# Social Media and Artificial Intelligence for Sustainable Cities and
Societies: A Water Quality Analysis Use-case

Muhammad Asif Auyb  Muhammad Tayyab Zamir  Imran Khan  Hannia Naseem  Nasir
Ahmad  Kashif Ahmad  Department of Computer Systems Engineering, University of
Engineering and Technology, Peshawar, Pakistan.  Abbasyn University,
Islamabad.  University of Western Australia, Australia.  Department of
Computer Science, Munster Technological University, Cork, Ireland.
<EMAIL_ADDRESS>
###### Abstract

This paper focuses on a very important societal challenge of water quality
analysis. Being one of the key factors in the economic and social development
of society, the provision of water and ensuring its quality has always
remained one of the top priorities of public authorities. To ensure the
quality of water, different methods for monitoring and assessing the water
networks, such as offline and online surveys, are used. However, these surveys
have several limitations, such as the limited number of participants and low
frequency due to the labor involved in conducting such surveys. In this paper,
we propose a Natural Language Processing (NLP) framework to automatically
collect and analyze water-related posts from social media for data-driven
decisions. The proposed framework is composed of two components, namely (i)
text classification, and (ii) topic modeling. For text classification, we
propose a merit-fusion-based framework incorporating several Large Language
Models (LLMs) where different weight selection and optimization methods are
employed to assign weights to the LLMs. In topic modeling, we employed the
BERTopic library to discover the hidden topic patterns in the water-related
tweets. We also analyzed relevant tweets originating from different regions
and countries to explore global, regional, and country-specific issues and
water-related concerns. We also collected and manually annotated a large-scale
dataset, which is expected to facilitate future research on the topic.

###### keywords:

Smart Cities, Sustainable Cities , AI, NLP , Water Quality , Topic Modelling ,
Named Entity Recognition , Text Classification, LLMs,

††journal: Journal of LaTeX Templates

##  1 Introduction

Real-time monitoring and observation of resources and infrastructure is a
primary task towards a resilient infrastructure and sustainable cities [1].
This allows for taking appropriate recovery actions for the mitigation of
risks and damages. Crowd-sourcing is one of the effective ways used for real-
time monitoring and feedback on infrastructure [2]. One of the key methods,
widely explored in the literature, for crowdsourcing is conducting surveys to
obtain citizens’ feedback on different services, such as water quality, air
quality, roads, infrastructure, and other societal challenges. These surveys
can help in obtaining more detailed, contextual, and localized information
[3]. These surveys are either conducted by asking citizens to fill in an
online form or a questionnaire. More recently, mobile applications have also
been developed for conducting such surveys, where the participants were asked
to install and give feedback. However, these online and in-person surveys have
several limitations [4]. One of the key limitations of such surveys is the
limited scope, which means they can cover a limited number of people as people
are often reluctant to install such applications as we noticed during the
COVID-19 pandemic [5]. Moreover, it takes a lot of time to complete a survey
and also needs to involve several human and other resources, thus, the
frequency of such surveys is generally very low as it is costly and unfeasible
to frequently conduct such surveys.

These limitations of the current crowd-sourcing methods could be overcome by
extracting information from social media outlets, such as Twitter and
Facebook. Social media outlets have already been proven to be an effective
source of communication and information spreading [6, 7]. Their capabilities
to engage large volumes of audiences worldwide make them a preferred platform
for discussing and conveying concerns over different domestic and global
challenges. The literature already reports their effectiveness in a
diversified set of societal, environmental, and technological topics [8, 9].

In this work, we explore the potential of social media as a crowd-sourcing
source/medium of instant feedback on water quality. To this aim, an automatic
solution is proposed that is able to collect and analyze citizens’ feedback on
water quality. The proposed system will not only engage a large number of
participants, which is a key factor in meaningful feedback, but it will also
be a continuous process and will keep collecting people’s feedback
continuously. One of the key advantages of the system is collecting and
analyzing feedback without asking the citizens to fill in any online
form/survey rather it will keep filtering and analyzing relevant social media
posts in a privacy-preserving manner.

The proposed system is composed of (i) a crawler, which is responsible for
collecting social media posts (Tweets), (ii) a classification framework
employing several NLP algorithms in a merit-based fusion to differentiate
between water-related and irrelevant tweets, and (iii) topical analysis to
automatically analyze and extract key water-related issues discussed in the
tweets. For the training and evaluation of the text classification framework,
we also collected and annotated a large-scale benchmark dataset, which will be
made publicly available for further research in the domain.

The key contributions of the work can be summarized as follows:

  * 1.

We propose an automatic tool to collect, analyze, and extract meaningful
information from social media posts as a source of instant feedback on water
quality, as a first step towards a sustainable water network.

  * 2.

We propose a merit-based fusion framework by combining several transformers-
based NLP algorithms to differentiate between water-related and irrelevant
tweets.

  * 3.

We also collected and annotated a large-scale benchmark dataset containing
around 8,000 tweets.

  * 4.

We also perform topic modeling on the relevant tweets to automatically extract
key water-related issues discussed in the relevant tweets.

  * 5.

We also analyze the origin of the water-related tweets and provide region and
country-wise distribution of the water-related tweets collected by our system.
This analysis shows the growing concern over this important societal
challenge.

The rest of the paper is organized as follows. Section 2 provides an overview
of the related work. Section 3 discusses the proposed methodology. Section 4
covers the experimental setup, conducted experiments, and experimental
results. Finally, Section 5 concludes the paper.

##  2 Related Work

The literature already reports several interesting crowdsourcing-based
solutions, which are mostly based on offline or online surveys for
infrastructure monitoring and feedback on different public services [10]. The
majority of the recent solutions rely on smartphones and other handheld
devices by developing smart applications allowing users to give feedback on
the infrastructure and services. For instance, Rapousis et al. [11] proposed
QoWater, a client-to-server architecture-based mobile application allowing
mobile users to give feedback on water quality. Similarly, Santani et al. [12]
proposed CommuniSense, a mobile phone application for crowdsourcing to monitor
road conditions in Nairobi. However, several challenges are associated with
such applications for crowdsourcing [13]. One of the key limitations of such
surveys is the limited scope, which means they can cover a limited number of
people as, generally, people are found reluctant to install and use such
mobile applications. A prime example of people’s reluctance to such mobile
applications is observed during COVID-19 when people showed concerns over such
applications in terms of privacy, difficulty in usage, and battery consumption
[5]. Moreover, it takes a lot of time to complete a survey and also needs to
involve several human and other resources, thus, the frequency of such surveys
is generally very low as it is costly and unfeasible to frequently conduct
such surveys. These challenges could be overcome by extracting people’s
feedback on infrastructure and public services. The literature already
provides some hints on the effectiveness of social media for real-time
monitoring and instant feedback on different services. For instance, Want et
al. [14] explore the potential of social media as a source of feedback on
government services by analyzing citizens’ opinions in a social media text.

Water quality analysis is one of the key applications that recently got the
attention of the community. To this aim, several interesting frameworks have
been introduced. The majority of the existing works aim at sentiment analysis
of social media posts to extract people’s opinions on water quality. For
instance, Lambert [15] proposed a sentiment analysis framework for analyzing
users’ feedback and perception of tap water quality. Similarly, Li et al. [16]
performed sentiment analysis on social media posts about recycled water in
China. Jiang et al. [17], on the other hand, analyzed the public’s opinion on
large hydro projects by performing sentiment analysis on relevant social media
posts. To this aim, three different hydro projects in China are considered,
and mixed opinions were noticed for the projects.

More recently, water quality analysis from social media posts has also been
introduced in MediaEval 2021 [18]. The task involved the retrieval of relevant
multimedia content describing water quality in an Italian region. A couple of
interesting solutions, incorporating different types of available information,
are proposed in response to the task. For instance, Hanif et al. [19] fine-
tuned exiting pre-trained deep-learning models namely VGGNet and BERT for
retrieving relevant visual and textual content, respectively. Overall better
results are reported for textual content. Ayub et al. [20] rather focused on
textual content by employing three different NNs models including BERT,
RoBERTa, and a custom LSTM both individually and jointly in a naive late
fusion scheme.

Despite the initial efforts in the domain, several interesting aspects of
water quality analysis and automatic analysis of people’s feedback on public
services and infrastructure, in general, are unexplored. For instance, the
majority of the initial efforts are based on sentiment analysis without
extracting meaningful information from the content itself. The domain also
lacks a large-scale benchmark dataset. To this aim, in this work, we collect
and annotate a large-scale benchmark dataset on water quality analysis. We
also extend the text classification framework with topic modeling to
automatically extract key water-related issues discussed in social media.

##  3 Methodology

Figure 1 provides the block diagram of the proposed system. As can be seen,
the proposed system is composed of five steps. In the first step, a large
number of Tweets have been collected. In the next step, these tweets are
annotated in a crowd-sourcing study. The annotated dataset is then used to
train/finetune Large Language Models (LLMs) for the classification of tweets
into relevant and non-relevant tweets. In the fourth step, several merit-based
fusion techniques are used to combine the classification scores obtained with
the individual models. In the final step, topic modeling techniques are used
to identify topics in the relevant tweets. In the next subsections, we provide
a detailed description of each step.

Figure 1: A block diagram of the proposed methodology.

###  3.1 Data Collection, Cleaning, and Annotation

For data collection, we developed a crawler able to continuously collect data
from different outlets of social media. As proof of concept, in the current
implementation, data is collected from Twitter only. To this aim, we used a
Python package namely Tweepy111https://www.tweepy.org/ with different relevant
keywords, such as waterpollution, water, watercrisis, watersmell,
drinkingwater, watercolour, cleanwater, waterquality, plasticpollution,
drinkingwater, watercrisis, savewater, waterislife, cleantheocean,
plasticocean, endplasticpollution. The list of keywords is prepared in a data-
driven manner by picking the keywords used in social media posts, blogs,
newspapers etc., . We tried to include as many as possible keywords to the
list to collect relevant and quality tweets. This resulted in a large
collection of tweets, which were saved in a CSV file. After data collection,
all the collected tweets are manually annotated by involving multiple
volunteers in a crowd-sourcing activity. Before the annotation, the collected
data is manually checked to remove less informative tweets. For example, we
removed very short tweets without sufficient text or containing tags only. We
also removed duplicate entries in the file.

During the crowd-sourced activity, we manually analyzed a total of 8,000
tweets, which are annotated as relevant or non-relevant. To ensure, the
quality of the annotated data, each sample is checked by three different
annotators and is labeled based on the majority votes. The participants of the
crowd-sourcing activity are postgraduate students with sufficient knowledge of
the domain.

###  3.2 Text Classification

For text classification, we employed several LLMs both individually and
jointly in a merit-based fusion technique to differentiate between relevant
and non-relevant tweets. In the next subsections, we provide a detailed
description of the classification and fusion process.

####  3.2.1 Classification Via Individual Models

In this work, we mainly rely on state-of-the-art transformer-based NLP models
for the classification of tweets. In total, six different models are used.
These models include the original BERT model, RoBERTa, ALBERT, DistilBERT,
GPT, and Meta-LLAMA. The selection of these models is motivated by their
proven performances in similar tasks, and we believe the evaluation of these
models will provide a baseline for future work in the domain. A brief overview
of these models is provided below.

  * 1.

BERT: It is one of the state-of-the-art NLP algorithms that have been widely
used for a diversified list of NLP applications. Its ability to read/learn in
both directions makes it a preferred choice in different text-processing
applications. Several implementations of BERT are available. In this work, we
used Tensorflow implementation. The model is composed of 12 layers and
attention heads, and 110 million parameters. Our loss function is based on the
Binary Cross entropy loss function while the Adaptive Moments (Adam) optimizer
is used in the experiments.

  * 2.

RoBERTa: RoBERTa is another state-of-the-art transformer-based NLP model, and
it uses self-attention for processing and generating contextualized
representations of input text. One of the key advantages of RoBERTa over BERT
is its training on a larger dataset and the use of a dynamic masking technique
allowing the model to learn robust and generalizable representations of words.
In this work, we fine-tuned the model on our dataset by using the Adam
optimizer with a binary cross-entropy loss function.

  * 3.

ALBERT: It is a modified version of BERT with fewer memory requirements.
ALBERT has a reduced number of parameters mainly due to factorized embedding
parameterization and cross-layer parameter sharing. In this first technique,
the large vocabulary embedding matrix is decomposed into two small matrices,
separating the size of the hidden layers from the size of the vocabulary
embedding. The cross-layer parameter sharing, on the other hand, prevents an
increase in the number of parameters with the depth of the model.

  * 4.

DistilBERT: DistilBERT is another variant of the BERT model aiming at
applications with less computational and memory requirements. The concept of
knowledge distillation is adopted during pre-training allowing a significant
reduction in parameters without a significant impact on the performance of the
model.

  * 5.

GPT: Generative Pre-trained transformer (GPT) models represent a family of
Neural Network (NNs)-based language prediction models built on the Transformer
architecture [21]. These models are pre-trained on a huge volume of diverse
text data. Currently, GPT is available in different versions. However, the
first version of the model was introduced in 2018 by Open AI [21]. In this
work, we used GPT version 3.5 turbo. It is composed of 175 billion parameters,
which is significantly higher than the number of parameters used in its
previous versions and other transformers, such as BERT. In this work, we used
prompt engineering for the classification of tweets through GPT 3.5.

  * 6.

Meta-LLAMA: Large Language Model Meta AI (LLMA) is also a family of pre-
trained LLMs. Similar to GPT, multiple versions of LLAMA are available having
7B to 70B parameters. In this work, we used LLAMA 2, which is an improved
version of the base model LLAMA. Similar to the base model, LLAMA 2 is built
on the Google transformer architecture with several interesting changes and
improvements. For example, the RMSNorm pre-normalization, a SwiGLU activation
function, and multi-query attention instead of multi-head attention and AdamW
optimizer. The key differences between LLAMA 2 and the original LLAMA include
a higher context length (i.e., 4096 compared to 2048 tokens) and grouped-query
attention instead of multi-query attention. Similar to GPT 3.5, we used the
prompt engineering method for text classification with LLAMA.

####  3.2.2 Fusion of the Models

Our fusion methods are based on a late fusion scheme, where the
scores/posterior probabilities of the individual models are accumulated for
the final decision using equation 1. In the equation,
$S_{m1},S_{m2},S_{m3},...S_{mn}$ represent the scores/posterior probabilities
obtained through the $1$st, $2$nd, , $3$rd, and $n$th model, respectively
while $W_{1},W_{2},W_{3},...W_{n}$ are the corresponding weights assigned to
these models.

$S_{f}=W_{1}S_{m1}+W_{2}S_{m2}+W_{3}S_{m3}+....+W_{n}S_{mn}$ (1)

The weights are assigned to the models on the basis of their performances. To
this aim, several weight optimization/selection methods, including PSO, Nelder
Mead, BFGS, and Powell method, are employed. These methods seek a set of
variable values (i.e., $W_{1},W_{2},W_{3},...W_{n}$ in our case) optimizing an
objective function under a set of constraints. In this case, the
fitness/objective function is based on accumulative classification error
obtained on a validation set using equation 2. In the equation, $A_{acc}$
represents the accumulative accuracy computed on the validation set. In this
work, our goal is to find a set of weights to be assigned to the models that
minimize the classification error.

$e=1-A_{acc}$ (2)

We note that the same fitness function is used by all the weight optimization
methods employed in this work. These methods use different mechanisms and have
their own pros and cons. A brief overview of each method is provided below.

  * 1.

PSO: Particle Swarm Optimization (PSO), which is a heuristic approach, has
been widely used in the literature for different tasks. For instance, in
several works, PSO has been used for the optimization of hyper-parameters of
ML algorithms, such as the number of layers, batch size, number of neurons,
etc., in LSTMs and CNNs [22, 23]. Similarly, it has been also used for the
hyper-parameter optimization of Federated Learning (FL) algorithms [24]. The
literature also reports the effectiveness of the optimization technique in
late fusion where the algorithm is used to assign optimal weights to the
classifiers [25, 26]. The algorithm solves the optimization problem in three
steps, iteratively; starting from a random set of candidate solutions, where
each candidate solution is called a particle. At each iteration, each particle
keeps track of its personal and global best solution in the swarm. The
particles adjust two parameters namely (i) velocity and (ii) the position. The
velocity of a particle is adjusted based on its own experience and the
information shared by the other particles in the swarm. The position of
particles is adjusted based on their current position, velocity, and distances
between their current positions and personal and global best. The process
continues until a global optimum is obtained. The key limitations of the
method include a slow convergence rate, especially in high dimensional
problems, and entrapment in local minima. Being one of the key optimization
algorithms, PSO implementation is available in several libraries. In this
work, we used the open-source library namely
pyswarm222https://pyswarms.readthedocs.io/en/latest/ for the implementation of
the algorithm.

  * 2.

Nelder Mead Method: Similar to PSO, the Nelder Mead method has also been
widely explored for different optimization tasks. For instance, Takenaga et
al. [27] employed the method for computationally expensive optimization
problems. Similarly, Ozaki et al. [28] used the algorithm for the hyper-
parameter selection/optimization of a CNN model. The method has also been
widely used for the fusion of classification algorithms in different visual
and NLP applications [29, 8]. The method optimizes a set of variables leading
to a minimum or maximum value of an objective function in a multidimensional
space. To this aim, it uses a set of $n+1$ test points (solutions), which are
arranged as a simplex. The method then estimates the behavior of the objective
function at each test point for new test points, which replace the old ones in
an iterative manner. In this work, we used a Python open-source library,
namely, SciPy333https://scipy.org/ for the implementation of the method.

  * 3.

Limited-memory Broyden Fletcher Goldfarb Shanno Algorithm (BFGS): Similar to
PSO and Neldar Mead, BFGS and its variants have been proven very effective in
different tasks, such as optimization hyper-parameters of deep learning models
and fusion. For instance, Saputro et al. [30] employed the algorithm for
parameter estimation on a geographically weighted ordinal logistic regression
model. Maria et al. [31] employed the method along with other optimization
techniques for the fusion of inducers’ scores for media interestingness
prediction. BFGS, which is a local search optimization algorithm, belongs to
the Quasi-Newton optimization family and aims at the optimization of the
second-order derivative of the objective function. To obtain a set of optimal
values, the algorithm computes the inverse of the Hessian matrix used for
multivariate functions. To this aim, the algorithm approximates the inverse
using a gradient that eliminates the need for inverse calculation at each
step. One of the key limitations of the algorithm is its large memory
requirement, and it becomes impractical to compute the inverse of the Hessian
matrix with a larger number of input parameters. To overcome this limitation,
several variations of the algorithms have been proposed. For instance, Limited
BFGS/LBFGS [32] is one of the variants of the algorithms with fewer memory
requirements. In this work, though we don’t have a large number of inputs, we
used the LBFGS implementation of the method.

  * 4.

Powell Method: Powell method is another interesting optimization method that
has been widely used for similar tasks. For instance, Maria et al. [31] and
[8] employed the method for merit-based late fusion of classifiers for media
interestingness and water quality analysis, respectively. Similar to PSO,
several variations of the algorithm have been proposed in the literature. The
algorithm seeks the local minima of the objective function. The objective
function, which is a real-valid function with multiple inputs, doesn’t need to
be differentiable. The algorithm finds the minima in several steps starting
with a random selection and evaluation of initial points/solutions. A list of
parameters is then randomly selected. A subset of the initial points with
minimum error is then selected as parents to produce children for the next
step for the next generation. The children/new points are then evaluated in
the fifth step and the process is repeated again from the third step until a
global minima is found.

###  3.3 Regions Extraction

In this phase, we define different regions based on the locations associated
with the tweets. This allows us to analyze the water quality or water-related
issues in different regions of the world as each region may have specific
issues. We note that this step is added to facilitate in region-wise topic
modeling, where we aim to extract keywords used in water-related tweets from
different parts of the world. To this aim, the location addresses associated
with each tweet are fed into Chat GPT to identify the corresponding countries
by mapping the addresses to the respective countries. To ensure the quality of
the mapping, the identified countries and the associated addresses are
meticulously verified. To further enhance the accuracy of the data, we applied
filtering techniques to specific locations. For example, in cases where the
user’s location included the address ’Florida, FL,’ we replaced it with ’USA’.
This replacement was applied wherever the specified keyword was encountered.
As a result, we successfully extracted and verified 4707 accurate locations.
The countries list is then provided to Chat GPT to expand the geographical
scope by translating the unique countries into regions using Chat GPT.

###  3.4 Topic Modeling

The final component of the methodology is based on BERTopic [33], which is a
state-of-the-art topic modeling technique. One of the key advantages of topic
modeling is its ability to quickly discover the hidden topical patterns
present in the data. These hidden patterns could result in meaningful insights
leading to useful data-driven decisions. In this work, we aim to automatically
extract the hidden topical patterns in the water-related tweets to identify
the key water-related issues and concerns expressed over the water quality in
the tweets.

The algorithm used in this work extracts topics from Tweets in three different
steps starting from converting the tweets into embeddings, then reducing the
dimensionality and clustering, and finally converting them into topics. The
embeddings are obtained by a pre-trained model namely Sentence-BERT. The
dimensionality reduction and clustering are carried out through Uniform
Manifold Approximation and Projection(UMAP) and HDBSCAN (Hierarchical DBSCAN),
respectively. Finally, topics are extracted from the clustering using a
modified form of TF-IDF (Term Frequency-Inverse Document Frequency) namely
c-TF-IDF.

The algorithm brings several advantages. For instance, it clusters documents
based on both lexical and semantic similarities. Moreover, BERTopic provides a
library with several packages allowing more accurate and better visualization
of the clusters, topics, and probabilities. It also comes with a few
limitations. For instance, its assumption that each document/tweet contains
only one topic is its main limitation, though it is possible to have Tweets
with multiple topics. We note that we also performed some pre-processing in
addition to the data cleaning before topic modeling. For instance, we removed
short and stop words, numbers, and alphanumeric characters. This allows us to
remove irrelevant frequently used words.

##  4 Experiments and Results

###  4.1 Dataset

Our final dataset, after removing less informative tweets during the manual
analysis and annotation, contains a total of 7,930 tweets. Among these, 5,728
tweets are annotated as irrelevant while the remaining 2,202 tweets were
classified as relevant. The dataset has been divided into three subsets namely
(i) training, (ii) test, and (iii) validation set using a ratio of 70%, 20%,
and 10%, respectively. The validation set is used for the computation of the
classification error for the fitness function of the fusion methods. Table 1
provides some sample relevant and irrelevant tweets from the dataset.

Table 1: Sample Tweets from the dataset. Relevant Samples |  Irrelevant Samples
---|---
We have been receiving water of the worst quality from past 6 months. I want to bring this situation to your notice and solve this problem ASAP. Water is a basic need. Area : Adarsh Nagar, Bahadurgarh |  One of the most popular urban beaches in Gran Canaria, Las Canteras is a two-kilometer ribbon of sand caressed by warm and calm water.
Drinking contaminated water can transmit diseases and back in 2017 nearly 1.6 million people died from diarrheal diseases. 1/3 of those were children under the age of 5. #climatecrisis #water |  Wondering about #books about #water sports (canoeing, sailing, yachting, scuba diving, etc.)? Check out call number range
The landmark research blames chemical #pollution from plastics, farm fertilisers and pharmaceuticals in the #water. Previously, it was thought the amount of #plankton had halved since the 1940s, but the #evidence gathered by the Scots suggests 90% has now vanished. |  Hope people leave water out in their gardens or balcony in any containers for all the beautiful wildlife x #water #wildlife #thirsty #animals x
In face of recurring drought, cities seek security in wastewater recycling projects #security #projects #recycling #wastewater #water Removing pollution from water using water shaping tech #sketchup #depollution #watershaping #waterpollution |  In a larger portion of cases, #carpet #damage is treated efficiently and all the defects are repaired. Professional services take care of all the #Water #Damage #Restoration Sunshine Coast.
The privatisation of water and power has been one of the biggest rip-offs of the British public in modern times. Time to jail those profiteering through pollution of our rivers and waterways! #water #corporategreed #utilities |  The theme this time is ”Water from Japanese restaurants”. Is it true that there are many paid shops outside Japan? The popular article has exceeded 650pv Is water free at Japanese restaurants?
Life without water is impossible. Save water. Save life. With every little drop, a day less to live on Earth. |  Your body depends on #water to survive. Every cell, tissue, and organ in your body needs water to work properly and for overall good health. Learn how to ensure you stay hydrated, and why it is important to do so, here in familydoctor
Drinking contaminated #water can be harmful to one’s health. #Cholera, #diarrhea, #dysentery, and #typhoid are just a few of the ailments it can induce. |  We’ve worked with TheMixUK to explain what support is available for those struggling to pay for the increasing price of #Fuel and #Water bills. Take a read of the article here
Clean Water is a necessity to daily life. Empower economically disadvantaged small communities to develop and sustain clean water supplies. |  Visit Central Florida Water Ski Sweepstakes

###  4.2 Experimental Results

The objectives of this work are multi-fold. On one side, we aim to extract
flood-related tweets, and on the other hand, we want to automatically extract
keywords from the relevant tweets. It is very possible that each
country/region may have different water-related issues than others, thus, we
are also interested in keywords/topics of tweets tweeted from a specific
country/region. To achieve these objectives, we perform the following
experiments.

  * 1.

Evaluation of the performance of several state-of-the-art LLMs individually.

  * 2.

Fusion of the classification scores obtained through the individual models in
a merit-based fusion framework by employing several weight
selection/optimization methods.

  * 3.

Topic modeling on all the relevant tweets. This will allow us to highlight key
global water-related issues.

  * 4.

Topic modeling of the collection of tweets tweeted from a specific
country/region. This will allow us to highlight the water-related issues
specific to a particular region.

In the next subsections, we provide a detailed analysis of the results of all
the experiments.

####  4.2.1 Text Classification Results

Table 2 provides the results of our first experiments, where we evaluate the
performance of several LLMs in the application. We note that for GPT and
LLAMA-2 we use the prompt engineering method with a few-shot (5-shot and
10-shot) classification setting without fine-tuning the models. As can be
seen, overall similar results are obtained for BERT and its different variants
and XLNET. However, the lowest results are observed with Meta-LLAMA-2. One of
the potential reasons for the lowest performance of the model is the few-shot
learning as the model may have limited generalization to classify the samples
from the seen examples.

Table 2: Experimental results of the individual LLMs. LLM | F1-score
---|---
BERT | 0.7686
ALBERT | 0.7636
DistilBERT | 0.7491
ROBERTA | 0.7541
XLNET | 0.76
Gpt-3.5 (5-shot) | 0.7146
Gpt-3.5 (10-shot) | 0.7246
Meta-LLAMA2 (5-shot) | 0.5832
Meta-LLAMA2 (10-shot) | 0.5876

Table 3 reports the results of our fusion experiment, where combine the
classification scores of the best-performing individual models in a merit-
based fusion scheme. In this experiment, we considered two experimental
settings. In the first case, we combined the classification scores obtained
with the top 5 performing models including BERT, RoBERTa, DistilBERT, ALBERT,
and XLNET while in the second experiment, we considered the top 2 models
namely BERT and ALBERT.

Overall there is a slight improvement in the results of the fusion compared to
the best-performing individual model. Generally, the fusion results in an
improvement in the F1 score, however, the less improvement in this case could
be the complexity of the dataset or the fewer variations in the individual
models’ results. As far as the comparison of the fusion methods is concerned,
no significant differences have been observed. However, the performance of all
the methods is higher when the top 2 best-performing models are used in the
fusion compared to the top 5 models. In the case of top models, though there
is no significant difference, the slightly lower performance could be due to
the low-performing models that could adversely affect the performance of the
fusion methods.

Table 3: Evaluation of the fusion methods. Fusion Method | F1-score
---|---
Top 2 (BERT and ALBERT) | Top 5
Simple Averaging | 0.770 | 0.7630
PSO | 0.772 | 0.770
Nelder Mead Method | 0.776 | 0.771
Powell Method | 0.7713 | 0.7680
BFGS | 0.77 | 0.7687

####  4.2.2 Location Extraction and Topic Modeling Analysis

In the topic modeling, we conducted two different experiments. In the first
case, we analyzed and tried to discover hidden topical patterns in the
complete collection of water-related tweets in the test set. Figure 2 provides
the top 10 topics and the corresponding words extracted from the collection of
relevant tweets through BERTopic. As can be seen, most of the topics and
associated words are very relevant. The issues highlighted by the algorithm
from the tweet collection include sanitation & access to water, plastic
pollution in water reservoirs, saving and utilization of rainwater, irrigation
& drought issues, environmental factors, filtering drinking water, heatwaves &
heatwave, chemicals and tap water, etc.,

Figure 2: Top 10 hidden topic patterns extracted from the complete collection
of relevant tweets.

In the second experiment, the collected relevant tweets were divided into
regions, which allowed us to discover topics in tweets relevant to or tweeted
from certain regions. This experiment helps to discover people’s concerns
about this important topic of water-related issues including both local and
global issues. As a first step, we extracted country names from the addresses
associated with relevant tweets using ChatGPT. This resulted in a long list of
countries from where water-related tweets were tweeted. We observed that very
few tweets were recorded from certain countries. For example, our collection
of relevant tweets contains a single tweet from Slovenia, Mozambique, El
Salvador, and Grenada. To ensure a sufficient number of tweets from each
country, we considered only those countries from which at least 70 tweets were
tweeted. We note that there is no scientific reason behind this threshold
(i.e., min 70 tweets per country), we simply wanted to make sure a sufficient
number of countries in our list at the same time to ensure a sufficient number
of tweets for our analysis from each country. Figure 3 provides the country-
wise distribution of the relevant tweets in our dataset. A large portion of
the tweets originated from the United States and the United Kingdom. This also
indicates the interest of the people from these countries in this important
societal challenge.

Figure 3: Country-wise distribution of the origin of the water-related
tweets..

Figure 4 provides the list of topics extracted from the tweets originating
from different countries. Topic 0 to Topic 6 show the group of topics
extracted from the tweet collections for Australia, Canada, India, Pakistan,
South Africa, the United States, and the United Kingdom, respectively. Some of
the topics and the associated words are less relevant compared to the others.
For example, most of the words associated with topic 0, which is extracted
from tweets originating from Australia, are not very relevant to water-related
issues. However, on the other side, Topics 2 to Topic 6 are very relevant and
helpful in highlighting the issues. For instance, Topic 2 stresses the careful
usage of water in general and rainwater in particular. Topic 3 and Topic are
about drinking water in one of the provinces of Pakistan and South Africa,
respectively. Similarly, Topic 5 is based on heatwaves, wildlife, and water
pollution. Finally, Topic 6 also includes relevant keywords, such as clean
water and droughts.

Figure 4: Country-wise topic modeling of the relevant tweets. Topic 0 to topic
6 represents topics extracted from tweets originating from Australia, Canada,
India, Pakistan, South Africa, the United States, and the United Kingdom,
respectively.

We also performed topic modeling on different geographic regions by combining
tweets from all the countries in the region. These regions are formed on the
basis of the geographic locations of the countries. To this aim, the list of
countries is provided to Chat GPT resulting in five regions including Asia,
Africa, America, Oceania, and Europe. Similar to country-wise topic modeling,
we included the regions having at least 70 tweets. Figure 5 provides the
distribution of relevant tweets from each region. As can be seen in the
figure, overall, a higher number of tweets originated from America, Europe,
and Asia.

Figure 5: A region-wise distribution of the origin of the water-related
tweets. Figure 6: Region-wise topic modeling of the relevant tweets. Topic 0
to topic 4 represent topics extracted from tweets originating from Africa,
America, Asia, Europe, and Oceania, respectively.

Figure 6 provides the summary of the topics extracted from the tweets
originating from different regions. Topic 0 to topic 4 represent topics
extracted from tweets originating from Africa, America, Asia, Europe, and
Oceania, respectively. The majority of the topics and the associated keywords
are very relevant to water quality except the topic extracted from the Oceania
region. The topics are similar to what has been observed in the country-wise
topic modeling, which indicates that the regions mostly have similar types of
water-related issues or at least the topics/concerns are similar.

##  5 Conclusion

In this paper, we presented our solution to automatic water quality analysis
and identification of hidden topic patterns in water quality-related tweets.
We also analyzed the tweets originating from different countries and regions
allowing global and regional water-related issues and concerns. In the text
classification part, we employed several LLMs both individually and jointly in
a merit-based fusion framework. No significant difference has been observed in
the performance of the individual models. However, Overall, better results are
obtained with the models are jointly utilized. The topic modeling techniques
also efficiently extracted relevant topics from the complete collection of
tweets as well as tweets originating from specific regions and countries.
Moreover, we observed a large portion of the relevant tweets are from certain
countries, such as the United States and the United Kingdom.

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#  Sourcing the Kerr geometry

Ram Brustein(1), A.J.M. Medved(2,3)
${}^{\textrm{\normalsize(1)\ Department of Physics, Ben-Gurion University,
Beer-Sheva 84105, Israel}}$ ${}^{\textrm{\normalsize(2)\ Department of Physics
\& Electronics, Rhodes University, Grahamstown 6140, South Africa}}$
${}^{\textrm{\normalsize(3)\ National Institute for Theoretical Physics
(NITheP), Western Cape 7602, South Africa}}$<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Kerr metric is a vacuum solution of the Einstein equations outside of a
rotating black hole, but what interior matter is actually rotating and
sourcing the Kerr geometry? Here, we describe a rotating exotic matter which
can source the Kerr geometry for the entire acceptable range of its spin
parameter and be shown to saturate the radial null-energy condition at every
point in the interior, while being free of any obvious pathologies. We do so
by introducing the rotating frozen star, whose compactness is controlled by a
perturbative parameter and whose outer surface can be arbitrarily close to the
horizon of a Kerr black hole. The interior geometry modifies Kerr’s such that
there is neither an inner ergosphere nor an inner horizon, and the metric and
Einstein tensors are regular everywhere except for a mild, removable
singularity at the center of the star. The geometry of each radial slice of
the interior is a nearly null surface with the same geometry, but different
radial size, as that of the would-be horizon on the outermost slice. Moreover,
the metric limits to that of the static frozen star as the spin is taken to
zero. The integral of the energy density leads to a rest mass that is equal to
the irreducible mass of a Kerr black hole, and the integral of the angular-
momentum density confirms that the ratio of the angular momentum to the mass
is equal to the Kerr spin parameter. Including the rotational energy in the
standard way, we then obtain the total gravitational mass and angular momentum
of a Kerr black hole with the same mass and spin parameters. We propose that
the rotating frozen star provides a classical description of the highly
quantum final state of gravitationally collapsed matter, which features a
maximally entropic, stringy fluid as described by our polymer model for the
black hole interior.

##  1 Introduction

Most astrophysical black holes (BHs) are rapidly rotating objects, whose
exterior geometry is believed to be described by the Kerr solution. But what
is the interior matter that is actually rotating and sourcing the Kerr
geometry?

This question has attracted much attention over the years, starting at the
time that the Kerr solution was found [1] up to the present day. However, many
of the numerous attempts at modeling a regularized mimicker as an alternative
to a general-relativistic BH discuss the static case (see [2] for a
comprehensive catalogue). Attempts at including rotation are, in many
instances, limited to slowly rotating versions of already-existent static
models or put some limitations on the possibility of finding rotating
mimickers that can maintain regularity. Other models allow some pathologies,
such as violations of the averaged null-energy condition or strong
instabilities, with the hope that such ailments will be cured by quantum
effects. A review of the earliest attempts can be found in [3] and a more up-
to-date review can be found in [4]. A partial list of relevant discussions can
be found in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27].

Here, we propose that a possible answer to this question is a rotating frozen
star. This is an “upgrade” of our proposed model for a BH mimicker [28, 29],
recently dubbed as the frozen star [30] (also see [31, 32, 33]), although
there is no claim of originality in the use of this name [34]. This model is
meant to represent what the interior, classical geometry of a regular, non-
rotating BH might look like. Moreover, it is meant to provide an effective
geometric description of the equilibrium state of a collection of highly
excited, long, closed, fundamental strings. Meanwhile, our original proposal
for describing the quantum state of such strings [35] — the polymer model —
has been shown to exhibit all of the classical [36] and quantum [37]
properties of an unperturbed Schwarzschild BH. What would distinguish the
polymer from an actual BH is its out-of-equilibrium behavior, as this cannot
be explained without some knowledge of the object’s fundamental composition
[38, 32].

The underlying polymer model is based on a sequence of well-motivated
assumptions [35]: (1) To avoid contradictions and paradoxes, a BH must deviate
significantly from its general-relativistic description. This can either be in
the form of a so-called firewall that is localized at its surface [39] or on
horizon-sized length scales within. (2) The latter option requires a strongly
non-classical state of matter throughout the interior, which could evolve from
normal matter if it undergoes a quantum-gravitational phase transition or
tunneling event, as described in [40, 33]. (3) Extreme non-classicality is
synonymous with maximal entropy [41]. (4) A fluid with maximal entropy density
can only be realized by a theory of highly excited, long, closed strings, as
this matter alone is on the cusp of violating causality, $\;p=\rho\;$ [42],
meaning that the entropy density $s$ at fixed temperature $T$ and fixed energy
density $\rho$ cannot be made any higher, $\;s=\frac{p+\rho}{T}\;.$ Here and
onward, $p$ refers to the radial pressure as a theory of long, closed strings
is effectively $1+1$-dimensional.

Back to our classical solution, maximally negative radial pressure
$\;p=-\rho\;$, was found to be a key ingredient in attaining the condition of
regularity at a BH horizon [43, 44] and then, by extension, throughout the
interior of the frozen star. This condition corresponds to the saturation of
the radial component of the null-energy condition (RNEC), which is central to
the singularity theorems [45, 46] and essential for circumventing the closely
related bounds that limit the compactness of gravitationally collapsing
systems [47, 48, 49, 50, 51].

The link between the quantum and classical pictures now becomes clear.
Maximally positive pressure and, thus, maximal entropy tells us how far
causality can be pushed before spacetime ceases to have a semiclassical
description. Maximally negative pressure and, thus, the saturation of the RNEC
also tells us how far causality can be pushed but from a different
perspective. The former viewpoint considers the actual matter content while
the latter is stated in geometric terms. Thanks to the saturation of their
respective bounds, the two models share another important feature: Each radial
slice of the interior is itself a nearly null surface, just like the would-be
horizon on the outermost slice.

Our own rotating solution of the Einstein equations will be required to have
the following properties in addition to regularity: (1) The solution should,
to leading order in a perturbative parameter, saturate the RNEC throughout the
interior for a class of probe observers that can formally reach the inside of
the star and (2) it should be able to reproduce the total mass and angular
momentum of a Kerr BH. The former requirement ensures its connection to the
polymer model, while the latter guarantees that the interior “fluid” sources
the exterior Kerr geometry. We will, for the sake of consistency, also assume
that the solution agrees with the static frozen star in the zero-rotation
limit.

The paper is organized as follows: The rotating frozen star is introduced in
Section 2. To this end, we start from the Kerr metric in an ADM-like, near-
horizon version of Boyer–Lindquist coordinates. This form of the metric is
then transformed into another that is appropriate for a zero-angular-momentum
observer (ZAMO). We also make use of a closely related coordinate system from
[44] for which the RNEC is exactly saturated on the Kerr horizon. In our case,
however, the same condition is saturated, to leading order in the perturbative
parameter, throughout the interior of the rotating star. This can be seen
after finally imposing the transformation from the ZAMO–Kerr geometry to that
of the rotating frozen star. The existence of a Killing tensor for the
rotating star interior is established in the second major part of Section 2.
The Einstein tensor for the just-discussed metric is calculated in Section 3;
leading in particular to the energy density and radial pressure from the
perspective of a locally corotating observer. We integrate this energy density
over the volume of the star and show that the star’s rest mass is equal to the
irreducible mass of the corresponding Kerr BH. We then integrate the angular-
momentum density and find that the ratio of the angular momentum to the rest
mass is exactly equal to the Kerr spin parameter. The rotational energy is
included in the standard way to obtain the total gravitational mass, which
matches with that of its Kerr BH counterpart. The Kerr value for the angular
momentum immediately follows. We further show that the energy density is
formally negative in a certain region of the interior. However, from the
perspective of the class of probe observers that can access this region, the
energy density actually vanishes. The energy density is also mildly singular
near the center of the star; however, we expect that it can be regularized by
a procedure which is similar to that used to regularize non-rotating frozen
stars [31]. The paper then ends with a brief overview.

##  2 A rotating frozen star: The geometry

###  2.1 The metric

Our starting point is the Boyer–Lindquist metric for the Kerr spacetime; see,
for example, [52]. Near the horizon, this metric can be reexpressed in an ADM-
like form [44] (see the Appendix for details),

$ds^{2}\;=\;-\left(N^{2}-\omega^{2}g_{\phi\phi}\right)dt^{2}+\frac{dr^{2}}{N^{2}}-2\omega
g_{\phi\phi}dtd\phi+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}\;.$ (1)

Here, $\omega=-\frac{g_{\phi t}}{g_{\phi\phi}}$ can be interpreted as a local
measure of the angular velocity and $N$ is a standard lapse function in that
$\;N^{2}=g^{rr}=0\;$ at the horizon or surface of “infinite redshift”. Let us
recall a few properties of the Kerr geometry. The horizon at $\;r=r_{+}\;$ is
defined by $\;\Delta=r^{2}-2Mr+a^{2}=0\;$ and is located at
$\;r_{+}=m+\sqrt{M^{2}-a^{2}}\;$, where $M$ is the ADM mass of the BH and $a$
is the dimensional spin parameter, $\;0\leq a\leq M\;$. The lapse function is
$N^{2}=\Delta/\Sigma$, with
$\Sigma=g_{\theta\theta}=r^{2}+a^{2}\cos^{2}\theta$. Also,
$\;N^{2}=\omega^{2}g_{\phi\phi}\;$ at the ergosphere which is situated at
$\;r=r_{ES}=M+\sqrt{M^{2}-a^{2}\cos^{2}{\theta}}\;$. It is worth recalling
that $\;r_{+}<r_{ES}<r_{S}=2M\;$. The inner horizon and inner ergosphere are
of no interest here, as these are vanquished for the rotating frozen star.

Next, we transform to ZAMO coordinates, which are those for a special class of
observers for which the locally measured angular momentum vanishes [52, 53].
This amounts to adapting the coordinates with respect to the Killing vector
$\;\chi^{a}=\xi^{a}+\omega\psi^{a}\;=\;(1,0,0,\omega)\;,$ where $\xi^{a}$ and
$\psi^{a}$ are the usual timelike and axially symmetric Killing vectors for
the Kerr spacetime.

In terms of the adapted time coordinate, the line element in the near-horizon
region becomes,

$ds^{2}\;\to\;-N^{2}d\chi^{2}+\frac{dr^{2}}{N^{2}}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}\;.$
(2)

This is precisely the horizon limit of the metric obtained in [44],

$ds^{2}\;=\;-\left[N^{2}-\left(\Omega_{H}-\omega\right)^{2}g_{\phi\phi}\right]d\chi^{2}+\frac{dr^{2}}{N^{2}}+\left(\Omega_{H}-\omega\right)g_{\phi\phi}d\chi
d\phi+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}\;,$ (3)

with $\Omega_{H}=\frac{a}{r_{+}^{2}+a^{2}}=\frac{a}{2Mr_{+}}$ being the
horizon value of $\omega$ — the only angular velocity with which observers on
the horizon are allowed to move.

The above metric was used in [44] to show that the RNEC is exactly saturated
on the horizon of a rotating BH. This happens because, as the horizon is
approached, $\;\Omega_{H}-\omega\;$ has to go to zero at least as fast as the
square of the lapse function, as this ensures the regularity of scalar
invariants involving the Riemann curvature. So that, in this limit, one
obtains the diagonal metric (2) for which the elements of the Einstein tensor
satisfy $\;G^{\chi}_{\;\;\chi}=G^{r}_{\;\;r}\;$ by virtue of the
temporal–radial sector of the Einstein tensor being directly proportional to
the metric. In other words, $\;\rho+p=0\;$.

We now turn our attention to the interior, as the spacetime external to the
horizon will be left alone (but see Footnote 2). The near-horizon form of the
metric (1) is sufficient for our purposes because one of the premises of the
frozen star model is that, up to perturbatively small corrections, every
radial slice of the interior is an almost-null surface whose geometry is like
that of a BH horizon. Introducing the perturbatively small parameter
$\varepsilon^{2}=\dfrac{\Delta}{\Sigma}$, we thus write,

$ds^{2}\;=\;-\varepsilon^{2}d\chi^{2}+\frac{dr^{2}}{\varepsilon^{2}}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}\;,$
(4)

where $\;\varepsilon^{2}\ll 1\;$. 111What we now call $\varepsilon^{2}$ had,
until recently, been called $\varepsilon$. The change was made to emphasize
its positivity. To get some idea of just how small, a recent paper used
empirical data to argue that $\;\varepsilon^{2}\lesssim 10^{-22}\;$ [54].

To complete the interior metric, we set the angular components of the metric
equal to their Boyer–Lindquist counterparts but with one caveat: The component
$g_{\phi\phi}$ depends on the ADM mass but, for the static frozen star, the
mass was redefined by a scaling relation $\;M\to
M(r)=\frac{r}{2}(1-\varepsilon^{2})\;$. The natural generalization for the
rotating case is $\;M(r)=\frac{r^{2}+a^{2}}{2r}(1-\varepsilon^{2})\;$, which
can be used to rewrite $g_{\phi\phi}$ in a suitable “near-horizon” form. This
leads to

$ds^{2}\;\to\;-\varepsilon^{2}d\chi^{2}+\frac{dr^{2}}{\varepsilon^{2}}+\Sigma
d\theta^{2}+\frac{(r^{2}+a^{2})^{2}}{\Sigma}\sin^{2}{\theta}d\phi^{2}\;.$ (5)

where we kept only the leading term in $g_{\phi\phi}$. Our subsequent
calculations are, consequently, valid only to leading order in
$\varepsilon^{2}$, but we do expect that it is possible to obtain results that
are valid to higher order by adding $\varepsilon^{2}$ corrections to Eq. (5).

By design and as an explicit calculation confirms, the RNEC is saturated, to
leading order in $\varepsilon^{2}$, throughout the interior of the rotating
star given this choice of metric. A consequence of this saturated condition is
that every surface of fixed radial coordinate $r$ has, to leading order, the
same geometry as the horizon of a Kerr BH, except that the redshift is large
but finite rather than infinite. The area of this surface is $A=4\pi
R^{2}=4\pi(r^{2}+a^{2})$, with $R$ being the Kerr areal coordinate.

Besides ensuring the saturation of the RNEC, the metric (5) limits to that of
the static frozen star as $\;a\to 0\;$ and, perhaps most importantly, is
regular throughout the interior except for a mild, integrable singularity at
the center of the star. But the constraint that uniquely fixes the form of the
metric is that its corresponding Einstein tensor is able to reproduce the mass
and angular momentum of a Kerr BH, as will be shown in Section 3.

Before moving on to discuss symmetries, there are a few pertinent points to be
made.

  1. a)

Given the form of the interior metric (5), one might wonder if there is a
natural way to limit the value of $a$ like there is for the original Kerr
solution. However, it must be the case that $\;a\left(r\lesssim
r_{+}\right)\leq a\left(r\gtrsim r+\right)\;$; otherwise, the star will be
shedding mass until an equilibrium state is attained. It then follows that
$a_{interior}\leq a_{exterior}\leq M\;$.

  2. b)

Matching the interior and exterior solutions will require a narrow
transitional layer (width $\;\lambda\ll r_{+}$), which allows for the interior
metric and its first two derivatives to be continuously deformed so as to
match up with the exterior geometry, just as for the static frozen star [30].
222In this regard and also including corrections of order $\varepsilon^{2}$,
the interior geometry formally stops at
$\;r_{+}+\varepsilon^{2}-\frac{\lambda}{2}\;$ and the exterior geometry
formally begins at $\;r_{+}+\varepsilon^{2}+\frac{\lambda}{2}\;$ and neither
at $r_{+}$ as implied earlier. We do not expect the process to be
significantly different from that used in [30] for the static case. We also do
not expect the inclusion of the layer to modify calculations such as the mass
in the next section, at least not at leading order.

  3. c)

On the other hand, as can be seen in Section 3, the curvature and all non-
vanishing components of the Einstein tensor remain regular throughout the
interior, except for a mild singularity near the center of the star, as in the
static case [31]. We also expect, similarly to the static case, that a
regularization zone can be introduced near the center of the rotating star
that will affect the star’s mass and angular momentum by negligibly small
amounts.

  4. d)

The property that ensures the ultrastability of the static frozen star,
$\;G^{t}_{\;\;t}=G^{r}_{\;\;r}\;$, remains intact, and so we anticipate that
the interior of the rotating frozen star is similarly ultrastable.

###  2.2 Killing vectors and tensor

It is clear that the timelike and axially symmetric Killing vectors survive
for the revised interior, but what about the Kerr Killing tensor? This is, to
some extent, a moot issue inasmuch as any material body will be torn apart as
it approaches the (would be) horizon. So that, when discussing the symmetries
of the solution, we need formally to consider probe observers; for example, a
photon that propagates through the interior.

Let us further point out that any such “observer” must be moving on a radial,
null trajectory in the frozen star interior. To understand why, one can
consider a convenient form of a particle’s equation of motion,

$k\;=\;-\varepsilon^{2}(u^{t})^{2}+\frac{1}{\varepsilon^{2}}(u^{r})^{2}+g_{\theta\theta}(u^{\theta})^{2}+g_{\phi\phi}(u^{\phi\phi})^{2}\;,$
(6)

where $u^{a}$ is a component of the $4$-velocity, while $\;k=1\;$ for a
timelike and $\;k=0\;$ for a null trajectory.

As the asymptotic conserved energy per unit mass is $\;{\cal
E}=-g_{tt}u^{t}=\varepsilon^{2}u^{t}\;$, Eq. (6) can be recast as

${\cal
E}^{2}-(u^{r})^{2}\;=\;\varepsilon^{2}\left[g_{\theta\theta}(u^{\theta})^{2}+g_{\phi\phi}(u^{\phi\phi})^{2}+k\right]\;,$
(7)

where $\;{\cal E}\gtrsim\Omega_{H}L\;$ for a ZAMO such that
$\;L=g_{\phi\phi}u^{\phi}\;$ is the axial angular momentum per unit mass,
which is also conserved.

One can now see that the angular components of velocity, as well as the
distinction between null and timelike motion, are suppressed by a factor of
$\varepsilon^{2}\;$ throughout the frozen star interior,

${\cal E}^{2}-(u^{r})^{2}\;=\;{\cal O}[\varepsilon^{2}]\;,$ (8)

where the correction represents a perturbatively small deviation from a
perfectly null, radial path. Further note that objects with large angular
velocities at infinity would not get through the external potential barrier,
so that the right-hand side will not have large-enough numbers to compensate
for the factor of $\varepsilon^{2}$.

With all this in mind, let us recall the Killing tensor’s standard form [55],

$K^{ab}\;=\;\Sigma\;l^{\left(a\right.}n^{\left.b\right)}+r^{2}g^{ab}\;,$ (9)

where $l^{a}$ and $n^{a}$ are null and mostly radial vectors with
normalization $\;l^{a}n_{a}=-2\;$. In the ZAMO case,
$\;l^{a}=\left(\frac{1}{\varepsilon^{2}},1,0,0\right)\;$,
$\;n^{a}=\left(1,-\varepsilon^{2},0,0\right)\;$ and the Killing tensor does
not quite satisfy the requisite identity,

$I^{abc}\;\equiv\;\nabla^{\left(a\right.}K^{\left.bc\right)}\;=\;0\;,$ (10)

in all instances. It does only fail for two particular cases (and permutations
thereof):
$\;I^{r\phi\phi}=4\varepsilon^{2}\frac{ra^{2}\Sigma}{(r^{2}+a^{2})^{3}}\;$ and
$I^{rr\theta}=\varepsilon^{2}\frac{a^{2}\sin{2\theta}}{\Sigma}\;$. Both
violations are of order $\varepsilon^{2}$, but cannot immediately be dismissed
on this basis since their covariant forms $I_{abc}$ are not similarly
suppressed. We will rather argue that these apparent violations do not change
the integrability of the Hamilton–Jacobi equations nor the constancy of the
Carter constant, again, to leading order in $\varepsilon^{2}$.

Let us recall the Carter constant

$C=K_{ab}u^{a}u^{b}$ (11)

and that it is one of the four conserved quantities for motion in a Kerr
spacetime [55]. It is, essentially, the square of the total angular momentum
of a propagating particle plus a contribution that can be attributed to a
distortion in its radial propagation when $\;a\neq 0\;$ [56].

In the case of an exactly radial, null trajectory in the frozen star interior,
one finds from Eq. (9) that

$C\;=\;K_{ab}u^{a}u^{b}\;=\;0\;.$ (12)

The Carter constant in this case is not only zero but identically vanishing,
and so its derivatives are also identically vanishing. This result is a
consequence of the structure of the $\chi$–$r$ sector of the geometry, thus
rendering $I^{r\phi\phi}$ and $I^{rr\theta}$ as irrelevant to the constancy of
the Carter constant. Our conclusion is that a physically relevant observer
cannot detect violations of the Killing identity in the frozen star spacetime
up to $\varepsilon^{2}$ corrections. Similarly, one can show that, to leading
perturbative order, the Carter constant is indeed constant. The condition for
the Carter constant being constant is $\;u^{a}\nabla_{a}C=0\;$, as this is
what gives rise to the Killing identity (10).

We have also verified that the Carter constant is the separation constant in
the Hamilton–Jacobi equation, $\;p^{a}p_{a}+k=0\;$. This leads to the
following pair of separated equations,

$\displaystyle\varepsilon^{2}p_{r}^{2}+\frac{r^{2}}{(a^{2}+r^{2})^{2}}L^{2}-\frac{1}{\varepsilon^{2}}{\cal
E}^{2}+k$ $\displaystyle=$ $\displaystyle-{\widetilde{C}}\;,$ (13)
$\displaystyle\frac{1}{\Sigma}p^{2}_{\theta}+\frac{a^{2}\cos^{2}{\theta}}{(a^{2}+r^{2})^{2}}L^{2}$
$\displaystyle=$ $\displaystyle+{\widetilde{C}}\;,$ (14)

where $p_{a}$ is a component of 4-momentum per unit mass and ${\widetilde{C}}$
is the Carter constant per square of the unit mass. The latter equation is
telling us that ${\widetilde{C}}$ is typically a number of order unity, which
allows the former equation to be reexpressed (using $\;p_{a}=g_{ab}u^{b}$)
into Eq. (8).

##  3 A rotating frozen star: The matter

###  3.1 The Einstein tensor

The Einstein tensor $\;G_{\mu\nu}={\cal R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{\cal
R}\;$ or, equivalently, the stress tensor for the interior metric (5) takes a
relatively simple form to leading order in $\varepsilon^{2}$. The diagonal
components at this order go as (with $\chi$ now relabeled as $t$)

$G^{t}_{\;\;t}\;=\;G^{r}_{\;\;r}\;=\;-\frac{\left(r^{2}+a^{2}\right)\left(r^{2}-3a^{2}\cos^{2}{\theta}\right)}{\left(r^{2}+a^{2}\cos^{2}{\theta}\right)^{3}}\;,$
(15) $G^{\theta}_{\;\;\theta}\;\;,\;\;G^{\phi}_{\;\;\phi}\;=\;{\cal
O}[\varepsilon^{2}]\;.$ (16)

As expected, the first two diagonal components are equal, meaning that
$\;p=-\rho\;$, while the angular components are perturbatively small. Although
this latter pair depend differently on $r$ and $\theta$, they both scale as
$a^{2}$ and thus go to zero in the non-rotating limit, as they must to agree
with the static case. Both $G^{t}_{\;\;t}$ and $\;G^{r}_{\;\;r}$ are mildly
divergent in the region $r\to 0$. This is similar to the non-rotating case
[31] and, as previously discussed, can readily be dealt with by using an
appropriate regularization procedure.

Meanwhile, there is a single non-vanishing off-diagonal element,

$G_{r\theta}=\frac{2\leavevmode\nobreak\
a^{2}r\sin{2\theta}}{\left(r^{2}+a^{2}\cos{\theta}\right)^{2}}\;,$ (17)

which correctly goes to 0 as $a^{2}$ does. If one chooses to diagonalize the
$r$–$\theta$-sector, the result is that the diagonal components are unchanged,
up to $\varepsilon^{2}$ corrections, so that the off-diagonal component has
minimal physical impact.

The Einstein tensor, although formally a four-dimensional tensor, is
determined to leading order by only the two-dimensional angular part of the
metric $\;g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}\;$. A related,
interesting feature of the Einstein tensor is that it implies that
$\;\rho=-\frac{1}{8\pi}G^{t}_{\;\;t}\;$ can be negative for small-enough
values of $r$ because of the factor of $\;r^{2}-3a^{2}\cos^{2}{\theta}\;$ in
Eq. (15). This apparent negative-energy region corresponds to parts of the
interior where the extrinsic curvature on radial slices becomes negative. A
system, even a classical one, with negative energy density is not necessarily
a problem unto itself provided that the NEC remains satisfied [57]. This is
certainly true for our special class of infalling observers, for which the
physically relevant measure of energy is, in fact, the radial null energy.
Ironically enough, since the radial null energy vanishes throughout the
interior, these observers would believe that our non-vacuum source for the
Kerr exterior is, in fact, a vacuum.

Although the RNEC is not in jeopardy, what about the angular components of
this condition? These are clearly violated in the negative-energy region since
the transverse pressures (or angular components of $G^{a}_{\;\;\;b}$) go as
$\;p_{\theta}\;,\;p_{\phi}\sim\varepsilon^{2}\;$ and, therefore, cannot
possibly compensate for a negative value of $\rho$. Nevertheless, this is not
a problem for the following reason: No physically relevant observer could ever
observe a local violation of the NEC to leading order in $\varepsilon^{2}$. To
elaborate, from Eq. (7), the squares of the (effective) angular velocities are
at most order $\varepsilon^{2}$ in comparison to the squares of the leading
velocities ${\cal E}$ and $u^{r}$. For instance, allowing $(u^{r})^{2}$ to be
smaller than ${\cal E}^{2}$ by $\varepsilon^{2}$, compensating for this change
with $\;(u^{\theta})^{2}=\frac{\varepsilon^{2}}{\Sigma}\;$ and contracting the
Einstein tensor twice with the modified velocity, one then obtains

$G^{a}_{\;\;b}u_{a}u^{b}\;=\;-\left(G^{t}_{\;\;t}-G^{r}_{\;\;r}\right)\;-\;\varepsilon^{2}G^{r}_{\;\;r}\;+\;2{\cal
E}\sqrt{\frac{\varepsilon^{2}}{\Sigma}}G_{r\theta}\;+\;\varepsilon^{4}G^{\theta}_{\;\;\theta}\;,$
(18)

showing that, after diagonalization of the $r-\theta$ components of $G$, any
such violation is indeed $\varepsilon^{2}$ suppressed.

###  3.2 Mass and angular momentum

We will next calculate both the mass and the angular momentum of a frozen star
using the standard expressions [58, 59, 52], which are also valid for an
approximate, isolated horizon [60, 61, 62, 63, 64], such as the outer surface
of a frozen star. To fix the ambiguity in the lapse $N$, we adopt the areal
radius $R$, as is done in the just-cited discussions on isolated horizons.
Then the 3-volume $d^{3}V$ on the horizon can be decomposed as
$\;d^{3}V=dR\leavevmode\nobreak\ R^{2}d\Omega\;$. We further simplify our
expressions by working in units in which
$\;R_{+}^{2}=r_{+}^{2}+a^{2}=\dfrac{1}{4\pi}A\;$, is set to unity (but restore
the units at the end of the calculations).

In these units, the irreducible mass of a Kerr BH is [65, 66, 67, 52]

$M_{irr}\;=\;\frac{1}{2}\sqrt{\frac{A}{4\pi}}=\frac{1}{2}\;,$ (19)

which is the mass of a Schwarzschild BH with surface area $\;A=4\pi
R_{+}^{2}\;$ and obviously in agreement with the Schwarzschild mass in the
$\;a\to 0\;$ limit.

Let us begin by using the Einstein equation
$\;-\frac{1}{8\pi}G^{t}_{\;\;t}=\rho\;$, to determine the mass of the frozen
star. 333The mass calculation is from the perspective of an external observer,
and so there is no contradiction with the notion that $\;\rho+p=0\;$ is the
energy density from a local ZAMO’s perspective. The first step will be to
calculate what we identify as the star’s intrinsic rest mass $M_{0}$, which
will turn out to be equal to $M_{irr}$,

$\displaystyle 8\pi M_{0}$ $\displaystyle=$
$\displaystyle-2\pi\int_{0}^{1}dR\int_{0}^{\pi}d\theta\sqrt{g_{\theta\theta}g_{\phi\phi}}\;G^{t}_{\;\;t}\;.$
(20)

Note that one does not include $\sqrt{g_{rr}}$ in the measure as its positive
contribution is canceled off by the negative contribution of the gravitational
potential energy [59, 52, 60, 61, 62, 63, 64].

Performing the above integral,

$\displaystyle 2M_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{1}dR\;R^{4}\int_{0}^{\pi}d\theta\sin{\theta}\frac{\left(r^{2}-3a^{2}\cos^{2}{\theta}\right)}{\left(r^{2}+a^{2}\cos^{2}{\theta}\right)^{3}}\;=\;1\;,$
(21)

and then restoring the units, we find that

$M_{0}\;=\;\frac{1}{2}R_{+}\;=\;M_{irr}\;.$ (22)

The intrinsic angular momentum is similarly calculated via the standard
techniques,

$8\pi
J_{0}\;=\;-2\pi\int_{0}^{1}dR\int_{0}^{\pi}d\theta\sqrt{g_{\theta\theta}g_{\phi\phi}}\leavevmode\nobreak\
G^{t}_{\leavevmode\nobreak\ \phi}\;.$ (23)

However, a ZAMO finds that this angular momentum vanishes because
$\;G^{t}_{\leavevmode\nobreak\ \phi}=0\;$, as expected. So that, to calculate
the angular momentum, we need to turn to a “Boyer–Lindquist observer”, for
whom the frozen star is rigidly rotating at the horizon frequency
$\Omega_{H}$, which in our current units is $\;\Omega_{H}=a\;$. In which case,
the metric contains a non-trivial off-diagonal element,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{a^{2}R^{4}\sin^{2}\theta}{\Sigma}-\varepsilon^{2}\right)dt^{2}\;-\;2a\sin^{2}\theta\frac{R^{4}}{\Sigma}dtd\phi\;+\;\frac{1}{\varepsilon^{2}}dr^{2}\;+\;\Sigma
d\theta^{2}\;+\;\frac{R^{4}}{\Sigma}\sin^{2}\theta d\phi^{2}\;.$

The $t$–$t$ component of the Einstein tensor is unchanged, while the desired
$t$–$\phi$ component of the Einstein tensor is found to satisfy

$G^{t}_{\leavevmode\nobreak\ \phi}\;=\;a\;G^{t}_{\leavevmode\nobreak\ t}\;.$
(25)

We can then use Eqs. (20), (23) and then (21) to infer that

$J_{0}\;=\;a\;M_{0}\;=\;\frac{1}{2}a\;.$ (26)

Or, with units restored,

$J_{0}\;=\;a\;M_{irr}\;.$ (27)

The conclusion is that $\;a=J_{0}/M_{0}\;$ is indeed the spin parameter of the
frozen star.

Finally, following the usual procedure, one is instructed to include the
rotational energy $E_{rot}$, to obtain the total gravitational mass $M$, of
the rotating frozen star. The standard prescription for this is to integrate
the relation $\;dE_{rot}=\Omega dJ\;$ and then sum the squares of the two
energy contributions [66, 52], which leads to the result

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\;M_{irr}^{2}+\frac{1}{4}\frac{J^{2}}{M_{irr}^{2}}\;.$ (28)

Since the irreducible mass and the spin parameter of the frozen star are the
same as the corresponding Kerr BH, its total mass is also the same. The total
angular momentum of the BH follows directly, albeit trivially, as
$\;J\;=\;aM\;$.

And so we find that the total gravitational mass and angular momentum of our
rotating frozen star are precisely the same as those of a Kerr BH with the
same size and spin parameter. Our conclusion is that the interior matter in
the frozen star is a possible source for the geometry of the Kerr exterior
metric.

##  4 Conclusion

We have identified a possible source for the Kerr geometry by incorporating
rotation into our model for a regular BH mimicker — the frozen star. Key
properties of our solution include the regularity of the metric and the
Einstein tensor throughout the interior, except for a mild removable
singularity at the center of the star, as well as the saturation of the RNEC,
again, throughout the interior. Our calculations were done to leading order in
$\varepsilon^{2}$; however, we expect that they can be extended to higher
order if necessary.

The above properties and the fact that the rotating star limits to the static
star as the rotation is switched off are all important consistency checks. The
intrinsic mass of our solution was calculated and found to be equal to the
irreducible mass of the corresponding Kerr BH. We also calculated the angular
momentum of the solution and found that the ratio of the angular momentum to
the mass is the spin parameter $a$, just as it is for the corresponding Kerr
BH. Together these findings mean that the total gravitational mass and the
total angular momentum of our solution are equal to those of a Kerr BH of the
same size and with the same spin parameter for any acceptable value of $a$.

Having a rotating frozen star solution at hand is a key step toward our
longer-range goal of calculating the gravitational-wave spectrum for the fluid
modes of a deformed frozen star; for the analogous static treatment, see [32].
Rotation is of course essential if one wants to use the current and soon-to-be
available observational data to seriously test the validity of any alternative
BH model. The lesson from [32] is that the key to generating gravitational
waves in an otherwise ultrastable compact object is to deform the star in a
way that modifies the saturating condition $\;G^{t}_{\;\;t}=G^{r}_{\;\;r}\;$.
We expect to report on the spectra for the rotating model in the near future,
and then find the corresponding Love numbers and spectrum of ringdown modes.

## Acknowledgments

We thank Badri Krishnan and Yoav Zigdon for discussions. The research is
supported by the German Research Foundation through a German-Israeli Project
Cooperation (DIP) grant “Holography and the Swampland” and by VATAT (Israel
planning and budgeting committee) grant for supporting theoretical high energy
physics. The research of AJMM received support from an NRF Evaluation and
Rating Grant 119411 and a Rhodes Discretionary Grant SD07/2022. AJMM thanks
Ben Gurion University for their hospitality during his visit.

##  Appendix A The ADM form of Boyer-Lindquist

Here, we want to verify that the ADM form of the time–time component in the
metric (1) does indeed agree with its Boyer–Lindquist counterpart at least for
the near-horizon solution, which is the relevant region in our analysis. Let
us then recall the Boyer–Lindquist metric,

$\displaystyle\left.ds^{2}\right|_{BL}$ $\displaystyle=$
$\displaystyle-\frac{\Delta-a^{2}\sin^{2}{\theta}}{\Sigma}dt^{2}\;+\;\frac{\Delta}{\Sigma}dr^{2}\;-\;\frac{4Mr\,a\sin^{2}{\theta}}{\Sigma}dtd\phi$
(29) $\displaystyle+$ $\displaystyle\Sigma
d\theta^{2}\;+\;\left[r^{2}+a^{2}+\frac{2Mra^{2}\sin^{2}{\theta}}{\Sigma}\right]\sin^{2}{\theta}d\phi^{2}\;,$

with the time–time component displayed in a way which makes it clear that the
objective is to show that

$\omega^{2}g_{\phi\phi}\;=\;\frac{g_{\phi
t}^{2}}{g_{\phi\phi}}\;=\;\frac{a^{2}\sin^{2}{\theta}}{\Sigma}\;.$ (30)

Let us now substitute the relevant components of the Boyer–Lindquist metric
into the middle term in the above equation,

$\frac{g_{\phi
t}^{2}}{g_{\phi\phi}}\;=\;\frac{4M^{2}r^{2}a^{2}\sin^{2}{\theta}}{\Sigma\left[\left(r^{2}+a^{2}\right)\left(r^{2}+a^{2}\cos^{2}{\theta}\right)+2Mra^{2}\sin^{2}{\theta}\right]}\;.$
(31)

One can then use the relation $\;\Delta=r^{2}+a^{2}-2Mr\;$ along with
$\;\sin^{2}{\theta}+\cos^{2}{\theta}=1\;$ to rewrite this as

$\frac{g_{\phi
t}^{2}}{g_{\phi\phi}}\;=\;\frac{\left(r^{2}+a^{2}-\Delta\right)^{2}a^{2}\sin^{2}{\theta}}{\Sigma\left[\left(r^{2}+a^{2}\right)^{2}-\Delta
a^{2}\sin^{2}{\theta}\right]}\;.$ (32)

Now one finds that taking $\;\Delta\to 0\;$ on the right-hand side leads to
the desired outcome,

$\frac{g_{\phi
t}^{2}}{g_{\phi\phi}}\;=\;\frac{a^{2}\sin^{2}{\theta}}{\Sigma}\;.$ (33)

To further see that the location of the ergosphere also comes out correctly,
one can similarly substitute $\;\Delta=a^{2}\sin^{2}{\theta}\;$ on the right-
hand side of Eq. (32) to again obtain Eq. (33).

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