-Harmonic Oscillator in Higher Dimensions: Classical and Quantum
Aspects.
Subir Ghosh and Bibhas Ranjan Majhi
J. Phys. A41, 065306 (2008) [arXiv:0709.4325].

  3. 3.

Quantum Tunneling and Back Reaction.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B662, 62 (2008) [arXiv:0801.0200].

  4. 4.

Noncommutative Black Hole Thermodynamics.
Rabin Banerjee, Bibhas Ranjan Majhi and Saurav Samanta
Phys. Rev. D77, 124035 (2008) [arXiv:0801.3583].

  5. 5.

Noncommutative Schwarzschild Black Hole and Area Law.
Rabin Banerjee, Bibhas Ranjan Majhi and Sujoy Kumar Modak
Class. Quant. Grav. 26, 085010 (2009) [arXiv:0802.2176].

  6. 6.

Quantum Tunneling Beyond Semiclassical Approximation.
Rabin Banerjee and Bibhas Ranjan Majhi
JHEP 0806, 095 (2008) [arXiv:0805.2220].

  7. 7.

Quantum Tunneling and Trace Anomaly.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B674, 218 (2009) [arXiv:0808.3688].

  8. 8.

Fermion Tunneling Beyond Semiclassical Approximation.
Bibhas Ranjan Majhi
Phys. Rev. D79, 044005 (2009) [arXiv:0809.1508].

  9. 9.

Connecting anomaly and tunneling methods for Hawking effect through chirality.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Rev. D79, 064024 (2009) [arXiv:0812.0497].

  10. 10.

Hawking Radiation due to Photon and Gravitino Tunneling.
Bibhas Ranjan Majhi and Saurav Samanta
Annals Phys. 325, 2410 (2010) [arXiv:0901.2258].

  11. 11.

Hawking black body spectrum from tunneling mechanism.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B675, 243 (2009) [arXiv:0903.0250].

  12. 12.

Quantum tunneling and black hole spectroscopy.
Rabin Banerjee, Bibhas Ranjan Majhi and Elias C. Vagenas
Phys. Lett. B686, 279 (2010) [arXiv:0907.4271].

  13. 13.

Hawking radiation and black hole spectroscopy in Horava-Lifshitz gravity.
Bibhas Ranjan Majhi
Phys. Lett. B686, 49 (2010) [arXiv:0911.3239] .

  14. 14.

New Global Embedding Approach to Study Hawking and Unruh Effects.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B690, 83 (2010) [arXiv:1002.0985].

  15. 15.

Statistical Origin of Gravity.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Rev. D81, 124006 (2010) [arXiv:1003.2312].

  16. 16.

A Note on the Lower Bound of Black Hole Area Change in Tunneling Formalism.
Rabin Banerjee, Bibhas Ranjan Majhi and Elias C. Vagenas
Europhys. Lett. 92, 20001 (2010) [arXiv:1005.1499].

  17. 17.

Quantum gravitational correction to the Hawking temperature from the Lemaitre-
Tolman-Bondi model.
Rabin Banerjee, Claus Kiefer and Bibhas Ranjan Majhi
Phys. Rev. D82, 044013 (2010) [arXiv:1005.2264].

  18. 18.

Killing Symmetries and Smarr Formula for Black Holes in Arbitrary Dimensions.
Rabin Banerjee, Bibhas Ranjan Majhi, Sujoy Kumar Modak and Saurav Samanta
Phys. Rev. D82, 124002 (2010) [arXiv:1007.5204].

This thesis is based on the papers numbered by [3,4,6,9,11,12,14,15] whose
reprints are attached at the end of the thesis.

##

QUANTUM TUNNELING

IN

BLACK HOLES

###### Contents

  1. 1  Introduction
    1. 1.1 Overview
    2. 1.2  Outline of the thesis
  2. 2 The tunneling mechanism
    1. 2.1 Hamilton-Jacobi method
      1. 2.1.1 Schwarzschild like coordinate system
      2. 2.1.2 Painleve coordinate system
    2. 2.2 Radial null geodesic method
    3. 2.3 Calculation of Hawking temperature
      1. 2.3.1 Schwarzschild black hole
      2. 2.3.2 Kerr black hole
    4. 2.4 Discussions
  3. 2.A Ingoing and outgoing modes
  4. 3 Null geodesic approach
    1. 3.1 Back reaction effect
    2. 3.2 Inclusion of noncommutativity
      1. 3.2.1 Schwarzschild black hole in noncommutative space
      2. 3.2.2 Noncommutative Hawking temperature, tunneling rate and entropy in the presence of back reaction
    3. 3.3 Discussions
  5. 3.A Incomplete gamma function
  6. 3.B Some useful formulas
  7. 4 Tunneling mechanism and anomaly
    1. 4.1 Metric and null coordinates
    2. 4.2 Chirality conditions
    3. 4.3 Chirality, gravitational anomaly and Hawking flux
    4. 4.4 Chirality, quantum tunneling and Hawking temperature
    5. 4.5 Discussions
  8. 5 Black body spectrum from tunneling mechanism
    1. 5.1 Black body spectrum and Hawking flux
    2. 5.2 Discussions
  9. 6 Global embedding and Hawking-Unruh effect
    1. 6.1 Reduced global embedding
      1. 6.1.1 Schwarzschild metric
      2. 6.1.2 Reissner-Nordstr$\ddot{\textrm{o}}$m metric
      3. 6.1.3 Schwarzschild-AdS metric
    2. 6.2 Kerr-Newman metric
    3. 6.3 Conclusion
  10. 6.A Dimensional reduction technique
  11. 7 Quantum tunneling and black hole spectroscopy
    1. 7.1 Near horizon modes
    2. 7.2 Entropy and area spectrum
    3. 7.3 Discussions
  12. 8 Statistical origin of gravity
    1. 8.1 Partition function and the relation $S_{bh}=\frac{E}{2T_{H}}$
    2. 8.2 Identification of $E$ in Einstein’s gravity
    3. 8.3 Discussions
  13. 9 Conclusions

##  Chapter 1  Introduction

###  1.1 Overview

This is a short overview of the vast subject of black holes. It specifically
highlights those issues which are relevant for the present thesis. The search
for a theory of quantum gravity drives a great deal of research in theoretical
physics today, and much has been learned along the way, but convincing success
remains elusive. There are two parts of general relativity: the framework of
space-time curvature and its influence on matter, and the dynamics of the
metric in response to energy-momentum (as described by Einstein’s equation).
Lacking the true theory of quantum gravity, we may still take the first part
of GR - the idea that matter fields propagate on a curved space-time
background - and consider the case where those matter fields are quantum
mechanical. In other words, we take the metric to be fixed, rather than
obeying some dynamical equations, and study quantum field theory in the curved
space-time.

Classical solutions of Einstein’s equation gives several metrics of space-time
in absence (Schwarzschild metric) or in presence (e.g. Reissner-Nordstrom
metric) of matter fields. Both of these solutions show there exists a region
of space-time in which information can enter, but nothing can come out from
it. The partition that separates this region (known as black hole) is usually
called the event horizon. The black holes are usually formed from the collapse
of star etc. According to the No Hair theorem, collapse leads to a black hole
endowed with small number of macroscopic parameters (mass, charge, angular
momentum) with no other free parameters. All these are classical pictures.

Hawking showed that the area of a black hole never decreases \- known as area
theorem [1]. This fact attracted Bekenstein a lot. A simple thought experiment
led him to associate entropy with the black hole. Then he [2] proposed that a
black hole has an entropy $S_{bh}$ which is some finite multiple $\eta$ of its
area of the event horizon $A$. He was not able to determine the exact value of
$\eta$, but gave heuristic arguments for conjecturing that it was
$\frac{ln2}{8\pi}$. Also, several investigations reveled that classical black
hole mechanics can be summarized by the following three basic laws [3],

  1. 1.

Zeroth law : The surface gravity $\kappa$ of a black hole is constant on the
horizon.

  2. 2.

First law : The variations in the black hole parameters, i.e mass $M$, area
$A$, angular momentum $L$, and charge $Q$, obey

$\delta M=\frac{\kappa}{8\pi}\delta A+\Omega\delta L-V\delta Q$ (1.1)

where $\Omega$ and $V$ are the angular velocity and the electrostatic
potential, respectively.

  3. 3.

Second law : The area of a black hole horizon $A$ is nondecreasing in time
[1],

$\delta A\geq 0.$ (1.2)

These laws have a close resemblance to the corresponding laws of
thermodynamics. The zeroth law of thermodynamics says that the temperature is
constant throughout a system in thermal equilibrium. The first law states that
in small variations between equilibrium configurations of a system, the
changes in the energy and entropy of the system obey equation (1.1), if the
surface gravity $\kappa$ is replaced by a term proportional to temperature of
the system (other terms on the right hand side are interpreted as work terms).
The second law of thermodynamics states that, for a closed system, entropy
always increases in any (irreversible or reversible) process. Therefore from
Bekenstein’s argument and the first law of black hole mechanics one might say
$T_{H}=\epsilon\kappa$ and $S_{bh}=\eta A$ with $8\pi\eta\epsilon=1$.
Bekenstein proposed that $\eta$ is finite and it is equal to
$\frac{ln2}{8\pi}$. Then one would get $\epsilon=\frac{1}{ln2}$ and so
$T_{H}=\frac{\kappa}{ln2}$.

Later on, the study of QFT in curved space-time by Hawking in 1974-75 [4, 5]
showed that black holes are not really black, instead emit thermal radiation
at temperature ($T_{H}$) proportional to surface gravity ($\kappa$) of black
hole - popularly known as Hawking effect. The exact expression was found to be
[5]:

$\displaystyle T_{H}=\frac{\hbar c\kappa}{2\pi k_{B}},$ (1.3)

where $c$, $\hbar$ and $k_{B}$ are respectively the velocity of light, plank
constant and Boltzmann constant. This is known as Hawking temperature
111Although in (1.3) we keep all the fundamental constants explicitly, for
later analysis, whenever any particular unit will be chosen, that will be
mentioned there.. For the Schwarzschild black hole $\kappa=\frac{c^{2}}{4GM}$
where $M$ is the mass of the black hole and $G$ is the gravitational constant.
All these reflects the fact that Hawking effect incorporates quantum
mechanics, gravity as well as thermodynamics. The key idea behind quantum
particle production in curved space-time is that the definition of a particle
is vacuum dependent. It depends on the choice of reference frame. Since the
theory is generally covariant, any time coordinate, possibly defined only
locally within a patch, is a legitimate choice with which to define positive
and negative frequency modes. Hawking considered a massless quantum scalar
field moving in the background of a collapsing star. If the quantum field was
initially in the vacuum state (no particle state) defined in the asymptotic
past, then at late times it will appear as if particles are present in that
state. Hawking showed [5], by explicit computation of the Bogoliubov
coefficients (see also [6, 7] for detailed calculation of Bogoliubov
coefficients) between the two sets of vacuum states defined at asymptotic past
and future respectively, that the spectrum of the emitted particles is
identical to that of black body with the temperature (1.3). This remarkable
discovery indeed helps us to get various physical information about the
classically forbidden region inside the horizon. Since then people thought
that the black holes may play a major role in the attempts to shed some light
on the nature of quantum theory of gravity as the role played by atoms in
early development of quantum mechanics. Hence QFT on curved space-time and
Hawking effect attracted the physicists for their beauty and usefulness in
various aspects.

Hawking then realised that Bekenstein’s idea was consistent. In fact, since
the black hole temperature is given by (1.3), $\epsilon=\frac{1}{2\pi}$ and
hence $\eta=\frac{1}{4}$. This leads to the famous Bekenstein-Hawking area law
for entropy of black hole

$\displaystyle S_{bh}=\frac{A}{4},$ (1.4)

where $A$ is the area of the event horizon 222Here all the fundamental
constants are chosen to be unity.. This astonishing result is obtained using
the approximation that the matter field behaves quantum mechanically but the
gravitational field (metric) satisfy the classical Einstein equation. This
semi-classical approximation holds good for energies below the Planck scale
[5]. Although it is a semi-classical result, Hawking’s computation is
considered an important clue in the search for a theory of quantum gravity.
Any theory of quantum gravity that is proposed must predict black hole
evaporation.

Apart from Hawking’s original calculation there are other semi-classical
approaches. We summarise these briefly. S. Hawking and G. Gibbons, in 1977 [8]
developed an approach based on the Euclidean quantum gravity. In this approach
they computed an action for gravitational field, including the boundary term,
on the complexified space-time. The purely imaginary values of this action
gives a contribution of the metrics to the partition function for a grand
canonical ensemble at Hawking temperature (1.3). Using this, they were able to
show that the entropy associated with these metrics is always equal to (1.4).
Almost at the same time, Christensen and Fulling [9], by exploiting the
structure of trace anomaly, were able to obtain the expectation value for each
component of the stress tensor $\langle T_{\mu\nu}\rangle$, which eventually
lead to the Hawking flux. This approach is exact in $(1+1)$ dimensions,
however in $3+1$ dimensions, the requirements of spherical symmetry, time
independence and covariant conservation are not sufficient to fix completely
the flux of Hawking radiation in terms of the trace anomaly [6, 9]. There is
an additional arbitrariness in the expectation values of the angular
components of the stress tensor.

Later on, S. Robinson and F. Wilczek [10, 11, 12] gave a new approach to
compute the Hawking flux from a black hole. This approach is based on gauge
and gravitational or diffeomorphism anomalies. Basic and essential fact used
in their analysis is that the theory of matter fields (scalar or fermionic) in
the $3+1$ dimensional static black hole background can effectively be
represented, in the vicinity of event horizon, by an infinite collection of
free massless $1+1$ dimensional fields, each propagating in the background of
an effective metric given by the $r-t$ sector of full $3+1$ dimensional metric
333Such a dimensional reduction of matter fields has been already used in the
analysis of [13, 14] to compute the entropy of $2+1$ dimensional $BTZ$ black
hole.. By definition the horizon is a null surface and hence the region inside
it is causally disconnected from the exterior. Thus, in the region near to the
horizon the modes which are going into the black hole do not affect the
physics outside the horizon. In other words, the theory near the event horizon
acquires a definite chirality. Any two dimensional chiral theory in general
curved background possesses both gauge and gravitational anomaly [15]. This
anomaly is manifested in the nonconservation of the current or the stress
tensor. The theory far away from the event horizon is $3+1$ dimensional and
anomaly free and the stress tensor in this region satisfies the usual
conservation law. Consequently, the total energy-momentum tensor, which is a
sum of two contribution from the two different regions, is also anomalous.
However, it becomes anomaly free once we take into account the contribution
from classically irrelevant ingoing modes. This imposes restrictions on the
structure of the energy-momentum tensor and is ultimately responsible for the
Hawking radiation [10]. The expression for energy-momentum flux obtained by
this anomaly cancellation approach is in exact agreement with the flux from
the perfectly black body kept at Hawking temperature [10]. In this approach
they used consistent expression for anomaly (satisfying Wess-Zumino
consistency condition) but used a covariant boundary condition.

Recently, a technically simple (only one Ward identity) and conceptually
cleaner (covariant expression for anomaly with covariant BC) derivation of
Hawking flux was introduced by Banerjee and Kulkarni [16, 17]. In addition to
this, a new method [18], to obtain the Hawking flux using chiral effective
action, was put forwarded by them. In all these approaches, the covariant
boundary condition is applied by hand. Later on, it was shown again by them
that such a boundary condition is compatible with the choice of Unruh vacuum
[19]. The connection of the diffeomorphism anomaly approach with the earlier
trace anomaly approach [9] was also elaborated [20, 21].

Interestingly, none of the existing approaches to study Hawking effect,
however, corresponds directly to one of the heuristic pictures that visualises
the source of radiation as tunneling, first stated in [5]. Later on, this
picture was mathematically introduced to discuss the Hawking effect [22, 23].
This picture is similar to an electron-positron pair creation in a constant
electric field. The idea is that pair production occurs inside the event
horizon of a black hole. One member of the pair corresponds to the ingoing
mode and other member corresponds to the outgoing mode. The outgoing mode is
allowed to follow classically forbidden trajectories, by starting just behind
the horizon onward to infinity. So this mode travels back in time, since the
horizon is locally to the future of the external region. The actual physical
picture is that the tunneling occures by the shrinking of the horizon so that
the particle effectively moves out. Thus the classical one particle action
becomes complex and so the tunneling amplitude is governed by the imaginary
part of this action for the outgoing mode. However, the action for the ingoing
mode must be real, since classically a particle can fall behind the horizon.
This is an important point of this mechanism as will be seen later. Also,
since it is a near horizon theory and the tunneling occures radially, the
phenomenon is effectively dominated by the two dimensional ($t-r$) metric.
This follows form the fact that near the horizon all the angular part can be
neglected and the solution of the field equation corresponds to angular
quantum number $l=0$ which is known as $s$-wave [22]. Hence, the essence of
tunneling based calculations is, thus, the computation of the imaginary part
of the action for the process of $s$-wave emission across the horizon, which
in turn is related to the Boltzmann factor for the emission at the Hawking
temperature. It also reveals that the presence of the event horizon is
necessary and the Hawking effect is a completely quantum mechanical
phenomenon. There are two different methods in literature to calculate the
imaginary part of the action: one is by Srinivasan et al [22] \- the Hamilton-
Jacobi (HJ) method 444For more elaborative discussions and further development
on HJ method see [24, 25, 26]. and another is radial null geodesic method
which was first given by Parikh - Wilczek [23] 555To find the basis of this
method see [27, 28, 29].. Both these approaces will be discussed in this
thesis.

Historically, another phenomenon was discovered by Unruh [30] \- Known as
Unruh effect \- in an attempt to understand the physics underlying the Hawking
effect [5]. The basic idea of the Unruh effect is based on the equivalence
principle \- locally gravitational effect can be ignored by choosing a
uniformly accelerated frame and the observers with different notions of
positive and negative frequency modes will disagree on the particle content of
a given state. A uniformly accelerated observer on the Minkowski space-time
percives a horizon. The space-time seen by the observer is known as Rindler
space-time and so the observer is usually called as the Rindler observer.
Although, an inertial observer would describe the Minkowski vacuum as being
completely empty, the Rindler observer will detect particles in that vacuum. A
detailed calculation tells that the emission spectrum exactly matches with
that of the black body with the temperature given by [30],

$\displaystyle T=\frac{{\hbar{\tilde{a}}}}{2\pi}$ (1.5)

where $a$ is the accleration of the Rindler observer. The similarity with
Hawking temperature is obvious with $a\rightarrow\kappa$. It is now well
understood that Hawking effect is related to the event horizon of a black hole
intrinsic to the space-time geometry while Unruh effect connects the horizon
associated with a uniformly accelerated observer on the Minkowski space-time.

A unified description of them was first put forward by Deser and Levin [31,
32] followed from an earlier attempt [33]. This is called the global embedding
Minkowskian space (GEMS) approach. In this approach, the relevant detector in
curved space-time (namely Hawking detector) and its event horizon map to the
Rindler detector in the corresponding higher dimensional flat embedding space
[34, 35] and its event horizon. Then identifying the acceleration of the Unruh
detector and using (1.5), the Unruh temperature (or local Hawking temperature)
was calculated. Finally, use of the Tolman relation [36] yields the Hawking
temperature. Subsequently, this unified approach to determine the Hawking
temperature using the Unruh effect was applied for several black hole space-
times [37, 38, 39]. However the results were confined to four dimensions and
the calculations were done case by case, taking specific black hole metrics.
It was not clear whether the technique was applicable to complicated examples
like the Kerr-Newman metric which lacks spherical symmetry.

In the mean time, after the discovery of Hawking effect, it was believed that
the black holes may give some hints to find the quantum theory of gravity. It
is then natural to consider quantization of a black hole. This was first
pioneered by Bekenstein [40, 41]. The idea was based on the remarkable
observation that the horizon area of a non-extremal black hole behaves as a
classical adiabatic invariant quantity. In the spirit of the Ehrenfest
principle, any classical adiabatic invariant corresponds to a quantum entity
with discrete spectrum, Bekenstein conjectured that the horizon area of a non-
extremal black hole should have a discrete eigenvalue spectrum. To elucidate
the spacing of the area levels he used Christodoulou’s reversible process [42]
\- the assimilation of a neutral point particle by a non-extremal black hole.
Bekenstein pointed out that the limit of a point particle is not a legal one
in quantum theory. Because, according to the Heisenberg’s uncertainty
principle, the particle cannot be both at the horizon and at a turning point
of its motion. Considering a finite size of the particle - not smaller than
the Compton wavelength - he found a lower bound on the increase in the black
hole surface area [2, 43]:

$\displaystyle(\Delta A)_{min}=8\pi l_{p}^{2}$ (1.6)

where $l_{p}=(\frac{G\hbar}{c^{3}})^{1/2}$ is the Planck length (we use
gravitational units in which $G=c=1$). The independence of the black hole
parameters in the lower bound shows its universality and hence it is a strong
evidence in favor of a uniformly spaced area spectrum for a quantum black
holes.

These ideas led to a new research direction; namely the derivation of the area
and thus the entropy spectrum of black holes utilizing the quasinormal modes
(QNM) of black holes [44]. According to this method, since QNM frequencies are
the characteristic of the black hole itself, the latter must have an adiabatic
invariant quantity. Its form is given by energy of the black hole divided by
this frequency, as happens in classical mechanics. Hod showed for
Schwarzschild black hole that if one considers the real part of the QNM
frequency only, then this adiabatic invariant quantity is actually related to
area of the black hole horizon. Now use of Bohr-Sommerfield quantization rule
gives the spectrum for the area which is equispaced. Then by the well known
Bekenstein-Hawking area law, the entropy spectrum is obtained. In this case
the spacing of this entropy spectrum is given by $\Delta S_{bh}=\ln 3$.
Another significant attempt was to fix the Immirzi parameter in the framework
of Loop Quantum Gravity [45] but it was unsuccessful [46]. Later on Kunstatter
[47] gave an explicit form of the adiabatic invariant quantity for the black
hole:

$\displaystyle I_{adiab}=\int\frac{dW}{\Delta f(W)},\,\,\,\ \Delta
f=f_{n+1}-f_{n}$ (1.7)

where ‘$W$’ and ‘$f$’ are the energy and the frequency of the QNM
respectively. The Borh-Sommerfield quantization rule is given by,

$\displaystyle I_{adiab}=n\hbar$ (1.8)

which is valid for semi-classical (large $n$) limit. For the real part of the
frequency of QNM, (1.7) can be shown to be related to black hole entropy
which, ultimately by (1.8), yields the entropy spectrum. For Schwarzschild
black hole it yields the same spacing as obtained by Hod [44]. This, however,
disagrees with Bekenstein’s result, $\Delta S_{bh}=2\pi$ [2]. In a recent work
[48], Maggiore told that a black hole behaves like a damped harmonic
oscillator whose frequency is given by
$f=(f_{R}^{2}+f_{I}^{2})^{\frac{1}{2}}$, where $f_{R}$ and $f_{I}$ are the
real and imaginary parts of the frequency of the QNM. In the large $n$ limit
$f_{I}>>f_{R}$. Consequently one has to use $f_{I}$ rather than $f_{R}$ in the
adiabatic quantity (1.7). It then leads to Bekenstein’s result. With this new
interpretation, entropy spectrum for the most general black hole has been
calculated in [49], which leads to an identical conclusion. In addition, it
has been tested that the entropy spectrum is equidistance even for more
general gravity theory (e.g. Einstein-Gauss-Bonnet theory), but that of area
is not alaways equispaced, particularly, if the entropy is not proportional to
area [50]. In this sense quantization of entropy is more fundamental than that
of area.

A universal feature for black hole solutions, in a wide class of theories, is
that the notions of entropy and temperature can be attributed to them [2, 4,
3, 51]. Also, of all forces of nature gravity is clearly the most universal.
Gravity influences and is influenced by everything that carries an energy, and
is intimately connected with the structure of space-time. The universal nature
of gravity is also demonstrated by the fact that its basic equations closely
resemble the laws of thermodynamics [3, 51, 52, 53]. So far, there has not
been a clear explanation for this resemblance. Gravity is also considerably
harder to combine with quantum mechanics than all the other forces. The quest
for unification of gravity with these other forces of nature, at a microscopic
level, may therefore not be the right approach. It is known to lead to many
problems, paradoxes and puzzles. Many physicists believe that gravity and
space-time geometry are emergent. Also string theory and its related
developments have given several indications in this direction. Particularly
important clues come from the AdS/CFT correspondence. This correspondence
leads to a duality between theories that contain gravity and those that don’t.
It therfore provides evidence for the fact that gravity can emerge from a
microscopic description that doesn’t know about its existence 666Such a
prediction was first given long ago by Sakharov [54].. The universality of
gravity suggests that its emergence should be understood from general
principles that are independent of the specific details of the underlying
microscopic theory.

###  1.2  Outline of the thesis

This thesis, based on the work [55, 56, 57, 58, 59, 60, 61, 62], is focussed
towards the applications of field theory, classical as well as quantum, to
study black holes – mainly the Hawking effect. This is discussed by the
quantum tunneling mechanism. Here we give a general frame work of the existing
tunneling mechanism, both the radial null geodesic and Hamilton – Jacobi
methods. On the radial null geodesic method side, we study the modifications
to the tunneling rate, Hawking temperature and the Bekenstein-Hawking area law
by including the back reaction as well as non-commutative effects in the
space-time.

A major part of the thesis is devoted to the different aspects of the
Hamilton-Jacobi (HJ) method. A reformulation of this method is first
introduced. Based on this, a close connection between the quantum tunneling
and the gravitational anomaly mechanisms to discuss Hawking effect, is put
forwarded. An interesting advantage of this reformulated HJ method is that one
can get directly the emission spectrum from the event horizon of the black
hole, which was missing in the earlier literature. Also, the quantization of
the entropy and area of a black hole is discussed in this method.

Another part of the thesis is the introduction of a new type of global
embedding of curved space-time to higher dimensional Minkowskian space-time
(GEMS). Using this a unified description of the Hawking and Unruh effects is
given. Advantage of this approach is, it simplifies as well as generalises the
conventional embedding. In addition to the spherically symmetric space-times,
the Kerr-Newman black hole is exemplified.

Finally, following the above ideas and the definition of partition function
for gravity, it is shown that extremization of entropy leads to the Einstein’s
equations of motion. In this frame work, a relation between the entropy,
energy and the temperature of a black hole is given where energy is shown to
be the Komar expression. Interestingly, this relation is the generalized Smarr
formula. In this analysis, the GEMS method provides the law of equipartition
of energy as an intermediate step.

The whole thesis is consists of $9$ \- chapters, including this introductory
part. Chapter wise summary is given below.

Chapter -2: The tunneling mechanism: In this chapter, we present a general
framework of tunneling mechanism within the semi-classical approximation. The
black hole is considered to be a general static, spherically symmetric one.
First, the HJ method is discussed both in Schwarzschild like coordinates and
Painleve coordinates. Then a general methodology of the radial null geodesic
method is presented. Here the tunneling rate, which is related to the
imaginary part of the action, is shown to be equal to the exponential of the
entropy change of the black hole. In both the methods, a general expression
for Hawking temperature is obtained, which ultimately reduces to the Hawking
expression (1.3). Finally, using this general expression, calculation of
Hawking temperature for some particular black hole metrics, is explicitly
done. Chapter -3: Null geodesic approach: In this chapter, we provide an
application of the general frame work, discussed in the previous chapter, for
the radial null geodesic method, to incorporate back reaction as well as
noncommutative effects in the space-time. Here the main motivation is to find
their effects on the thermodynamic quantities. First, starting from a modified
surface gravity of a black hole due to one loop back reaction effect, the
tunneling rate is obtained. From this, the temperature and the area law are
derived. The semi-classical Hawking temperature is altered. Interestingly, the
leading order correction to the area law is logarithmic of the horizon area of
the black hole while the non-leading corrections are the inverse powers of the
area. The coefficient of the logarithmic term is related to the trace anomaly.
Similar type of corrections were also obtained earlier [63, 64, 65, 66, 67,
68, 69, 70, 71] by different methods.

Next, we shall apply our general formulation to discuss various thermodynamic
properties of a black hole defined in a noncommutative Schwarzschild space
time where back reaction is also taken into account. In particular, we are
interested in the black hole temperature when the radius is very small. Such a
study is relevant because noncommutativity is supposed to remove the so called
“information paradox” where for a standard black hole, temperature diverges as
the radius shrinks to zero. The Hawking temperature is obtained in a closed
form that includes corrections due to noncommutativity and back reaction.
These corrections are such that, in some examples, the “information paradox”
is avoided. Expressions for the entropy and tunneling rate are also found for
the leading order in the noncommutative parameter. Furthermore, in the absence
of back reaction, we show that the entropy and area are algebraically related
in the same manner as occurs in the standard Bekenstein-Hawking area law.

Chapter -4: Tunneling mechanism and anomaly: Several existing methods to study
Hawking effect yield similar results. The universality of this phenomenon
naturally tempts us to find the underlying mechanism which unifies the
different approaches. Recently, two widely used approaches – gravitational
anomaly method and quantum tunneling method – can be described in a unified
picture, since these two have several similarities in their techniques. One of
the most important and crucial step in the tunneling approach (in both the
methods) is that the tunneling of the particle occurs radially and its a near
horizon phenomenon. This enforces that only the near horizon ($t-r$) sector of
the original metric is relevant. Also, the ingoing mode is completely trapped
inside the horizon. Similar step is also invoked in the gravitational (chiral)
anomaly approach [10, 11, 12, 16]. Here, since near the event horizon the
theory is dominated by the two dimensional, ($t-r$) sector of the metric, and
the ingoing mode is trapped inside the horizon, the theory is chiral. Hence
one should has the gravitational anomaly in the quantum level. Therefore, one
might thought that these two approaches - quantum tunneling and anomaly
methods - can be discussed in an unified picture.

We begin this exercise by introducing the chirality conditions on the modes
and the energy-momentum tensor in chapter-4. The Klein-Gordon equation under
the effective ($t-r$) sector of the original metric shows that the there exits
a general solution which is a linear combination of two solutions. One is left
moving and function of only one null tortoise coordinate ($v$) while other is
right moving which is function of the other null tortoise coordinate ($u$).
From this information it is easy to find the chirality conditions. Then use of
these conditions on the usual expressions for the anomaly in the non-chiral
theory in two dimensions leads to the chiral anomaly expression. Finally,
following the approach by Banerjee et al [16], it is easy to find the
expression for the Hawking flux. Another portion of this chapter is dedicated
to show that the same chirality conditions are enough to find the Hawking
temperature in quantum tunneling method. First, the Hamilton-Jacobi equations
are obtained from these conditions, which are derived in the usual analysis
from the field equations. Then a reformulation of tunneling method is given in
which the trapping of the left mode is automatically satisfied. The right mode
tunnels through the horizon with a finite probability which is exactly the
Boltzmann factor. This immediately leads to the Hawking temperature. Thus,
this analysis reflects the crucial role of the chirality to give a unified
description of both the approaches to discuss Hawking effect.

Chapter -5: Black body spectrum from tunneling mechanism: So far, in the
tunneling mechanism only the Hawking temperature was obtained by comparing the
tunneling rate with the Boltzmann factor. The discussion of the emission
spectrum is absent and hence it is not clear whether this temperature really
corresponds to the emission spectrum from the black hole event horizon. This
shortcoming is addressed in chapter -5. Following the modified tunneling
approach, introduced in the previous chapter, the reduced density matrix for
the outgoing particles, as seen from the asymptotic observer, is constructed.
Then determination of the average number of outgoing particles yields the Bose
or Fermi distribution depending on the nature of the particles produced inside
the horizon. The distributions come out to be exactly similar to those in the
case of black body radiation. It is now easy to identify the temperature
corresponding to the emission spectrum. The temperature here we obtain is just
the Hawking expression. Thereby we provide a complete description of the
Hawking effect in the tunneling mechanism.

Chapter -6: Global embedding and Hawking - Unruh Effect: After Hawking’s
discovery, Unruh showed that an uniformly accelerated observer on the
Minkowski space-time sees a thermal radiation from the Minkowski vacuum. Later
on, Levin and Deser gave a unified picture of these two effects by using the
globally embedding of the curved space-time in the higher dimensional
Minkowski space-time. Such an interesting analysis was done using the
embedding of the full curved metric and was confined within the spherically
symmetric black hole space-time. The main difficulty to discuss for more
general space-times is the finding of the embeddings.

This issue is addressed in chapter - 6. Since, the thermodynamic quantities of
a black hole are determined by the horizon properties and near the horizon the
effective theory is dominated by the two dimensional ($t-r$) metric, it is
sufficient to consider the embedding of this two dimensional metric.
Considering this fact, a new type of global embedding of curved space-times in
higher dimensional flat ones is introduced to present a unified description of
Hawking and Unruh effects. Our analysis simplifies as well as generalises the
conventional embedding approach.

Chapter -7: Quantum tunneling and black hole spectroscopy: The entropy-area
spectrum of a black hole has been a long-standing and challenging problem. In
chapter - 7, based on the modified tunneling mechanism, introduced in the
previous chapters, we obtain the entropy spectrum of a black hole. In
Einstein’s gravity, we show that both entropy and area spectrum are evenly
spaced. But in more general theories (like Einstein-Gauss-Bonnet gravity),
although the entropy spectrum is equispaced, the corresponding area spectrum
is not. In this sense, quantization of entropy is more fundamental than that
of area.

Chapter -8: Statistical origin of gravity: Based on the above conceptions and
findings, we explore in chapter - 8 an intriguing possibility that gravity can
be thought as an emergent phenomenon. Starting from the definition of entropy,
used in statistical mechanics, we show that it is proportional to the gravity
action. For a stationary black hole this entropy is expressed as
$S_{bh}=E/2T_{H}$, where $T_{H}$ is the Hawking temperature and $E$ is shown
to be the Komar energy. This relation is also compatible with the generalised
Smarr formula for mass.

Chapter -9: Conclusions: Finally, in chapter-9 we present our conclusion and
outlook.

##  Chapter 2 The tunneling mechanism

Classical general relativity gives the concept of black hole from which
nothing can escape. This picture was changed dramatically when Hawking [4, 5]
incorporated the quantum nature into this classical problem. In fact he showed
that black hole radiates a spectrum of particles which is quite analogous with
a thermal black body radiation, popularly known as Hawking effect. Thus
Hawking radiation emerges as a nontrivial consequence of combining gravity and
quantum mechanics. People then started thinking that this may give some
insight towards quantum nature of gravity. Since the original derivation,
based on the calculation of Bogoliubov coefficients in the asymptotic states,
was technically very involved, several derivations of Hawking radiation were
subsequently presented in the literature to give fresh insights. For example,
Path integral derivation [8], Trace anomaly approach [9] and chiral
(gravitational) anomaly approach [10, 11, 12, 16, 17, 18, 19], each having its
merits and demerits.

Interestingly, none of the existing approaches to study Hawking effect,
however, corresponds directly to one of the heuristic pictures that visualises
the source of radiation as tunneling. This picture is similar to an electron-
positron pair creation in a constant electric field. The idea is that pair
production occurs inside the event horizon of a black hole. One member of the
pair corresponds to the ingoing mode and other member corresponds to the
outgoing mode. The outgoing mode is allowed to follow classically forbidden
trajectories, by starting just behind the horizon onward to infinity. So this
mode travels back in time, since the horizon is locally to the future of the
external region. Unitarity is not violated since physically it is possible to
envisage the tunneling as the shrinking of the horizon forwarded in time
rather than the particle travelling backward in time [23]. The classical one
particle action becomes complex and so the tunneling amplitude is governed by
the imaginary part of this action for the outgoing mode. However, the action
for the ingoing mode must be real, since classically a particle can fall
behind the horizon. Another essential fact is that tunneling occurs radially
and it is a near horizon phenomenon where the theory is driven by only the
effective ($t-r$) metric [22]. Under this circumstance the solution of a field
equation corresponds to $l=0$ mode which is actually the $s$ \- wave. These
are all important points of this mechanism as will be seen later. The essence
of tunneling based calculations is, thus, the computation of the imaginary
part of the action for the process of $s$-wave emission across the horizon,
which in turn is related to the Boltzmann factor for the emission at the
Hawking temperature. Also, it reveals that the presence of the event horizon
is necessary and the Hawking effect is a completely quantum mechanical
phenomenon, determined by properties of the event horizon.

There are two different methods in the literature to calculate the imaginary
part of the action: one is by Parikh-Wilczek [23] \- radial null geodesic
method and another is the Hamilton-Jacobi (HJ) method which was first used by
Srinivasan et. al. [22]. Later, many people [72, 73] used the radial null
geodesic method as well as HJ method to find out the Hawking temperature for
different space-time metrics. Also, several issues and aspects of these
methods have been discussed extensively [74, 75, 76, 77, 78, 79, 80, 81, 82,
83, 84, 85, 86, 87, 88].

In this chapter, we will give a short review of both the HJ and radial null
geodesic methods. While most of the material is available in the rather
extensive literature [22, 23, 24, 25, 26, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 88, 89, 90, 91] on tunneling, there are some new insights
and clarifications. The organization of the chapter is the following. First we
will discuss the HJ method within a semi-classical approximation to find the
Hawking temperature both in Schwarzschild like coordinate system and Painleve
coordinate system. A general static, spherically symmetric black hole metric
will be considered. In the next section, the radial null geodesic method will
be introduced. A general derivation of the Hawking temperature of this black
hole will be presented. Both these expressions will be shown identical. Then
using this obtained expression, the Hawking temperature will be explicitly
calculated for some known black hole metrics. Final section will be devoted
for the concluding remarks.

###  2.1 Hamilton-Jacobi method

Usually, calculations of the Hawking temperature, based on the tunneling
formalism, for different black holes conform to the general formula
$T_{H}=\frac{\hbar\kappa}{2\pi}$. This relation is normally understood as a
consequence of the mapping of the second law of black hole thermodynamics
$dM=\frac{\kappa}{8\pi}dA$ with $dE=T_{H}dS_{bh}$, coupled with the
Bekenstein-Hawking area law $S_{bh}=\frac{A}{4\hbar}$.

Using the tunneling approach, we now present a derivation of
$T_{H}=\frac{\hbar\kappa}{2\pi}$ where neither the second law of black hole
thermodynamics nor the area law are required. In this sense our analysis is
general.

In this section we will briefly discuss about the HJ method [22] to find the
temperature of a black hole using the picture of Hawking radiation as quantum
tunneling. The analysis will be restricted to the semi-classical limit.
Equivalent results are obtained by using either the standard Schwarzschild
like coordinates or other types, as for instance, the Painleve ones. We
discuss both cases in this section.

####  2.1.1 Schwarzschild like coordinate system

First, we consider a general class of static (i.e. invariant under time
reversal as well as stationary), spherically symmetric space-time of the form

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}d\Omega^{2}$ (2.1)

where the horizon $r=r_{H}$ is given by $f(r_{H})=g(r_{H})=0$.

Let us consider a massless particle in the space-time (2.1) described by the
massless Klein-Gordon equation

$\displaystyle-\frac{\hbar^{2}}{\sqrt{-g}}\partial_{\mu}[g^{\mu\nu}\sqrt{-g}\partial_{\nu}]\phi=0~{}.$
(2.2)

Since tunneling across the event horizon occurs radially, only the radial
trajectories will be considered here. Also, it is an near horizon phenomenon
and so the theory is effectively dominated by the two dimensional ($r-t$)
sector of the full metric. Here the modes corresponds to angular quantum
number $l=0$, which is actually the $s$ \- wave [22, 10, 11]. In this regard,
only the $(r-t)$ sector of the metric (2.1) is important. Therefore under this
metric the Klein-Gordon equation reduces to

$\displaystyle-\frac{1}{\sqrt{f(r)g(r)}}\partial^{2}_{t}\phi+\frac{1}{2}\Big{(}f^{\prime}(r)\sqrt{\frac{g(r)}{f(r)}}+g^{\prime}(r)\sqrt{\frac{f(r)}{g(r)}}\Big{)}\partial_{r}\phi+\sqrt{f(r)g(r)}\partial_{r}^{2}\phi=0~{}.$
(2.3)

The semi-classical wave function satisfying the above equation is obtained by
making the standard ansatz for $\phi$ which is

$\displaystyle\phi(r,t)={\textrm{exp}}\Big{[}-\frac{i}{\hbar}S(r,t)\Big{]},$
(2.4)

where $S(r,t)$ is a function which will be expanded in powers of $\hbar$.
Substituting into the wave equation (2.3), we obtain

$\displaystyle\frac{i}{\sqrt{f(r)g(r)}}\Big{(}\frac{\partial S}{\partial
t}\Big{)}^{2}-i\sqrt{f(r)g(r)}\Big{(}\frac{\partial S}{\partial
r}\Big{)}^{2}-\frac{\hbar}{\sqrt{f(r)g(r)}}\frac{\partial^{2}S}{\partial
t^{2}}+\hbar\sqrt{f(r)g(r)}\frac{\partial^{2}S}{\partial r^{2}}$
$\displaystyle+\frac{\hbar}{2}\Big{(}\frac{\partial f(r)}{\partial
r}\sqrt{\frac{g(r)}{f(r)}}+\frac{\partial g(r)}{\partial
r}\sqrt{\frac{f(r)}{g(r)}}\Big{)}\frac{\partial S}{\partial r}=0~{}.$ (2.5)

Expanding $S(r,t)$ in a powers of $\hbar$, we find,

$\displaystyle S(r,t)$ $\displaystyle=$ $\displaystyle S_{0}(r,t)+\hbar
S_{1}(r,t)+\hbar^{2}S_{2}(r,t)+...........$ (2.6) $\displaystyle=$
$\displaystyle S_{0}(r,t)+\sum_{i}\hbar^{i}S_{i}(r,t).$

where $i=1,2,3,......$. In this expansion the terms from ${\cal{O}}(\hbar)$
onwards are treated as quantum corrections over the semi-classical value
$S_{0}$. Here, as mentioned earlier, we will restrict only upto the semi-
classical limit, i.e. $\hbar\rightarrow 0$. The effects due to inclusion of
higher order terms are discussed in [55, 82, 83, 84, 85, 86, 87, 88] 111For
extensive literature on the discussion of the higher order terms see [93, 94].

Substituting (2.6) in (2.5) and taking the semi-classical limit
$\hbar\rightarrow 0$, we obtain the following equation:

$\displaystyle\frac{\partial S_{0}}{\partial
t}=\pm\sqrt{f(r)g(r)}\frac{\partial S_{0}}{\partial r}~{}.$ (2.7)

This is the usual semi-classical Hamilton-Jacobi equation [22]. Now, to obtain
a solution for $S_{0}(r,t)$, we will proceed in the following manner. Since
the metric (2.1) is static it has a time-like Killing vector. Thus we will
look for a solution of (2.7) which behaves as

$\displaystyle S_{0}=\omega t+\tilde{S}_{0}(r),$ (2.8)

where $\omega$ is the conserved quantity corresponding to the time-like
Killing vector. This ultimately is identified as the energy of the particle as
seen by an observer at infinity. Substituting this in (2.7) and then
integrating we obtain,

$\displaystyle\tilde{S_{0}}(r)=\pm\omega\int\frac{dr}{\sqrt{f(r)g(r)}}$ (2.9)

where the limits of the integration are chosen such that the particle just
goes through the horizon $r=r_{H}$. So the one can take the range of
integration from $r=r_{H}-\epsilon$ to $r=r_{H}+\epsilon$, where $\epsilon$ is
a very small constant. The $+(-)$ sign in front of the integral indicates that
the particle is ingoing ($L$) (outgoing ($R$)) (For elaborate discussion to
determine the nature of the modes, see Appendix 2.A). Using (2.9) in (2.8) we
obtain

$\displaystyle S_{0}(r,t)=\omega t\pm\omega\int\frac{dr}{\sqrt{f(r)g(r)}}~{}.$
(2.10)

Therefore the ingoing and outgoing solutions of the Klein-Gordon equation
(2.2) under the back ground metric (2.1) is given by exploiting (2.4) and
(2.10),

$\displaystyle\phi^{(L)}={\textrm{exp}}\Big{[}-\frac{i}{\hbar}\Big{(}\omega
t+\omega\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$ (2.11)

and

$\displaystyle\phi^{(R)}={\textrm{exp}}\Big{[}-\frac{i}{\hbar}\Big{(}\omega
t-\omega\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}.$ (2.12)

In the rest of the analysis we will call $\phi^{(L)}$ as the left mode and
$\phi^{(R)}$ as the right mode.

A point we want to mention here that if one expresses the above modes in terms
of null coordinates ($u,v$), then $\phi^{(L)}$ becomes function of “$v$” only
while $\phi^{(R)}$ becomes that of “$u$”. These are call holomorphic modes.
Such modes satisfies chirality condition. This will be elaborated and used in
the later discussions.

Now for the tunneling of a particle across the horizon the nature of the
coordinates change. The time-like coordinate $t$ outside the horizon changes
to space-like coordinate inside the horizon and likewise for the outside
space-like coordinate $r$. This indicates that ‘$t$’ coordinate may have an
imaginary part on crossing the horizon of the black hole and correspondingly
there will be a temporal contribution to the probabilities for the ingoing and
outgoing particles along with the spacial part. This has similarity with [78]
where they show for the Schwarzschild metric that two patches across the
horizon are connected by a discrete imaginary amount of time.

The ingoing and outgoing probabilities of the particle are, therefore, given
by,

$\displaystyle
P^{(L)}=|\phi^{(L)}|^{2}={\textrm{exp}}\Big{[}\frac{2}{\hbar}(\omega{\textrm{Im}}~{}t+\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(2.13)

and

$\displaystyle
P^{(R)}=|\phi^{(R)}|^{2}={\textrm{exp}}\Big{[}\frac{2}{\hbar}\Big{(}\omega{\textrm{Im}}~{}t-\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(2.14)

Now the ingoing probability $P^{(L)}$ has to be unity in the classical limit
(i.e. $\hbar\rightarrow 0$) - when there is no reflection and everything is
absorbed - instead of zero or infinity [89].Thus, in the classical limit,
(2.13) leads to,

$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}~{}.$
(2.15)

It must be noted that the above relation satisfies the classical condition
$\frac{\partial S_{0}}{\partial\omega}=$ constant. This is understood by the
following argument. Calculating the left side of this condition from (2.10) we
obtain,

$\displaystyle t={\textrm{constant}}\mp\int\frac{dr}{\sqrt{f(r)g(r)}}$ (2.16)

where $-(+)$ sign indicates that the particle is ingoing ($L$) (outgoing
($R$)). So for an ingoing particle this condition immediately yields (2.15)
considering that “constant” is always real. On the other hand a naive
substitution of ‘Im$~{}t$’ in (2.14) from (2.16) for the outgoing particle
gives $P^{(R)}=1$. But it must be noted that according to classical general
theory of relativity, a particle can be absorbed in the black hole, while the
reverse process is forbidden. In this regard, ingoing classical trajectory
exists while the outgoing classical trajectory is forbidden. Hence use of the
classical condition for outgoing particle is meaningless.

Now to find out ‘Im$~{}t$’ for the outgoing particle we will take the help of
the Kruskal coordinates which are well behaved throughout the space-time. The
Kruskal time ($T$) and space ($X$) coordinates inside and outside the horizon
are defined in terms of Schwarzschild coordinates as [95]

$\displaystyle T_{in}=e^{\kappa r^{*}_{in}}\cosh\\!\left(\kappa
t_{in}\right)~{}~{};\hskip 17.22217ptX_{in}=e^{\kappa
r^{*}_{in}}\sinh\\!\left(\kappa t_{in}\right)$ (2.17) $\displaystyle
T_{out}=e^{\kappa r^{*}_{out}}\sinh\\!\left(\kappa t_{out}\right)~{}~{};\hskip
17.22217ptX_{out}=e^{\kappa r^{*}_{out}}\cosh\\!\left(\kappa t_{out}\right)$
(2.18)

where $\kappa$ is the surface gravity defined by

$\displaystyle\kappa=\frac{1}{2}\sqrt{f^{\prime}(r_{H})g^{\prime}(r_{H})}~{}.$
(2.19)

Here ‘$in(out)$’ stands for inside (outside) the event horizon while $r^{*}$
is the tortoise coordinate, defined by

$\displaystyle r^{*}=\int\frac{dr}{\sqrt{f(r)g(r)}}~{}.$ (2.20)

These two sets of coordinates are connected through the following relations

$\displaystyle t_{in}=t_{out}-i\frac{\pi}{2\kappa}$ (2.21) $\displaystyle
r^{*}_{in}=r^{*}_{out}+i\frac{\pi}{2\kappa}$ (2.22)

so that the Kruskal coordinates get identified as $T_{in}=T_{out}$ and
$X_{in}=X_{out}$. This indicates that when a particle travels from inside to
outside the horizon, ‘$t$’ coordinate picks up an imaginary term
$-\frac{\pi}{2{\kappa}}$. This fact will be used elaborately in later
chapters. Below we shall show that this is precisely given by (2.15). Near the
horizon one can expand $f(r)$ and $g(r)$ about the horizon $r_{H}$:

$\displaystyle f(r)=f^{\prime}(r_{H})(r-r_{H})+{\cal{O}}((r-r_{H})^{2})$
$\displaystyle g(r)=g^{\prime}(r_{H})(r-r_{H})+{\cal{O}}((r-r_{H})^{2})~{}.$
(2.23)

Substituting these in (2.15) and using (2.19) we obtain,

$\displaystyle{\textrm{Im}}~{}t=-\frac{1}{2\kappa}{\textrm{Im}}~{}\int_{r_{H}-\epsilon}^{r_{H}+\epsilon}\frac{dr}{r-r_{H}}~{}.$
(2.24)

Here we explicitly mentioned the integration limits. Now to evaluate the above
integration we make a substitution $r-r_{H}=\epsilon e^{i\theta}$ where
$\theta$ runs from $\pi$ to $2\pi$. Hence,

$\displaystyle{\textrm{Im}}~{}t=-\frac{1}{2\kappa}{\textrm{Im}}~{}\int_{\pi}^{2\pi}id\theta=-\frac{\pi}{2\kappa}~{}.$
(2.25)

For the Schwarzschild space-time, since $\kappa=\frac{1}{4M}$, one can easily
show that ${\textrm{Im}}~{}t=-2\pi M$ which is precisely the value given in
[78].

Therefore, substituting (2.15) in (2.14), the probability of the outgoing
particle is

$\displaystyle
P^{(R)}={\textrm{exp}}\Big{[}-\frac{4}{\hbar}\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}.$
(2.26)

Now using the principle of “detailed balance” [22]

$\displaystyle
P^{(R)}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{H}}\Big{)}P^{(L)}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{H}}\Big{)},$
(2.27)

we obtain the temperature of the black hole as

$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{(}{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}^{-1}~{}.$
(2.28)

This is the standard semi-classical Hawking temperature of the black hole.
Using this expression and knowing the metric coefficients $f(r)$ and $g(r)$
one can easily find out the temperature of the corresponding black hole.

Some comments are now in order. The first point is that (2.28) yields a novel
form of the semi-classical Hawking temperature. Also, we will later show that
(2.28) can be applied to non-spherically symmetric metrics. This will be
exemplified in the case of the Kerr metric. For a spherically symmetric metric
it is possible to show that (2.28) reproduces the familiar form

$\displaystyle T_{H}=\frac{\hbar\kappa}{2\pi}~{}.$ (2.29)

This can be done in the following way. The near horizon expansions for $f(r)$
and $g(r)$ are given by (2.23). Inserting these in (2.28) and performing the
contour integration, as done earlier, (2.29) is obtained. Note that this form
is the standard Hawking temperature found [90, 91] by the Hamilton-Jacobi
method. There is no ambiguity regarding a factor of two in the Hawking
temperature as reported in the literature [90, 91, 92]. This issue is
completely avoided in the present analysis where the standard expression for
the Hawking temperature is reproduced.

The other point is that the form of the solution (2.8) of (2.7) is not unique,
since any constant multiple of ‘$S_{0}$’ can be a solution as well. For that
case one can easily see that the final expression (2.28) for the temperature
still remains unchanged. It is only a matter of rescaling the particle energy
‘$\omega$’. This shows the uniqueness of the expression (2.28) for the Hawking
temperature.

####  2.1.2 Painleve coordinate system

Here we will discuss the Hamilton-Jacobi method in Painleve coordinates and
explicitly show how one can obtain the standard Hawking temperature. Consider
a metric of the form (2.1), which describes a general class of static,
spherically symmetric space time. There is a coordinate singularity in this
metric at the horizon $r=r_{H}$ where $f(r_{H})=g(r_{H})=0$. This singularity
is avoided by the use of Painleve coordinate transformation [96],

$\displaystyle dt\to dt-\sqrt{\frac{1-g(r)}{f(r)g(r)}}dr~{}.$ (2.30)

Under this transformation, the metric (2.1) takes the following form,

$\displaystyle
ds^{2}=-f(r)dt^{2}+2f(r)\sqrt{\frac{1-g(r)}{f(r)g(r)}}dtdr+dr^{2}+r^{2}d\Omega^{2}.$
(2.31)

Note that the metric (2.1) looks both stationary and static, whereas the
transformed metric (2.31) is stationary but not static which reflects the
correct nature of the space time.

As before, consider a massless scalar particle in the spacetime metric (2.31)
described by the Painleve coordinates. Since the Klein-Gordon equation (2.2)
is in covariant form, the scalar particle in the background metric (2.31) also
satisfies (2.2). Therefore under this metric the Klein-Gordon equation reduces
to

$\displaystyle-$
$\displaystyle(\frac{g}{f})^{\frac{3}{2}}\partial^{2}_{t}\phi+\frac{2g\sqrt{1-g}}{f}\partial_{t}\partial_{r}\phi-\frac{gg^{\prime}}{2f\sqrt{1-g}}\partial_{t}\phi+g\sqrt{\frac{g}{f}}\partial^{2}_{r}\phi$
(2.32) $\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{\frac{g}{f}}(3g^{\prime}-\frac{f^{\prime}g}{f})\partial_{r}\phi=0.$

As before, substituting the standard ansatz (2.4) for $\phi$ in the above
equation, we obtain,

$\displaystyle-(\frac{g}{f})^{\frac{3}{2}}\Big{[}-\frac{i}{\hbar}\Big{(}\frac{\partial
S}{\partial t}\Big{)}^{2}+\frac{\partial^{2}S}{\partial
t^{2}}\Big{]}+\frac{2g\sqrt{1-g}}{f}\Big{[}-\frac{i}{\hbar}\frac{\partial
S}{\partial t}\frac{\partial S}{\partial r}+\frac{\partial^{2}S}{\partial
r\partial t}\Big{]}-\frac{gg^{\prime}}{2f\sqrt{1-g}}\frac{\partial S}{\partial
t}$
$\displaystyle+g\sqrt{\frac{g}{f}}\Big{[}-\frac{i}{\hbar}\Big{(}\frac{\partial
S}{\partial r}\Big{)}^{2}+\frac{\partial^{2}S}{\partial
r^{2}}\Big{]}+\frac{1}{2}(3g^{\prime}-\frac{f^{\prime}g}{f})\frac{\partial
S}{\partial r}=0.$ (2.33)

Substituting (2.6) in the above and then neglecting the terms of order $\hbar$
and greater we find to the lowest order,

$\displaystyle(\frac{g}{f})^{\frac{3}{2}}\Big{(}\frac{\partial S_{0}}{\partial
t}\Big{)}^{2}-\frac{2g\sqrt{1-g}}{f}\frac{\partial S_{0}}{\partial
t}\frac{\partial S_{0}}{\partial r}-g\sqrt{\frac{g}{f}}\Big{(}\frac{\partial
S_{0}}{\partial r}\Big{)}^{2}=0.$ (2.34)

It has been stated earlier that the metric (2.31) is stationary. Therefore
following the same argument as before it has a solution of the form (2.8).
Inserting this in (2.34) yields,

$\displaystyle\frac{d\tilde{S}_{0}(r)}{dr}=\omega\sqrt{\frac{1-g(r)}{f(r)g(r)}}\Big{(}-1\pm\frac{1}{\sqrt{1-g(r)}}\Big{)}$
(2.35)

Integrating,

$\displaystyle\tilde{S}_{0}(r)=\omega\int\sqrt{\frac{1-g(r)}{f(r)g(r)}}\Big{(}-1\pm\frac{1}{\sqrt{1-g(r)}}\Big{)}dr.$
(2.36)

The $+(-)$ sign in front of the integral indicates that the particle is
ingoing (outgoing). Therefore the actions for ingoing and outgoing particles
are

$\displaystyle S_{0}^{(L)}(r,t)=\omega
t+\omega\int\frac{1-\sqrt{1-g}}{\sqrt{fg}}dr$ (2.37)

and

$\displaystyle S_{0}^{(R)}(r,t)=\omega
t-\omega\int\frac{1+\sqrt{1-g}}{\sqrt{fg}}dr$ (2.38)

Since in the classical limit (i.e. $\hbar\rightarrow 0$) the probability for
the ingoing particle ($P^{(L)}$) has to be unity, $S_{0}^{(L)}$ must be real.
Following identical steps employed in deriving (2.15) we obtain, starting from
(2.37), the analogous condition,

$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int\frac{1-\sqrt{1-g}}{\sqrt{fg}}dr$
(2.39)

Substituting this in (2.38) we obtain the action for the outgoing particle:

$\displaystyle
S_{0}^{(R)}(r,t)=\omega{\textrm{Re}}~{}t-\omega{\textrm{Re}}\int\frac{1+\sqrt{1-g}}{\sqrt{fg}}dr-2i\omega{\textrm{Im}}\int\frac{dr}{\sqrt{fg}}$
(2.40)

Therefore the probability for the outgoing particle is

$\displaystyle
P^{(R)}=|e^{-\frac{i}{\hbar}S_{0}^{(R)}}|^{2}=e^{-\frac{4}{\hbar}\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}}$
(2.41)

Now using the principle of “detailed balance” (2.27) we obtain the same
expression (2.28) for the standard Hawking temperature which was calculated in
Schwarzschild like coordinates by the Hamilton-Jacobi method.

###  2.2 Radial null geodesic method

So far, we gave a general discussion on the HJ method both in Scwarzschild
like coordinates as well as Painleve coordinates and obtained the expression
of the temperature for a static, spherically symmetric black hole. Also, this
has been reduced to the famous Hawking expression - temperature is
proportional to the surface gravity.

In this section, we will give a general discussion on the radial null geodesic
method. A derivation of the Hawking temperature by this method will be
explicitly performed for the metric (2.1).

In this method, the first step is to find the radial null geodesic. To do that
it is necessary to remove the apparent singularity at the event horizon. This
is done by going to the Painleve coordinates. In these coordinates, the metric
(2.1) takes the form (2.31). Then the radial null geodesics are obtained by
setting $ds^{2}=d\Omega^{2}=0$ in (2.31),

$\displaystyle\dot{r}\equiv\frac{dr}{dt}=\sqrt{\frac{f(r)}{g(r)}}\Big{(}\pm
1-\sqrt{1-g(r)}\Big{)}$ (2.42)

where the positive (negative) sign gives outgoing (incoming) radial geodesics.
At the neighbourhood of the black hole horizon, the trajectory (2.42) of an
outgoing shell is written as,

$\displaystyle\dot{r}=\frac{1}{2}\sqrt{f^{\prime}(r_{H})g^{\prime}(r_{H})}(r-r_{H})+{\mathcal{O}}((r-r_{H})^{2})$
(2.43)

where we have used the expansions (2.23) of the functions $f(r)$ and $g(r)$.
Now we want to write (2.43) in terms of the surface gravity of the black hole.
The reason is that in some cases, for example in the presence of back
reaction, one may not know the exact form of the metric but what one usually
knows is the surface gravity of the problem. Also, the Hawking temperature is
eventually expressed in terms of the surface gravity. The form of surface
gravity for the transformed metric (2.31) at the horizon is given by,

$\displaystyle\kappa=\Gamma{{}^{0}}{{}_{00}}|_{r=r_{H}}=\frac{1}{2}\Big{[}\sqrt{\frac{1-g(r)}{f(r)g(r)}}g(r)\frac{df(r)}{dr}\Big{]}|_{r=r_{H}}.$
(2.44)

Using the Taylor series (2.23), the above equation reduces to the familiar
form of surface gravity (2.19). This expression of surface gravity is used to
write (2.43) in the form,

$\displaystyle\dot{r}=\kappa(r-r_{H})+{\mathcal{O}}((r-r_{H})^{2}).$ (2.45)

We consider a positive energy shell which crosses the horizon in the outward
direction from $r_{{\textrm{in}}}$ to $r_{{\textrm{out}}}$. The imaginary part
of the action for that shell is given by [23],

$\displaystyle\textrm{Im}~{}{\cal{S}}=\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}p_{r}dr=\textrm{Im}\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}dp_{r}^{\prime}dr.$
(2.46)

Using the Hamilton’s equation of motion $\dot{r}=\frac{dH}{dp_{r}}|_{r}$ the
last equality of the above equation is written as,

$\displaystyle\textrm{Im}~{}{\cal{S}}$ $\displaystyle=$
$\displaystyle\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\int_{0}^{H}\frac{dH^{\prime}}{\dot{r}}dr$
(2.47)

where, instead of momentum, energy is used as the variable of integration.

Now we consider the self gravitation effect [27] of the particle itself, for
which (2.45) and (2.47) will be modified. Following [23], under the $s$\- wave
approximation, we make the replacement $M\rightarrow M-\omega$ in (2.45) to
get the following expression

$\dot{r}=(r-r_{H})\kappa[M-\omega]$ (2.48)

where $\omega$ is the energy of a shell moving along the geodesic of space-
time 222Here $\kappa[M-\omega]$ represents that $\kappa$ is a function of
($M-\omega$). This symbol will be used in the later part of the chapter for a
similar purpose..

Now we use the fact [23], for a black hole of mass $M$, the Hamiltonian
$H=M-\omega$. Inserting in (2.47) the modified expression due to the self
gravitation effect is obtained as,

$\displaystyle\textrm{Im}~{}{\cal{S}}=\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\int_{M}^{M-\omega}\frac{d(M-\omega^{\prime})}{\dot{r}}dr=-\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\int_{0}^{\omega}\frac{d\omega^{\prime}}{\dot{r}}dr$
(2.49)

where in the final step we have changed the integration variable from
$H^{\prime}$ to $\omega^{\prime}$. Substituting the expression of $\dot{r}$
from (2.48) into (2.49) we find,

$\displaystyle\textrm{Im}~{}{\cal{S}}=-\textrm{Im}\int_{0}^{\omega}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\frac{dr}{r-r_{H}}~{}.$
(2.50)

The $r$-integration is done by deforming the contour. Ensuring that the
positive energy solutions decay in time (i.e. into the lower half of
$\omega^{\prime}$ plane and $r_{{\textrm{in}}}>r_{{\textrm{out}}}$) we have
after $r$ integration333One can also take the contour in the upper half plane
with the replacement $M\rightarrow M+\omega$ [27].,

$\displaystyle\textrm{Im}~{}{\cal{S}}=\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.51)

To understand the ordering $r_{{\textrm{in}}}>r_{{\textrm{out}}}$ \- which
supplies the correct sign, let us do the following analysis. For simplicity,
we consider the Schwarzschild black hole whose surface gravity is given by
$\kappa[M]=\frac{1}{4M}$. Substituting this in (2.51) and performing the
$\omega^{\prime}$ integration we obtain

$\displaystyle{\textrm{Im}}~{}{\cal{S}}=4\pi\omega(M-\frac{\omega}{2}).$
(2.52)

Now let us first perform the $\omega^{\prime}$ integration before $r$
integration in (2.50). For Schwarzschild black hole this will give

$\displaystyle{\textrm{Im}}~{}{\cal{S}}=4~{}{\textrm{Im}}~{}\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}dr\int_{M}^{M-\omega}\frac{M^{\prime}}{r-2M^{\prime}}dM^{\prime}$
(2.53)

where substitution of $M^{\prime}=M-\omega^{\prime}$ has been used. Evaluation
of $M^{\prime}$ integration and then $r$ integration in the above yields,

$\displaystyle{\textrm{Im}}~{}{\cal{S}}=\frac{\pi}{2}(r_{\textrm{in}}^{2}-r_{\textrm{out}}^{2})$
(2.54)

Hence (2.52) and (2.54) to be equal we must have $r_{\textrm{in}}=2M$ and
$r_{\textrm{out}}=2(M-\omega)$, which clearly shows that
$r_{{\textrm{in}}}>r_{{\textrm{out}}}$.

The tunneling amplitude following from the WKB approximation is given by,

$\displaystyle\Gamma\sim
e^{-\frac{2}{\hbar}{\textrm{Im}}~{}{\cal{S}}}=e^{\Delta S_{bh}}$ (2.55)

where the result is expressed more naturally in terms of the black hole
entropy change [23]. To understand the last identification ($\Gamma=e^{\Delta
S_{bh}}$), consider a process where a black hole emits a shell of energy. We
denote the initial state and final state by the levels $i$ and $f$. In thermal
equilibrium,

$\displaystyle\frac{dP_{i}}{dt}=P_{i}P_{i\rightarrow f}-P_{f}P_{f\rightarrow
i}=0$ (2.56)

where $P_{a}$ denotes the probability of getting the system in the macrostate
$a(a=i,f)$ and $P_{a\rightarrow b}$ denotes the transition probability from
the state $a$ to $b$ ($a,b=i,f$). According to statistical mechanics, the
entropy of a given state (specified by its macrostates) is a logarithmic
function of the total number of microstates ($S_{bh}={\textrm{log}}\Omega$).
So the number of microstates $\Omega$ for a given black hole is
$e^{S_{{bh}}}$. Since the probability of getting a system in a particular
macrostate is proportional to the number of microstates available for that
configuration, we get from (2.56),

$\displaystyle
e^{S_{i}}P_{{\textrm{emission}}}=e^{S_{f}}P_{{\textrm{absorption}}}$ (2.57)

where $P_{{\textrm{emission}}}$ is the emission probability $P_{i\rightarrow
f}$ and $P_{{\textrm{absorption}}}$ is the absorption probability
$P_{f\rightarrow i}$. So the tunneling amplitude is given by,

$\displaystyle\Gamma=\frac{P_{{\textrm{emission}}}}{P_{{\textrm{absorption}}}}=e^{S_{f}-S_{i}}=e^{\Delta
S_{{bh}}}$ (2.58)

thereby leading to the correspondence,

$\displaystyle\Delta S_{bh}=-\frac{2}{\hbar}{\textrm{Im}}~{}{\cal{S}}$ (2.59)

that follows from (2.55). We mention that the above relation (2.59) has also
been shown using semi-classical arguments based on the second law of
thermodynamics [97] or on the assumption of entropy being proportional to area
[29, 98]. But such arguments are not used in our derivation. Rather our
analysis has some points of similarity with the physical picture suggested in
[23] leading to a general validity of (2.58). This implies that when quantum
effects are taken into consideration, both sides of (2.59) are modified
keeping the functional relationship identical. In our analysis we will show
that self consistency is preserved by (2.59).

In order to write the black hole entropy in terms of its mass alone we have to
substitute the value of $\omega$ in terms of $M$ for which the black hole is
stable i. e.

$\displaystyle\frac{d(\Delta S_{bh})}{d\omega}=0~{}.$ (2.60)

Using (2.51) and (2.59) in the above equation we get,

$\displaystyle\frac{1}{{\kappa}[M-\omega]}=0~{}.$ (2.61)

The roots of this equation are written in the form

$\displaystyle\omega=\psi[M]$ (2.62)

which means

$\displaystyle\frac{1}{{\kappa}[M-\psi[M]]}=0.$ (2.63)

This value of $\omega$ from eq. (2.62) is substituted back in the expression
of $\Delta S_{bh}$ to yield,

$\displaystyle\Delta
S_{bh}=-\frac{2\pi}{\hbar}\int_{0}^{\psi[M]}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.64)

Having obtained the form of entropy change, we are now able to give an
expression of entropy for a particular state. We recall the simple definition
of entropy change

$\displaystyle\Delta S_{bh}=S_{{\textrm{final}}}-S_{{\textrm{initial}}}~{}.$
(2.65)

Now setting the black hole entropy at the final state to be zero we get the
expression of entropy as

$\displaystyle S_{bh}=S_{{\textrm{initial}}}=-\Delta
S_{bh}=\frac{2\pi}{\hbar}\int_{0}^{\psi[M]}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.66)

From the law of thermodynamics, we write the inverse black hole temperature
as,

$\displaystyle\frac{1}{T_{H}}$ $\displaystyle=$
$\displaystyle\frac{dS_{bh}}{dM}$ (2.67) $\displaystyle=$
$\displaystyle\frac{2\pi}{\hbar}\frac{d}{dM}\int_{0}^{\psi[M]}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.68)

Using the identity,

$\displaystyle\frac{dF[x]}{dx}=f[x,b[x]]b^{\prime}[x]-f[x,a[x]]a^{\prime}[x]+\int_{a[x]}^{b[x]}\frac{\partial}{\partial
x}f[x,t]dt$ (2.69)

for,

$\displaystyle F[x]=\int_{a[x]}^{b[x]}f[x,t]dt$ (2.70)

we find,

$\displaystyle\frac{1}{T_{H}}=\frac{2\pi}{\hbar}\big{[}\frac{1}{{\kappa}[M-\psi[M]]}\psi^{\prime}[M]-\int_{0}^{\psi[M]}\frac{1}{[{\kappa}[M-\omega^{\prime}]]^{2}}\frac{\partial{\kappa}[M-\omega^{\prime}]}{\partial(M-\omega^{\prime})}d\omega^{\prime}\big{]}~{}.$
(2.71)

Making the change of variable $x=M-\omega^{\prime}$ in the second integral we
obtain,

$\displaystyle\frac{1}{T_{H}}=\frac{2\pi}{\hbar}\big{[}\frac{\psi^{\prime}[M]-1}{{\kappa}[M-\psi[M]]}+\frac{1}{{\kappa}[M]}\big{]}~{}.$
(2.72)

Finally, making use of (2.63), the cherished expression (2.29) for the Hawking
temperature follows.

For a consistency check, consider the second law of thermodynamics which is
now written as,

$\displaystyle
dM=d\omega^{\prime}=T_{h}dS_{bh}=\frac{\hbar{\kappa}[M]}{2\pi}dS_{bh}~{}.$
(2.73)

Inserting in (2.51), yields,

$\displaystyle{\textrm{Im}}~{}{\cal
S}=\frac{\hbar}{2}\int_{S_{bh}[M]}^{S_{bh}[M-\omega]}dS_{bh}=-\frac{\hbar}{2}\Delta
S_{bh}$ (2.74)

thereby reproducing (2.59). This shows the internal consistency of the
tunneling approach.

###  2.3 Calculation of Hawking temperature

In this section we will consider some standard metrics to show how the semi-
classical Hawking temperature can be calculated from (2.28). For instance we
consider a spherically symmetric space-time, the Schwarzschild metric and a
non-spherically symmetric space-time, the Kerr metric.

####  2.3.1 Schwarzschild black hole

The spacetime metric is given by

$\displaystyle
ds^{2}=-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(2.75)

So the metric coefficients are

$\displaystyle f(r)=g(r)=(1-\frac{r_{H}}{r});\,\,\,r_{H}=2M.$ (2.76)

Since this metric is spherically symmetric we use the formula (2.28) to
compute the semi-classical Hawking temperature. This is found to be,

$\displaystyle T_{H}=\frac{\hbar}{4\pi r_{H}}=\frac{\hbar}{8\pi M}.$ (2.77)

which is the standard expression (2.29) where the surface gravity, calculated
by (2.19), is $\kappa=1/4M$.

####  2.3.2 Kerr black hole

This example provides a nontrivial application of our formula (2.28) for
computing the semi-classical Hawking temperature. Here the metric is not
spherically symmetric, invalidating the use of (2.29).

In Boyer-Linquist coordinates the form of the Kerr metric is given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\Big{(}1-\frac{2Mr}{\rho^{2}}\Big{)}dt^{2}-\frac{2Mar~{}{\textrm{sin}}^{2}\theta}{\rho^{2}}(dtd\phi+d\phi
dt)$ (2.78) $\displaystyle+$
$\displaystyle\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{{\textrm{sin}}^{2}\theta}{\rho^{2}}~{}\Big{[}(r^{2}+a^{2})^{2}-a^{2}\Delta~{}{\textrm{sin}}^{2}\theta\Big{]}d\phi^{2}$

where

$\displaystyle\Delta(r)$ $\displaystyle=$ $\displaystyle
r^{2}-2Mr+a^{2};\,\,\,\rho^{2}(r,\theta)=r^{2}+a^{2}~{}{\textrm{cos}}^{2}\theta$
$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{J}{M}$ (2.79)

and $J$ is the Komar angular momentum. We have chosen the coordinates for Kerr
metric such that the event horizons occur at those fixed values of $r$ for
which $g^{rr}=\frac{\Delta}{\rho^{2}}=0$. Therefore the event horizons are

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}.$ (2.80)

This metric is not spherically symmetric and static but stationary. So it must
have time-like Killing vectors. Subtleties in employing the tunneling
mechanism for such (rotating) black holes were first discussed in [90, 91]. In
the present discussion we will show that, although the general formulation was
based only on the static, spherically symmetric metrics, it is still possible
to apply this methodology for such a metric. The point is that for radial
trajectories, the Kerr metric simplifies to the following form

$\displaystyle
ds^{2}=-\Big{(}\frac{r^{2}+a^{2}-2Mr}{r^{2}+a^{2}}\Big{)}dt^{2}+\Big{(}\frac{r^{2}+a^{2}}{r^{2}+a^{2}-2Mr}\Big{)}dr^{2}$
(2.81)

where, for simplicity, we have taken $\theta=0$ (i.e. particle is going along
$z$-axis). This is exactly the form of the $(r-t)$ sector of the metric (2.1).
Since in our formalism only the $(r-t)$ sector is important, our results are
applicable here. In particular if the metric has no terms like $(drdt)$ then
we can apply (2.28) to find the semi-classical Hawking temperature. Here,

$\displaystyle f(r)=g(r)=\Big{(}\frac{r^{2}+a^{2}-2Mr}{r^{2}+a^{2}}\Big{)}$
(2.82)

Substituting these in (2.28) we obtain,

$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{(}{\textrm{Im}}\int\frac{r^{2}+a^{2}}{(r-r_{+})(r-r_{-})}\Big{)}^{-1}.$
(2.83)

The integrand has simple poles at $r=r_{+}$ and $r=r_{-}$. Since we are
interested only with the event horizon at $r=r_{+}$, we choose the contour as
a small half-loop going around this pole from left to right. Integrating, we
obtain the value of the semi-classical Hawking temperature as

$\displaystyle T_{H}=\frac{\hbar}{4\pi}\frac{r_{+}-r_{-}}{r_{+}^{2}+a^{2}}.$
(2.84)

which is the result quoted in the literature [99]. This can also be expressed
in standard expression (2.29) where
$\kappa=\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}$.

###  2.4 Discussions

In this chapter, we introduced the tunneling method to study the Hawking
effect within the semi-classical limit (i.e. $\hbar\rightarrow 0$),
particularly to find the familiar form of the semi-classical Hawking
temperature. There exist two approaches: Hamilton-Jacobi method [22] (HJ) and
radial null geodesic method [23]. For simplicity, a general form of the
static, spherically symmetric black hole metric was considered.

First, discussions on HJ method in both the Schwarzschild like coordinates and
Painleve coordinates have been done. In both coordinate systems, we obtained
identical results. A general expression (2.28) for the semi-classical Hawking
temperature was obtained. For the particular case of a spherically symmetric
metric, our expression reduces to the standard form (2.29). The factor of two
problem in the Hawking temperature has been taken care of by considering the
contribution from the imaginary part of the temporal coordinate since it
changes its nature across the horizon. Also, this method is free of the rather
ad hoc way of introducing an integration constant, as reported in [89]. Our
approach, on the other hand, is similar in spirit to [78] where it has been
shown that ‘$t$’ changes by an imaginary discrete amount across the horizon.
Indeed, the explicit expression for this change, in the case of Schwarzschild
metric, calculated from our general formula (2.15), agrees with the findings
of [78]. Then, a general discussion on the other method, the radial null
geodesic method, was given. In this method, again the standard form of the
Hawking temperature was obtained. Finally, as an application, we calculated
the semi-classical temperature of the Schwarzschild black hole from the
general expression (2.28). Also, use of this expression to find the
temperature of a non-spherically symmetric metric, for instance Kerr metric,
has been shown.

As a final remark, we want to mention that our derivation of Hawking
temperature in terms of the surface gravity by considering the action of an
outgoing particle crossing the black hole horizon due to quantum mechanical
tunneling is completely general. The expression of temperature was known long
before [100, 3, 2, 43] from a comparison between two classical laws. One is
the law of black hole thermodynamics which states that the mass change is
proportional to the change of horizon area multiplied by surface gravity at
the horizon. The other is the area law according to which the black hole
entropy is proportional to the surface area of the horizon. The important
point of our derivation is that it is not based on either of these two
classical laws. The only assumption is that the metric is static and
spherically symmetric. Hence it is useful to apply this method to study the
Hawking effect for the black holes which incorporates both the back reaction
and noncommutative effects but still are in static, spherically symmetric
form. This will be done in the next chapter.

## Appendix

##  Appendix 2.A Ingoing and outgoing modes

Our convention is such that, a mode will be called ingoing (outgoing) if its
radial momentum eigenvalue is negative (positive). For a wave function $\phi$,
the momentum eigenvalue equation is

$\displaystyle{\hat{p}_{r}}\phi=p_{r}\phi,$ (2a.1)

where ${\hat{p}_{r}}=-i\hbar\frac{\partial}{\partial r}$. So according to our
convention, if $p_{r}<0$ for a mode, then it is ingoing and vice versa.

Now the mode solutions are given by (2.11) and (2.12). So according to (2a.1),
the momentum eigenvalue for $\phi^{(L)}$ is
$p_{r}^{(L)}=-\frac{\omega}{\sqrt{fg}}$ which is negative. So this mode is
ingoing. Similarly, the momentum eigenvalue for $\phi^{(R)}$ mode comes out to
be positive and hence it is a outgoing mode.

##  Chapter 3 Null geodesic approach

In the previous chapter, a systematic analysis on tunneling mechanism, both by
HJ and radial null geodesic methods, to find the Hawking temperature has been
presented. The temperature was found to be proportional to the surface gravity
of a black hole represented by a general static, spherically symmetric metric.
This indicates that such an analysis can be extended to the cases in which the
space-time metric is modified by effects like back reaction and
noncommutivity, provided these are still in the static, spherically symmetric
form.

To investigate the last stage evolution of black hole evaporation back
reaction in space-time has a significant influence. An approach to this
problem is to solve the semiclassical Einstein equations in which the matter
fields including the graviton, are quantized at the one-loop level and coupled
to (c -number) gravity through Einstein’s equation. The space-time geometry
$g_{\mu\nu}$, generates a non-zero vacuum expectation value of the energy-
momentum tensor ($<T_{\mu\nu}>$) which in turn acts as a source of curvature
(this is the so-called ”back-reaction problem”). With this energy momentum
tensor and an ansatz for the metric, the solutions of Einstein’s equation
yields the metric solution, which is static and spherically symmetric [63].
Using the conformal anomaly method the modifications to the space-time metric
by the one loop back reaction was computed [101, 63]. Later it was shown [64,
65] that the Bekenstein-Hawking area law was modified, in the leading order,
by logarithmic corrections. Similar conclusions were also obtained by using
quantum gravity techniques [66, 71, 102]. Likewise, corrections to the semi-
classical Hawking temperature were derived [67, 68, 69, 70].

It is known that for the usual cases, the Hawking temperature diverges as the
radius of the event horizon decreases. This uncomfortable situation leads to
the “information paradox”. To avoid this one of the attempts is inclusion of
the noncommutative effect in the space-time. There exits two methods: (i)
directly take the space-time as noncommutative,
$[x_{\mu},x_{\nu}]=i\theta_{\mu\nu}$ and use Seibarg-Witten map to recast the
gravitational theory (in noncommutative space) in terms of the corresponding
theory in usual (commutative space) variables, and (ii) incorporate the effect
of noncommutativity in the mass term of the gravitating object.

In this chapter, we shall include the back reaction as well as noncommutative
effects in the space-time metric. Following the radial null geodesic method
presented in the previous chapter, the thermodynamic entities will be
calculated. Although there have been sporadic attempts in this direction [74,
75] a systematic, thorough and complete analysis was lacking.

The organization of the chapter is as follows. In the first section, we
compute the corrections to the semi-classical tunneling rate by including the
effects of self gravitation and back reaction. The usual expression found in
[23], given in the Maxwell-Boltzmann form $e^{-\frac{\omega}{T_{H}}}$, is
modified by a prefactor. This prefactor leads to a modified Bekenstein-Hawking
entropy. The semi-classical Bekenstein-Hawking area law connecting the entropy
to the horizon area is altered. As obtained in other approaches [64, 65, 66,
67, 68, 69, 70, 71], the leading correction is found to be logarithmic while
the nonleading one is a series in inverse powers of the horizon area (or
Bekenstein-Hawking entropy). We also compute the appropriate modification to
the Hawking temperature. Explicit results are given for the Schwarzschild
black hole.

Next, we shall apply our general formulation to discuss various thermodynamic
properties of a black hole defined in a noncommutative Schwarzschild space
time where back reaction is also taken into account. A short introduction of
the noncommutative Schwarzschild black hole is presented at the beginning of
the section (3.2). In particular we are interested in the black hole
temperature when the radius is of the order $\sqrt{\theta}$, where $\theta$ is
the noncommutative parameter. Such a study is relevant because
noncommutativity is supposed to remove the so called information paradox where
for a standard black hole, temperature diverges as the radius shrinks to zero.
The Hawking temperature is obtained in a closed form that includes corrections
due to noncommutativity and back reaction. These corrections are such that, in
some examples, the information paradox is avoided. Expressions for the entropy
and tunneling rate are also found for the leading order in the noncommutative
parameter. Furthermore, in the absence of back reaction, we show that the
entropy and area are algebraically related in the same manner as occurs in the
standard Bekenstein-Hawking area law.

###  3.1 Back reaction effect

In this section we shall derive the modifications in the Hawking temperature
and Bekenstein-Hawking area law due to the one loop back reaction effect in
the space-time. Back reaction is essentially the effect of the Hawking
radiation on the horizon. For simplicity, only the Schwarzschild black hole
will be considered. One way to include the back reaction effect into the
problem is to solve Einstein’s equation with an appropriate source. In this
case one considers the renormalized energy-momentum tensor due to one loop
back reaction effect on the right hand side of the Einstein’s equation. Then
solution of this equation gives the black hole metric given by the form (2.1)
[63]. Therefore it is feasible to apply the tunneling method developed in the
previous chapter for this case to find the modifications to the usual
thermodynamical entities. Here our discussions will be based on the radial
null geodesic method.

Here our starting point is the expression for the imaginary part of the action
(2.51), since in the present problem the form of the modified surface gravity
of the black hole is known. The modified surface gravity due to one loop back
reaction effects is given by [63],

$\displaystyle\kappa[M]=\kappa_{0}[M]\Big{(}1+\frac{\alpha}{M^{2}}\Big{)}$
(3.1)

where $\kappa_{0}$ is the classical surface gravity at the horizon of the
black hole. Such a form is physically dictated by simple scaling arguments. As
is well known, a loop expansion is equivalent to an expansion in powers of the
Planck constant $\hbar$. Therefore, the one loop back reaction effect in the
surface gravity is written as,

$\displaystyle\kappa=\kappa_{0}+\xi\kappa_{0}$ (3.2)

where $\xi$ is a dimensionless constant having magnitude of the order $\hbar$.
Now, in natural units $G=c=k_{B}=1$, Planck lenght $l_{p}=$ Planck mass
$M_{p}=\sqrt{\hbar}$ 111Planck length $l_{p}=\sqrt{\frac{\hbar G}{c^{3}}}$,
Planck mass $M_{p}=\sqrt{\frac{\hbar c}{G}}$. On the other hand, for
Schwarzschild black hole, mass $M$ is the only macroscopic parameter.
Therefore, $\xi$ must be function of $\frac{M_{p}}{M}$ which vanishes in the
limit $M_{p}<<M$. Since, as stated earlier, $\xi$ is a dimensionless constant
with magnitude of order $\hbar$, the leading term has the following quadratic
form,

$\displaystyle\xi=\beta\frac{M_{p}^{2}}{M^{2}}~{}.$ (3.3)

In the above, $\beta$ is a pure numerical factor. Taking $\alpha=\beta
M_{p}^{2}$ and then substituting (3.3) in (3.2) we obtain (3.1). The constant
$\beta$ is related to the trace anomaly coefficient taking into account the
degrees of freedom of the fields [103, 63, 64]. Its explicit form is given by
[103, 64],

$\displaystyle\beta=\frac{1}{360\pi}\Big{(}-N_{0}-\frac{7}{4}N_{\frac{1}{2}}+13N_{1}+\frac{233}{4}N_{\frac{3}{2}}-212N_{2})$
(3.4)

where $N_{s}$ denotes the number of fields with spin ‘$s$’.

For the classical Schwarzschild space-time the metric coefficients are given
by (2.76) and so by equation (2.19) the value of $\kappa_{0}[M]$ is

$\displaystyle\kappa_{0}[M]=\frac{f^{\prime}(r_{H}=2M)}{2}=\frac{1}{4M}~{}.$
(3.5)

Substituting (3.1) with $\kappa_{0}$ is given by (3.5) in (2.51) and then
integrating over $\omega^{\prime}$ we have

$\displaystyle
Im~{}{\cal{S}}=4\pi\omega(M-\frac{\omega}{2})+2\pi\alpha\ln{\Big{[}\frac{(M-\omega)^{2}+\alpha}{M^{2}+\alpha}\Big{]}}~{}.$
(3.6)

Now according to the WKB-approximation method the tunneling probability is
given by (2.55). So the modified tunneling probability due to back reaction
effects is,

$\displaystyle\Gamma\sim\Big{[}1-\frac{2\omega(M-\frac{\omega}{2})}{M^{2}+\alpha}\Big{]}^{-\frac{4\pi\alpha}{\hbar}}e^{-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})}$
(3.7)

The exponential factor of the tunneling probability was previously obtained by
Parikh and Wilczek [23]. The factor before the exponential is new. It is
actually due the effect of back reaction. It will eventually give the
correction to the Bekenstein-Hawking entropy, area law and the Hawking
temperature as will be shown below.

It was shown in the previous chapter and also in the literature [23, 80, 97]
that a change in the Bekenstein-Hawking entropy due to the tunneling through
the horizon is related to $Im~{}{\cal{S}}$ by the relation (2.59). Therefore
the corrected change in Bekenstein-Hawking entropy is

$\displaystyle\Delta
S_{bh}=-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})-\frac{4\pi\alpha}{\hbar}\ln\Big{[}(M-\omega)^{2}+\alpha\Big{]}+\frac{4\pi\alpha}{\hbar}\ln(M^{2}+\alpha)$
(3.8)

Next using the stability criterion $\frac{d(\Delta S_{bh})}{d\omega}=0$ for
the black hole, one obtains the following condition

$\displaystyle(\omega-M)^{3}=0$ (3.9)

which gives the only solution as $\omega=M$. Substituting this value of
$\omega$ in (3.8) we will have the change in entropy of the black hole from
its initial state to final state:

$\displaystyle S_{final}-S_{initial}=-\frac{4\pi
M^{2}}{\hbar}+\frac{4\pi\alpha}{\hbar}\ln{(\frac{M^{2}}{\alpha}+1)}~{}.$
(3.10)

Setting $S_{final}=0$, the Bekenstein-Hawking entropy of the black hole with
mass $M$ is

$\displaystyle S_{bh}=S_{initial}$ $\displaystyle=$ $\displaystyle\frac{4\pi
M^{2}}{\hbar}-4\pi\beta\ln{(\frac{M^{2}}{\beta\hbar}+1)}$ (3.11)

where we have substituted $\alpha=\beta M_{p}^{2}=\beta\hbar$.

Now the area of the black hole horizon given by

$\displaystyle A=4\pi r^{2}_{H}=16\pi M^{2}~{}.$ (3.12)

Putting (3.12) in (3.11) and expanding the logarithm, we obtain the final
form,

$\displaystyle S_{bh}$ $\displaystyle=$
$\displaystyle\frac{A}{4\hbar}-4\pi\beta\ln\frac{A}{4\hbar}-64\pi^{2}\hbar\beta^{2}\Big{[}\frac{1}{A}-\frac{16\pi\hbar\beta}{2A^{2}}+\frac{(16\pi\hbar\beta)^{2}}{3A^{3}}-.....\Big{]}$
(3.13) $\displaystyle+$ $\displaystyle\textrm{const.(independent~{} of~{}
$A$)}~{}.$

The first term is the usual semi-classical area law [2, 5] and other terms are
the corrections due to the one loop back reaction effect. The leading
correction is the well known logarithmic correction [64, 65, 66, 67, 68, 69,
70, 71]. Quantum gravity calculations lead to a prefactor $-\frac{1}{2}$ for
the $\ln\frac{A}{4\hbar}$ term which would correspond to choosing
$\beta=\frac{1}{8\pi}$. But here on the contrary $\beta$ is given by (3.4).
Also, the nonleading corrections are found to be expressed as a series in
inverse powers of $A$, exactly as happens in quantum gravity inspired analysis
[66, 71]. Now using the first law of black hole mechanics, $T_{H}dS_{bh}=dM$,
or the relation (2.29) between the Hawking temperature and surface gravity, we
can find the corrected form of the Hawking temperature $T_{H}$ due to back
reaction. This is obtained from (3.1) as,

$\displaystyle T_{H}=T_{0}\Big{(}1+\frac{\beta\hbar}{M^{2}}\Big{)}$ (3.14)

where $T_{0}=\frac{\hbar\kappa_{0}}{2\pi}=\frac{\hbar}{8\pi M}$ is the semi-
classical Hawking temperature and the other term is the correction due to the
back reaction. A similar expression was obtained previously in [64] by the
conformal anomaly method.

It is also possible to obtain the corrected Hawking temperature (3.14) in the
standard tunneling method to leading order [23] where this temperature is read
off from the coefficient of ‘$\omega$’ in the exponential of the probability
amplitude (3.7). Recasting this amplitude as,

$\displaystyle\Gamma\sim
e^{-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})-\frac{4\pi\alpha}{\hbar}\ln(1-\frac{2\omega(M-\frac{\omega}{2})}{M^{2}+\alpha})}$
(3.15)

and retaining terms upto leading order in $\omega$, we obtain,

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle e^{-\frac{8\pi
M\omega}{\hbar}+4\pi\beta(\frac{2M\omega}{M^{2}+\beta\hbar})}$ (3.16)
$\displaystyle=$ $\displaystyle e^{-(\frac{8\pi
M^{3}}{\hbar(M^{2}+\beta\hbar)})\omega}=e^{-\frac{\omega}{T_{H}}}.$

The inverse Hawking temperature, indentified with the coefficient of
‘$\omega$’, reproduces (3.14).

The above analysis showed how the effects of back reaction in the space-time
can be discussed in a general frame work of tunneling mechanism. The only
assumption was that the modified metric must be static, spherically symmetric.
In particular, the modifications to the temperature and entropy for the
Schwarzschild case were explicitly evaluated. The results agree with earlier
findings by different methods. Next, we shall discuss the noncommutative
effect in addition to the back reaction effect in the space-time using our
general frame work.

###  3.2 Inclusion of noncommutativity

Here we shall apply our previous formulations to find the modifications to the
Hawking temperature and Bekenstein-Hawking area law due to noncommutative as
well as back reaction effects. In the vanishing limit of noncommutative
parameter, the results reduce to those obtained in the previous section. First
a short introduction on the noncommutative Schwarzschild black hole will be
given. Then the modifications to the thermodynamic entities will be
calculated.

####  3.2.1 Schwarzschild black hole in noncommutative space

The fact is that gravitation is a manifestation of the structure of spacetime
as dictated by the presence of gravitating objects. Therefore, inclusion of
noncommutative effects in gravity can be done in two ways. Directly take the
spacetime as noncommutative, $[x_{\mu},x_{\nu}]=i\theta_{\mu\nu}$ and use the
Seibarg-Witten map to recast the gravitational theory (in noncommutative
space) in terms of the corresponding theory in usual (commutative space)
variables. This leads to correction terms (involving powers of
$\theta{\mu\nu}$) in the various expressions like the metric, Riemann tensor
etc. This approach has been adopted in [104, 105, 106, 107, 108] 222For a
detailed discussions of this approach and a list of references see [109]..
Alternatively, incorporate the effect of noncommutativity in the mass term of
the gravitating object. Here the mass density, instead of being represented by
a Dirac delta function, is replaced by a Gaussian distribution. This approach
has been adopted in [110, 111, 112, 113, 114, 115] 333For a review and list of
references, see [116].. The two ways of incorporating noncommutative effects
in gravity are, in general, not equivalent. Here we follow the second
approach, for our investigation on the computation of thermodynamic entities
and area law for the noncommutative Schwarzschild black hole.

The usual definition of mass density in terms of the Dirac delta function in
commutative space does not hold good in noncommutative space because of the
position-position uncertainty relation. In noncommutative space mass density
is defined by replacing the Dirac delta function by a Gaussian distribution of
minimal width $\sqrt{\theta}$ in the following way [110]

$\displaystyle\rho_{\theta}(r)=\dfrac{M}{{(4\pi\theta)}^{3/2}}e^{-{\frac{r^{2}}{4\theta}}};\,\,\,\
{\displaystyle{Lim}}_{\theta\rightarrow
0}\rho_{\theta}(r)=\frac{M\delta(r)}{4\pi r^{2}}$ (3.17)

where the noncommutative parameter $\theta$ is a small ($\sim$ Plank length2)
positive number. This mass distribution is inspired from the coherent state
approach, where one has to consider the Voros star product instead of the
Moyal star product [88]. Using this expression one can write the mass of the
black hole of radius $r$ in the following way

$\displaystyle m_{\theta}(r)=\int_{0}^{r}{4\pi r^{\prime
2}\rho_{\theta}(r^{\prime})dr^{\prime}}=\frac{2M}{\sqrt{\pi}}\gamma(3/2,r^{2}/4\theta)$
(3.18)

where $\gamma(3/2,r^{2}/4\theta)$ is the lower incomplete gamma function,
which is discussed in the appendix. In the limit $\theta\rightarrow 0$ it
becomes the usual gamma function $(\Gamma_{{\textrm{total}}})$. Therefore
$m_{\theta}(r)\rightarrow M$ is the commutative limit of the noncommutative
mass $m_{\theta}(r)$.

To find a solution of Einstein equation with the noncommutative mass density
of the type (3.17), the temporal component of the energy momentum tensor
${(T_{\theta})}_{\mu}^{\nu}$ is identified as,
${(T_{\theta})}_{t}^{t}=-\rho_{\theta}$. Now demanding the condition on the
metric coefficients ${(g_{\theta})}_{tt}=-{(g_{\theta})}^{rr}$ for the
noncommutative Schwarzschild metric and using the covariant conservation of
energy momentum tensor ${(T_{\theta})}_{\mu}^{\nu}~{}_{;\nu}=0$, the energy
momentum tensor can be fixed to the form,

$\displaystyle{(T_{\theta})}_{\mu}^{\nu}={\textrm{diag}}{[-\rho_{\theta},p_{r},p^{\prime},p^{\prime}]},$
(3.19)

where, $p_{r}=-\rho_{\theta}$ and
$p^{\prime}=p_{r}-\frac{r}{2}\partial_{r}\rho_{\theta}$. This form of energy
momentum tensor is different from the perfect fluid because here $p_{r}$ and
$p^{\prime}$ are not same,

$\displaystyle
p^{\prime}=\Big{[}\frac{r^{2}}{4\theta}-1\Big{]}\frac{M}{(4\pi\theta)^{\frac{3}{2}}}e^{-\frac{r^{2}}{4\theta}}$
(3.20)

i.e. the pressure is anisotopic.

The solution of Einstein equation (in $c=G=1$ unit)
${(G_{\theta})}^{\mu\nu}=8\pi{{(T_{\theta})}^{\mu\nu}}$, using (3.19) as the
matter source, is given by the line element [110],

$\displaystyle
ds^{2}=-f_{\theta}(r)dt^{2}+\frac{dr^{2}}{f_{\theta}(r)}+r^{2}d\Omega^{2};\,\,\,f_{\theta}(r)=-{(g_{\theta})}_{tt}=\left(1-\frac{4M}{r\sqrt{\pi}}\gamma(\frac{3}{2},\frac{r^{2}}{4\theta})\right)~{}.$
(3.21)

Incidentally, this is same if one just replaces the mass term in the usual
commutative Schwarzschild space-time by the noncommutative mass
$m_{\theta}(r)$ from (3.18). Also observe that for $r>>\sqrt{\theta}$ the
above noncommutative metric reduces to the standard Schwarzschild form.

It is interesting to note that the noncommutative metric (3.21) is still
stationary, static and spherically symmetric as in the commutative case. One
or more of these properties is usually violated for other approaches [105,
106, 107] of introducing noncommutativity, particularly those based on
Seiberg-Witten maps that relate commutative spaces with noncommutative ones.

The event horizon can be found where $g^{rr}(r_{H})=0$, that is

$\displaystyle
r_{H}=\frac{4M}{\sqrt{\pi}}\gamma\Big{(}\frac{3}{2},\frac{r_{H}^{2}}{4\theta}\Big{)}.$
(3.22)

This equation cannot be solved for $r_{H}$ in a closed form. In the large
radius regime ($\frac{r_{H}^{2}}{4\theta}>>1$) we use the expanded form of the
incomplete $\gamma$ function given in the Appendix (eq. (3A.4)) to solve eq.
(3.22) by iteration. Keeping upto the order
$\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}$, we find

$\displaystyle r_{H}\simeq
2M\Big{(}1-\frac{2M}{\sqrt{\pi\theta}}e^{-\frac{M^{2}}{\theta}}\Big{)}.$
(3.23)

####  3.2.2 Noncommutative Hawking temperature, tunneling rate and entropy in
the presence of back reaction

Here the one loop back reaction effect on the space-time will be considered.
As explained earlier the modified surface gravity will be of the form given by
(3.2). But in this case since the only macroscopic parameter is $m_{\theta}$,
$\xi$ will has the following structure:

$\displaystyle\xi=\beta\frac{M_{\textrm{p}}^{2}}{m^{2}_{\theta}}$ (3.24)

where, as earlier, $\beta$ is a pure numerical factor. In the commutative
picture $\beta$ is known to be related to the trace anomaly coefficient [63,
64]. Putting this form of $\xi$ in (3.2) we get,

$\displaystyle\kappa=\kappa_{0}[r_{H}]\Big{(}1+\beta\frac{M_{\textrm{p}}^{2}}{m^{2}_{\theta}}\Big{)}.$
(3.25)

Equation (3.25) is recast as,

$\displaystyle\kappa=\kappa_{0}[r_{H}]\Big{(}1+\frac{\alpha}{m^{2}_{\theta}(r_{h})}\Big{)}$
(3.26)

where $\alpha=\beta M_{\textrm{p}}^{2}$. Since as mentioned already, the
noncommutative parameter $\theta$ is of the order of $l_{\textrm{p}}^{2}$,
$\alpha$ and $\theta$ are of the same order. This fact will be used later when
doing the graphical analysis.

In order to calculate the right hand side of (3.26), we need to obtain an
expression for noncommutative classical surface gravity at the horizon of the
black hole $(\kappa_{0}[r_{H}])$. This is done by using (2.19). For the
classical noncommutative Schwarzschild spacetime the metric coefficients are
given by (3.21). The value of $\kappa_{0}[r_{H}]$ is thus found to be,

$\displaystyle\kappa_{0}[r_{H}]=\frac{f^{\prime}(r_{H})}{2}=\frac{1}{2}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}.$
(3.27)

Inserting (3.27) in (3.26) we get,

$\displaystyle\kappa=\frac{1}{2}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}\Big{(}1+\frac{\alpha}{m^{2}_{\theta}(r_{H})}\Big{)}.$
(3.28)

In order to write the above equation completely in terms of $r_{H}$ we have to
express the mass $m_{\theta}$ in terms of $r_{H}$. For that we compare
equations (3.18) and (3.22) to get,

$\displaystyle m_{\theta}(r_{H})=\frac{r_{H}}{2}.$ (3.29)

This relation is the noncommutative deformation of the standard radius-mass
relation for the usual (commutative space) Schwarzschild black hole.
Expectedly in the limit $\theta\rightarrow 0$ eq. (3.29) reduces to its
commutative version $r_{H}=2M$.

Substituting (3.29) in (3.28) we get the value of modified noncommutative
surface gravity

$\displaystyle\kappa=\frac{1}{2}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}\Big{(}1+\frac{4\alpha}{r^{2}_{H}}\Big{)}.$
(3.30)

So from (2.29), the modified noncommutative Hawking temperature including the
effect of back reaction is given by,

$\displaystyle
T_{H}=\frac{\hbar\kappa}{2\pi}=\frac{\hbar}{4\pi}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}\Big{(}1+\frac{4\alpha}{r^{2}_{H}}\Big{)}.$
(3.31)

If the back reaction is ignored (i. e. $\alpha=0$), the expression for the
Hawking temperature agrees with that given in [110]. Also for the
$\theta\rightarrow 0$ limit, one can recover the standard result (3.14) [63,
64].

In the standard (commutative) case $T_{H}$ diverges as $M\rightarrow 0$ and
this puts a limit on the validity of the conventional description of Hawking
radiation. Against this scenario, temperature (3.31) includes noncommutative
and back reaction effects which are relevant at distances comparable to
$\sqrt{\theta}$. The behaviour of the temperature $T_{H}$ as a function of
horizon radius $r_{H}$ is plotted in fig.(3.1) (with positive $\alpha$) and in
fig.(3.2) (with negative $\alpha$).

Figure 3.1: $T_{H}$ Vs. $r_{H}$ plot (Here $\alpha=\theta$, $\alpha$ and
$\theta$ are positive).
$r_{H}$ is plotted in units of $\sqrt{\theta}$ and $T_{H}$ is plotted in units
of $\frac{1}{\sqrt{\theta}}$.
Red curve: $\alpha\neq 0,\theta=0$.
Blue curve: $\alpha=0,\theta=0$.
Black curve: $\alpha\neq 0,\theta\neq 0$.
Yellow curve: $\alpha=0,\theta\neq 0$.

Fig.(3.1) shows that in the region $r_{H}\simeq\sqrt{\theta}$, the effect of
noncommutativity significantly changes the nature of commutative space curves.
Interestingly two noncommutative curves, whether including back reaction or
not are qualitatively same. Both of them attain a maximum value at
$r_{H}={\tilde{r}}_{0}\simeq 4.7\sqrt{\theta}$ and then sharply drop to zero
forming an extremal black hole. In the region $r_{H}<r_{0}\simeq
3.0\sqrt{\theta}$ there is no black hole, because physically $T_{H}$ cannot be
negative. The only difference between them is that the back reaction effect
increases the maximum temperature by $20\%$. Infact, in the commutative space
also, back reaction effect increases the value of Hawking temperature. But
quite contrary to the noncommutative curves, both of them diverge as
$r_{H}\rightarrow 0$. As easily observed, the Hawking paradox is circumvented
by noncommutativity, with or without back reaction. This was also noted in
[110] where, however, the quantitative effects of back reaction were not
considered.

On the other hand fig.(3.2) shows that if any of the two effects (i.e. either
noncommutativity or back reaction) is present $T_{H}$ drops to zero. For
$\alpha=0,\theta\neq 0$ (yellow curve) $T_{H}$ becomes zero at
$r_{H}=r_{0}\simeq 3.0\sqrt{\theta}$ and for $\alpha\neq 0,\theta=0$ (red
curve) it becomes zero at $r_{H}=r_{0}\simeq 2.0\sqrt{\theta}$ . These cases
therefore bypass the Hawking paradox. But for noncommutative black hole with
back reaction ($\alpha\neq 0,\theta\neq 0$), $T_{H}$ is zero for two values of
$r_{H}$: $r_{H}\simeq 3.0\sqrt{\theta}$ and $r_{H}=2.0\sqrt{\theta}$ and then
it diverges towards positive infinity. This is not physically possible since
after entering the forbidden zone it resurfaces on the allowed sector. So for
both noncommutativity and back reaction effect, $\alpha$ can never be
negative.

Figure 3.2: $T_{H}$ Vs. $r_{H}$ plot (Here $|\alpha|=\theta$, $\alpha$ is
negative but $\theta$ is positive).
$r_{H}$ is plotted in units of $\sqrt{\theta}$ and $T_{H}$ is plotted in units
of $\frac{1}{\sqrt{\theta}}$.
Red curve: $\alpha\neq 0,\theta=0$.
Blue curve: $\alpha=0,\theta=0$.
Black curve: $\alpha\neq 0,\theta\neq 0$.
Yellow curve: $\alpha=0,\theta\neq 0$.

Having obtained the Hawking temperature of the black hole we calculate the
Bekenstein-Hawking entropy. The expression of entropy can be obtained from the
second law of thermodynamics. But instead of using it we employ the formula
(2.59) to calculate the entropy. Using (3.23) the modified surface gravity
(3.30) can be approximately expressed in terms of $M$. To the leading order,
we obtain,

$\displaystyle\kappa(M)$ $\displaystyle=$
$\displaystyle\frac{M^{2}+\alpha}{4M^{3}}\Big{[}1-\frac{4M^{5}}{(M^{2}+\alpha)\theta\sqrt{\pi\theta}}e^{-\frac{M^{2}}{\theta}}\Big{]}+{\cal{O}}(\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}).$
(3.32)

Substituting this in (2.51) and then integrating over $\omega^{\prime}$ we
have,

$\displaystyle{\textrm{Im}}~{}{\cal{S}}$ $\displaystyle=$ $\displaystyle
4\pi\omega(M-\frac{\omega}{2})+2\pi\alpha\ln{\Big{[}\frac{(M-\omega)^{2}+\alpha}{M^{2}+\alpha}\Big{]}}-8\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}$
(3.33) $\displaystyle+$ $\displaystyle
8\sqrt{\frac{\pi}{\theta}}(M-\omega)^{3}e^{-\frac{(M-\omega)^{2}}{\theta}}$
$\displaystyle+$ $\displaystyle\textrm{const.(independent of $M$)}+{\cal
O}(\sqrt{\theta}e^{-\frac{M^{2}}{\theta}}).$

So by the relation (2.55) the modified tunneling probability due to
noncommutativity and back reaction effects is,

$\displaystyle\Gamma$ $\displaystyle\sim$
$\displaystyle\Big{[}1-\frac{2\omega(M-\frac{\omega}{2})}{M^{2}+\alpha}\Big{]}^{-\frac{4\pi\alpha}{\hbar}}\textrm{exp}\Big{[}\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}-\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}(M-\omega)^{3}e^{-\frac{(M-\omega)^{2}}{\theta}}$
(3.34) $\displaystyle+$ $\displaystyle\textrm{const.(independent of
$M$)}\Big{]}\textrm{exp}\Big{[}-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})\Big{]}.$

The last exponential factor of the tunneling probability was previously
obtained by Parikh and Wilczek [23] where neither noncommutativity nor back
reaction effects were considered. The factors before this exponential are
actually due the effect of back reaction and noncommutativity. It will
eventually give the correction to the Bekenstein-Hawking entropy and the
Hawking temperature as will be shown below. Taking $\theta\rightarrow 0$ limit
we can immediately reproduce the commutative tunneling rate for Schwarzschild
black hole with back reaction effect [57].

We are now in a position to obtain the noncommutative deformation of the
Bekenstein-Hawking area law. The first step is to compute the entropy change
$\Delta S_{bh}$. Using (2.55) and (3.34) we obtain, to the leading order,

$\displaystyle\Delta S_{bh}=S_{final}-S_{initial}$ $\displaystyle\simeq$
$\displaystyle-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})-\frac{4\pi\alpha}{\hbar}\ln{\Big{[}\frac{(M-\omega)^{2}+\alpha}{M^{2}+\alpha}\Big{]}}+\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}$
(3.35) $\displaystyle-$
$\displaystyle\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}(M-\omega)^{3}e^{-\frac{(M-\omega)^{2}}{\theta}}+\textrm{const.(independent
of $M$)}.$

Next using the stability criterion $\frac{d(\Delta S_{bh})}{d\omega}=0$ for
the black hole, one obtains the only physically possible solution for $\omega$
as $\omega=M$. Substituting this value of $\omega$ in (3.35) and setting
$S_{{\textrm{final}}}=0$ we have the Bekenstein-Hawking entropy

$\displaystyle S_{bh}=S_{{\textrm{initial}}}$ $\displaystyle\simeq$
$\displaystyle\frac{4\pi
M^{2}}{\hbar}-\frac{4\pi\alpha}{\hbar}\ln{(\frac{M^{2}}{\alpha}+1)}$ (3.36)
$\displaystyle-$
$\displaystyle\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}+\textrm{const.(independent
of $M$)}.$

Neglecting the back reaction effect ($\alpha=0$) the above expression of black
hole entropy is written as

$\displaystyle S_{bh}\simeq\frac{4\pi
M^{2}}{\hbar}-\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}.$
(3.37)

Now in order to write the above equation in terms of the noncommutative
horizon area ($A_{\theta}$), we use (3.23) to obtain,

$\displaystyle A_{\theta}=4\pi r_{H}^{2}=16\pi
M^{2}-64\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}+{\cal{O}}(\sqrt{\theta}e^{-\frac{M^{2}}{\theta}}).$
(3.38)

Comparing equations (3.37) and (3.38) we see that at the leading order the
noncommutative black hole entropy satisfies the area law

$\displaystyle S_{bh}=\frac{A_{\theta}}{4\hbar}.$ (3.39)

This is functionally identical to the Bekenstein-Hawking area law in the
commutative space.

Considering $\theta\rightarrow 0$ limit in (3.36) we have the corrected form
of Bekenstein-Hawking entropy for commutative Schwarzschild black hole with
back reaction effect [64, 57]. The well known logarithmic correction [71] is
reproduced (see equation (3.13)).

Now using the second law of thermodynamics (2.67) we can find the corrected
form of the Hawking temperature $T_{H}$ due to back reaction. This is obtained
from (3.36) as,

$\displaystyle\frac{1}{T_{H}}=\frac{dS_{bh}}{dM}=\frac{8\pi
M^{3}}{\hbar(M^{2}+\alpha)}+\frac{32}{\hbar}\frac{\sqrt{\pi}}{\theta^{\frac{3}{2}}}M^{4}e^{-\frac{M^{2}}{\theta}}+{\cal{O}}(\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}).$
(3.40)

Therefore the back reaction corrected noncommutative Hawking temperature is
given by

$\displaystyle T_{H}$ $\displaystyle=$
$\displaystyle\frac{\hbar(M^{2}+\alpha)}{8\pi M^{3}}-\frac{\hbar
M^{2}}{2(\pi\theta)^{\frac{3}{2}}}e^{-\frac{M^{2}}{\theta}}+{\cal{O}}(\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}).$
(3.41)

We now provide a simple consistency check on the relation (3.31). The Hawking
temperature is recalculated using this relation and showing that it reproduces
(3.41). For the large radius limit, (3.31) takes the value,

$\displaystyle
T_{H}\simeq\frac{\hbar}{4\pi}\big{[}\frac{1}{r_{H}}-\frac{r_{H}^{2}}{2\sqrt{\pi}\theta^{3/2}}e^{-\frac{r_{H}^{2}}{4\theta}}\big{]}\big{(}1+\frac{4\alpha}{r_{H}^{2}}\big{)}.$
(3.42)

Now the approximated form of $r_{H}$ in terms of $M$ (3.23) is substituted in
(3.42) to get the relation (3.41) upto the leading order in the noncommutative
parameter. This shows the self consistency of our calculation.

For $\alpha=\theta=0$, the expression (3.41) reduces to the usual Hawking
temperature $T_{H}=\frac{\hbar}{8\pi M}$ for a Schwarzschild black hole. Also,
keeping the back reaction ($\alpha$) but taking $\theta\rightarrow 0$ limit,
we reproduce the commutative Hawking temperature (3.14) [63, 64, 57].

###  3.3 Discussions

We have considered self-gravitation and (one loop) back reaction effects in
tunneling formalism for Hawking radiation. The modified tunneling rate was
computed. From this modification, corrections to the semiclassical expressions
for entropy and Hawking temperature were obtained. Also, the logarithmic
correction to the semiclassical Bekenstein-Hawking area law was reproduced.

The other significant part of this chapter was the application of our
formulation to a noncommutative Schwarzschild metric, keeping in mind the
consequence of back reaction. Several thermodynamic entities like the
temperature and entropy were computed. The tunneling rate was also derived.
The temperature, in particular, was obtained in a closed form. This result was
analyzed in detail using two graphical representations. We gave particular
attention to the small scale behaviour of black hole temperature where the
effects of both noncommutativity and back reaction are highly nontrivial. The
graphs presented here are naturally more general than [110, 63], because in
[110] the effect of back reaction was not included and in [63] space time was
taken to be commutative in nature. Expectedly in suitable limits, the results
of our paper reduced to that of [110, 63], but the combination of
noncommutativity and back reaction, as shown here, gave new results at small
scale. In particular, it was shown that in the presence of both
noncommutativity and back reaction, the back reaction parameter $\alpha$
cannot be negative. Interestingly, even for the commutative case, arguments
based on quantum geometry [57, 71, 66] fix a positive value for $\alpha$.

In the noncommutative analysis, with positive $\alpha$, (Fig 3.1), the maximum
Hawking temperature got enhanced in the presence of back reaction. However,
the Hawking paradox was avoided whether or not the back reaction is included.

Apart from the temperature, other variables like the tunneling rate and
entropy were given upto the leading order in the noncommutative parameter. The
entropy was expressed in terms of the area. The result was a noncommutative
deformation of the Bekenstein-Hawking area law, preserving the usual
functional form. Since both $T_{H}=\frac{\hbar\kappa}{2\pi}$ and the area law
retained their standard forms it suggests that the laws of noncommutative
black hole thermodynamics are a simple noncommutative deformation of the usual
laws. However, it must be remembered this result was obtained only in the
leading order approximation. For $r\sim\sqrt{\theta}$ this approximation is
expected to be significant.

As a final remark we mention that although our results are presented for the
Schwarzschild metric, the formulation is resilient enough to discuss both back
reaction and noncommutativity in other types of black holes.

## Appendix

##  Appendix 3.A Incomplete gamma function

The lower incomplete gamma function is given by

$\displaystyle\gamma(a,x)=\int_{0}^{x}t^{a-1}e^{-t}dt$ (3A.1)

whereas the upper incomplete gamma function is

$\displaystyle\Gamma(a,x)=\int_{x}^{\infty}t^{a-1}e^{-t}dt$ (3A.2)

and they are related to the total gamma function through the following
relation

$\displaystyle\Gamma_{{\textrm{total}}}(a)=\gamma(a,x)+\Gamma(a,x)=\int_{0}^{\infty}t^{a-1}e^{-t}dt.$
(3A.3)

Furthermore, for large $x$, i.e. $x>>1$, the asymptotic expansion of the lower
incomplete gamma function is given by

$\displaystyle\gamma(\frac{3}{2},x)$ $\displaystyle=$
$\displaystyle\Gamma_{{\textrm{total}}}(\frac{3}{2})-\Gamma(\frac{3}{2},x)$
(3A.4) $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{\pi}}{2}\Big{[}1-e^{-x}\sum_{p=0}^{\infty}\frac{x^{\frac{1-2p}{2}}}{\Gamma_{{\textrm{total}}}(\frac{3}{2}-p)}\Big{]}.$

Using the definition (3A.1) and then integrating by parts we have

$\displaystyle\gamma(a+1,x)=\int_{0}^{x}t^{a}e^{-t}dt$ $\displaystyle=$
$\displaystyle-t^{a}e^{-t}|_{0}^{x}+a\int_{o}^{x}t^{a-1}e^{-t}dt$ (3A.5)
$\displaystyle=$ $\displaystyle-x^{a}e^{-x}+a\gamma(a,x).$

Similarly by the definition (3A.2) one can show

$\displaystyle\Gamma(a+1,x)=x^{a}e^{-x}+a\Gamma(a,x).$ (3A.6)

##  Appendix 3.B Some useful formulas

$\displaystyle I_{1}=\int_{a}^{b}e^{-\alpha
x^{2}}dx=\frac{1}{2{\alpha}^{\frac{1}{2}}}\Big{[}\sqrt{\pi}-\gamma(\frac{1}{2},\alpha
a^{2})-\Gamma(\frac{1}{2},\alpha b^{2})\Big{]}$ (3B.1) $\displaystyle
I_{2}=\int_{a}^{b}x^{2}e^{-\alpha
x^{2}}dx=\frac{1}{2{\alpha}^{\frac{3}{2}}}\Big{[}\frac{\sqrt{\pi}}{2}-\gamma(\frac{3}{2},\alpha
a^{2})-\Gamma(\frac{3}{2},\alpha b^{2})\Big{]}$ (3B.2) $\displaystyle
I_{3}=\int_{a}^{b}x^{4}e^{-\alpha
x^{2}}dx=\frac{1}{2{\alpha}^{\frac{5}{2}}}\Big{[}\frac{3\sqrt{\pi}}{4}-\gamma(\frac{5}{2},\alpha
a^{2})-\Gamma(\frac{5}{2},\alpha b^{2})\Big{]}$ (3B.3)

##  Chapter 4 Tunneling mechanism and anomaly

Ever since Hawking’s original observation [4, 5] that black holes radiate,
there have been several derivations [8, 9, 22, 23, 10, 11, 16, 17, 18] of this
effect. A common feature in these derivations is the universality of the
phenomenon; the Hawking radiation is determined by the horizon properties of
the black hole leading to the same answer. This, in the absence of direct
experimental evidence, definitely reinforces Hawking’s original conclusion.
Moreover, it strongly suggests that there is some fundamental mechanism which
could, in some sense, unify the various approaches.

In this chapter we show that chirality is the common property which connects
the tunneling formalism [22, 23] and the anomaly method [9, 10, 11, 16, 17,
18, 19, 20, 21, 117, 118] in studying Hawking effect. Apart from being among
the most widely used approaches, interest in both the anomaly and tunneling
methods has been revived recently leading to different variations and
refinements in them [16, 18, 20, 21, 117, 118, 72, 73, 74, 78, 55, 119]. The
calculation will be performed using a family of metrics that includes a subset
of the stationary, spherically symmetric space-times which are asymptotically
flat. Also, the results are derived using mostly physical reasoning and do not
require any specific technical skill.

Before commencing on our analysis we briefly recapitulate the basic tenets of
the tunneling and anomaly methods. The idea of a tunneling description, quite
akin to what we know in usual quantum mechanics where classically forbidden
processes might be allowed through quantum tunneling, dates back to 1976
[120]. Present day computations generally follow either the null geodesic
method [23] or the Hamilton-Jacobi method [22], both of which rely on the
semi-classical WKB approximation yielding equivalent results. The essential
idea, as explained earlier, is that a particle-antiparticle pair forms close
to the event horizon. The ingoing negative energy mode is trapped inside the
horizon while the outgoing positive energy mode is observed at infinity as the
Hawking flux.

Although the notion of an anomaly, which represents the breakdown of some
classical symmetry upon quantisation, is quite old, its implications for
Hawking effect were first studied in [9]. It was based on the conformal
(trace) anomaly but the findings were confined only to two dimensions. However
it is possible to apply this method to general dimensions. Recently a new
method was put forward in [10, 11] where a general (any dimensions) derivation
was given. It was based on the well known fact that the effective theory near
the event horizon is a two dimensional conformal theory. The ingoing modes are
trapped within the horizon and cannot contribute to the effective theory near
the horizon. Thus the near horizon theory becomes a two dimensional chiral
theory. Such a chiral theory suffers from a general coordinate
(diffeomorphism) anomaly manifested by a nonconservation of the stress tensor.
Using this gravitational anomaly and a suitable boundary condition the Hawking
flux was obtained. A covariant version of this method, that was also
technically simpler, was given in [16]. This was followed by another, new,
effective action based approach in [18, 17].

The first step in our procedure is to derive the two dimensional gravitational
anomaly using the notion of chirality. This is a new method of obtaining the
gravitational anomaly. Once this anomaly is obtained, the flux is easily
deduced. Exploiting the same notion of chirality the probability of the
outgoing mode in the tunneling approach will be computed. The Hawking
temperature then follows from this probability. At an intermediate stage of
this computation we further show that the chiral modes obtained in the
tunneling formalism reproduce the gravitational anomaly thereby completing the
circle of arguments regarding the connection of the two approaches.

###  4.1 Metric and null coordinates

Consider a black hole characterised by a spherically symmetric, static space-
time and asymptotically flat metric of the form (2.1). For simplicity we
consider here $f(r)=g(r)=F(r)$ and hence the event horizon $r=r_{H}$ is
defined by $F(r_{H})=0$. Now it is well known [10, 11, 14, 121] that near the
event horizon the effective theory reduces to a two dimensional conformal
theory whose metric is given by the ($r-t$) sector of the original metric
(2.1).

It is convenient to express (2.1) in the null tortoise coordinates which are
defined as,

$\displaystyle u=t-r^{*},\,\,\,v=t+r^{*};$ (4.1)

where $r^{*}$ is defined by the relation (2.20). Under these set of
coordinates the relevant ($r-t$)-sector of the metric (2.1) takes the form,

$\displaystyle ds^{2}=-\frac{F(r)}{2}(du~{}dv+dv~{}du)$ (4.2)

Chiral conditions, to be discussed in the next section, are most appropriately
described in these coordinates.

###  4.2 Chirality conditions

Consider the Klein-Gordon (KG) equation (2.2) for a massless scalar particle
governed by the metric (4.2). Then the KG equation reduces to the following
form:

$\displaystyle 2\partial_{u}\partial_{v}\phi(u,v)=0.$ (4.3)

The general solution of this can be taken as
$\phi(u,v)=\phi^{(R)}(u)+\phi^{(L)}(v)$ where $\phi^{(R)}(u)$ and
$\phi^{(L)}(v)$ are the right (outgoing) and left (ingoing) modes (see
Appendix 2.A) satisfying

$\displaystyle\nabla_{v}\phi^{(R)}=0,\,\,\nabla_{u}\phi^{(R)}\neq 0;\,\,\,\
\nabla_{u}\phi^{(L)}=0,\,\,\,\nabla_{v}\phi^{(L)}\neq 0.$ (4.4)

These equations are expressed simultaneously as,

$\displaystyle\nabla_{\mu}\phi=\pm\bar{\epsilon}_{\mu\nu}\nabla^{\nu}\phi=\pm\sqrt{-g}\epsilon_{\mu\nu}\nabla^{\nu}\phi$
(4.5)

where $+(-)$ stand for left (right) mode and $\epsilon_{\mu\nu}$ is the
numerical antisymmetric tensor with $\epsilon_{uv}=\epsilon_{tr}=-1$. This is
the chirality condition 111In analogy with studies in 2d CFT this condition is
usually referred as holomorphy condition and the chiral modes $\phi^{(L,R)}$
are called the holomorphic modes.. In fact the condition (4.5) holds for any
chiral vector $J_{\mu}$ in which case
$J_{\mu}=\pm{\bar{\epsilon}}_{\mu\nu}J^{\nu}$. Likewise, the chirality
condition for the energy-momentum tensor is [19],

$\displaystyle
T_{\mu\nu}=\pm\frac{1}{2}(\bar{\epsilon}_{\mu\sigma}T^{\sigma}_{\nu}+\bar{\epsilon}_{\nu\sigma}T^{\sigma}_{\mu})+\frac{1}{2}g_{\mu\nu}T^{\alpha}_{\alpha}$
(4.6)

The $+$ ($-$) sign corresponds to the left (right) mode satisfying,

$\displaystyle T^{(R)}_{vv}=0,\,\,\,\,T^{(R)}_{uu}\neq 0$ (4.7) $\displaystyle
T^{(L)}_{uu}=0,\,\,\,\,T^{(L)}_{vv}\neq 0~{}.$ (4.8)

These are the analogous of (4.4). They manifest the symmetry under the
interchange $u\leftrightarrow v$ and $L\leftrightarrow R$. In the next
section, using these chirality conditions we will derive the explicit form for
the gravitational anomaly that reproduces the Hawking flux.

###  4.3 Chirality, gravitational anomaly and Hawking flux