# Multi-modal unsupervised brain image registration using edge maps

###### Abstract

Diffeomorphic deformable multi-modal image registration is a challenging task
which aims to bring images acquired by different modalities to the same
coordinate space and at the same time to preserve the topology and the
invertibility of the transformation. Recent research has focused on leveraging
deep learning approaches for this task as these have been shown to achieve
competitive registration accuracy while being computationally more efficient
than traditional iterative registration methods. In this work, we propose a
simple yet effective unsupervised deep learning-based multi-modal image
registration approach that benefits from auxiliary information coming from the
gradient magnitude of the image, i.e. the image edges, during the training.
The intuition behind this is that image locations with a strong gradient are
assumed to denote a transition of tissues, which are locations of high
information value able to act as a geometry constraint. The task is similar to
using segmentation maps to drive the training, but the edge maps are easier
and faster to acquire and do not require annotations. We evaluate our approach
in the context of registering multi-modal (T1w to T2w) magnetic resonance (MR)
brain images of different subjects using three different loss functions that
are said to assist multi-modal registration, showing that in all cases the
auxiliary information leads to better results without compromising the
runtime.

Index Terms—  multi-modal registration, inter-subject, gradient magnitude,
deep-learning registration, unsupervised learning

##  1 Introduction

Image registration is an important clinical application for many medical
imaging workflows. Its goal is to find corresponding anatomical or functional
locations in two or more images from the same or different subjects, acquired
by one or more imaging modalities [1]. In clinical applications, it is often
beneficial to fuse information from multiple image modalities for diagnosis or
intervention guidance due to the fact that each imaging modality depicts
different aspects of the underlying anatomy and pathology.

Fig. 1: Network architecture. Given a moving T1w and a fixed T2w image, we
first calculate their corresponding edge maps. Then, we map the fixed and
moving images and their edge maps to the parameters of the transformation
using a U-Net architecture which outputs the deformation field to warp the
images. The loss function is a weighted combination of a loss function for
image registration, a loss function for edge map registration and a
regularisation loss that penalises illegal transformation configurations.

The process of image registration involves finding the optimal geometric
transformation which maximizes the correspondences across the images, i.e.
given a target image $T$ and a source image $S$, image registration aims to
find the optimal spatial transformation $\phi$ which maps a location
${\mathbf{x}}$ of $S$ to the location with corresponding tissue or structure
of $T$ [2]. In traditional iterative approaches, the image alignment is
achieved via a an optimisation process which normally requires many iterations
to be effective. Thus, it can be time consuming. On the other hand, more
recent works are exploring data-driven approaches and the incorporation of
deep learning to tackle the problem [3], [4], [5], [6], [7]. Although
sometimes training can be time consuming, the registration step during
inference is faster and thus more efficient compared to the iterative
optimisation methods.

Recent works focus on training networks in an unsupervised manner, i.e.
without supervising the task with ground truth transformations. Most of these
works only consider the unimodal registration task and make use of intensity
based similarity metrics such us Mean Squared Error (MSE) or Cross Correlation
(CC) [8] as image matching criterion [3], [7], which make specific assumptions
(identity or linear relationship) about the image intensities. However, these
metrics are normally not suitable for the challenging problem of multi-modal
image registration due to the unknown relationships between the intensity
distributions and the potential differences in the geometry that each modality
can capture.

To tackle this problem, previous multi-modal approaches [4], [6] rely on
information theoretic measures such as mutual information (MI) [9]. However,
such measures might not be suitable because they are agnostic to spatial
dependency between adjacent voxels. Another approach is to reduce the multi-
modal registration problem to a unimodal one [5]. Techniques like the latter
are very time consuming to train and hard to control.

In this paper, we propose a novel unsupervised multi-modal image registration
approach to tackle the problem of registering images of different subjects
(inter-subject) acquired by two different imaging modalities (T1w, T2w MR
images). We are integrating ideas from the work of Pluim et al. [10] that
combines mutual information with a term based on the magnitude and direction
of the image gradients to rigidly register 3D MR, CT and PET images, the work
of Qin et al. [11] that uses segmentation maps as auxiliary information to
guide the task of motion estimation and the work of Zhe Xu at al. [12] that
performs unsupervised multimodal CT-MR image registration leveraging the
deformation fields estimated from a branch aiming to align the original fixed
and moving images and a branch aiming to align the corresponding gradient
intensity maps.

Similarly, we propose to make use of the edge maps extracted the images, as a
complementary signal, to aid the training of registration network. The main
assumption is that since the anatomy depicted by the two modalities is very
similar, the edge maps can serve as a geometrical constraint. At the same
time,

the edges are less modality dependent, a property that is very useful in the
case of multi-modal registration due to the fact that we do not have to deal
with the complicated intensity relationship between the modalities. This
simple, yet, effective technique shows that the edge signal can be beneficial
for multi-modal image registration.

Different from the typical one-branch unsupervised image registration network,
our approach leverages the deformation field estimated from two branches that
perform image and edge registration simultaneously. This is very similar to
the work of Zhe Xu at al. [12], but instead of trying to fuse the deformation
fields resulting from the image and edge branches respectively, we try to
estimate only one deformation field for both branches at the same time. The
network architectures that we incorporated and extended are CNN networks based
on Voxelmorph [3] and on MIDIR [6]. The whole pipeline of our method is
depicted in 1. The network consists of 2 encoders: one for encoding the
features of the images and the other for encoding the features of the edge
maps. Then the encoded features are passed through a decoder that outputs
directly a dense stationary velocity field (SVF), in the case of Voxelmorph,
or the velocities of the B-spline control points from which we can compute the
transformation via B-spline tensor product and Squaring and Scaling in the
case of MIDIR.

To train both networks we employ as loss functions: local normalised cross
correlation (LNCC) [8] as implemented in [3], normalised mutual information
(NMI) [13] as implemented in [6] and normalised gradient field (NGF) [14], an
edge-based loss function suitable for multi-modal registration. Our
experimental results prove that employing the edge maps leads to better
registration accuracy without a huge increase in the memory requirements and
the runtime.

##  2 METHOD

###  2.1 Edge Maps

As stated above, the edge maps extracted from multi-modal images can be a
useful auxiliary information to guide multi-modal image registration.
Specifically, the edge map of a 2D (but easily extensible to 3D) image can be
easily obtained by calculating the central differences between adjacent
pixels.

We decided to only use the gradient magnitude because it is sufficient to
depict the underlying geometry and discard the information relating to the
edge direction. The part of the network that handles the edge maps is more
sensitive in capturing the structural dependencies between the two images,
rather than the complex intensity relationships, and hence serves as a spatial
constraint assisting the task of image registration. In practice, the edge
maps are given as input to an encoder which adopts the same architecture as
the one that handles the images.

###  2.2 Network Architectures and Transformation

In our effort to tackle the problem of multi-modal deformable image
registration after the affine alignment, we decided to extend two different
existing network architectures ([3] and [6]). The generic network architecture
is depicted in Figure 1 and it is in general a convolutional U-Net [15]
architecture that consists of two separate encoders without shared weights.
The first encoder takes as input the T1w and T2w images, while the second one
takes the edge maps of those. Both encoders reduce the spatial dimensions in
half at each layer, operating on coarser representations. The features coming
from the two encoders are then concatenated and passed to a common decoder
which consists of upsampling, convolutions and concatenation of skip
connections and outputs a dense velocity field or the B-spline control points.
The velocity field can be a direct output of the network as described by [16],
or can be determined using the B-spline control points from which the
diffeomorphic transformation can be computed via B-spline tensor product and
Squaring and Scaling, as described by [6]. All convolution layers use a kernel
of size $3$ in all spatial dimensions and a stride of 2. The non-linearities
are LeakyReLUs with a slope of $-0.2$.

###  2.3 Loss Functions

During the training, the moving image $M$ and its corresponding edges $E_{M}$
are warped using the transformation $\phi$ via linear interpolation of the
image intensities to generate the moving image $M\circ\phi$ and moving edge
image $E_{M}\circ\phi$ respectively. These are then used to compute the image
similarity loss $\mathcal{L}_{\operatorname{im-sim}}(F,M\circ\phi)$ and the
edge similarity loss $\mathcal{L}_{\operatorname{ed-
sim}}(E_{F},E_{M}\circ\phi)$. The main goal is to find the optimal network
parameters that minimise the loss function which is comprised by a term for
image similarity, a term for edge image similarity and a term for the
transformation regularisation on the velocity field.

In order to compute the edge similarity loss, we chose to use Local Normalised
Cross Correlation (LNCC) which is a local metric suitable for tasks with
intensity variations close to liner [8]. On the other hand LNCC is also
incorporated as an image similarity metric along with the differentiable
version of Normalised Mutual Information [13] as introduced by [6] and the
edge-based metric Normalised Gradient Field (NGF) [14]. The
$\mathcal{L}_{\operatorname{Reg}}$ is a regularisation on the velocity field
to ensure smoothness and diffeomorphism [17].

The overall loss can be written as:

$\displaystyle\mathcal{L}_{1}=\lambda_{1}\mathcal{L}_{\operatorname{im-
sim}}(F,M\circ\phi),$ (1)
$\displaystyle\mathcal{L}_{2}=\lambda_{2}\mathcal{L}_{\operatorname{ed-
sim}}(E_{F},E_{M}\circ\phi),$ (2)
$\displaystyle\mathcal{L}_{3}=\lambda_{3}\mathcal{L}_{\operatorname{Reg}},$
(3) $\displaystyle\mathcal{L}=\mathcal{L}_{1}+\mathcal{L}_{2}+\mathcal{L}_{3}$
(4)

where $\lambda_{1},\lambda_{2},\lambda_{3}\geq 0$ are the weighting factors
that are optimally tuned using a validation set, while considering the balance
between registration accuracy and transformation regularity.

##  3 Experiments

###  3.1 Implementation Details

We used PyTorch to implement the deep learning networks and loss functions.
The Adam [18] optimiser is used with an initial learning rate $1e^{-4}$ and
the learning rate decay is set to 0.1 every 50 epochs. The traditional SVFFD
method, which we use as a baseline, is implemented by the MIRTK (Medical Image
Registration ToolKit) [19]. All the time measurements are being performed on a
workstation with an AMD RyzenTM ThreadripperTM 3960X CPU and NVIDIA® Quadro
RTX 8000 GPU.

###  3.2 Tasks and Data

We evaluate our work on the task of inter-subject brain MRI registration on
T1w-T2w images. For this purpose, we decided to use the Cambridge Centre for
Ageing and Neuroscience (CamCAN) dataset [20], [21]. The dataset consist T1w
and T2w MR 3D volumetric images of 310 subjects, with $1\mbox{mm}^{3}$
isotropic spatial resolution. In our work we only use 2D slicse of the
volumetric image cropped to the size of $192\times 192$. Although this work
could directly be transferred in 3D, we use 2D images as a proof-of-concept,
due to the large computational requirements when using 3D images. After affine
alignment to a common MNI space using ANTs [22], the images are skull-stripped
using ROBEX [23] and bias-field corrected using the N4 algorithm in SimpleITK
[24]. For evaluation, we also acquired the segmentation of 138 cortical and
sub-cortical structures (grouped into 5 groups) automatically using MALPEM
[25].

###  3.3 Evaluation Metrics

In order to evaluate our method, we choose to both evaluate the accuracy and
the regularity of the registration. The accuracy is being evaluated using the
Dice score to measure the overlap between anatomical segmentations of fixed
and moving images. On the other hand, the Jacobian determinant of the
transformation is an index of the registration regularity, i.e. the ratio of
points with $J<0$ evaluates the amount of folding of the transformation and
the magnitude of the gradient of the Jacobian ($|\nabla_{J}|$) determinant
evaluates the spatial smoothness of the transformation.

###  3.4 Comparison to state-of-the-art Registration

To present the effectiveness of the incorporation of the edge information in
the training process we are comparing our method with three baseline methods.
The first comparison is with MIRTK, an iterative method based on SVFFD
transformation [19]. Regarding the state-of-the-art deep learning methods, we
decided to compare with Voxelmorph [3] and MIDIR [6] using the loss functions
stated above. Additionally, our reference is the evaluation metrics of the
initial affine registration.

###  3.5 Results

In Table 1 we present the quantitative evaluation of all models and loss
functions on the inter-subject brain T1w-T2w registration task. The network
architectures that end with ED depict the ones that were trained with the
incorporation of the edge maps. We have trained the edge branch using LNCC and
MSE as a loss function, second and third column of 1 respectively. In the 2D
case images trained without the edge information, we found that LNCC performs
most of the times better than NMI and NGF. Although cross correlation is not
usually beneficial for multi-modal registration, its local implementation
manages to outperform the other metrics. The reason for that may be that T1w
and T2w images are locally closely related and hence the LNCC performs
particularly well in our case. We do not expect this behaviour though when we
have to combine features from two very different modalities e.g. MR (Magnetic
Resonance) - CT (Computed Tomography).

| LNCC as edge loss | MSE as edge loss
---|---|---
Methods | Dice | $J<0$ | $|\nabla_{J}|$ | Dice | $J<0$ | $|\nabla_{J}|$
Affine | $0.643\pm 0.087$ | - | - | - |  |
SVFFDLNCC | $0.749\pm 0.042$ | $5.3e^{-4}$ | $0.025$ | $0.749\pm 0.042$ | $5.3e^{-4}$ | $0.025$
VMLNCC | $0.733\pm 0.055$ | $22e^{-4}$ | $0.073$ | $0.733\pm 0.055$ | $22e^{-4}$ | $0.073$
MIDIRLNCC | $0.691\pm 0.063$ | $13e^{-4}$ | $0.0187$ | $0.691\pm 0.063$ | $13e^{-4}$ | $0.0187$
VM_EDLNCC | $\mathbf{0.754\pm 0.043}$ | $63e^{-4}$ | $0.044$ | $\mathbf{0.748\pm 0.065}$ | $37e^{-4}$ | $0.074$
MIDIR_EDLNCC | $0.732\pm 0.070$ | $0.37e^{-4}$ | $0.02$ | $0.691\pm 0.049$ | $0.67e^{-4}$ | $0.01$
SVFFDNMI | $0.7393\pm 0.081$ | $2.5e^{-4}$ | $0.024$ | $0.7393\pm 0.081$ | $2.5e^{-4}$ | $0.024$
VMNMI | $0.71\pm 0.076$ | $0.3e^{-4}$ | $0.032$ | $0.71\pm 0.076$ | $0.3e^{-4}$ | $0.032$
MIDIRNMI | $0.699\pm 0.073$ | $0.1e^{-4}$ | $0.013$ | $0.699\pm 0.073$ | $0.1e^{-4}$ | $0.013$
VM_EDNMI | $\mathbf{0.759\pm 0.093}$ | $0.1e^{-4}$ | $0.033$ | $\mathbf{0.732\pm 0.093}$ | $0.28e^{-4}$ | $0.06$
MIDIR_EDNMI | $0.721\pm 0.0731$ | $0.5e^{-4}$ | $0.015$ | $0.69\pm 0.068$ | $0.4e^{-4}$ | $0.06$
VMNGF | $0.672\pm 0.11$ | $0.16e^{-4}$ | $0.0035$ | $0.672\pm 0.11$ | $0.16e^{-4}$ | $0.0035$
MIDIRNGF | $0.652\pm 0.07$ | $1.2e^{-4}$ | $0.0042$ | $0.652\pm 0.07$ | $1.2e^{-4}$ | $0.0042$
VM_EDNGF | $\mathbf{0.694\pm 0.087}$ | $0.1e^{-4}$ | $0.021$ | $0.645\pm 0.098$ | $0.1e^{-4}$ | $0.005$
MIDIR_EDNGF | $0.681\pm 0.101$ | $0.23e^{-4}$ | $0.011$ | $\mathbf{0.711\pm 0.064}$ | $0.04e^{-4}$ | $0.028$

Table 1: Quantitative results for T1w-T2w inter-subject image registration. We
report the average Dice score of the different anatomical structures, the
Jacobian determinant of the transformation ($J<0$) whose higher number has the
meaning of more folding and the Jacobian gradient magnitude ($|\nabla_{J}|$)
whose lower values denote the smoother transformations.

Moreover, we observe that Voxelmorph attains better average Dice score for
almost every loss function. Nevertheless, MIDIR achieved better transformation
regularity, while the traditional iterative method achieved the better Dice
score outperforming the deep learning ones. The intuition behind that could be
that MIRTK used SVFFD which results to a very detailed optimisation for each
pair of images and a multi-resolution approach, while the deep learning
methods perform one pass predictions without using multiple resolutions. On
the other hand, the deep learning methods with the incorporation of the edge
maps during the training managed to outperform the methods without the edges
in each experiment. Interestingly, these methods also result in better Dice
scores than the SVFFD method, with Voxelmorph to achieve the best results with
every loss. These results confirm our hypothesis and actually strongly
supports that something very simple to calculate as the gradient of the image
edges can be useful as auxiliary information to drive the registration.

(a) Moving

(b) Fixed

(c) $M\circ\phi$

(d) Moving edges

(e) Fixed edges

(f) $E_{M}\circ\phi$

Fig. 2: T1w and T2w images with the corresponded extracted edge maps and the
outcome of the deformable registration using our method.

##  4 Conclusion

In this work we present a simple yet effective way to improve the results in
the challenging task of the multi-modal inter-subject brain registration. We
propose a registration network architecture that benefits from the usage of
the magnitude of the gradient of the image, i.e. the edge maps, during its
training phase. Our experimental evaluations with different loss functions and
network architectures and comparing against the state of the art demonstrate
that the suggested geometrical constraint can successfully assist us in our
effort to improve the image registration accuracy.

##  5 Compliance with ethical standards

This work uses the CamCAN dataset which was obtained in compliance with the
Helsinki Declaration, and has been approved by the local ethics committee,
Cambridgeshire 2 Research Ethics Committee (reference: 10/H0308/50).

##  6 Acknowledgments

No funding was received for conducting this study. The authors have no
relevant financial or non-financial interests to disclose.

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# URBAN CHANGE DETECTION USING A DUAL-TASK SIAMESE NETWORK AND SEMI-SUPERVISED
LEARNING

###### Abstract

In this study, a Semi-Supervised Learning (SSL) method for improving urban
change detection from bi-temporal image pairs was presented. The proposed
method adapted a Dual-Task Siamese Difference network that not only predicts
changes with the difference decoder, but also segments buildings for both
images with a semantics decoder. First, the architecture was modified to
produce a second change prediction derived from the semantics predictions.
Second, SSL was adopted to improve supervised change detection. For unlabeled
data, we introduced a loss that encourages the network to predict consistent
changes across the two change outputs. The proposed method was tested on urban
change detection using the SpaceNet7 dataset. SSL achieved improved results
compared to three fully supervised benchmarks. Code for the paper is available
at https://github.com/SebastianHafner/SiameseSSL.git.

Index Terms—  semi-supervised learning, urban change detection, siamese
network, remote sensing

##  1 INTRODUCTION

Urbanization is progressing at an unprecedented rate in many places around the
world. Earth observation is a crucial tool to map land cover changes
associated with urbanization. Change detection is typically conducted by
comparing images acquired at different times that cover the same geographical
area. Therefore, change detection is considered a binary classification
problem [1].

In computer vision, the process of linking each pixel in an image to a class
is referred to as semantic segmentation. Fully convolutional networks have
become the state-of-the-art method for segmentation tasks by leveraging large
collections of examples (i.e., labeled data). Fully convolutional networks
such as the encoder-decoder architecture U-Net [2] can easily be adopted for
change detection by concatenating bi-temporal image pairs along the channel
axis, also referred to as Early Fusion (EF) [3]. However, network
architectures tailored to Earth observation change detection tasks may be more
promising than off-the-shelf architectures adopted from computer vision. Daudt
et al. [3] proposed two Siamese network architectures that pass images
separately through two encoders with shared weights, before fusing the
extracted information in a decoder. Several improvements to Siamese networks
have since been proposed by, for example, learning more discriminative
features from the input images [4].

Despite these efforts, current research on change detection with Siamese
networks is predominately based on fully supervised learning; and thus,
dependent on large collections of labeled data. However, change labels are
scarce and manually annotating data is costly and time-consuming. On the other
hand, unlabeled satellite data is plentiful. Therefore, it is desirable to
investigate Semi-Supervised Learning (SSL) to improve supervised change
detection by incorporate readily available unlabeled satellite data, alongside
labeled satellite data, into network training [5].

In this paper, we expand on a Siamese network architecture developed by others
[3, 4] and propose a new learning task for unlabeled data. In combination with
labeled data, we train the network using SSL and demonstrate its effectiveness
by comparing it to several fully supervised benchmarks on the SpaceNet7
dataset [6].

##  2 METHODOLOGY

###  2.1 The proposed method

The proposed method consists of a new network architecture and a loss function
to train the network in a semi-supervised fashion. The proposed network
architecture (Fig. 1) is an extension of the Siamese Difference (Siam-Diff)
architecture introduced by Daudt et al. [3]. The Siam-Diff network processes
images separately in encoders with shared weights (red arrows) to extract
corresponding features ($f_{1}$–$f_{5}$) from image $t1$ and $t2$. The
temporal features are then being forwarded via skip connections (black arrows)
to the respective level of the difference decoder where they are subtracted
from one another before being passed through subsequent layers. We
incorporated the Dual-Task concept from Liu et al. [4] into our Siam-Diff
network by adding decoders with shared weights for the semantic segmentation
of buildings in images $t1$ and $t2$. Liu et al. [4] successfully used the
Dual-Task concept to extract more discriminative features from input images
with a Siamese network, which, in turn, improved change detection results.
However, we use the building segmentation task not only to learn more
discriminative features but also for SSL. To this end, the concatenated
building outputs are passed through a 1x1 conv layer, generating an additional
change output ($p_{cs}$). In total, our network produces four outputs:
buildings segmentations for image $t1$ ($p_{s}^{t1}$) and $t2$ ($p_{s}^{t2}$),
a change output from the building segmentation ($p_{cs}$) and the change
output from the difference decoder ($p_{c}$).

Fig. 1: Diagram of the proposed Dual-Task Siam-Diff network for urban change
detection with semi-supervised learning. Diagram style was adopted from [3]
where blue, yellow, red and purple blocks denote the operations convolution,
max pooling, concatenation and transpose convolution, respectively. Red arrows
illustrate shared weights.

The four network outputs are used to train the network in a semi-supervised
fashion using a loss function composed of two supervised terms for labeled
samples, namely for semantics ($\mathcal{L}_{s}$) and change
($\mathcal{L}_{c}$), and a consistency term ($\mathcal{L}_{cons}$) for
unlabeled samples. The loss function is formally defined as follows:

$\displaystyle\mathcal{L}_{s}$
$\displaystyle=\mathcal{L}(p^{t1}_{s},y^{t1}_{s})+\mathcal{L}(p^{t2}_{s},y^{t2}_{s})$
(1) $\displaystyle\mathcal{L}_{c}$
$\displaystyle=\mathcal{L}(p_{c},y_{c})+\mathcal{L}(p_{cs},y_{c})$
$\displaystyle\mathcal{L}_{cons}$
$\displaystyle=\mathcal{L}(p_{c},y_{c})+\mathcal{L}(p_{cs},y_{c})$
$\displaystyle\mathcal{L}_{sample}$
$\displaystyle=\begin{cases}\mathcal{L}_{s}+\mathcal{L}_{c},&\text{if }y\text{
exists}\\\ \varphi*\mathcal{L}_{cons},&\text{otherwise}\end{cases}$

where $s$ and $c$ denote semantics and change, respectively. Accordingly, $cs$
denotes change derived from the semantics outputs. For labeled samples, ground
truth exists for the buildings at $t1$ ($y^{t1}_{s}$) and $t2$ ($y^{t2}_{s}$),
and the derived change ($y_{c}$). The consistency loss term encourages change
predictions to agree on unlabeled samples. Hyper-parameter $\varphi$ is used
to tune its impact on the overall loss. Power Jaccard was used for all loss
terms [7].

###  2.2 Dataset

The proposed method was tested on the multi-temporal SpaceNet7 dataset [6]. It
consists of temporal stacks of monthly Planet composites, including
corresponding manually annotated building footprints, ranging from the
beginning of 2018 to 2020. An overview of the 60 SpaceNet7 train sites, split
into training, validation and test set, and the 40 SpaceNet7 test sites used
as unlabelled data is shown in Fig. 2. Timestamps affected by clouds were
removed from the dataset.

Fig. 2: Overview of the SpaceNet7 sites constituting our training, validation
and test set.

###  2.3 Network training and performance assessment

Training samples were generated by randomly selecting two timestamps from the
time series of a site. The building labels for these timestamps, obtained from
rasterizing the building footprints, were used to compute the change label.
Twenty patches of size 256x256 pixels were then randomly cropped from the
change label, before assigning each patch a probability according to its
change pixel percentage, including a base probability for patches with no
change pixels. A single patch was chosen based on those probabilities to
oversample change pixels during training. Per epoch, one hundred samples were
drawn from each site. The network was trained for 100 epochs with batch size 8
and an initial learning rate of 10-4, using AdamW as optimizer. Horizontal and
vertical flips and rotations ($k*90^{\circ}$, where $k\in\\{0,1,2,3\\}$) were
used as data augmentations. Network training, implemented in PyTorch, was done
on a NVIDIA GeForce RTX 3090 GPU. For the quantitative performance assessment,
F1 score, precision (P) and recall (R) were used, defined as follows:

$\displaystyle P=\frac{TP}{TP+FP}\>R=\frac{TP}{TP+FN}\>F1=2*\frac{P*R}{P+R}$
(2)

where TP, FP and FN are true positives, false positives and false negatives,
respectively.

##  3 RESULTS AND DISCUSSION

Loss curves for the change, semantics and consistency term are shown in Fig.
3. At the beginning of training, losses for change and semantics rapidly
decreased but then slowed down. In contrast, consistency loss first went up,
before decreasing slowly. After approximately 50 epochs, all losses converged
and remained relatively stable.

Fig. 3: Loss terms comprising the loss function for the proposed change
detection method using semi-supervised learning.

The performance of our proposed method was compared with three fully
supervised benchmarks: EF U-Net [3], Siam-Diff [3] and Siam-Diff + Dual-Task
[4]. To run a fair comparison, we adopted the concept from the papers but used
encoder and decoder architectures identical to the ours (Fig. 1), as well as
an identical training setup (Section 2.3). In terms of quantitative results
(Table 1), our method achieved the highest F1 score on the test set (0.559),
followed by Siam-Diff + Dual-Task (0.529). The lowest F1 score was obtained
from Siam-Diff (0.484). For all networks, recall is good (0.650 +) but
precision is relatively low with our method achieving the highest value
(0.490). In contrast, the benchmarks did not exceed precision values of 0.450.

Table 1: Quantitative change detection comparison of our method with benchmarks replicated using our encoder and decoder architectures and training setup. | F1 score | Precision | Recall
---|---|---|---
EF U-Net [3] | 0.525 | 0.440 | 0.651
Siam-Diff [3] | 0.484 | 0.368 | 0.704
Siam-Diff + Dual-Task [4] | 0.529 | 0.440 | 0.664
Siam-Diff + Dual-Task + SSL | 0.559 | 0.490 | 0.651

A qualitative comparison was done for sites in Mexico, India and China (Fig. 4
from top to bottom). Generally, change detection ability of our network is
better than that of the benchmark networks, indicated by fewer FNs (purple).
However, a considerable amount of building constructions at small scale was
not detected by any of the networks. Incorrectly detected changes (green) are
mostly located around correctly detected change areas.

Fig. 4: Qualitative comparison between the results obtained by (d) our Siam-
Diff + Dual-Task + SSL method, (e) EF U-Net [3], (f) Siam-Diff [3] and (g)
Siam-Diff + Dual-Task [4]. Images for $t1$ and $t2$ and the ground truth are
shown in (a), (b) and (c), respectively. Legend for results: white is true
positive, black is true negative, green is false positive and purple is false
negative.

Despite improvements upon the supervised benchmarks, the proposed method is
limited by the fact that both change outputs are based on the same underlying
feature extraction. Therefore, encouraging consistency across change output
fails to considerably improve the feature extraction; and thus, the learning
potential from unlabeled data using the proposed consistency task may be
limited. We assume that a more promising approach is to incorporate fusion of
radar and optical data into SSL, since consistency across these data
modalities is a powerful SSL task. Moreover, the different but complementary
information in radar and optical data proved to be promising for urban change
detection in combination with deep learning [8].

##  4 CONCLUSION

This study presents an improved method for urban change detection using a
modified Dual-Task Siamese Difference network for SSL. In the proposed method,
the two change outputs of our network are harnessed for a consistency learning
task on unlabeled data. Experiments on the SpaceNet7 dataset showed that the
proposed method improves upon fully supervised benchmarks. Future work will
focus on incorporating multi-modal data into the proposed method.

## References

  * [1] Ban, Y. and Yousif, O.A.,  “Multitemporal spaceborne sar data for urban change detection in china,”  IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 5, no. 4, pp. 1087–1094, 2012.
  * [2] Ronneberger, O. et al,  “U-net: Convolutional networks for biomedical image segmentation,”  in International Conference on Medical image computing and computer-assisted intervention. Springer, 2015, pp. 234–241.
  * [3] Daudt, R.C. et al,  “Fully convolutional siamese networks for change detection,”  in 2018 25th IEEE International Conference on Image Processing (ICIP). IEEE, 2018, pp. 4063–4067.
  * [4] Liu, Y. et al,  “Building change detection for remote sensing images using a dual-task constrained deep siamese convolutional network model,”  IEEE Geoscience and Remote Sensing Letters, vol. 18, no. 5, pp. 811–815, 2020.
  * [5] Zhu, X. and Goldberg, A.B.,  “Introduction to semi-supervised learning,”  Synthesis lectures on artificial intelligence and machine learning, vol. 3, no. 1, pp. 1–130, 2009.
  * [6] Van Etten, A. et al,  “The multi-temporal urban development spacenet dataset,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 6398–6407.
  * [7] Duque-Arias, D. et al,  “On power jaccard losses for semantic segmentation,”  in VISAPP 2021: 16th International Conference on Computer Vision Theory and Applications, 2021.
  * [8] Hafner, S. et al,  “Sentinel-1 and sentinel-2 data fusion for urban change detection using a dual stream u-net,”  IEEE Geoscience and Remote Sensing Letters, vol. 19, pp. 1–5, 2022\.


# Quantum Tunneling Beyond Semiclassical Approximation

Rabin Banerjee, Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract:
Hawking radiation as tunneling by Hamilton-Jacobi method beyond semiclassical
approximation is analysed. We compute all quantum corrections in the single
particle action revealing that these are proportional to the usual
semiclassical contribution. We show that a simple choice of the
proportionality constants reproduces the one loop back reaction effect in the
spacetime, found by conformal field theory methods, which modifies the Hawking
temperature of the black hole. Using the law of black hole mechanics we give
the corrections to the Bekenstein-Hawking area law following from the modified
Hawking temperature. Some examples are explicitly worked out.

##  1 Introduction

Classical general relativity gives the concept of a black hole from which
nothing can escape. In 1974 Hawking [1] startled the physics community by
proving that black holes are not black; they radiate energy continuously.
Later on in 1975, he [2] showed that the radiation of the black hole perfectly
matches with the black body radiation whose temperature is
$T=\frac{{\cal{K}}}{2\pi}$, where ${\cal{K}}$ is the surface gravity of the
black hole. His calculation was completely based on quantum field theory.

Since the original analysis was technically very involved, several derivations
of Hawking radiation were subsequently presented in the literature [3, 4, 5].
None of them, 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 the energy of the particle changes sign as it crosses the horizon, so
that a pair created just inside or outside the horizon can materialise with
zero total energy, after one member of the pair has tunneled to the opposite
side. In this method [6, 7], the particles are allowed to follow classically
forbidden trajectories, by starting just behind the horizon onward to
infinity. The particles then travel back in time, since the horizon is locally
to the future of the external region. 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 particle. However, the action for the
ingoing particle must be real, since classically a particle can fall behind
the horizon. This is an important point of our calculations 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. There are two different methods to
calculate the imaginary part of the action: one is by Parikh-Wilczek [6] \-
radial null geodesic method and another is the Hamilton-Jacobi method which
was first used by Srinivasan et. al. [7]. Later, many people [8] used the
radial null geodesic method to find out the Hawking temperature for different
spacetime metrics. Recently [9], tunneling of a Dirac particle through the
event horizon was also studied. All these computations are, however, confined
to the semiclassical approximation only. The issue of quantum corrections is
generally not discussed. Inspite of some sporadic attemps [10, 11] a
systemetic, thorough and complete analysis is lacking. In our previous work
[12] we found out the corrections to the temperature and entropy by including
the effects of back reaction knowing the modified surface gravity of the black
hole due to one loop back reaction for the Schwarzschild case by radial null
geodesic method. As an extension we [13] also applied this method for a
noncommutative Schwarzschild metric. Recently, a problem in this approach has
been discussed in [14, 15, 16, 17, 18] which corresponds to a factor two
ambiguity in the original Hawking temperature.

In this paper, we formulate the Hamilton-Jacobi method beyond the
semiclassical approximation by considering all the terms in the expansion of
the one particle action. We show that the higher order terms are proportional
to the semiclassical contribution. By dimensional argument the form of these
proportionality constants, upto some dimensionless parameters, are determined.
In particular, for Schwarzschild and Anti-de Sitter Schwarzschild spacetimes,
these are given by the inverse powers of the square of the mass of the black
hole, because in these cases the only macroscopic parameter is mass. By using
the principle of “detailed balance” the Hawking temperature is identified. It
involves a correction to the usual semiclassical approximation. Incidentally,
the form of the semiclassical Hawking temperature found here is general enough
to consider examples of metrics that need not be spherically symmetric. We
also show that this temperature reduces to the standard form for spherically
symmetric metrics. The modified Hawking temperature which includes corrections
to the semiclassical structure is also derived. We show that for an
appropriate choice of the dimensionless parameters, it is possible to
reproduce the one loop back reaction effects [20, 21] in the spacetime or the
quantum corrections, obtained by conformal field theory techniques, due to
trace anomaly. Also, we analyse the two loop corrections in the surface
gravity. The Hamilton-Jacobi method is then discussed in other coordinates
like Painleve. Here we show how the standard form of the Hawking temperature
can be recovered. There are no factor two ambiguities which are usually
reported in dealing with tunneling methods [15, 18]. Using the law of black
hole mechanics the corrections to the Bekenstein-Hawking area law for
Schwarzschild and AdS Schwarzschild black holes are explicitly calculated.
Interestingly the leading order correction to the Bekenstein-Hawking entropy
is the famous logarithmic term which was found earlier in [22, 23, 24, 25, 26,
27, 28, 29, 30]. We also analyse the Kerr metric which is not spherically
symmetric and static, but stationary.

The organisation of our paper is as follows. In the second section we give a
brief review of the radial null geodesic method and its limitations. Section 3
is devoted for the Hamilton-Jacobi method beyond the semiclassical
approximation where we derive a general expression for Hawking temperature and
then show how one can reproduce the loop corrections to the surface gravity.
Also, the Hamilton-Jacobi method in Painleve coordinates is discussed here. In
the next two sections the derivations of corrected temperature and entropy for
some well known spacetime metrics is given. The well known logarithmic
corrections to the area law are reproduced. The final section is for
conclusions.

##  2 Radial null geodesic method: a brief review

In this section we will briefly discuss about the radial null geodesic method
[6] to find the temperature of a black hole using the picture of Hawking
radiation as quantum tunneling. Then we will show how one can include the loop
corrections due to the back reaction effect in the spacetime. We also point
out about the possible limitations of the method.

This method involves calculating the imaginary part of the action for the
(classically forbidden) process of s-wave emission across the horizon which in
turn is related to the Boltzmann factor for emission at the Hawking
temperature. We consider a general class of static (i.e. invariant under time
reversal as well as stationary), spherically symmetric spacetime of the form

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

where the horizon $r=r_{H}$ is given by $f(r_{H})=g(r_{H})=0$. The coordinate
singularity at the horizon is removed by going to Painleve coordinates [31].
Under the transformation

$\displaystyle dt\rightarrow dt-\sqrt{\frac{1-g(r)}{f(r)g(r)}}dr$ (2)

the metric (1) takes the form

$\displaystyle
ds^{2}=-f(r)dt^{2}+2f(r)\sqrt{\frac{1-g(r)}{f(r)g(r)}}dtdr+dr^{2}+r^{2}d\Omega^{2}$
(3)

This metric is stationary but not static. It has a number of interesting
features. At any fixed time the spatial geometry is flat and for any fixed
radius the boundary geometry is the same as that of the metric (1).

The basic idea of this method is to find the radial null geodesics
($ds^{2}=d\Omega^{2}=0$) for the metric (3):

$\displaystyle\dot{r}=\sqrt{\frac{f(r)}{g(r)}}(\pm 1-\sqrt{1-g(r)})$ (4)

where $+(-)$ sign gives outgoing (ingoing) null radial geodesics. Using this
one has to calculate the imaginary part of the action for a shell of energy
$\omega$. In the original work [6], the imaginary part of the action is
defined as,

$\displaystyle{\textrm{Im}}S$ $\displaystyle=$
$\displaystyle{\textrm{Im}}\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}p_{r}dr={\textrm{Im}}\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}\int_{0}^{p_{r}}dp^{\prime}_{r}dr$
(5) $\displaystyle=$
$\displaystyle{\textrm{Im}}\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}\int_{0}^{H}\frac{dH^{\prime}}{\dot{r}}dr$

where in the last step we multiply and divide the integrand by the two sides
of Hamilton’s equation $\dot{r}=\frac{dH}{dp_{r}}|_{r}$.

Near the horizon one can expand $f(r)$ and $g(r)$ about the horizon $r_{H}$.

$\displaystyle f(r)=f^{\prime}(r_{H})(r-r_{H})+{\cal{O}}((r-r_{H})^{2})$
$\displaystyle g(r)=g^{\prime}(r_{H})(r-r_{H})+{\cal{O}}((r-r_{H})^{2})$ (6)

Substituting these in (4) $\dot{r}$ can be approximately expressed as

$\displaystyle\dot{r}\simeq\frac{1}{2}\sqrt{f^{\prime}(r_{H})g^{\prime}(r_{H})}(r-r_{H})$
(7)

Using this in (5) the imaginary part of the action is calculated. Now taking
the tunneling probability as $\Gamma\sim e^{-\frac{2}{\hbar}{\textrm{Im}}S}$
and equating it with the Boltzmann factor $e^{-\frac{\omega}{T}}$ we find the
Hawking temperature as

$\displaystyle
T_{H}=\frac{\omega\hbar}{2{\textrm{Im}}S}=\frac{\hbar\sqrt{f^{\prime}(r_{H})g^{\prime}(r_{H})}}{4\pi}.$
(8)

It is easy to confirm that for a Schwarzschild black hole the correct result
of $T_{H}=\frac{\hbar}{8\pi M}$ follows.

Recently, a problem in this approach has been pointed out in [14].
Particularly, it has been shown that
$2{\textrm{Im}}\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}p_{r}dr$ is not
canonically invariant and thus it is not a proper observable. The object which
is canonically invariant is ${\textrm{Im}}\oint p_{r}dr$ where the closed path
goes across the horizon and back. For ordinary tunneling problems where the
tunneling amplitude is the same whether one tunnels from left to right or
right to left one finds that $\oint
p_{r}dr=2\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}p_{r}dr$. However for the
Painleve coordinates there is only a barrier to a particle coming from inside
the horizon to outside. Particles going from outside to inside do not see a
barrier and thus the two expressions are not equivalent. Consequently $\oint
p_{r}dr\neq 2\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}p_{r}dr$ thereby
invalidating the analysis leading to (8). Rather if one uses the invariant
definition $\Gamma\sim e^{-\frac{1}{\hbar}{\textrm{Im}}\oint p_{r}dr}$, the
Hawking temperature is found to be twice the original temperature. This
ambiguity has been mentioned in more details in [15, 17, 18, 19, 34, 35].
However our analysis will be based on Hamilton-Jacobi method which is free of
this factor two discrepancy. This will be shown in the next section.

Situations in which there is a back reaction have to be handled separately,
as, for instance, discussed for the Schwarzschild case [12]. But it is still
not clear how to handle other metrics. For example, in the case of Reissner-
Nordstrom metric one has to consider the tunneling of a charged particle. In
that case the invariant tunneling rate $\Gamma\sim
e^{-\frac{1}{\hbar}{\textrm{Im}}\oint p_{r}dr}$ will change. There are some
papers [32, 33] in which this case is analysed. But these are completely based
on the semiclassical approximation. No quantum effects have been included.
Also, in this method it is necessary to work in Painleve coordinates to avoid
the singularity at the horizon. So one is not allowed to perform the
calculations in the original metric coordinates, at least in the standard
formulation. Furthermore, this method corresponds to radial null geodesics and
so it is valid only for massless particles.

##  3 Hamilton-Jacobi method beyond the semiclassical approximation

We next consider an alternate method for calculating the imaginary part of the
action, making use of the Hamilton-Jacobi equation [7, 34, 35], from which we
will calculate the Hawking temperature. The analysis goes beyond the
semiclassical approximation by including all possible quantum corrections. The
semiclassical Hawking temperature is thereby appropriately altered. 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.

###  3.1 Schwarzschild like coordinate system

Let us consider a massless particle 111Though we consider only the massless
particle, this method is valid for massive case also. It has been shown
earlier [7] that ultimately the final expressions are same. Therefore for
simplicity we consider only the massless case. in the spacetime (1) described
by the Klein-Gordon equation

$\displaystyle-\frac{\hbar^{2}}{\sqrt{-g}}\partial_{\mu}[g^{\mu\nu}\sqrt{-g}\partial_{\nu}]\phi=0.$
(9)

For radial trajectories only the $(r-t)$ sector of the metric (1) is
important. Therefore under this metric the Klein-Gordon equation reduces to

$\displaystyle-\frac{1}{\sqrt{f(r)g(r)}}\partial^{2}_{t}\phi+\frac{1}{2}\Big{(}f^{\prime}(r)\sqrt{\frac{g(r)}{f(r)}}+g^{\prime}(r)\sqrt{\frac{f(r)}{g(r)}}\Big{)}\partial_{r}\phi+\sqrt{f(r)g(r)}\partial_{r}^{2}\phi=0.$
(10)

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

$\displaystyle\phi(r,t)={\textrm{exp}}\Big{[}-\frac{i}{\hbar}S(r,t)\Big{]},$
(11)

where $S(r,t)$ is a function which will be expanded in powers of $\hbar$.
Substituting into the wave equation (10), we obtain

$\displaystyle\frac{i}{\sqrt{f(r)g(r)}}\Big{(}\frac{\partial S}{\partial
t}\Big{)}^{2}-i\sqrt{f(r)g(r)}\Big{(}\frac{\partial S}{\partial
r}\Big{)}^{2}-\frac{\hbar}{\sqrt{f(r)g(r)}}\frac{\partial^{2}S}{\partial
t^{2}}+\hbar\sqrt{f(r)g(r)}\frac{\partial^{2}S}{\partial r^{2}}$
$\displaystyle+\frac{\hbar}{2}\Big{(}\frac{\partial f(r)}{\partial
r}\sqrt{\frac{g(r)}{f(r)}}+\frac{\partial g(r)}{\partial
r}\sqrt{\frac{f(r)}{g(r)}}\Big{)}\frac{\partial S}{\partial r}=0.$ (12)

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

$\displaystyle S(r,t)$ $\displaystyle=$ $\displaystyle S_{0}(r,t)+\hbar
S_{1}(r,t)+\hbar^{2}S_{2}(r,t)+...........$ (13) $\displaystyle=$
$\displaystyle S_{0}(r,t)+\sum_{i}\hbar^{i}S_{i}(r,t).$

where $i=1,2,3,......$. In this expansion the terms from ${\cal{O}}(\hbar)$
onwards are treated as quantum corrections over the semiclassical value
$S_{0}$. Substituting (13) in (12) and equating the different powers of
$\hbar$ on both sides, we obtain the following set of equations:

$\displaystyle\hbar^{0}~{}:~{}$ $\displaystyle\Big{(}\frac{\partial
S_{0}}{\partial t}\Big{)}^{2}-f(r)g(r)\Big{(}\frac{\partial S_{0}}{\partial
r}\Big{)}^{2}=0,$ $\displaystyle\hbar^{1}~{}:~{}$ $\displaystyle
2i\frac{\partial S_{0}}{\partial t}\frac{\partial S_{1}}{\partial
t}-2if(r)g(r)\frac{\partial S_{0}}{\partial r}\frac{\partial S_{1}}{\partial
r}-\frac{\partial^{2}S_{0}}{\partial
t^{2}}+f(r)g(r)\frac{\partial^{2}S_{0}}{\partial r^{2}}$
$\displaystyle+\frac{1}{2}\Big{(}f^{\prime}(r)g(r)+f(r)g^{\prime}(r)\Big{)}\frac{\partial
S_{0}}{\partial r}=0,$ $\displaystyle\hbar^{2}~{}:~{}$ $\displaystyle
i\Big{(}\frac{\partial S_{1}}{\partial t}\Big{)}^{2}+2i\frac{\partial
S_{0}}{\partial t}\frac{\partial S_{2}}{\partial
t}-if(r)g(r)\Big{(}\frac{\partial S_{1}}{\partial
r}\Big{)}^{2}-2if(r)g(r)\frac{\partial S_{0}}{\partial r}\frac{\partial
S_{2}}{\partial r}-\frac{\partial^{2}S_{1}}{\partial
t^{2}}+f(r)g(r)\frac{\partial^{2}S_{1}}{\partial r^{2}}$
$\displaystyle+\frac{1}{2}\Big{(}f^{\prime}(r)g(r)+f(r)g^{\prime}(r)\Big{)}\frac{\partial
S_{1}}{\partial r}=0,$ . . . (14)

so on. Now it is interesting to note that any equation in the above set can be
simplified by using the equations coming before it. This leads to an identical
set of relations,

$\displaystyle\hbar^{0}~{}:~{}\frac{\partial S_{0}}{\partial
t}=\pm\sqrt{f(r)g(r)}\frac{\partial S_{0}}{\partial r},$ (15)
$\displaystyle\hbar^{1}~{}:~{}$ $\displaystyle\frac{\partial S_{1}}{\partial
t}=\pm\sqrt{f(r)g(r)}\frac{\partial S_{1}}{\partial r},$
$\displaystyle\hbar^{2}~{}:~{}$ $\displaystyle\frac{\partial S_{2}}{\partial
t}=\pm\sqrt{f(r)g(r)}\frac{\partial S_{2}}{\partial r},$ . . .

and so on; i.e. the functional form of the above set of linear differential
equations is same. Therefore the solutions of these equations are not
independent and $S_{i}$’s are proportional to $S_{0}$. Since $S_{0}$ has the
dimension of $\hbar$ the proportionality constants should have the dimension
of inverse of $\hbar^{i}$. Again in the units $G=c=k_{B}=1$ the Planck
constant $\hbar$ is of the order of square of the Planck Mass $M_{P}$ and so
from dimensional analysis the proportionality constants have the dimension of
$M^{-2i}$ where $M$ is the mass of black hole. Specifically, for Schwarzschild
type black holes having mass as the only macroscopic parameter, these
considerations show that the most general expression for $S$, following from
(13), is given by,

$\displaystyle S(r,t)$ $\displaystyle=$ $\displaystyle
S_{0}(r,t)+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}S_{0}(r,t)$ (16)
$\displaystyle=$
$\displaystyle\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}S_{0}(r,t).$

where $\beta_{i}$’s are dimensionless constant parameters.

To obtain a solution for $S(r,t)$ it is therefore enough to solve for
$S_{0}(r,t)$ which satisfies (15). In fact the standard Hamilton-Jacobi
solution determined by this $S_{0}(r,t)$ is just modified by a prefactor to
yield the complete solution for $S(r,t)$. Since the metric (1) is stationary
it has timelike Killing vectors. Thus we will look for solutions of (15) which
behave as

$\displaystyle S_{0}=\omega t+\tilde{S}_{0}(r),$ (17)

where $\omega$ is the energy of the particle. Substituting this in (15) and
then integrating we obtain,

$\displaystyle\tilde{S_{0}}(r)=\pm\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}$
(18)

where the limits of the integration are chosen such that the particle goes
through the horizon $r=r_{H}$. The $+(-)$ sign in front of the integral
indicates that the particle is ingoing (outgoing). Using (17) and (18) in (16)
we obtain

$\displaystyle
S(r,t)=\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}\Big{[}\omega
t\pm\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}.$ (19)

Therefore the ingoing and outgoing solutions of the Klein-Gordon equation (9)
under the back ground metric (1) is given by exploiting (11) and (19),

$\displaystyle\phi_{{\textrm{in}}}={\textrm{exp}}\Big{[}-\frac{i}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega
t+\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$ (20)

and

$\displaystyle\phi_{{\textrm{out}}}={\textrm{exp}}\Big{[}-\frac{i}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega
t-\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}.$ (21)

Now for the tunneling of a particle across the horizon the nature of the
coordinates change. The sign of the metric coefficients in the $(r-t)$ sector
is altered. This indicates that ‘$t$’ coordinate has an imaginary part for the
crossing of the horizon of the black hole and correspondingly there will be a
temporal contribution to the probabilities for the ingoing and outgoing
particles. This has similarity with[16] 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,

$\displaystyle
P_{{\textrm{in}}}=|\phi_{{\textrm{in}}}|^{2}={\textrm{exp}}\Big{[}\frac{2}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega{\textrm{Im}}~{}t+\omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(22)

and

$\displaystyle
P_{{\textrm{out}}}=|\phi_{{\textrm{out}}}|^{2}={\textrm{exp}}\Big{[}\frac{2}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega{\textrm{Im}}~{}t-\omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(23)

Now the ingoing probability $P_{\textrm{in}}$ 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 [36].Thus, in the classical limit,
(22) leads to,

$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}.$
(24)

From the above one can easily show that ${\textrm{Im}}~{}t=-2\pi M$ for the
Schwarzschild spacetime which is precisely the imaginary part of the
transformation $t\rightarrow t-2i\pi M$ when one connects the two regions
across the horizon as shown in [16]. Therefore the probability of the outgoing
particle is

$\displaystyle
P_{{\textrm{out}}}={\textrm{exp}}\Big{[}-\frac{4}{\hbar}\omega\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}.$
(25)

Now using the principle of “detailed balance” [7, 13]

$\displaystyle
P_{{\textrm{out}}}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{h}}\Big{)}P_{\textrm{in}}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{h}}\Big{)}$
(26)

we obtain the temperature of the black hole as

$\displaystyle T_{h}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{4}\Big{[}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}){\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}^{-1}$
(27) $\displaystyle=$ $\displaystyle
T_{H}\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}^{-1}$

where

$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{(}{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}^{-1}$
(28)

is the standard semiclassical Hawking temperature of the black hole and other
terms are the corrections due to the quantum effect. 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 (28) yields a novel
form of the semiclassical Hawking temperature. For instance, it can be used
for metrics that need not be spherically symmetric. Later, in section $4.3$,
this will be exemplified in the case of the Kerr metric. For a spherically
symmetric metric it is possible to show that (28) reproduces the familiar form
(8). Inserting the near horizon expansion (6) in (28) and performing the
contour integration, (8) is obtained. Note that this form is the standard
Hawking temperature found [34, 35] by the Hamilton-Jacobi method. There is no
ambiguity regarding a factor of two in the Hawking temperature as reported in
the literature [34, 35]. 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 (17) of (15) 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 (27) for the temperature
still remains unchanged. It is only a matter of rescaling the particle energy
‘$\omega$’. This shows the uniqueness of the expression (27) for the Hawking
temperature.

We will now show that various choices of the coefficients $\beta_{i}$ in (27)
correspond to higher order loop corrections to the surface gravity of the
black hole, obtained by including back reaction effects [20, 21] or by
accounting for the trace anomaly [22]. To see this note that the standard
relation between the surface gravity (${\cal{K}}$) and the Hawking temperature
($T_{h}$) is

$\displaystyle T_{h}=\frac{{\cal{K}}}{2\pi}.$ (29)

Hence the modified form of the surface gravity of the black hole following
from (27), is given by

$\displaystyle{\cal{K}}={\cal{K}}_{0}\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}^{-1}.$
(30)

where ${\cal{K}}_{0}=2\pi T_{H}$ is the standard semiclassical surface
gravity.

Now if we choose the dimensionless parameters $\beta_{i}$’s in terms of a
single dimensionless parameter $\alpha$ in the following way,

$\displaystyle\beta_{i}=\alpha^{i}$ (31)

then the expression within the parenthesis in (30) is simplified to

$\displaystyle 1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}$ $\displaystyle=$
$\displaystyle
1+\Big{(}\frac{\alpha\hbar}{M^{2}}+\frac{\alpha^{2}\hbar^{2}}{M^{4}}+\frac{\alpha^{3}\hbar^{3}}{M^{6}}+\frac{\alpha^{4}\hbar^{4}}{M^{8}}+........\Big{)}=(1-\frac{\alpha\hbar}{M^{2}})^{-1}.$
(32)

Therefore under this choice of the parameters the surface gravity (30)
simplifies to the following form

$\displaystyle{\cal{K}}={\cal{K}}_{0}\Big{(}1-\frac{\alpha\hbar}{M^{2}}\Big{)}.$
(33)

This expression was found earlier [20, 21] by considering the one loop back
reaction effects in the spacetime. Moreover, the coefficient $\alpha$ is
related to the trace anomaly. Using conformal field theory techniques it was
shown [22] that the one loop quantum correction to the surface gravity, for
the Schwarzschild black hole, is given by (33) where,

$\displaystyle\alpha=-\frac{1}{360\pi}\Big{(}-N_{0}-\frac{7}{4}N_{\frac{1}{2}}+13N_{1}+\frac{233}{4}N_{\frac{3}{2}}-212N_{2}\Big{)}$
(34)

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

Likewise, the higher order loop corrections in the surface gravity can also be
reproduced by this method. The only important part is to choose the expansion
coefficients $\beta_{i}$’s suitably. For instance, if $\beta_{i}$’s are chosen
as below,

$\displaystyle\beta_{i}=\sum_{k=0}^{\infty}(i-k)_{C_{k}}\alpha^{(i-2k)}\gamma^{k}$
(35)

with $i\geq 2k$, then substituting this in (30) and simplifying, one can show
that

$\displaystyle{\cal{K}}={\cal{K}}_{0}\Big{(}1-\frac{\alpha\hbar}{M^{2}}-\frac{\gamma\hbar^{2}}{M^{4}}\Big{)}$
(36)

which is nothing but the corrected form of the surface gravity of a black hole
due to two loop back reaction effects in the spacetime. $`\gamma$’ is a
dimensionless parameter corresponding to the contribution from the second
loop. This indicates that one can reproduce the all loop back reaction effects
in the spacetime by keeping all the terms in the expansion of $S(r,t)$ (13)
and suitably choosing the expansion coefficients.

###  3.2 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. As
before, consider a massless scalar particle in the spacetime metric (3)
described by the Painleve coordinates. Since the Klein-Gordon equation (9) is
in covariant form, the scalar particle in the background metric (3) also
satisfies (9). Therefore under this metric the Klein-Gordon equation reduces
to

$\displaystyle-(\frac{g}{f})^{\frac{3}{2}}\partial^{2}_{t}\phi+\frac{2g\sqrt{1-g}}{f}\partial_{t}\partial_{r}\phi-\frac{gg^{\prime}}{2f\sqrt{1-g}}\partial_{t}\phi+g\sqrt{\frac{g}{f}}\partial^{2}_{r}\phi+\frac{1}{2}\sqrt{\frac{g}{f}}(3g^{\prime}-\frac{f^{\prime}g}{f})\partial_{r}\phi=0.$
(37)

As before, substituting the standard ansatz (11) for $\phi$ in the above
equation, we obtain,

$\displaystyle-(\frac{g}{f})^{\frac{3}{2}}\Big{[}-\frac{i}{\hbar}\Big{(}\frac{\partial
S}{\partial t}\Big{)}^{2}+\frac{\partial^{2}S}{\partial
t^{2}}\Big{]}+\frac{2g\sqrt{1-g}}{f}\Big{[}-\frac{i}{\hbar}\frac{\partial
S}{\partial t}\frac{\partial S}{\partial r}+\frac{\partial^{2}S}{\partial
r\partial t}\Big{]}-\frac{gg^{\prime}}{2f\sqrt{1-g}}\frac{\partial S}{\partial
t}$
$\displaystyle+g\sqrt{\frac{g}{f}}\Big{[}-\frac{i}{\hbar}\Big{(}\frac{\partial
S}{\partial r}\Big{)}^{2}+\frac{\partial^{2}S}{\partial
r^{2}}\Big{]}+\frac{1}{2}(3g^{\prime}-\frac{f^{\prime}g}{f})\frac{\partial
S}{\partial r}=0.$ (38)

Neglecting the terms of order $\hbar$ and greater we find to the lowest order,

$\displaystyle(\frac{g}{f})^{\frac{3}{2}}\Big{(}\frac{\partial S}{\partial
t}\Big{)}^{2}-\frac{2g\sqrt{1-g}}{f}\frac{\partial S}{\partial
t}\frac{\partial S}{\partial r}-g\sqrt{\frac{g}{f}}\Big{(}\frac{\partial
S}{\partial r}\Big{)}^{2}=0.$ (39)

It has been stated earlier that the metric (3) is stationary. Therefore
following the same argument as before it has a solution of the form (17).
Inserting this in (39) yields,

$\displaystyle\frac{d\tilde{S}_{0}(r)}{dr}=\omega\sqrt{\frac{1-g(r)}{f(r)g(r)}}\Big{(}-1\pm\frac{1}{\sqrt{1-g(r)}}\Big{)}$
(40)

Integrating,

$\displaystyle\tilde{S}_{0}(r)=\omega\int_{0}^{r}\sqrt{\frac{1-g(r)}{f(r)g(r)}}\Big{(}-1\pm\frac{1}{\sqrt{1-g(r)}}\Big{)}dr.$
(41)

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

$\displaystyle S_{\textrm{in}}(r,t)=\omega
t+\omega\int_{0}^{r}\frac{1-\sqrt{1-g}}{\sqrt{fg}}dr$ (42)

and

$\displaystyle S_{\textrm{out}}(r,t)=\omega
t-\omega\int_{0}^{r}\frac{1+\sqrt{1-g}}{\sqrt{fg}}dr$ (43)

Since in the classical limit (i.e. $\hbar\rightarrow 0$) the probability for
the ingoing particle ($P_{\textrm{in}}$) has to be unity, $S_{\textrm{in}}$
must be real. Following identical steps employed in deriving (24) we obtain,
starting from (42), the analogous condition,

$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int_{0}^{r}\frac{1-\sqrt{1-g}}{\sqrt{fg}}dr$
(44)

Substituting this in (43) we obtain the action for the outgoing particle:

$\displaystyle
S_{\textrm{out}}(r,t)=\omega{\textrm{Re}}~{}t-\omega{\textrm{Re}}\int_{0}^{r}\frac{1+\sqrt{1-g}}{\sqrt{fg}}dr-2i\omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{fg}}$
(45)

Therefore the probability for the outgoing particle is

$\displaystyle
P_{\textrm{out}}=|e^{-\frac{i}{\hbar}S_{\textrm{out}}}|^{2}=e^{-\frac{4}{\hbar}\omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}}$
(46)

Now using the principle of “detailed balance” (26) we obtain the same
expression (28) for the standard Hawking temperature which was calculated in
Schwarzschild like coordinates by the Hamilton-Jacobi method. Inclusion of
higher order terms is straightforward and leads to the same relation as (27).

##  4 Calculation of Hawking temperature

In this section we will consider some standard metrics to show how the
semiclassical Hawking temperature can be calculated from (28). For the
Schwarzschild and Anti-de Sitter Schwarzschild black hole the modified form of
the Hawking temperatures due to the corrections beyond semiclassical
approximation will be explicitly shown by using (27).

###  4.1 Schwarzschild black hole

The spacetime metric is given by

$\displaystyle
ds^{2}=-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(47)

So the metric coefficients are

$\displaystyle f(r)=g(r)=(1-\frac{r_{H}}{r});\,\,\,r_{H}=2M.$ (48)

Since this metric is spherically symmetric we use the formula (8) to compute
the semiclassical Hawking temperature. This is found to be,

$\displaystyle T_{H}=\frac{\hbar}{4\pi r_{H}}=\frac{\hbar}{8\pi M}.$ (49)

which is the standard expression. Now using (27) it is easy to write the
corrected Hawking temperature:

$\displaystyle T_{h}=\frac{\hbar}{8\pi
M}\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar}{M^{2i}}\Big{)}^{-1}.$ (50)

In particular, for the choice (31), this yields,

$\displaystyle T_{h}=\frac{\hbar}{8\pi
M}\Big{(}1-\frac{\alpha\hbar}{M^{2}}\Big{)}$ (51)

which is the modified form of the Hawking temperature. Such a structure was
obtained earlier [12] in radial null geodesic approach by explicitly taking
into account the one loop back reaction effect. Also, as stated earlier, such
a form follows from conformal field theory techniques [22] where $\alpha$ is
given by (34).

###  4.2 Anti-de Sitter Schwarzschild black hole

The AdS-Schwarzschild metric is given by

$\displaystyle
ds^{2}=-\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{l^{2}}\Big{)}dt^{2}+\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{l^{2}}\Big{)}^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(52)

Here,

$\displaystyle f(r)=g(r)=\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{l^{2}}\Big{)}.$
(53)

Since the metric is spherically symmetric we employ similar steps as before to
obtain the semiclassical Hawking temperature,

$\displaystyle T_{H}=\frac{\hbar}{4\pi}\frac{3r_{+}^{2}+l^{2}}{l^{2}r_{+}}.$
(54)

Also, by (27), the corrected form of the Hawking temperature due to quantum
effects, is

$\displaystyle
T_{h}=\frac{\hbar}{4\pi}\frac{3r_{+}^{2}+l^{2}}{l^{2}r_{+}}\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}^{-1}.$
(55)

Once again, for the choice (31), this yields,

$\displaystyle
T_{h}=\frac{\hbar}{4\pi}\frac{3r_{+}^{2}+l^{2}}{l^{2}r_{+}}\Big{(}1-\frac{\alpha\hbar}{M^{2}}\Big{)}$
(56)

which reproduces the corrected Hawking temperature by including the one loop
back reaction effect [21].

###  4.3 Kerr black hole

This example provides a nontrivial application of our formula (28) for
computing the semiclassical Hawking temperature. Here the metric is not
spherically symmetric, invalidating the use of (8).

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

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\Big{(}1-\frac{2Mr}{\rho^{2}}\Big{)}dt^{2}-\frac{2Mar~{}{\textrm{sin}}^{2}\theta}{\rho^{2}}(dtd\phi+d\phi
dt)$ (57) $\displaystyle+$
$\displaystyle\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{{\textrm{sin}}^{2}\theta}{\rho^{2}}~{}\Big{[}(r^{2}+a^{2})^{2}-a^{2}\Delta~{}{\textrm{sin}}^{2}\theta\Big{]}d\phi^{2}$

where

$\displaystyle\Delta(r)$ $\displaystyle=$ $\displaystyle
r^{2}-2Mr+a^{2};\,\,\,\rho^{2}(r,\theta)=r^{2}+a^{2}~{}{\textrm{cos}}^{2}\theta$
$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{J}{M}$ (58)

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

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}.$ (59)

This metric is not spherically symmetric and static but stationary. So it must
have timelike Killing vectors. Although in our general formulation we consider
only 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

$\displaystyle
ds^{2}=-\Big{(}\frac{r^{2}+a^{2}-2Mr}{r^{2}+a^{2}}\Big{)}dt^{2}+\Big{(}\frac{r^{2}+a^{2}}{r^{2}+a^{2}-2Mr}\Big{)}dr^{2}$
(60)

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 (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 (28) to find the semiclassical Hawking temperature. Here,

$\displaystyle f(r)=g(r)=\Big{(}\frac{r^{2}+a^{2}-2Mr}{r^{2}+a^{2}}\Big{)}$
(61)

Substituting these in (28) we obtain,

$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{(}{\textrm{Im}}\int\frac{r^{2}+a^{2}}{(r-r_{+})(r-r_{-})}\Big{)}^{-1}.$
(62)

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 semiclassical Hawking temperature as

$\displaystyle T_{H}=\frac{\hbar}{4\pi}\frac{r_{+}-r_{-}}{r_{+}^{2}+a^{2}}.$
(63)

This agrees with results quoted in the literature [37].

##  5 Bekenstein-Hawking entropy and area law

The semiclassical Bekenstein-Hawking area law [38, 39, 2] is given by

$\displaystyle S_{\textrm{BH}}=\frac{A}{4\hbar}$ (64)

where ‘$A$’ is the area of the horizon. This is altered when quantum effects
come into play. Here, using the modified form of the temperature for
Schwarzschild and AdS-Schwarzschild black hole derived in the previous
section, we will explicitly show the corrections to the Bekenstein-Hawking
area law with the help of the second law of thermodynamics. This law of black
hole thermodynamics which expresses the conservation of energy by relating the
change in the black hole mass $M$ to the changes in its entropy
$S_{\textrm{bh}}$, angular momentum $J$ and electric charge $Q$, is given by

$\displaystyle dM=T_{h}dS_{\textrm{bh}}+\Phi dQ+\Omega dJ$ (65)

where the angular velocity $\Omega$ and the electrostatic potential
$\Phi=\frac{\partial M}{\partial Q}$ are constant over the event horizon of
any stationary black hole. From this conservation law the entropy is computed
as,

$\displaystyle S_{\textrm{bh}}=\int\frac{1}{T_{h}}(dM-\Phi dQ-\Omega dJ)$ (66)

###  5.1 Schwarzschild black hole

It has no charge and spin. Hence (66) simplifies to

$\displaystyle S_{\textrm{bh}}=\int\frac{dM}{T_{h}}$ (67)

Substituting the value of temperature from (50) in (67) we obtain,

$\displaystyle S_{\textrm{bh}}$ $\displaystyle=$ $\displaystyle\frac{4\pi
M^{2}}{\hbar}+8\pi\beta_{1}\ln
M-\frac{4\pi\hbar\beta_{2}}{M^{2}}+{\textrm{higher order terms in $\hbar$}}$
(68) $\displaystyle=$ $\displaystyle\frac{\pi
r_{H}^{2}}{\hbar}+8\pi\beta_{1}\ln
r_{H}-\frac{16\pi\hbar\beta_{2}}{r_{H}^{2}}+{\textrm{higher order terms in
$\hbar$}}$

The area of the event horizon is

$\displaystyle A=4\pi r_{H}^{2}$ (69)

so that,

$\displaystyle S_{\textrm{bh}}$ $\displaystyle=$
$\displaystyle\frac{A}{4\hbar}+4\pi\beta_{1}\ln
A+\frac{64\pi^{2}\hbar\beta_{2}}{A}+......................$ (70)

It is noted that the first term is the usual semiclassical area law (64). The
other terms are the quantum corrections. Now it is possible to express the
quantum corrections in terms of $S_{\textrm{BH}}$ by eliminating $A$:

$\displaystyle S_{\textrm{bh}}=S_{\textrm{BH}}+4\pi\beta_{1}\ln
S_{\textrm{BH}}+\frac{16\pi^{2}\beta_{2}}{S_{\textrm{BH}}}+.........$ (71)

Interestingly the leading order correction is logarithmic in $A$ or
$S_{\textrm{BH}}$ which was found earlier in [22, 23] by field theory
calculations and later in [24, 29] with $\beta_{1}=-\frac{1}{8\pi}$ by quantum
geometry method. The higher order corrections involve inverse powers of $A$ or
$S_{\textrm{BH}}$.

###  5.2 Anti-de Sitter Schwarzschild black hole

Since this black hole also does not have charge and spin, the appropriate
equation defining the entropy is given by (67). The first point to observe is
that the event horizon $r=r_{+}$ is defined by

$\displaystyle\Big{(}1-\frac{2M}{r_{+}}+\frac{r_{+}^{2}}{l^{2}}\Big{)}=0.$
(72)

This shows that it is possible to interpret $M$ as a function of
$r_{+}$;$M=M(r_{+})$. Hence $dM=\frac{\partial M}{\partial r_{+}}dr_{+}$.
Calculating $\frac{\partial M}{\partial r_{+}}$ from (72) we obtain

$\displaystyle dM=\frac{3r_{+}^{2}+l^{2}}{2l^{2}}dr_{+}.$ (73)

Substituting this and (55) in (67) and integrating, we obtain the corrected
Bekenstein-Hawking entropy as

$\displaystyle S_{\textrm{bh}}$ $\displaystyle=$ $\displaystyle\frac{\pi
r_{+}^{2}}{\hbar}+8\pi\beta_{1}\ln r_{+}+.........$ (74)

Here also the area $A$ satisfies (69). Re-expressing (74) in terms of the
semiclassical expression $S_{\textrm{BH}}$ (64), we obtain,

$\displaystyle S_{\textrm{bh}}=S_{\textrm{BH}}+4\pi\beta_{1}\ln
S_{\textrm{BH}}+.............$ (75)

The leading order correction is again logarithmic as found earlier in [26, 27]
by a statistical method with $\beta_{1}=-\frac{1}{4\pi}$.

##  6 Conclusions

We have given a general expression (28) for the semiclassical Hawking
temperature by the Hamilton-Jacobi method which corresponds directly to the
picture that visualises the source of radiation as tunneling. For the
particular case of a spherically symmetric metric, our expression reduces to
the standard form (8). Going beyond the semiclassical approximation, the one
particle action is computed by including all higher order corrections. From
this action, the modified Hawking temperature for the Schwarzschild and AdS-
Schwarzschild black holes are given. The Hamilton-Jacobi method is also
studied in other coordinates like Painleve. Here also the Hawking temperature
is explicitly calculated which exactly matches with that evaluated in
Schwarzschild like coordinates. In all these cases there is no ambiguity
regarding a factor of two in the temperature, as reported in [15, 18, 34, 35].

In this paper, 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 [36]. Our approach, on the other hand, is similar to
in spirit [16] 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 (24) agrees with the findings of [16].

The other significant point of this paper is that for an appropriate choice of
the dimensionless parameters appearing in the single particle action, it is
possible to reproduce the one loop results obtained by including back reaction
effects [20, 21] or those based on conformal field theory techniques [22].
Apart from the one loop case, we have also discussed the nature of two loop
corrections to the surface gravity of the black hole. By using a law of black
hole mechanics, the corrections to the Bekenstein-Hawking area law are given
for Schwarzschild and AdS-Schwarzschild black holes. Interestingly, the
leading order correction is a logarithmic function of the horizon area $A$ or
the semiclassical entropy $S_{\textrm{BH}}$, as reported earlier in [22, 23,
24, 28, 29, 25, 26, 27, 30].

An important part of the analysis is that this is done in four dimensions and
the expansion coefficients, apart from some undetermined parameters, are
identified by a dimensional analysis. Also, explicit higher order computations
were presented for Schwarzschild type black holes that have mass as the only
macroscopic parameter. As was discussed, this simplified the dimensional
analysis allowing for a compact expression for the modified one particle
action. It remains an open issue to extend this analysis to include higher
order corrections for other black hole geometries.

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