# Entity Tracking via Effective Use of Multi-Task Learning Model and
Mention-guided Decoding

Janvijay Singh♣ Fan Bai♣ Zhen Wang♢
$\clubsuit$ School of Interactive Computing, Georgia Institute of Technology
$\diamondsuit$ Department of Computer Science and Engineering, The Ohio State
University
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

Cross-task knowledge transfer via multi-task learning has recently made
remarkable progress in general NLP tasks. However, entity tracking on the
procedural text has not benefited from such knowledge transfer because of its
distinct formulation, i.e., tracking the event flow while following structural
constraints. State-of-the-art entity tracking approaches either design
complicated model architectures or rely on task-specific pre-training to
achieve good results. To this end, we propose MeeT, a Multi-task learning-
enabled entity Tracking approach, which utilizes knowledge gained from general
domain tasks to improve entity tracking. Specifically, MeeT first fine-tunes
T5, a pre-trained multi-task learning model, with entity tracking-specialized
QA formats, and then employs our customized decoding strategy to satisfy the
structural constraints. MeeT achieves state-of-the-art performances on two
popular entity tracking datasets, even though it does not require any task-
specific architecture design or pre-training.111Our code and data are
available at https://github.com/iamjanvijay/MeeT.

##  1 Introduction

Figure 1: Overview of MeeT (Multi-task learning-enabled entity Tracking). MeeT
utilizes the multi-task learning in T5 to boost entity tracking performance,
with a customized decoding strategy addressing the structural constraints in
state prediction (e.g., "move" cannot happen after "destroy").

Pre-trained language models have revolutionized the NLP field in recent years
(Devlin et al., 2019; Liu et al., 2019; Brown et al., 2020) and also become
more versatile with the novel encoder-decoder architecture (Raffel et al.,
2020; Lewis et al., 2020), which allows them to handle different types of NLP
tasks without further architectural changes. This versatility inherently
facilitates cross-task knowledge transfer via multi-task learning (Raffel et
al., 2020; Aribandi et al., 2022), and thus helps push the boundary of many
popular NLP tasks such as question answering (Khashabi et al., 2020) and
semantic parsing (Xie et al., 2022). However, entity tracking, which tracks
the states and locations of an entity throughout the procedural text, like
scientific processes or recipes, has not been impacted by this multi-task
learning wave for two main reasons. First, entity tracking requires the model
to make step-wise predictions while satisfying structural constraints (e.g.,
an entity cannot be "moved" after being "destroyed" in the previous steps).
This requirement is usually tackled by designing task-specific architectures
(Gupta and Durrett, 2019b; Tang et al., 2020; Huang et al., 2021), and those
generic multi-task models with the encoder-decoder architecture cannot address
it easily. Second, understanding procedural text requires domain-specific
knowledge, which usually does not exist in general domain tasks that multi-
task learning models are trained on, so it is not clear how effective the
knowledge transfer will be given this domain gap (Zhang et al., 2021; Bai et
al., 2021; Shi et al., 2022).

In this paper, we study how entity tracking can benefit from the current
multi-task learning paradigm and present MeeT, a Multi-task learning-enabled
entity Tracking approach. This approach includes two parts. The first part
fine-tunes T5 (Raffel et al., 2020), a model that has been pre-trained on a
diverse set of NLP tasks and has shown great cross-task generalizability.
Here, we design entity tracking-specialized QA formats to accommodate the need
to make step-specific predictions, while facilitating effective knowledge
transfer from T5. The second part resolves conflicted state predictions under
structural constraints. We use a customized offline CRF inference algorithm,
where the main idea is to emphasize the predictions of steps, in which the
query entity is explicitly mentioned, because the fine-tuned model performs
better in those cases (Table 5). On two benchmark datasets, ProPara (Dalvi et
al., 2018) and Recipes (Bosselut et al., 2018), our MeeT outperforms previous
state-of-the-art methods, which require extra domain-specific pre-training or
data augmentation. We verify the importance of multi-task learning in T5 and
our proposed decoding strategy through careful analyses and ablation studies.

To sum up, our contributions are three-fold: (1) Our work is the first to
explore cross-task knowledge transfer for entity tracking on procedural text;
(2) Our proposed approach, MeeT, effectively uses the off-the-shelf pre-
trained multi-task learning model T5 with a customized decoding strategy, and
thus achieves state-of-the-art performance on two benchmark datasets; (3) Our
comprehensive analyses verify the benefits of multi-task learning on entity
tracking.

##  2 Related Work

Tracking the progression of an entity within procedural text, such as cooking
recipes (Bosselut et al., 2018) or scientific protocols (Tamari et al., 2021;
Le et al., 2022; Bai et al., 2022), is challenging as it calls for a model to
understand both superficial and intrinsic dynamics of the process. Recent work
on entity tracking can be divided into two lines. One focuses on designing
task-specific fine-tuning architectures to ensure that the model makes step-
grounded predictions while following the structural constraints. For instance,
Rajaby Faghihi and Kordjamshidi (2021) introduce time-stamp embeddings into
RoBERTa Liu et al. (2019) to encode the index of the query step. Gupta and
Durrett (2019b) frame entity tracking as a structured prediction problem and
use a CRF layer to promote global consistency under those structural
constraints. In our case, we show that, with QA formulation, simply appending
the index of the query step to the question and indexing the procedure
produces step-specific predictions. Moreover, we propose a customized offline
CRF-decoding strategy for structural constraints to compensate for the fact
that it is hard to jointly train T5, our backbone LM, with a CRF layer, like
in previous methods.

The other line of work focuses on domain-specific knowledge transfer (Zhang et
al., 2021; Bai et al., 2021; Shi et al., 2022; Ma et al., 2022). Concretely,
LEMon Shi et al. (2022) achieves great performance by performing in-domain
pre-training on 1 million procedural paragraphs. CGLI (Ma et al., 2022) shows
that adding high-quality pseudo-labeled data (generated via self-training)
during fine-tuning can also boost the model performance. In contrast, our work
explores how entity tracking can benefit from out-of-domain knowledge via
using off-the-shelf pre-trained multi-task learning models.

##  3 Method

In this section, we present MeeT, a Multi-task learning-enabled entity
Tracking approach. Here, we first review the problem definition, and then lay
out the details of MeeT.

###  3.1 Problem Definition

Entity tracking aims at monitoring the status of an entity throughout a
procedure. The input of this task contains two items: 1) a procedural
paragraph $P$, composed of a sequence of sentences
$\\{s_{1},s_{2},...,s_{T}\\}$; and 2) a procedure-specific query entity $e$.
Given the input, our goal is to predict the state and location of the query
entity at each timestamp of the procedure (see an example from the ProPara
dataset in Figure 1).

###  3.2 MeeT

MeeT includes two parts, task-specific fine-tuning with our proposed QA
formats and the mention-guided conflict-resolve decoding.

#### Task-spefic Fine-tuning

We formulate the two sub-tasks of entity tracking, state prediction and
location prediction, as multi-choice and extractive QA problems respectively
(see §4.2 for comparison with other task formulations), and fine-tune T5 to
make independent predictions for every step in the procedure. Given a query
entity $e$ and procedure $P$, to predict the entity state at step $t$, the
input sequence is formatted as the concatenation of the template question
“What is the state of $e$ in step $t$?”, candidate states (e.g., create, move
and destroy), and the full procedure with step index prepended. The output is
just one of the candidate states. For location prediction, the input sequence
is the concatenation of the question “Where is $e$ located in step $t$?” and
the indexed procedure, with the snippet “Other locations: none, unknown.”
appended. This is because entity locations sometimes are not explicitly
mentioned in the procedure. The output is a text span, indicating the location
of the query entity after step $t$. Examples of both tasks can be found in
Appendix A.

#### Conflict-resolve Decoding

Entity tracking places unique structural constraints on state predictions
(e.g., move cannot happen after destroy). Similar to Gupta and Durrett
(2019a), we run an offline CRF-decoding method (Viterbi decoding) to resolve
conflicting state predictions. We initialize CRF transition scores $T$ with
the transition statistics in the training data, following Ma et al. (2022).
For example, $T(p,q)$, the transition score between state $p$ and $q$, is
$log(1/10)$ if there is only one $p\Rightarrow q$ transition out of 10
transitions starting with the state $p$. We set the scores of all unseen
transitions to $-inf$. As for CRF emission scores, we use the state prediction
logits from T5. In contrast with previous methods, which treat each step
equally, we weigh the emission scores differently, depending on whether the
query entity $e$ is explicitly mentioned in the step:

$U_{i}^{\prime}=\begin{cases}\tau_{exp}\cdot U_{i},&\text{if }e\text{ is
mentioned in step }i,\\\ \tau_{imp}\cdot U_{i},&\text{otherwise}\\\
\end{cases}$

where $U_{i}^{\prime}$ represents the emission score of step $i$ after
weighing, and $\tau_{exp}$ and $\tau_{imp}$ are hyper-parameters, determined
by the grid search on the dev set. The intuition behind our approach is that,
as the fine-tuned model performs better on "explicitly mentioned" steps (Table
5), leaning toward those steps during decoding via controlled weights will
result in more accurate predictions.222After hyper-parameter tuning, the
optimal values for $\tau_{exp}$ and $\tau_{imp}$ are 0.6 and 0.7 respectively.

Model | P | R | F1
---|---|---|---
DynaPro (Amini et al., 2020) | 75.2 | 58.0 | 65.5
TSLM† (Faghini et al., 2021) | 68.4 | 68.9 | 68.6
KoaLa† (Zhang et al., 2021) | 77.7 | 64.4 | 70.4
LEMon†(Shi et al., 2022) | 74.8 | 69.8 | 72.2
CGLI† (Ma et al., 2022) | 75.7 | 70.0 | 72.7
MeeT (ours) | 80.3 | 67.1 | 73.1

Table 1:  Test set performance on ProPara. † indicates that the backbone
language model has been further pre-trained on either in-domain corpus or
auxiliary tasks. MeeT performs on par with SOTA models without pre-finetuning
on any in-domain corpus.

##  4 Experiments

#### Datasets

We experiment with two benchmark datasets of entity tracking: ProPara (Dalvi
et al., 2018) and Recipes Bosselut et al. (2018). ProPara contains 488
scientific process-based procedural paragraphs (Figure 1), and Recipes
includes 866 cooking recipes. Note that previous work experiments with
different splits of the Recipes dataset; in this paper, we follow the split of
Zhang et al. (2021)333 https://github.com/ytyz1307zzh/KOALA/issues/4 as it is
used in most of the recent work (Huang et al., 2021; Shi et al., 2022). More
dataset details are presented in Appendix B.

#### Evaluation

ProPara performances are evaluated in two levels: sentence-
level444https://github.com/Mayer123/CGLI/blob/main/src/evalQA.py (Dalvi et
al., 2018) and document-level555https://github.com/allenai/aristo-
leaderboard/blob/master/propara/evaluator (Tandon et al., 2018). Here, we
focus on the document-level evaluation because it provides a comprehensive
assessment of the model’s understanding of the overall procedure and serves as
the basis for the ProPara leaderboard rankings. The document-level evaluation
is conducted by comparing the input/output entities and their transformations
in the procedure with the gold answers. Further details regarding two
evaluations and the result of the sentence-level evaluation can be found in
Appendix C. For Recipes, following previous work (Zhang et al., 2021; Shi et
al., 2022), we evaluate the location changes of each ingredient throughout the
recipe.666
https://drive.google.com/drive/folders/1PYGLe7hSoCYfpKmpPumeTy6jmPyONGz4

#### Baselines

For ProPara, we compare MeeT with the top five approaches on its leaderboard.
Among these five approaches, DynaPro (Amini et al., 2020), TSLM (Rajaby
Faghihi and Kordjamshidi, 2021), and CGLI (Ma et al., 2022) design task-
specific fine-tuning architecture using off-the-shelf LMs while KoaLa (Zhang
et al., 2021) and LEMon (Shi et al., 2022) develop in-domain LMs for
procedural text. For Recipes, as mentioned previously, we compare MeeT with
methods that experiment on the same data split of Zhang et al. (2021). We
refer readers to the corresponding paper of each baseline for further details.

#### Implementation Details

Our approach MeeT is implemented using Huggingface Transformers (Wolf et al.,
2020). Given the limited computational resources, we choose T5-large as the
backbone of our MeeT. The fine-tuning process employs the AdamW optimizer with
a learning rate of $1\times 10^{-4}$ and a batch size of 16. To resolve any
potential conflict between state prediction and location prediction, we apply
the rules designed in Ma et al. (2022) to integrate the output from both
tasks.

###  4.1 Results

We present the test set results of ProPara and Recipes in Table 1 and Table 2,
respectively. Our MeeT outperforms the competitive baseline CGLI (Ma et al.,
2022) on the ProPara dataset with state-of-the-art performance despite the
fact that CGLI uses extra pseudo-labeled training data (generated by self-
training) for data augmentation. On Recipes, MeeT surpasses the previous best-
performing method LEMon (Shi et al., 2022) by a substantial margin of 4.9 F1.
It is noteworthy that the cooking recipes in the Recipes dataset were
collected from the web,777http://www.ffts.com/recipes.htm which may have been
included in the C4 corpus888https://www.tensorflow.org/datasets/catalog/c4
used for pre-training T5 and thus potentially contributes to the advantage of
our MeeT on Recipes.

###  4.2 Analysis & Ablation Study

#### Multi-task Learning

To investigate the impact of T5’s multi-task learning process on entity
tracking, we experiment with two variants of T5 as the backbone of MeeT: 1)
T5-v1.1,999https://huggingface.co/docs/transformers/model_doc/t5v1.1 a T5-like
LM (with slight architecture changes) whose pre-training does not include any
supervised tasks; 2) T5-v1.1QA-FT, the resulting LM after fine-tuning T5-v1.1
on the three QA datasets,101010The three datasets include MultiRC (Khashabi et
al., 2018), ReCoRD (Zhang et al., 2018), and BoolQ (Clark et al., 2019) which
T5 is pre-trained on. The performance of the three LMs (T5-large size) on the
ProPara dev set is presented in the top section of Table 3. We can see that T5
outperforms T5-v1.1 by a large margin, verifying that multi-task learning on
out-of-domain non-entity-tracking tasks can benefit entity tracking. In
addition, the advantage of T5 over T5-v1.1QA-FT indicates that knowledge
transfer can cross the task boundaries with T5’s encoder-decoder architecture.

Model | P | R | F1
---|---|---|---
NCET (Gupta and Durrett, 2019b) | 56.5 | 46.4 | 50.9
IEN (Tang et al., 2020) | 58.5 | 47.0 | 52.2
KoaLa (Zhang et al., 2021) | 60.1 | 52.6 | 56.1
Real (Huang et al., 2021) | 55.2 | 52.9 | 54.1
LEMon (Shi et al., 2022) | 56.0 | 67.1 | 61.1
MeeT (ours) | 64.2 | 78.0 | 66.0

Table 2:  Test set results on Recipes. MeeT achieves the state-of-the-art
performance, outperforming the previous SOTA LEMon by 4.9 F1.

| P | R | F1
---|---|---|---
MeeT (ours) | 77.3 | 71.1 | 74.1
Multi-task Learning
T5-v1.1 | 76.6 | 64.9 | 70.3
T5-v1.1QA-FT | 76.3 | 65.8 | 70.7
Task Formulation
Process-level | 89.3 | 30.2 | 45.1
Step-level | 76.7 | 61.3 | 68.1
Decoding Strategy & Model Size
CRF-normal | 75.0 | 72.8 | 73.8
T5-base | 76.8 | 68.2 | 72.2

Table 3:  Analysis and ablation study on ProPara (dev set results). Top:
Comparison of different backbone LMs to investigate the impact of multi-task
learning. Middle: Comparison of different task formulations. Bottom: Ablation
on decoding strategy and model size. Multi-task learning leads to a better
entity tracking model, especially with the QA formulation and mention-guided
decoding.

#### Task Formulation

We compare our QA formulation with two other task formulations, proposed in
recent work, for T5. The first formulation is called "step-input" (Gupta and
Durrett, 2019a; Amini et al., 2020), where each pair of the query entity $e$
and procedure step $t$ is formulated as one instance. Here, the state
prediction is formulated as a classification problem, where the entity name is
appended to the input, and no candidate answers are provided. Moreover, the
procedure is trimmed until step $t$ to specify the step index in the input.
The second formulation is called "process-input" (Zhang et al., 2021; Gupta
and Durrett, 2019b), where the model predicts entity states or locations in
all steps in one instance. The input is the concatenation of entity $e$ and
the full procedure, and the model decodes entity states and locations in all
steps sequentially. The results of two new formulations are presented in the
middle of Table 3. Our proposed QA formulation outperforms the other two
formulations by a large margin. Detailed analyses of formulation comparison
can be found in Appendix D.

#### Decoding Strategy & Model Size

The ablation study on decoding strategy and model size is shown at the bottom
section of Table 3. Clearly, our proposed "mention-guided" decoding strategy,
as well as using a larger LM as the backbone, contribute to the success of
MeeT.

##  5 Conclusion

We presented MeeT, a T5-based entity tracking approach. This approach includes
our newly proposed QA fine-tuning formats and a customized decoding strategy
so that it can effectively encode the flow of events in the procedural text
while following structural constraints. The state-of-the-art performances on
two benchmark datasets demonstrate the effectiveness of MeeT, and further
analyses verify that multi-task learning on out-of-domain tasks can be
beneficial for entity tracking.

## Limitations

This paper demonstrates that multi-task learning on a combination of general
domain datasets can effectively improve the model’s understanding of the
procedural text. However, the precise source dataset responsible for this
improvement remains uncertain, making it an avenue for future research to
investigate more efficient knowledge transfer through the identification of
the most pertinent source dataset. Moreover, the pipeline structure of MeeT
may limit its practical utilization. As such, future work could consider
incorporating our proposed mention-guided decoding strategy into the end-to-
end training of the multi-task learning model.

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##  Appendix A Fine-tuning Formats for T5

###  A.1 State Prediction (Multi-choice QA)

Input:

> What is the state of water in step 2?
>  (a) create (b) ... (f) move
>  step 1: Water flows downawards thanks to gravity. step 2: Enters the dam at
> high pressure. step 3: The moving water spins the turbines in the power
> plant ... step 6: The water leaves the dam at the bottom.

Output:

> move
>

###  A.2 Location Prediction (Extractive QA)

Input:

> Where is water located in step 2?
>  step 1: Water flows downwards thanks to gravity. step 2: Enters the dam at
> high pressure. step 3: The moving water spins the turbines in the power
> plant ... step 6: The water leaves the dam at the bottom. Other locations:
> none, unknown.

Output:

> dam
>

##  Appendix B Dataset

Dataset | Statistics | Train | Dev | Test | Total
---|---|---|---|---|---
Recipes | # procedures | 693 | 86 | 87 | 866
Avg. steps / proc. | 8.8 | 8.9 | 9.0 | 8.8
Avg. entities / proc. | 8.6 | 8.8 | 8.5 | 8.6
ProPara | # procedures | 391 | 43 | 54 | 488
Avg. steps / proc. | 6.8 | 6.7 | 6.9 | 6.8
Avg. entities / proc. | 3.8 | 4.1 | 4.4 | 3.9

Table 4: Statistics of Recipes and ProPara.

| P | R | F1
---|---|---|---
implicit | 37.4 | 23.2 | 28.3
explicit | 68.3 | 72.4 | 70.2

Table 5:  MeeT’s sentence-level performance (before applying offline CRF) on
implicit and explicit steps (where the query entity is explicitly mentioned).
Clearly, MeeT makes more accurate predictions on explicit steps.

For ProPara (Dalvi et al., 2018), following Ma et al. (2022), the state
prediction task includes six candidate states (Outside_Before, Create,
Destroy, Move, Exist and Outside_After). For Recipes Bosselut et al. (2018),
each ingredient has two possibles states (Exist or Absence) in each step of
the recipe. Full data statistics on two datasets are presented in Table 4.

##  Appendix C Evaluation

#### Sentence-level evaluation

This evaluation measures the following questions for each target entity:

  * •

Cat-1: Is entity created (destroyed, moved) in the process?

  * •

Cat-2: When (step #) is entity created (destroyed, moved)?

  * •

Cat-3: Where (location) is entity created (destroyed, moved to/from)?

Further, the F1 scores of the three questions are aggregated with micro/macro
averages.

#### Document-level evaluation

It measures the four questions below for each paragraph:

  * •

What are the input entities to the process?

  * •

What are the output entities of the process?

  * •

What entity conversions occur, when (step #), and where (location)?

  * •

What entity movements occur, when, and where?

The macro average of the F1 scores of these four questions will be used as the
final score.

Model | Document-level | Sentence-level
---|---|---
P | R | F1 | Cat-1 | Cat-2 | Cat-3 | macro | micro
NCET (Gupta and Durrett, 2019b) | 67.1 | 58.5 | 62.5 | 73.7 | 47.1 | 41.0 | 53.9 | 54.0
IEN Tang et al. (2020) | 69.8 | 56.3 | 62.3 | 71.8 | 47.6 | 40.5 | 53.3 | 53.0
DynaPro (Amini et al., 2020) | 75.2 | 58.0 | 65.5 | 72.4 | 49.3 | 44.5 | 55.4 | 55.5
ProGraph Zhong et al. (2020) | 67.3 | 55.8 | 61.0 | 67.8 | 44.6 | 41.8 | 51.4 | 51.5
TSLM (Faghini et al., 2021) | 68.4 | 68.9 | 68.6 | 78.8 | 56.8 | 40.9 | 58.8 | 58.4
KoaLa (Zhang et al., 2021) | 77.7 | 64.4 | 70.4 | 78.5 | 53.3 | 41.3 | 57.7 | 57.5
REAL Huang et al. (2021) | 81.9 | 61.9 | 70.5 | 78.4 | 53.7 | 42.4 | 58.2 | 57.9
LEMon (Shi et al., 2022) | 74.8 | 69.8 | 72.2 | 81.7 | 58.3 | 43.3 | 61.1 | 60.7
CGLI (Ma et al., 2022) | 75.7 | 70.0 | 72.7 | 80.8 | 60.7 | 46.8 | 62.8 | 62.4
MeeT (ours) | 80.3 | 67.1 | 73.1 | 77.5 | 61.0 | 49.6 | 62.7 | 62.4
Table 6:  Document-level and sentence-level evaluation results on ProPara test
set.

Table 6 provides a comprehensive comparison of past work on the ProPara
dataset, including both document-level and sentence-level evaluations.

##  Appendix D Analysis of Formulation Comparison

When compared with the "step-input" formulation, the QA formulation allows the
model to have the full context, and may take better advantage of LM’s pre-
training scheme (Li et al., 2019; Nagata et al., 2020). The "process-input"
formulation works the worst in this comparison. With qualitative analyses, we
find that it suffers from error propagation due to its autoregressive
decoding, so future work may explore incorporating structural decoding (Tandon
et al., 2018) into T5.


###### Abstract

Here we added an alternative derivation to get the main result third part of
the paper. We laso added the necessary references for it and changed the
”acknowledgement” section.. Here we continue studying the Wahlquist metric. We
know that the wave equation written for a zero mass scalar particle in the
background of this metric gives Heun type solutions. To be able to use the
existing literature on Heun functions, we try to put our wave equation to the
standard form for these functions. Then we calculate the reflection
coefficient of a wave coming from infinity and scattered at the center using
this formalism.

Wahlquist Metric Revisited

M. Hortaçsu, 111E-mail<EMAIL_ADDRESS>
Mimar Sinan Güzel Sanatlar Üniversitesi, Physics Depat. Istanbul, Turkey

##  1 Introduction

The work on Heun class solutions of wave equations written for particles in
the background metric of different metrics is still popular. One of the most
recent papers is given in [1]. Here we study a similar problem for a metric,
which was given a while ago.

The Wahlquist metric [2, 3, 4] is an exact interior solution for a finite
rotating body of perfect fluid to Einstein’s field equations with equation of
state corresponding to constant gravitational mass density. It is also given
in the celebrated book Exact Solutions to Einstein’s Field Equations [5]. This
metric is axially symmetric, stationary and is type D. As stated in [6],
quoting Wahlquist [2], it can be ”described as a superposition of a Kerr-NUT
metric [7, 8] and is a rigidly rotation perfect fluid in the same space-time
region”. Equation of the state for the perfect fluid is $\rho+3p=\mu_{0}$, a
constant.

The original metric written in [2], was slightly modified by Senovilla [9,
10], and put to new form by Mars [11], ”to show that the Kerr-de Sitter and
Kerr metrics are contained as sub cases”. Mars states in [11] that Kramer [12]
showed the vanishing of the Simon tensor [13] for this metric. Mars also
states that the space-time admits a Killing tensor as shown in [14]”.

A negative point about this metric is the paper, [15], where it was shown that
” the Wahlquist perfect fluid space-time can not be smoothly joined to an
exterior asymptotically flat vacuum region since the conditions for matching
the induced metrics and extrinsic curvatures are mutually contradictory”.
Thus, ” the Wahlquist metric can not describe an isolated rotating body”.
Bradley et al., however, showed that this obstruction is removed for a
limiting form of the Wahlquist solution [16], and also for every D-type metric
[17], if the requirement of an asymptotically flatness is removed.

More recent work in this field exists, ”where the existence of a rank-2
generalized closed conformal Killing-Yano tensor with a skew-symmetric torsion
” [18], and ”the separability of the Maxwell equation on the Wahlquist
spacetime” are shown [19].

Here we start with the metric as given in [18] and try to write the wave
equation, in the background of this metric, as given in [6], in the standard
form given by [20]. How to perform this task is described in [21], as quoted
by [22], and used meticulously by [23, 24]. The same method is recently used
by Vieira et al in [25] . We need the standard form of the wave equation to be
able to apply the information given in the existing literature to our work.

We will first summarize our previous work, [6]. Then we show how to convert
the wave equation to the standard form. In the third section, we give the
approximate reflection coefficient if a wave , coming from infinity is
scattered at the origin. We conclude with a few remarks.

##  2 Summary of former work

Here, we will summarize the work in [6] below. First, using the metric given
in [2], we try to calculate $\phi$ where it obeys the equation

$\frac{1}{\sqrt{g}}\partial_{\mu}g^{\mu\nu}\sqrt{g}\partial_{\nu}\phi=0.$ (1)

Here $g$ is the determinant of the metric coefficients $g_{\mu\nu}$.

We first write the metric as is given in [18], which is equivalent to the one
giving in [11]. We follow some of the work in [18] in the first part, and [6]
in the second part of this section.

Here ”the comoving, pseudoconfocal, spatial coordinates are used, which are
closely related to the oblate-spheroidal coordinates in Euclidean geometry”
[2]. We set $4\pi G=c=1$. Then, the metric reads

$ds^{2}=(v_{1}+v_{2})\big{(}\frac{dz^{2}}{U}+\frac{dw^{2}}{V}\big{)}+\frac{U}{v_{1}+v_{2}}(d\tau+v_{2}d\sigma)^{2}-\frac{V}{v_{1}+v_{2}}(d\tau-
v_{1}d\sigma)^{2}$ (2)

where

$\displaystyle U=Q_{0}+a_{1}{\frac{sinh(2\beta
z)}{2\beta}}-\frac{\nu_{0}}{\beta^{2}}{\frac{cosh(2\beta z)-1}{2\beta^{2}}}$
$\displaystyle-{\frac{\mu_{0}}{2\beta^{2}}}\big{[}{\frac{cosh(2\beta
z)-1}{2\beta^{2}}}-z{\frac{sinh(2\beta z)}{2\beta}}\big{]},$ (3)

$\displaystyle V=Q_{0}+a_{2}{\frac{sin(2\beta
w)}{2\beta}}+\frac{\nu_{0}}{\beta^{2}}{\frac{-cos(2\beta w)+1}{2\beta^{2}}}$
$\displaystyle-{\frac{\mu_{0}}{2\beta^{2}}}\big{[}{\frac{cos(2\beta
w)-1}{2\beta^{2}}}+w{\frac{sin(2\beta w)}{2\beta}}\big{]}$ (4)

where

$v_{1}=\frac{cosh(2\beta z)-1}{2\beta^{2}},v_{2}=\frac{-cos(2\beta
w)+1}{2\beta^{2}}.$ (5)

This metric has six real constants $Q_{0},a_{1},a_{2},\nu_{0},\mu_{0},\beta$.
Here $a_{1}$ is related to the NUT [26, 27] parameter; $a_{2}$ is related to
the mass parameter. One writes the other variables to express the energy
density, pressure and the fluid velocity of the perfect fluid. Here $\beta$ is
related to the scaling of $z$ and $w$, both space coordinates, in the linear
transformation of these variables are given in the original paper by Wahlquist
[2]. $U$ and $V$ are related to $h_{2},h_{1}$ in the original metric. All
these parameters are scaled so that they do not vanish in the $\beta$ going to
zero limit.

When we use the ansatz

$\phi=R(z)Y(w)T(\tau)S(\sigma)$ (6)

for the solution, the wave equation written in this metric separates easily.
We have two Killing vectors since the metric does not depend on $\tau$ and
$\sigma$, related to $t$ and $\theta$ in the original metric [2]. If we make a
Wick rotation which changes $w$ to $y=iw$, $a_{2}$ to $-ia_{2}$ [11] where $i$
is the square root of minus unity, the metric becomes symmetrical

$ds^{2}=(v_{1}+v_{2})\big{(}\frac{dz^{2}}{U}-\frac{dy^{2}}{V}\big{)}+\frac{U}{v_{1}+v_{2}}(d\tau-
v_{2}d\sigma)^{2}-\frac{V}{v_{1}+v_{2}}(d\tau+v_{1}d\sigma)^{2}.$ (7)

Then we have identical equations for $z$ and $y$. The equation for $z$ reads

$\partial_{z}(U\partial_{z})R(z)+\big{(}\frac{v_{1}^{2}}{U}\partial_{\tau}^{2}+2\frac{v_{1}}{U}\partial_{\tau}\partial_{\sigma}+\frac{1}{U}\partial_{\sigma}^{2}\big{)}R+\Lambda
R=0.$ (8)

We get exactly the same equation for the new variable $y$, with appropriate
changes like $U$ going to $V$ and $v_{1}$ going to $v_{2}$.

$\partial_{y}(V\partial_{y})S(y)+\big{(}\frac{v_{2}^{2}}{V}\partial_{\tau}^{2}+2\frac{v_{2}}{V}\partial_{\tau}\partial_{\sigma}+\frac{1}{V}\partial_{\sigma}^{2}\big{)}S(y)=\Lambda
S(y).$ (9)

Since the functions $v_{1},U$ ( similarly $v_{2},V$) don’t depend on $\tau$
and $\sigma$, the solutions for $\tau$ and $\sigma$ are just exponential
functions, giving us constants upon differentiation.

Here we will first try to write the differential equation for a zero mass
scalar field for the radial coordinate $z$, in the background of this metric.
Then, we make more independent variable changes to put our equation in the the
standard form , as given in [20].

We find that this is not an easy task. The presence of hyperbolic sine and
cosine functions in the wave equation prevents us from using standard
analytical methods in solving differential equations. We change our variables
to $x=exp(2\beta z)$ which makes it possible to write the hyperbolic sine and
cosine functions in terms of powers. $sinh{2\beta z}=1/2[x-1/x]$, $cosh{2\beta
z}-1=1/2[x+1/x-2]$. Then, however, exists the relation $z=ln{x}/(2\beta)$.

The codes we have to solve differential equations analytically, does not
recognize $ln{x}$. Thus, we can not give an exact solution for all values of
the independent variable $x$. We can get solutions only in one patch, by using
an algebraic expression approximating $lnx$ in this region.

Upon the variable change from $z$ to $x$, we get the new equation

$\frac{d^{2}\psi(x)}{dx^{2}}+\big{(}\frac{{\frac{dU}{dx}}}{U}+\frac{1}{x}\big{)}\frac{d}{dx}\psi(x)-\frac{1}{x^{2}U^{2}}\big{(}\omega^{2}v_{1}^{2}+2v_{1}\omega
s+s^{2}+\Lambda\big{)}\psi(x)=0.$ (10)

Here we used the solution to the $\tau$ equation in the form
$exp(-i\omega\tau)$ and the solution for the $i\sigma$ equation a solution in
the form $exp(-i\sigma s)$. Note that our metric had Killing vectors for both
$\tau$ and $\sigma$. Here

$U=Q_{0}+a_{1}\frac{(x-1}{4x\beta}-\big{[}\frac{\mu_{0}}{4\beta^{4}}+\frac{\nu_{0}}{4\beta^{3}}]\frac{(x-1)^{2}}{x}+\frac{\mu_{0}}{8\beta^{4}}\ln
x\frac{(x^{2}-1)}{x}\big{]},$ (11) $v_{1}=\frac{(x-1)^{2}}{4\beta^{2}x}$ (12)

,

$\frac{dU}{dx}=\big{[}\frac{a_{1}}{\beta}(1+\frac{1}{x^{2}})-(\frac{\nu_{0}}{\beta^{2}}+\frac{\mu_{0}}{\beta^{4}}(1-\frac{1}{x^{2}})+\frac{\mu_{0}}{8\beta^{4}x}(x-\frac{1}{x})+\frac{\mu_{0}}{8\beta^{4}}\ln{x}(1+\frac{1}{x^{2}})\big{]}.$
(13)

From here on, we will define $\mu_{0}^{\prime}=\frac{\mu_{0}}{(8\beta^{4})}$
and $\nu_{0}^{\prime}=\frac{\nu_{0}}{(4\beta^{3})}$,
$a^{\prime}_{1}=a_{1}/\beta$ and use these new constants in our equations.

By studying the equation written in terms of the variable $x$, we know that
there is a singularity near $x=1$. To study the solution around this
singularity, we expand $lnx$ in the neighborhood of this point and use $x-1$
instead of $lnx$ in the wave equation we use in our calculation, after we
differentiate $U$ with respect to $x$.

We find that, if we keep all our constants non zero, we get [6] the exact
solution in terms the general Heun function [20, 28, 29, 30] up to an
exponential and terms (powers of polynomials) multiplying this function. The
regular singularities are at zero and at three other finite points.

Then we try the next possible choice. We try to find the solution when we keep
all the terms in $U$ aside from $Q_{0}$, which we set equal to zero. Then, our
equation, again, has five regular singularities at $0,1$, infinity and at the
points

$x_{a}=-\frac{1}{2\mu^{\prime}_{0}}(M-[N]^{1/2}),$ (14)
$x_{b}=-\frac{1}{2\mu^{\prime}_{0}}(M+[N]^{1/2}),$ (15)

where

$M=(a^{\prime}_{1}-\nu^{\prime}_{0}-2\mu^{\prime}_{0}),$ (16)
$N^{2}=M^{2}-4\mu^{\prime}_{0}(a^{\prime}_{1}+\nu^{\prime}_{0}+\mu^{\prime}_{0}).$
(17)

The $HeunG$, General Heun form solution is retained when we set $Q_{0}$ to
zero. To further simplify our expressions, we define
$a^{\prime\prime}_{1}=\frac{a^{\prime}_{1}}{\mu^{\prime}_{0}},\nu^{\prime\prime}_{0}=\frac{\nu^{\prime}_{0}}{\mu^{\prime}_{0}}$.

##  3 Reduction to the standard form

How to reduce an equation with four regular singular points at finite values
to an equation with singularities at zero, unity, a finite point and infinity
is described in [22]. If the coefficients of the first derivatives satisfy a
linear relation, one just has to make a homographic substitution to bring
three of the original singularities $b_{1},b_{2},b_{4}$ to zero, unity and
infinity. The previous regular point at infinity is transformed to $-1+x_{b}$.
This point is decoupled from the equation as shown by [23]. This
transformation is

$u=\frac{x-b_{1}}{x-b_{3}}\frac{b_{2}-b_{3}}{b_{2}-b_{1}}.$ (18)

Then in our case $b_{1}=0,b_{1}=1,b_{3}=x_{b},b_{4}=x_{a}$, which gives
$u=\frac{x(1-x_{b})}{x-x_{b}}$. The singularity at $x_{b}$ is moved to
infinity, the singularity at $x_{a}$ is moved to
$\frac{-x_{a}(1-x_{b})}{x_{a}-x_{b}}$. Then our equation reads

$\displaystyle\frac{d^{2}\phi(u)}{du^{2}}+\big{(}\frac{1}{x_{a}x_{b}}-\frac{1}{u-1}+\frac{(1+x_{b})-x_{a}x_{b}\frac{1}{x_{a}}}{(1-x_{a})(u+\frac{x_{a}(1-x_{b})}{x_{b}-x_{a}})}\big{)}\frac{d\phi(u)}{du}+\frac{J}{2Tu}\phi(u)$
$\displaystyle+\frac{1}{4}\big{(}\frac{D}{T^{2}}+\frac{2s\omega}{EuT}+\frac{2s^{2}}{F(u-1)T}+\frac{4s^{2}}{H(u-1)^{2}}+\frac{\omega^{2}}{4(x_{a}x_{b}u)^{2}}+\Lambda\big{)}\phi(u)=0.$
(19) $T=(u+\frac{x_{a}(1-x_{b})}{x_{b}-x_{a}}),$ (20)
$D=\frac{1}{(x_{b}-x_{a})^{2}}(\frac{\omega(1-x_{a})}{2\beta^{2}x_{a}}-\frac{2s}{1-x_{a}})^{2},$
(21) $E=4x_{a}x_{b}(x_{b}-x_{a}),$ (22)
$F=(1-x_{a})^{2}(1-x_{b})(x_{b}-x_{a})$ (23) $H=(1-x_{a})^{2}(1-x_{b})^{2},$
(24)
$J=-\frac{\omega^{2}x_{b}(1-x_{a})}{\beta^{4}x_{a}^{2}x_{b}^{2}(x_{b}-x_{a})}$
(25)

The new equation is not in the standard General Heun form, though. One has to
remove the terms where the singularities are in quadratic form from the new
differential equation. One makes a F-homotopic transformation on the dependent
variable $\phi(u)$ in the form

$\phi(u)=(u)^{\lambda}(u-1)^{\xi}(u+\frac{x_{a}(1-x_{b})}{x_{b}-x_{a}})^{\kappa}g(u),$
(26)

which gives rather complicated expressions for these indices.

$\lambda=\frac{1}{2}[1\pm{\sqrt{1-\frac{w^{2}}{4\beta^{4}(x_{a}x_{b})^{2}}}}],$
(27)

$\xi=1\pm{\sqrt{1-\frac{4s^{2}}{(1-x_{a})^{2}(1-x_{b})2}}},$ (28)
$\kappa=-\frac{1}{2}[\frac{x_{a}}{x_{a}-x_{b}}-1\pm{\sqrt{(\frac{x_{a}}{x_{a}-x_{b}})^{2}-(\frac{w(x_{a}-1)}{x_{a}\beta^{2}}-\frac{4s}{(x_{a}-1)})^{2}}}].$
(29)

Now we can write the new wave equation as

$\frac{d^{2}g(u)}{du^{2}}+\big{(}\frac{c}{u}+\frac{d}{u-1}+\frac{e}{(u+\frac{x_{a}(1-x_{b})}{x_{b}-x_{a}})}\big{)}\frac{d}{du}g(u)+\frac{ABu+Q+\Lambda}{u(u-1)(u+\frac{x_{a}(1-x_{b})}{x_{b}-x_{a}})}g(u)=0.$
(30)

Here

$c=\big{[}1-{\sqrt{1-\frac{w^{2}}{4\beta^{4}(x_{a}x_{b})^{2}}}}\big{]},$ (31)
$d=1-2{\sqrt{1-\frac{s^{2}}{(1-x_{a})^{2}(1-x_{b})2}}},$ (32)
$e=1-{\sqrt{{(\frac{x_{a}}{x_{a}-x_{b}})^{2}-(\frac{w(x_{a}-1)}{x_{a}\beta^{2}}-\frac{4s}{(x_{a}-1)})^{2}}}}$
(33)

Using the relation

$c+d+e=A+B+1,$ (34)

which is valid when our equation has Heun form, we can also calculate $A$ and
$B$.

$A=\frac{1}{2}(c+d+e-1+\sqrt{(c+d+e-1)^{2}-4S}),$ (35)
$B=\frac{1}{2}(c+d+e-1-\sqrt{(c+d+e-1)^{2}-4S}).$ (36)

The expression inside the square root above simplifies

$\displaystyle(c+d+e-1)^{2}-4S=[\frac{2}{x_{a}x_{b}}+(-1+\frac{x_{a}x_{b}-1}{x_{b}(x_{b}-x_{a})})^{2}$
$\displaystyle-[\frac{\omega(1-x_{b})}{2\beta^{2}x_{b}(x_{b}-x_{a})}-\frac{2s}{(1-x_{b})(x_{b}-x_{a})}]^{2}].$
(37)

We give

$Q_{1}=-\frac{1}{2}[ec+\frac{x_{a}(1-x_{b})dc}{(x_{b}-x_{a})}+\frac{x_{a}(1-x_{b})}{2x_{a}x_{b}(x_{b}-x_{a})}+\frac{1+x_{b}-x_{a}x_{b}-\frac{1}{x_{a}}}{2(x_{b}-x_{a})(1-x_{a})x_{a}x_{b}}],$
(38)
$Q_{2}=\frac{\omega^{2}x_{b}(1-x_{a})}{2\beta^{4}x_{a}^{2}x_{b}^{2}(x_{b}-x_{a})}-\frac{s\omega}{2\beta^{2}x_{a}x_{b}(x_{b}-x_{a})}.$
(39)

In our expression for the wave equation and the solution, we then use $Q$
where $Q=Q_{1}+Q_{2}+\Lambda$,

##  4  Scattering at the origin

While changing our independent variable from $z$ to $x$, we overlooked one
point. Now $x$ corresponds to minus infinity for $z$. To scatter from origin ,
we need to take $x=1$. We, therefore, change our variables to $u-1=-y$. We
know that $u=1$ is the same point as $x=1$. Then our equation reads

$\frac{d^{2}g(y)}{dy^{2}}+\big{(}\frac{c}{y-1}+\frac{d}{y}+\frac{e}{(y-\frac{x_{b}(1-x_{a})}{x_{b}-x_{a}})}\big{)}\frac{d}{dy}g(y)+\frac{AB(y-1)-Q+\Lambda}{y(y-1)(y-\frac{x_{b}(1-x_{a})}{x_{b}-x_{a}})}g(y)=0,$
(40)

which gives as solution

$g(y)=H_{G}(\frac{x_{b}(1-x_{a})}{x_{b}-x_{a}},-Q-AB;A,B,d,c,e;y=1-u),$ (41)

where $H_{G}$ is the generalized Heun function, in the standard form[20]. We
had a singular point at infinity. We want to have our wave to come from
infinity and scatter at the origin. The other singular point at $x_{a}$ can be
arranged to be at a complex value.

For the scattering process, we will use the formula given by Dekar et al [31].
This formula is between two finite points. We bring the the point infinity to
unity by using the transformation $t=\frac{y}{y-1}$. Now the point where $y$
and $t$ are equal to zero coincide, and $y$ going to infinity is given by
$t=1$.

The new equation is not of the Heun form. To bring it back to the Heun form,
we multiply the solution $g((t)$ by a power

$g(t)=(t-1)^{A}h(t)$ (42)

and try to use $h(t)$ in our further calculations. Our new dependent variable
$h(t)$ satisfies the equation

$\frac{d^{2}h(t)}{dt^{2}}+\big{(}\frac{A+1-B}{t-1}+\frac{d}{t}+\frac{e}{t-\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})}}\big{)}\frac{d}{dt}h(t)+\big{(}\frac{(A(d+e-B)t+Q_{5}+\Lambda}{t(t-1)(t-\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})})}\big{)}h(t)=0.$
(43)

At this point, we make a change in our notation. We want to use the standard
Heun notation. This means we will redefine our parameters as
$A+1-B=\delta,d=\gamma,d+e-B=\beta$. The other parameters are defined as
$e=\epsilon,A=\alpha$ will remain as they are. Now, we can use the standard
notation, since our equations will be of the Heun form in the further part of
this work, and we will be able to use known relations for this function.

Now, the differential equation is written as

$\frac{d^{2}h(t)}{dt^{2}}+\big{(}\frac{\gamma}{t}+\frac{\delta}{t-1}+\frac{\epsilon}{t-\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})}}\big{)}\frac{d}{dt}h(t)+\big{(}\frac{(\alpha\beta)t+Q_{5}+\Lambda}{t(t-1)(\frac{t-x_{b}(1-x_{a})}{x_{a}(1-x_{b})})}\big{)}h(t)=0.$
(44) $Q_{5}=\frac{(x_{b}-x_{a})(\alpha(\gamma+\epsilon-\beta)-Q)-\gamma(\alpha
x_{b}(1-x_{a})}{x_{a}(1-x_{b})}.$ (45)

From now on, we will rename $h(t)$ as $y_{1}$. Now we have to write $y_{1}$ in
terms of the two linearly independent solutions, which we name $y_{3}(1-t)$
and $y_{4}(1-t)$, the two linearly independent solutions one gets when one
expands around $1-t=v$.

In other words, we want to write

$y_{1}(t)=C_{1}y_{3}(1-t)+C_{2}y_{4}(1-t).$ (46)

The right hand side gives us the two Heun solutions when the same equation is
expanded as a function of $1-t$. This expansion is easily done

$y_{3}=H_{G}\big{(}\frac{(x_{a}-x_{b})}{x_{a}(1-x_{b})},Q_{3}+\Lambda;\alpha,\beta;\delta,\gamma,\epsilon;v=1-t\big{)},$
(47)

where $Q_{3}=Q_{5}-\alpha\beta$, and which is the solution of the equation

$\frac{d^{2}y_{3}(v)}{dv^{2}}+\big{(}\frac{\gamma}{v-1}+\frac{\delta}{v}+\frac{\epsilon}{v-\frac{x_{a}-x_{b}}{x_{a}(1-x_{b})}}\big{)}\frac{d}{dv}y_{3}(v)+\big{(}\frac{(\alpha\beta)v+Q_{3}+\Lambda}{v(v-1)(v-\frac{(x_{a}-x_{b})}{x_{a}(1-x_{b})})}\big{)}y_{3}(v)=0.$
(48)

Then, we make the transformation

$y_{4}(v)=v^{(1-\delta)}j(v),$ (49)

where $j(v)$ satisfies the equation

$\frac{d^{2}j(v)}{dv^{2}}+\big{(}\frac{\gamma}{v-1}+\frac{2-\delta}{v}+\frac{\epsilon}{v-\frac{x_{a}-x_{b}}{x_{a}(1-x_{b})}}\big{)}\frac{d}{dv}j(v)+\big{(}\frac{(\gamma+\epsilon-\alpha)(\gamma+\epsilon-\beta)v+Q_{4}+\Lambda}{v(v-1)(v-\frac{(x_{a}-x_{b})}{x_{a}(1-x_{b})})}\big{)}j(v)=0.$
(50)

to give

$y_{4}=(1-t)^{(1-\delta)}(H_{G}(\frac{x_{a}-x_{b})}{x_{a}(1-x_{b})},Q_{4}+\Lambda;\gamma+\epsilon-\alpha),\gamma+\epsilon-\beta;2-\delta,\gamma,\epsilon;v=1-t),$
(51)

where
$Q_{4}=Q_{3}-(1-\delta)(\gamma\frac{x_{a}-x_{b}}{x_{a}(1-x_{b})}+\epsilon)$.
Note that this last transformation is one of the transformations which does
not change the Heun form of the differential equation. Furthermore,the
relation $1+\alpha+\beta=\gamma+\delta+\epsilon$ is valid in this part of our
calculations.

At this point we use the formula given in [31] for expanding $H_{G}(t)$ in
terms of two linearly independent solutions of the equation written for
$H_{G}(1-t=v)$, our equation (46).

$y_{1}(t)=C_{1}y_{3}(1-t)+C_{2}y_{4}(1-t).$ (52)

We find

$C_{1}=H_{G}(\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})},Q_{5}+\Lambda;\alpha,\beta;\gamma,\epsilon;1),$
(53)
$C_{2}=H_{G}\Big{(}\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})},Q_{5}+(\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})})\gamma(1-\epsilon);\gamma+\epsilon-\alpha,\gamma+\epsilon-\beta;\gamma,\epsilon;1\Big{)}.$
(54)

Note that the factor multiplying $j(1-t)$ in $y_{4}$ can be written as

$\exp[{(2i{\sqrt{-1+\frac{s^{2}}{(1-x_{a})^{2}(1-x_{b})2}}}\ln{1-t}}].$ (55)

Then we can write our eq. (46,52) as

$\displaystyle\exp{-i(V\ln(1-t)})y_{1}(t)$
$\displaystyle=C_{1}{\exp{-i(V\ln(1-t))}y_{3}(1-t)+C_{2}\exp{(iV\ln(1-t)})j(1-t)},$
(56)

where

$V={\sqrt{-1+\frac{s^{2}}{(1-x_{a})^{2}(1-x_{b})^{2}}}}.$ (57)

Then our reflection coefficient is given by

$R=|\frac{C_{2}}{C_{1}}|^{2}.$ (58)

This calculation is a formal one. We have to make sure that the Heun function,
found by expansion around zero, is also finite at the second singular point
for appropriate values of the parameters of our wave equation. A proper Leaver
analysis [32] shows that for $\frac{x_{a}(1-x_{b})}{x_{b}(1-x_{a})}<1$, these
terms are convergent when the independent varüable equals unity. There is a
catch here. Leaver analysis actually gives two roots for convergence for the
Heun function at unity. One is, as stated above,
$\frac{x_{a}(1-x_{b})}{x_{b}(1-x_{a})}<1$. The other root is unity,for which
the convergence when the variable equals unity can not be realized. In writing
our connection formula, we could not control which one of these two roots will
be taken by our solution. Here, we, therefore use a different method to fix
our constants in our connection equation, i.e. writing the solution in terms
of the solutions at unity. This different method was used by [33] and
Motohashi and Noda [34, 35] in their recent papers.

To use this method, we first have to find the second solutions of equation
[44]. We set

$h(t)=t^{(1-\gamma)}n(t)$ (59)

Then the new equation reads

$\frac{d^{2}n(t)}{dt^{2}}+\big{(}\frac{2-\gamma}{t}+\frac{\delta}{t-1}+\frac{\epsilon}{t-\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})}}\big{)}\frac{d}{dt}n(t)+\big{(}\frac{(\alpha+1-\gamma)(\beta+1-\gamma))t+Q_{6}+\Lambda}{t(t-1)(\frac{t-x_{b}(1-x_{a})}{x_{a}(1-x_{b})})}\big{)}n(t)=0.$
(60)
$Q_{6}=Q_{5}+(\gamma-1)(\frac{x_{b}\delta(1-x_{a})}{x_{a}(1-x_{b})}+\epsilon)$
(61)

From now on, we will rename $h(t)$ as $y_{1}$, $t^{1-\gamma}n(t)$ as $y_{2}$.
Now we have to write $y_{1},y_{2}$ in terms of the two linearly independent
solutions, which we name $y_{3}(1-t)$ and $y_{4}(1-t)$, the two linearly
independent solutions one gets when one expands around $1-t=v$. In other
words, we want to write

$y_{1}(t)=C_{1}y_{3}(1-t)+C_{2}y_{4}(1-t).$ (62)
$y_{2}(t)=C_{3}y_{3}(1-t)+C_{4}y_{4}(1-t).$ (63)

Note that we had already calculated $y_{3}$ and $y_{4}$ above. Using the
method given in references [33, 34, 35],

$C_{1}=\frac{W_{t}[y_{1},y_{4}]}{W_{t}[y_{3},y_{4}]},C_{2}=\frac{W_{t}[y_{1},y_{3}]}{W_{t}[y_{4},y_{3}]},C_{3}=\frac{W_{t}[y_{2},y_{4}]}{W_{t}[y_{3},y_{4}]},C_{4}=\frac{W_{t}[y_{2},y_{3}]}{W_{t}[y_{4},y_{3}]}.$
(64)

Here $W_{t}$ is the Wronskian, where the differentiations are performed with
respect to $t$, noting that $v=1-t$.

Note that $W_{t}[y_{3},y_{4}]$ may be calculated analytically and is given by

$W_{t}[y_{3},y_{4}]=(-1)^{(-\gamma-\epsilon-\delta+2)}(t)^{-\gamma}(t-1)^{-\delta+2}(t-\frac{x_{b}(1-x_{a})}{x_{a}(1-x_{b})})^{-\epsilon}.$
(65)

We still have to calculate the other four wronskians given in eq.s (64). It
takes a while to calculate these terms. At the mean time, we did not want to
leave this paper in the former form. We will give another revision to this
paper when this calculation is performed.

##  5 Conclusion

Here we put the wave equation for a zero mass scalar particle coupled to the
Wahlquist metric to a form to give the Heun solutions in the standard notation
and calculated the reflection coefficient for a particle coming from infinity
at the origin.

Note that this calculation has weak points. We made an approximation by taking
$lnx\approx x-1$. This is valid only when $x$ is close to unity. We made few
transformations on the independent variable. We can not guarantee that this
approximation is valid in the range we work. We have no choice, though, since
we can not solve differential equations whose coefficients are transcendental
functions.

An interesting point is that we can satisfy an important algebraic constraint,
given in [21, 20], in reducing the first equation to the standard form only if
we make this approximation after we calculate $\frac{dU}{dx}$ exactly, and
apply this approximation only after the differentiation is done.

##  6 Acknowledgement

This work was written to celebrate the 70th birthday of my dear colleague
Prof. Tekin Dereli, whom I met close to fifty years ago, in 1972, when he was
still studying for his doctoral degree and I was a fresh Ph.D. while I was
visiting M.E.T.U.in Ankara. We are close friends ever since.

In the preparation of this work, I am grateful to Prof. Nadir Ghazanfari for
giving us the key paper by Dekar et. al. I am also grateful to Prof. Barış
Yapışkan for important discussions and correcting a vital mistake in my
calculations. I thank to Proıf. Reyhan Kaya and Dr. H.T. Özçelik for doing
some numerical work. I also thank Dr. Oğuzhan Kaşıkçı for technical
assistance.

This work is morally supported by Science Academy, Istanbul, Turkey.

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