# Insider Threats Mitigation: Role of Penetration Testing

Krutarth Chauhan  Department of Computer Science
University of Guelph
Guelph, Canada
<EMAIL_ADDRESS>
###### Abstract

Conventional security solutions are insufficient to address the urgent
cybersecurity challenge posed by insider attacks. While a great deal of
research has been done in this area, our systematic literature analysis
attempts to give readers a thorough grasp of penetration testing’s role in
reducing insider risks. We aim to arrange and integrate the body of knowledge
on insider threat prevention by using a grounded theory approach for a
thorough literature review. This analysis classifies and evaluates the
approaches used in penetration testing today, including how well they uncover
and mitigate insider threats and how well they work in tandem with other
security procedures. Additionally, we look at how penetration testing is used
in different industries, present case studies with real-world implementations,
and discuss the obstacles and constraints that businesses must overcome. This
study aims to improve the knowledge of penetration testing as a critical part
of insider threat defense, helping to create more comprehensive and successful
security policies [1].

###### Index Terms:

case studies, integration difficulties, cybersecurity, insider threats,
penetration testing, mitigation tactics, security solutions.

##  I Introduction

The security of systems, intellectual property, and organizational data is
seriously jeopardized by insider threats. Individuals with authorized access
to vital systems and data within the company, such as workers, contractors, or
business partners, are the source of these dangers. Insider threats can have
disastrous effects, including monetary losses, harm to one’s reputation, and
legal repercussions. Because of this, firms in a variety of industries now
consider insider threat mitigation to be of utmost importance. Penetration
testing is a crucial tactic used to combat insider threats. The purpose of
this systematic review of the literature is to investigate how penetration
testing might help mitigate insider risks by examining the most recent
findings, useful applications, and research techniques.

###  I-A Background and Importance of Insider Threats

Insider threats refer to a broad category of malevolent actions such as
sabotage, fraud, espionage, and theft of intellectual property. Insiders can
be anyone with authorized access to an organization’s resources, including
partners, contractors, and employees. Insider threats have a variety of
motivations, from coercion and inadvertent mistakes to monetary gain and
retaliation. The Ponemon Institute (2021) analysis highlights the serious
financial implications of insider threat incidents, with an average cost of
$11.45 million. Furthermore, the study notes that during the past two years,
insider threat occurrences have increased 47%, indicating the growing
importance of this problem in the cybersecurity landscape [2].

Because insiders have authorized access, it is inherently difficult to
identify insider threats. Because they are primarily built to fight against
external threats, traditional security measures like firewalls and intrusion
detection systems are sometimes insufficient to recognize and stop insider
attacks. Penetration testing is an essential part of the more sophisticated
and focused strategies that are required to close this security gap.

###  I-B Penetration Testing: Overview

Penetration testing, also known as ethical hacking, is intentional
investigation of a company’s networks, systems, and applications to find
security flaws that could be used by hostile actors [3]. Penetration testing,
in contrast to traditional vulnerability assessments, takes things a step
further by aiming to exploit vulnerabilities that are found, giving a more
accurate picture of the possible consequences of security flaws. Finding and
fixing vulnerabilities before they can be used as an attack vector is the main
objective of penetration testing.

Penetration testing can be customized to mimic situations in which an insider
abuses their access privileges in the context of insider threats. Testing for
privilege escalation, illegal access to private data, and essential system
manipulation are included in this document. Penetration testing helps to
establish strong mitigation mechanisms and offers important insights into the
organization’s security posture by simulating the tactics, techniques, and
procedures (TTPs) that insiders might use.

##  II Prior Research

Numerous research works have examined how well penetration testing works to
reduce insider risks. Probst and Hunker (2019), for example, carried out a
thorough analysis of penetration testing techniques and how they are used in
insider threat situations. According to their findings, penetration testing
plays a crucial role in revealing hidden vulnerabilities that conventional
security solutions could miss. According to the study, to guarantee complete
security coverage, penetration testing procedures should include insider
threat scenarios [4].

AlHogail and Mirza (2021) emphasize the need to incorporate penetration
testing into a more comprehensive framework for mitigating insider threats.
Their study offers a framework that integrates access control methods, user
activity analytics, employee training initiatives, and penetration testing
with other security procedures [5]. By offering a comprehensive strategy for
mitigating insider threats, the framework seeks to strengthen the
organization’s defenses against internal threats. According to AlHogail and
Mirza (2021), conducting penetration tests on a regular basis can help to
monitor and improve the efficacy of mitigation techniques, thus reducing the
risk of insider threats [6].

Williams and Smith’s (2020) empirical study emphasizes the importance of
penetration testing in lowering the danger of insider threats. The study looks
at how frequent penetration testing affects an organization’s security posture
in a variety of industries. The results show that companies that regularly
perform penetration tests encounter fewer insider threat events than those
that don’t [7]. The reduction in insider exploits of security flaws is
attributed by the study to proactive detection and mitigation of
vulnerabilities [8].

##  III Methodologies in Penetration Testing for Insider Threats

While the literature provides a strong foundation for understanding the
importance and effectiveness of penetration testing, it is equally crucial to
delve into the specific methodologies employed. Methodologies for penetration
testing differ according to the objectives, scope, and available resources.
The approaches used to address insider threats can be broadly categorized as
follows: black-box, white-box, and gray-box testing.

###  III-A Black-Box Testing

With this method, testing is done without any prior understanding of the
system’s core operations. Testers act as low-privilege insiders or external
attackers looking to obtain unapproved access. Although black-box testing can
be helpful in locating security holes in publicly accessible systems and
services, it might not be able to reveal more serious insider-specific risks
[9].

###  III-B White-Box Testing

This method, which is often referred to as ”clear-box testing,” gives testers
complete access to source code, network diagrams, and configuration files, as
well as complete system knowledge. White-box testing makes it possible to
thoroughly examine the system from the point of view of an insider, revealing
flaws that an insider with a high privilege could exploit [10].

###  III-C Gray-Box Testing

With this method, which combines white-box and black-box testing, testers only
have a limited understanding of the system. By simulating situations in which
an insider has restricted but considerable access, gray-box testing seeks to
uncover vulnerabilities that might be taken advantage of via lateral network
movement or privilege escalation [11].

Every one of these approaches has benefits and drawbacks. Although black-box
testing can be helpful in locating external attack routes, it may not detect
specific risks for insiders. White-box testing offers a thorough understanding
of the system, but it necessitates substantial resources and expertise. Gray-
box testing simulates genuine insider threat scenarios with incomplete access,
providing a well-rounded approach.

##  IV Case Studies and Practical Applications

Building on the methodologies discussed, the practical application of these
penetration testing techniques can be observed in various case studies. These
real-world implementations provide valuable insight into the effectiveness of
penetration testing in different industry contexts.

A case study by Johnson et al. (2020), which looks at the application of
penetration testing in a financial institution, is one noteworthy example. The
report demonstrates how the organization was able to find and fix
vulnerabilities related to data leakage and access control through the use of
focused penetration tests [12]. The penetration testing enhanced the
institution’s overall security posture by providing useful insights through
the simulation of insider threats [13].

Lee and Kim (2018) have another case study that focuses on the healthcare
industry and employs penetration testing to assess the security of electronic
health record (EHR) systems [14]. According to the report, penetration testing
revealed a number of serious flaws, including inadequate audit logging and
erroneous access controls, which could be used by hostile insiders. By fixing
these weaknesses, the possibility of insider threats was greatly reduced,
protecting the privacy and accuracy of patient data [15].

A case study by Garcia and Lopez (2019) illustrates the role of penetration
testing in protecting intellectual property in the manufacturing industry
[16]. The report describes how penetration tests found vulnerabilities in the
company’s data storage and internal network. The company was able to stop
possible insider threats that would have stolen trade secrets and confidential
data by fixing these vulnerabilities [17].

##  V Research Goals and Contributions

The primary goal of this systematic literature review is to present a thorough
overview of penetration testing’s contribution to insider threat mitigation.
Specifically, this review aims to address the following research questions:

  * •

What are the current methodologies and frameworks used in penetration testing
to mitigate insider threats?

  * •

How effective are behavior modeling and machine learning techniques in
enhancing insider threat detection during penetration testing?

  * •

What are the key challenges and limitations faced in implementing penetration
testing for insider threat mitigation across different organizational
contexts?

  * •

What are the emerging trends and future directions in penetration testing
methodologies for combating insider threats?

  * •

How do regulatory frameworks and compliance requirements influence the
adoption and effectiveness of penetration testing for insider threat
mitigation?

By accomplishing these objectives, this review hopes to add to the body of
knowledge already available on insider threat mitigation and offer businesses
looking to fortify their security posture against insider threats practical
insights.

##  VI Research Methodology

Insider threats have become a major concern in today’s cybersecurity
environments, requiring thorough measures to reduce possible risks and protect
corporate assets. This section outlines a comprehensive process for carrying
out a systematic literature review (SLR) on insider threat mitigation with a
particular focus on penetration testing’s effectiveness as a preventative
security strategy.

Understanding the importance and background of the research approach used in
the systematic review of the literature (SLR) is based on the introduction.
Insider threats have become a significant concern for organizations in a
variety of industries. These dangers originate from people who have authorized
access to critical systems and information. These threats pose serious dangers
to the security, integrity, and resilience of organizations because they cover
a wide range of harmful behaviors such as fraud, sabotage, espionage, and data
exfiltration [18].

Insider threats are complicated and require a diversified approach to
successfully mitigate the risks involved. In contrast to external threats,
insiders are more difficult to identify and neutralize because they frequently
have in-depth knowledge of organizational systems, procedures, and weaknesses.
Therefore, proactive actions are essential to strengthen defenses and reduce
the effect of insider threats [19].

Penetration testing stands out as one of the most important proactive
strategies for locating vulnerabilities and assessing how well-performing
security policies are against insider threats. Penetration testing helps
businesses identify latent gaps in their defenses and prioritize remedial
efforts by replicating real-world attack scenarios and using the perspective
of possible attackers [20]. The introduction highlights the strategic
significance of penetration testing in enhancing organizational resilience and
its role as the cornerstone of successful insider threat mitigation plans
[21].

###  VI-A Selection of Primary Studies

The selection of primary studies for this systematic literature review was
conducted using a comprehensive search strategy across multiple academic
databases. The search strategy was formulated to identify relevant studies
that address the role of penetration testing in mitigating insider threats.
The Boolean operators AND and OR were employed to ensure a broad yet focused
search. The search terms used included combinations of keywords and phrases
such as:

  * •

”Insider Threats” OR ”Insider Threat Detection” OR ”Insider Attack”

  * •

AND ”Penetration Testing” OR ”Ethical Hacking” OR ”Security Assessment”

  * •

AND ”Cybersecurity” OR ”Information Security” OR ”Data Protection”

The databases and platforms searched were:

  * •

IEEE Xplore Digital Library

  * •

ScienceDirect

  * •

SpringerLink

  * •

ACM Digital Library

  * •

Google Scholar

The search was performed on titles, abstracts, and keywords depending on the
capabilities of each search platform. The initial search was conducted on
October 9, 2022, and included all studies published up to that date.

To ensure the comprehensiveness of the search, we followed a multi-step
process:

  * •

Initial Search: Using the defined search terms, we retrieved a large set of
potential studies from each database.

  * •

Screening: The titles and abstracts of the retrieved studies were screened to
exclude irrelevant papers.

  * •

Full-Text Review: The remaining studies were then subjected to a full-text
review to assess their relevance and quality based on predefined inclusion and
exclusion criteria.

Inclusion Criteria:

  * •

Studies specifically addressing penetration testing and its role in mitigating
insider threats.

  * •

Peer-reviewed journal articles, conference papers, and reputable whitepapers.

  * •

Studies published in English.

  * •

Studies providing empirical data, case studies, or theoretical insights into
the effectiveness of penetration testing for insider threat mitigation.

Exclusion Criteria:

  * •

Studies focusing solely on external threats without relevance to insider
threat scenarios.

  * •

Non-peer-reviewed articles, editorials, and opinion pieces.

  * •

Studies not providing sufficient methodological details or empirical data.

After the initial screening and full-text review, we applied a snowball
technique both forward and backward to identify additional relevant studies by
examining the references of selected papers and identifying studies that cited
these papers. This iterative process ensured that we captured a comprehensive
set of studies relevant to our research objectives.

The results of the selection process are presented in Table 1, summarizing the
number of studies identified, screened, and included at each stage of the
review.

This rigorous selection process ensured that only the most relevant and high-
quality studies were included in the systematic literature review, providing a
robust foundation for analyzing the role of penetration testing in mitigating
insider threats.

TABLE I: Selection process for primary studies Stage | Number of Studies
---|---
Initial search | 745
Title and abstract screening | 432
Full-text review | 189
Final included studies | 64

###  VI-B Research Objectives and Questions

This part expands on the introduction by outlining the main goals and research
questions that steer the SLR. The principal aim is to perform a thorough
examination of the extant literature pertaining to insider threat mitigation
with a particular emphasis on assessing the effectiveness of penetration
testing approaches [22]. In order to accomplish this goal, the following
important research topics are developed:

  * •

What are the prevailing patterns and obstacles in the tactics for mitigating
insider threats?

  * •

What role does penetration testing play in mitigating insider threats?

  * •

What are the most effective procedures and techniques for carrying out
penetration tests in order to counter insider threats?

  * •

What factual data is available to support the idea that penetration testing is
a useful tool for detecting and thwarting insider threats?

  * •

What are the limitations and gaps in the current literature on insider threat
mitigation and penetration testing?

These research questions serve as guiding principles for the systematic review
of the literature, informing the selection of relevant studies, the data
extraction process, and the synthesis of the findings.

###  VI-C Literature Search Strategy

The methodical process used to find pertinent academic publications on insider
threats and penetration tests is described in the literature search technique
[23]. Ensure a thorough coverage of relevant material by incorporating a wide
range of scholarly databases, electronic journals, and conference proceedings.
The methodical creation of search queries using pertinent keywords, synonyms,
and Boolean operators characterizes the search strategy to gather a variety of
viewpoints and thoughts on the topic.

Key components of the literature search strategy include:

  * •

Locating pertinent resources and repositories, including Web of Science, IEEE
Xplore, ACM Digital Library, PubMed, and Scopus.

  * •

Forming search queries with terms associated with penetration testing (e.g.,
”penetration testing,” ”ethical hacking,” ”security assessment”) and insider
threats (e.g., ”insider threat,” ”insider attack,” ”privileged access abuse”).

  * •

Synonyms and related phrases are included to guarantee thorough coverage of
pertinent material.

  * •

Using Boolean operators to improve precision and refine search queries, such
as AND, OR, and NOT.

  * •

Iteratively improving search tactics in response to subject matter experts’
input and initial search results.

The goal of the literature search method is to maximize the retrieval of
pertinent studies for the SLR while minimizing any biases and constraints.

###  VI-D Selection Results

This section provides an overview of the outcomes of the selection process
after the inclusion and exclusion criteria have been applied. It also provides
information on the total number of possible studies that were found, screened,
and eventually included in the SLR. It offers a thorough explanation of the
selection procedure, including the quantity of studies that were obtained from
every database, repository, and search engine, along with the justification
for each step’s inclusion or exclusion [24].

The selection outcomes highlight how stringent the procedure was in order to
guarantee the SLR’s comprehensiveness and applicability. The reader can assess
the reliability of the review procedure and the representativeness of the
chosen research by clearly stating the selection criteria and results.

###  VI-E Quality Assessment

To guarantee the validity and dependability of the results of the systematic
literature review (SLR), quality assessment is essential. To determine the
methodological robustness and relevance of the chosen studies to the study
aims, a thorough evaluation of the studies is conducted using predetermined
criteria. The quality assessment criteria cover a range of dimensions such as:

Methodological Rigor: Evaluating the validity of the study design, data
collection strategies, analytical procedures, and objective clarity used in
the primary studies, as well as the research methodology [25].

Relevance to Research Objectives: Guaranteeing congruence between the main
studies’ objectives and the general SLR objectives. This involves determining
whether the chosen articles adequately cover relevant aspects of penetration
testing effectiveness and insider threat mitigation [26].

Validity and Reliability: Examining the validity of the conclusions reached,
as well as the consistency of the results in many researches, to thoroughly
assess the reliability of the findings. This could entail evaluating the
reliability of the data analysis methods, sampling strategies, and measuring
instruments applied in the original research [27].

Ethical Considerations: Assessing how well primary research follows ethical
guidelines, such as getting informed consent from participants, maintaining
participant privacy, and reducing risk of harm [15].

Creating a quality appraisal checklist or rubric that specifies the precise
evaluation criteria is a common step in the quality evaluation process. After
that, each chosen study is thoroughly examined, considering these standards,
and the overall methodological rigor and applicability are evaluated by
assigning a quality score or rating [20].

It is standard procedure to include several independent reviewers who evaluate
the chosen studies separately in order to increase the validity of the quality
evaluation process. Any differences or disagreements in quality assessment are
resolved by consensus discussions.

The results of the quality assessment play a crucial role in guiding the
interpretation of the SLR data and establishing the degree of evidence needed
to justify the conclusions made. Research that is judged to be more
methodologically rigorous and relevant is given more weight when synthesizing
the results, while studies that have limitations or biases are evaluated
carefully to determine how they might affect the conclusions.

###  VI-F Data Extraction

In order to answer the research questions and objectives of the SLR, data
extraction comprises the methodical recovery and synthesis of pertinent
material from the chosen studies. Key data items must be identified and
categorized, such as the following:

Study Characteristics: Extracting the standard bibliographic data like the
name of the journal or conference, the study title, the year of publication,
the author(s), and the kind of publishing (e.g., conference paper, journal
article) [21].

Research Methodology: Providing an overview of the sample plan, data gathering
procedures, analytical approaches, and research design used in the original
studies [20].

Key Findings: Summarizing the main findings, results, and conclusions of each
study about the effectiveness of penetration testing and mitigation of insider
threats [25].

Perspectives on Penetration Testing Effectiveness: Gathering information from
case studies, empirical data, and practitioner perspectives regarding how well
penetration testing [26].

A structured data extraction form or template that specifies the precise
categories of information to be extracted is usually used as a guide for data
extraction. This ensures that the extraction process is uniform and
standardized. Moreover, depending on the type of data that are synthesized,
data extraction may require the use of quantitative or qualitative coding
systems or tools. Many independent reviewers who extract data independently
from the chosen research are advised to improve the accuracy and dependability
of data extraction. In data extraction, conflicts or discrepancies are settled
by consensus-building talks or by appointing a third-party arbiter. In order
to answer the research questions and goals of the SLR, the extracted data
provide the basis for further data analysis, which synthesizes, interprets,
and integrates the findings [22].

###  VI-G Data Analysis

The core of the SLR technique is data analysis, which is the synthesis,
interpretation, and integration of data that has been recovered in order to
answer research questions and produce significant insights [23]. The process
of data analysis involves a number of methods and strategies, such as:

Meta-Analysis: Quantitative synthesis of data from several studies to
determine the prevalence or overall effect size of phenomena related to the
effectiveness of penetration testing and mitigation of insider threats [26].
Statistical techniques are used in meta-analysis to combine effect sizes from
several studies, evaluate heterogeneity, and produce general findings.

Thematic Synthesis: A qualitative method of synthesis of data to find
recurring themes, correlations, and patterns in the chosen research [27]. The
process of thematic synthesis entails the coding and classification of
qualitative data in order to clarify theoretical constructs, empirical
conclusions, and underlying concepts.

Qualitative Content Analysis: A thorough investigation of textual material to
glean significant conclusions, interpretations, and implications concerning
the efficacy of penetration testing and insider threat mitigation [20]. In
qualitative content analysis, emergent themes, views, and discourses are
identified through the coding, categorization, and interpretation of textual
material.

Iterative and interactive, the process of data analysis involves continuously
improving and revising analytical frameworks and interpretations in light of
fresh information and discoveries. Sensitivity analysis may also be necessary
to evaluate the validity of findings and investigate research [25].

The primary research questions and SLR objectives serve as the framework for
the structured and logical presentation of the synthesized data analysis
findings. The implications of the findings are examined considering current
theories, empirical data, and real-world applications for penetration testing
and insider threat mitigation techniques [21].

###  VI-H Ethical Considerations

Given the sensitive nature of the subject matter and the potential hazards to
stakeholders, ethical issues are critical for research on insider threats and
cybersecurity. To maintain integrity, openness, and respect for the rights and
welfare of the participants, the SLR is conducted in accordance with ethical
norms. Important ethical factors include :

Protecting Participant Confidentiality: Ensuring the privacy and anonymity of
participants, especially in research involving sensitive or private material
[20]. Pseudonyms, secure data transmission and storage techniques, and
anonymization of data may be used to avoid unwanted access or exposure.

Informed Consent: Before allowing individuals to participate in the study,
informed consent must be obtained; this requires that participants be fully
informed about the goals, methods, risks, and rewards of their participation
[25]. By giving their informed consent, participants can decide whether or not
to participate in the study, ensuring that their participation is their
choice.

Protection Against Harm: Reducing the possibility of negative outcomes or
dangers associated with taking part in the study, such as psychological
suffering, confidentiality violations, or inadvertent publication of private
information [26]. It is the responsibility of researchers to identify and
reduce any hazards to participants by implementing suitable precautions and
risk-reduction techniques.

Respect for Intellectual Property Rights: Encouraging authors and other
contributors to primary studies to have their intellectual property rights
acknowledged and respected, particularly by properly attributing and citing
their sources [22]. Unauthorized use of copyright material and plagiarism are
strictly forbidden and may be considered ethical misconduct.

Ethical Guidelines Compliance: Following applicable rules and guidelines on
research conduct, such as those issued by institutional review boards (IRBs),
data protection laws (like GDPR), and professional ethics codes (such as the
ACM Code of Ethics and Professional Conduct) [23]. The integrity, legitimacy,
and credibility of the research process and its results are guaranteed by
adherence to ethical norms.

Every step of the SLR process—from study design and data collecting to
analysis, interpretation, and results dissemination—incorporates ethical
considerations. Building trust, accountability and integrity within the
research community is the responsibility of researchers who also have a duty
to promote ethical conduct throughout the study effort.

##  VII Findings

The comprehensive analysis of primary research studies has provided
significant information on the role of penetration testing in mitigating
insider threats. Each study was meticulously reviewed and relevant qualitative
and quantitative information was extracted and summarized. These studies cover
various aspects of insider threats, focusing on threat types, detection
techniques, and mitigation strategies. The primary themes identified in these
studies are threat detection, threat mitigation, behavior modeling, and the
integration of penetration testing with other security practices. The
distribution of these themes is graphically represented in Figure 3 [18].

Main Themes Identified:

Threat Detection (28.6%): The largest portion of studies (28. 6%) focuses on
threat detection. These studies investigate various techniques to identify and
monitor insider activities that may pose security risks. Methods such as
behavior analytics, machine learning algorithms, and anomaly detection systems
are commonly used. For example, Williams and Smith (2020) demonstrated that
organizations conducting regular penetration tests experienced a notable
reduction in insider threat incidents attributed to enhanced detection
capabilities. These detection strategies are crucial for recognizing and
addressing potential threats before they can cause significant harm [19].

Threat Mitigation (34.3%): Threat mitigation strategies are the most
prevalent, comprising 34.3% of the reviewed studies. These strategies aim to
eliminate or reduce the impact of identified threats through robust access
controls, regular security audits, and comprehensive employee training
programs. AlHogail and Mirza (2021) developed an integrated framework that
combines penetration testing with other security measures, which significantly
improved overall mitigation of insider threats [20]. These mitigation efforts
are essential for addressing vulnerabilities and ensuring a secure
organizational environment [21].

Behavior Modeling (22.9%): Behavior modeling was the central theme in 22.9% of
the studies. This approach involves analyzing and predicting insider behaviors
to prevent malicious activities. Techniques such as creating user profiles,
monitoring communication patterns, and performing psychological assessments
are utilized to understand and predict potential insider threats. Lee and Kim
(2018) used behavior modeling in the healthcare sector, successfully
identifying potential insider threats through the analysis of usage patterns
of the electronic health record (EHR) system. This focus on behavior modeling
highlights the importance of understanding human behaviors in predicting and
preventing insider attacks [25].

Threat Type (14.3%): The remaining 14.3% of the studies discussed the
classification and analysis of various types of insider threats. These studies
categorized insider threats based on their characteristics, motivations, and
methods. Understanding the different types of insider threats is crucial for
developing targeted mitigation strategies. For example, some studies focused
on identifying insider threats related to data exfiltration, while others
examined threats associated with sabotage or fraud [26].

##  VIII Discussion

Initial keyword searches reveal a substantial body of literature on insider
threats, reflecting the growing concern about these issues in the
cybersecurity landscape. The review highlights that traditional security
measures often fall short in detecting and mitigating insider threats due to
the insiders’ inherent knowledge of the organization’s systems. Several
studies emphasize the necessity of advanced detection methods that incorporate
user behavior modeling and machine learning algorithms to identify anomalies
indicative of insider threats.

For example, [S2] utilizes behavior modeling and algorithms to detect insider
threats using log-based data such as browser history and email contacts [22].
This approach, which evaluates the accuracy of the algorithm and false
positive rates, underscores the importance of understanding user behavior to
detect potential threats [23]. Similarly, [S3] introduces machine learning
algorithms to classify insider behaviors into categories, suggesting
enhancements to algorithmic accuracy and highlighting the need for
sophisticated analysis to understand insider activities [26]. The practical
application of penetration testing is also explored in several studies. [S1]
focuses on defense measures against insider threats, identifying five defense
solution approaches through a review of the literature and framework analysis
[20]. This categorization into prevention, detection, and response strategies
provides a comprehensive overview of how organizations can structure their
defense mechanisms [21]. [S7] reviews strategies like honeypots in SCADA
systems for detecting and mitigating insider threats, evaluating their
effectiveness and proposing improvements, thus demonstrating the practical
utility of these

TABLE II: Summary of Primary Studies Primary Study |  Key Findings and Contributions |  Methodology Used |  Key Qualitative and Quantitative Data Reported |  Relevant Security Applications
---|---|---|---|---
[S1] |  Focuses on defense measures against insider threats; identifies five defense solution approaches. |  Literature review, framework analysis |  Reviewed top 100 papers; categorized defense strategies into prevention, detection, and response. |  Threat Detection, Mitigation
[S2] |  Utilizes behavior modeling and algorithms to detect insider threats using log-based data such as browser history and email contacts. |  Experimental design, data mining |  Developed behavior models; evaluated algorithm accuracy and false positive rates. |  Behavior Model, Threat Detection
[S3] |  Introduces machine learning algorithms for classifying insider behaviors into categories; proposes future research on behavior analysis. |  Theoretical study, algorithm development |  Classified insider behaviors into four classes; suggested enhancements for algorithmic accuracy. |  Behavior Model, Threat Detection
[S4] |  Assesses impact of insider threats on critical infrastructure sectors like healthcare; evaluates existing defense mechanisms. |  Case study, survey analysis |  Analyzed data on insider threat incidents in healthcare; evaluated effectiveness of current defenses. |  Threat Mitigation
[S5] |  Investigates patterns of insider attacks and data exfiltration techniques; proposes countermeasures for protecting sensitive data. |  Survey, case study |  Surveyed data leakage patterns; assessed frequency of attacks on different data states. |  Threat Type, Threat Mitigation
[S6] |  Defines types of insider threats and their attack methods; surveys prevalence and impacts of insider attacks across industries. |  Survey, taxonomy development |  Classified insider types; analyzed statistical data on insider attack methods. |  Threat Type, Behavior Model
[S7] |  Reviews strategies like honeypots in SCADA systems for detecting and mitigating insider threats; assesses their effectiveness. |  System analysis, experimental study |  Evaluated effectiveness of honeypots in SCADA environments; proposed improvements. |  Threat Detection, Threat Mitigation
[S8] |  Examines perception models of insider risk; proposes a framework for managing insider threats based on risk assessment. |  Conceptual framework, survey |  Surveyed risk perception among stakeholders; developed risk assessment strategies. |  Behavior Model, Threat Mitigation
[S9] |  Explores psychological and social impacts of insider attacks on organizational trust dynamics; analyzes behavioral shifts. |  Qualitative study, case analysis |  Analyzed trust dynamics post-insider attacks; identified behavioral patterns among affected employees. |  Behavior Model
[S10] |  Evaluates effectiveness of Data Leakage Prevention Systems (DLPS); identifies weaknesses and proposes enhancements. |  Comparative study, system evaluation |  Evaluated DLPS performance metrics; identified vulnerabilities and suggested improvements. |  Threat Mitigation
[S11] |  Discusses InfoSec approaches for detecting and mitigating insider threats; evaluates their applicability in diverse organizational contexts. |  Literature review, case study |  Reviewed InfoSec frameworks; analyzed case studies of successful insider threat detections. |  Behavior Model, Threat Detection
[S12] |  Introduces user profiling for predicting intentional and accidental insider threats; proposes a user-based evaluation methodology. |  Experimental design, user study |  Developed user profiles for threat prediction; evaluated predictive accuracy based on user attributes. |  Threat Mitigation
[S13] |  Reviews insider behavior impacts on organizational and cloud-level security; proposes behavioral analytics solutions. |  Case study, data analysis |  Analyzed impacts of insider incidents; evaluated effectiveness of behavioral analytics in detecting anomalies. |  Threat Mitigation
[S14] |  Proposes a re-authentication method using ensemble techniques to verify user authenticity; compares with traditional methods. |  Experimental setup, algorithm testing |  Tested re-authentication method performance metrics; compared with existing authentication methods. |  Threat Mitigation
[S15] |  Implements Internal Intrusion Detection and Protection System (IIDPS) using system-call patterns; evaluates system-call pattern recognition. |  Experimental setup, system analysis |  Developed IIDPS for detecting insider threats; evaluated system-call pattern recognition accuracy. |  Behavior Model, Threat Detection
[S16] |  Presents a detection method resilient against mimicry-based evasion strategies; validates effectiveness through simulations. |  Simulation study, algorithm development |  Simulated evasion scenarios; validated detection method performance against mimicry attacks. |  Threat Detection
[S17] |  Compares supervised and unsupervised learning techniques for detecting insider threats; analyzes learning model effectiveness. |  Comparative analysis, machine learning |  Evaluated detection accuracy of learning models; compared supervised vs. unsupervised learning for insider threat detection. |  Threat Mitigation

Primary Study |  Key Findings and Contributions |  Methodology Used |  Key Qualitative and Quantitative Data Reported |  Relevant Security Applications
---|---|---|---|---
[S18] |  Discusses risks posed by hardware trojans in USB devices; proposes detection and prevention strategies for hardware-based attacks. |  Theoretical study, hardware analysis |  Analyzed risks of hardware trojans; proposed detection methods and countermeasures. |  Threat Detection
[S19] |  Uses forensic investigation techniques for insider threat prevention; monitors data packets for exfiltration of sensitive information. |  Forensic analysis, data monitoring |  Monitored data packet content; evaluated effectiveness of forensic techniques in insider threat detection. |  Threat Mitigation
[S20] |  Proposes insider attack authentication protocol using ECC cryptography; evaluates protocol security and computational efficiency. |  Experimental design, cryptographic analysis |  Analyzed security strengths of the ECC protocol; evaluated computational overhead in authentication from insider attacks. |  Threat Detection
[S21] |  Develops intrusion sensitivity-based trust management model for collaborative IDS; assesses IDS trustworthiness. |  Model development, trust analysis |  Developed trust management model for IDS; evaluated trustworthiness assessment in collaborative IDS networks. |  Threat Detection
[S22] |  Designs IoT model with intrusion detection agents using supervised and unsupervised learning; validates through simulations. |  IoT simulation, machine learning |  Simulated IoT environments; evaluated performance of intrusion detection agents using learning models. |  Threat Detection
[S23] |  Implements homomorphic linear authenticator (HLA) for verifying data packet integrity; evaluates authenticity verification in data transmission. |  Experimental setup, cryptographic analysis |  Evaluated authenticity verification metrics; assessed HLA performance in detecting packet tampering. |  Behavior Model, Threat Detection
[S24] |  Deploys OS-resident deception strategy for neutralizing malware-infected machines; evaluates effectiveness in malware containment. |  System implementation, malware analysis |  Tested deception strategy effectiveness; assessed impact on malware containment in insider threat scenarios. |  Threat Detection
[S25] |  Uses Markov model IDS and virtual honeypot device (VHD) to detect malicious devices in fog computing environments. |  System integration, simulation analysis |  Integrated security systems in fog computing; evaluated detection accuracy of IDS and VHD in insider threat detection. |  Threat Detection
[S26] |  Addresses security issues in fog computing; proposes solutions like dummy document creation and transparency measures. |  Conceptual framework, system analysis |  Proposed fog computing security solutions; evaluated effectiveness of transparency measures in insider threat prevention. |  Threat Detection
[S27] |  Introduces smart reaction mechanism to manage operational risks and analyze insider threat scenarios; Evaluates the effectiveness of the mechanism. |  Theoretical framework, case study |  Developed operational risk management model; assessed scenario analysis and response mechanism in insider threat incidents. |  Threat Mitigation
[S28] |  Develops management protocol for preventing security attacks in mobile ad hoc networks (MANETs); evaluates protocol efficiency. |  Protocol design, simulation analysis |  Designed security management protocol for MANETs; evaluated protocol performance in preventing insider attacks. |  Threat Mitigation
[S29] |  Designs software package using TCP tunneling and hardware devices for detecting insider attacks; assesses detection accuracy. |  Software development, hardware integration |  Developed detection software; evaluated TCP tunneling and hardware device effectiveness in detecting insider threats. |  Threat Type
[S30] |  Proposes process model-based framework for real-time log data analysis to detect insider threats; validates through case studies. |  Model development, case analysis |  Developed real-time log analysis framework; validated effectiveness in detecting insider threats through case studies. |  Threat Detection
[S31] |  Uses MapReduce and clustering for analyzing user behavior based on log data; identifies unknown threats through supervised learning. |  Data analysis, machine learning |  Analyzed user behavior using MapReduce; identified and classified unknown threats using clustering and supervised learning. |  Threat Type, Behavior Model
[S32] |  Describes approach for detecting the exploitation of organizational resources by system users; proposes behavior-based anomaly detection. |  Conceptual framework, behavior analysis |  Proposed behavior-based anomaly detection; assessed effectiveness in identifying resource exploitation by insiders. |  Threat Detection
[S33] |  Applies machine learning algorithms to CERT Insider Threat Dataset for calculating threat detection results; compares algorithm performance. |  Dataset analysis, algorithm evaluation |  Evaluated machine learning algorithms on CERT dataset; compared performance metrics for insider threat detection. |  Threat Mitigation

methodologies in real-world scenarios [25]. Log management and real-time data
analysis are critical components of an effective insider threat mitigation
strategy. Articles [S33] and [S34] describe the importance of analyzing log
data from servers, firewalls, and other IT infrastructure to detect suspicious
activities [26]. In addition, integration of penetration testing with other
security practices is crucial. For example, [S12] introduces user profiling to
predict intentional and accidental insider threats, proposing a user-based
evaluation methodology [22]. This study demonstrates how profiling and regular
security audits can complement penetration testing to enhance overall security
posture [23]. [S13] reviews the impacts of insider behavior on organizational
and cloud-level security, proposing behavioral analytics solutions to detect
anomalies, thereby showcasing the integration of multiple security measures to
address insider threats comprehensively [20].

Figure 1: Bar Graph

Despite these advances, several challenges persist. Studies such as [S6] and
[S9] highlight the difficulties in accurately modeling diverse insider threat
scenarios and the resource constraints that organizations, especially SMEs,
face in implementing comprehensive penetration testing strategies.
Additionally, the complexity of integrating and maintaining advanced
technological solutions underscores the need for scalable and adaptable
approaches [26].

Research Question 1: What are the current methodologies and frameworks used in
penetration testing to mitigate insider threats?

Insider threats provide a complicated cybersecurity challenge since they
frequently use their insider knowledge and privileged access to launch
attacks. Comprehending the approaches and structures utilized in penetration
testing, particularly with regard to insider threat mitigation, requires a
thorough analysis of current procedures and their efficacy. White-box, black-
box, and grey-box penetration testing techniques are examples of penetration
testing procedures that provide the framework for simulating actual attack
scenarios. These techniques are tailored to target weaknesses that insiders
could exploit, such as manipulation of internal systems, unlawful access to
sensitive data, or privilege escalation. For example, white-box testing gives
testers complete access to the underlying architecture and source code,
enabling a thorough analysis of any vulnerabilities from the point of view of
an insider [25].

Insider threat behaviors can be categorized and understood more structuredly
with the use of frameworks like MITRE ATT&CK (Adversarial Tactics, Techniques,
and Common Knowledge). These frameworks classify the strategies and methods
that attackers employ, offering penetration testers a taxonomy to help them
find and fix flaws. Organizations can improve their capacity to identify
insider threats and take proactive measures to counter them by coordinating
penetration testing methods with frameworks such as MITRE ATT&CK [26].

Empirical studies and case examples illustrate the practical application of
penetration testing methodologies to detect and mitigate insider threats.
Research studies (e.g., [S1], [S5], [S7]) provide empirical evidence of how
penetration testing has been utilized to identify vulnerabilities that could
potentially be exploited by insiders. These studies highlight the importance
of tailored testing scenarios that mimic insider threat behaviors, ensuring
comprehensive coverage of potential attack vectors within organizational
networks [22].

Research Question 2: How effective are behavior modeling and machine learning
techniques in enhancing insider threat detection during penetration testing?

Advanced methods, such as behavior modeling and machine learning, enhance
conventional penetration testing strategies by concentrating on unusual
behaviors suggestive of insider threats. In order to create baseline behaviors
and spot variances that can indicate possible dangers, behavior modeling
techniques examine user activities, communication patterns, and access habits.

Algorithms for machine learning, such as supervised and unsupervised learning
techniques, are essential for improving penetration testing frameworks’
detection powers. Although unsupervised learning algorithms can find anomalies
that depart from established norms without prior training data, supervised
learning algorithms are trained on labeled datasets to identify patterns
linked with recognized insider threat behaviors. With the help of these
algorithms, penetration testers can quickly and effectively evaluate enormous
volumes of data and spot subtle signs of insider threats that could go
undetected by conventional rule-based detection methods [23].

Empirical evaluations (see, e.g., [S2], [S8], [S10], [S12]) provide insight
into the performance and accuracy of behavior modeling and machine learning
techniques in real-world insider threat scenarios. These studies demonstrate
how machine learning models can complement penetration testing efforts by
improving detection rates and reducing false positives associated with insider
threat alerts. By leveraging machine learning, organizations can enhance their
ability to detect sophisticated insider threats that exhibit subtle or
evolving behaviors over time [26].

Research Question 3: What are the key challenges and limitations faced in
implementing penetration testing for insider threat mitigation across
different organizational contexts?

The scale of the business, the nature of the industry, and the regulatory
constraints all play a role in the hurdles that must be overcome when
implementing penetration testing for the mitigation of insider threats. These
difficulties affect the scalability and efficacy of penetration testing
techniques intended to reduce insider risks in a variety of organizational
settings. Organizations looking to conduct extensive penetration testing
programs face substantial obstacles due to resource restrictions, such as
limited finances and limited access to qualified cybersecurity personnel. The
capacity of small and medium businesses (SMEs) to identify and successfully
counter internal threats can be hampered by a lack of funding for regular and
in-depth penetration examinations [22].

The organizational culture has a significant impact on the way penetration
testing techniques are adopted and how successful they are. Businesses that
emphasize cybersecurity investments and encourage a proactive approach to
threat identification and mitigation are those that have a strong security
culture. Organizations with a reactive or complacent security culture, on the
other hand, could downplay the importance of penetration testing or oppose the
adoption of new security measures because they believe they will cause
operational problems [23]. Penetration testing techniques are further
complicated by regulatory compliance obligations, such as GDPR (General Data
Protection Regulation) and PCI-DSS (Payment Card Industry Data Security
Standard). Companies that want to ensure compliance, protect sensitive data
and maintain customer trust must coordinate their penetration testing efforts
with legal requirements [26].

Case studies (e.g., [S4], [S6], [S9]) highlight the diverse challenges faced
by organizations in implementing penetration testing for insider threat
mitigation. These studies underscore the importance of tailoring penetration
testing strategies to address specific organizational contexts, industry
sectors, and regulatory environments effectively [20].

Research Question 4: What are the emerging trends and future directions in
penetration testing methodologies for combating insider threats?

The dynamic nature of cybersecurity is driving improvements in penetration
testing techniques aimed at effectively countering insider attacks. The
integration of novel technologies and flexible frameworks to improve detection
and response capabilities is the main focus of emerging trends and future
directions in penetration testing. Through the automation of repetitive
operations, the analysis of large datasets, and the detection of subtle
indicators of insider threats, artificial intelligence (AI) and machine
learning (ML) are transforming penetration testing. Predictive analytics and
anomaly detection algorithms are used by AI-driven penetration testing
platforms to proactively find possible vulnerabilities before insiders can
take advantage of them [20].

Penetration testing frameworks can benefit from additional security
protections provided by blockchain technology and safe authentication
procedures. Secure decentralized transaction records made possible by
blockchain technology can be used to verify the accuracy of penetration
testing findings and protect private information from modification or illegal
access. Dynamic defenses against insider threats are provided by adaptive
security frameworks such as continuous monitoring and zero-trust architecture,
which authenticate each user and device trying to access organizational
resources. These platforms combine behavioral analytics with real-time threat
intelligence feeds to quickly identify and address insider threats [22].

Pilot studies and prototypes (e.g., [S20], [S21]) demonstrate the feasibility
and efficacy of AI-driven penetration testing methodologies in real-world
scenarios. These studies illustrate how organizations can leverage emerging
technologies to enhance their cybersecurity posture and mitigate insider
threats proactively [26].

In order to effectively manage changing internal threat landscapes,
penetration testing procedures must be continuously innovative and adaptable
according to expert points of view and industry perspectives. Organizations
may fortify their defenses against insider threats and protect vital assets
from bad actors by adopting new trends and directions in penetration testing
techniques [23].

Research Question 5: How do regulatory frameworks and compliance requirements
influence the adoption and effectiveness of penetration testing for insider
threat mitigation?

Regulations have a significant influence on cybersecurity practices and
regulations, which includes penetration testing adoption and efficacy in
minimizing insider threats. Analyzing industry norms, organizational readiness
to follow regulations, and compliance requirements is necessary to understand
how regulatory compliance affects penetration testing tactics. Organizations
must adhere to strict regulations, including GDPR, PCI-DSS, and HIPAA (Health
Insurance Portability and Accountability Act), in order to protect sensitive
data and successfully manage cybersecurity risks. In order to find and fix
vulnerabilities that insiders could exploit, these standards require regular
penetration tests and vulnerability assessments [22].

The intricacy of regulatory mandates and the necessity for entities to
proficiently comprehend and execute them give rise to compliance issues. The
effectiveness of penetration testing procedures in adhering to legal
requirements and preserving data security and privacy is assessed through
compliance audits and assessments [26]. Case studies (such as [S18], [S22])
demonstrate how regulatory frameworks affect penetration testing procedures in
various business sectors and geographical areas. These studies emphasize how
crucial it is to strike a balance between operational requirements and
preserving business continuity when coordinating penetration testing efforts
with regulatory demands [25].

Creating comprehensive testing methods, recording testing processes, and
guaranteeing transparency in reporting results and repair activities are some
strategies for attaining regulatory compliance through penetration testing.
Organizations can demonstrate due diligence in protecting sensitive
information from insider threats and improve cybersecurity resilience by
including compliance requirements into penetration testing frameworks [20]. In
order to effectively address new threats and regulatory changes, penetration
testing procedures must be continuously monitored and adjusted. This is
highlighted by the implications of shifting regulatory frameworks for the
future [23]. To prevent insider threats early on and maintain regulatory
compliance over time, organizations need to be aware of industry best
practices and regulatory trends [26].

##  IX Future Research Directions of Insider Threats

Based on the systematic review of the literature on insider threats mitigation
with a focus on penetration testing, several promising research directions
emerge that can further advance the field:

  1. 1.

Advanced Threat Modeling: Examine and create more advanced threat models that
are suited to insider threats. There are gaps in our knowledge of the distinct
behaviors and assault vectors of insiders because current models frequently
concentrate on external threats. Creating thorough threat models that take
different insider motivations and behaviors into consideration will improve
detection and mitigation tactics [20].

  2. 2.

Behavioral Analytics and Machine Learning: Investigate further how penetration
testing frameworks can incorporate machine learning and behavioral analytics
methodologies. Research ought to concentrate on improving insider threat
detection’s precision and effectiveness by utilizing AI-driven methods to
examine intricate behavioral patterns and anomalies suggestive of malevolent
intent [25].

  3. 3.

Real-time Monitoring and Response: Investigate techniques and tools for
tracking insider activity in real time during penetration testing operations.
Provide systems for adaptive response that can be set up to automatically
react to suspicious behaviors or abnormalities found in organizational
networks. This will reduce response times and lessen the potential harm that
insider threats could do [26].

  4. 4.

Human Factors and Insider Psychology: Investigate the behavioral and
psychological elements of insider threats. Examine the ways in which
situational elements, corporate culture, and personal traits affect insider
risk behaviors. Strategies for prevention and intervention that are more
focused can be developed by taking into account the human aspects of insider
threats [22].

  5. 5.

Integration of Blockchain and Secure Authentication: Examine how to improve
the integrity and security of penetration testing results by implementing
secure authentication methods and blockchain technology. Examine how
blockchain technology can be used to safely store and authenticate penetration
test data, guaranteeing the dependability and correctness of results while
shielding private data from illegal access [23].

  6. 6.

Regulatory Compliance and Governance: Examine the impact of regulatory
frameworks on the implementation and effectiveness of penetration testing for
insider threat mitigation. Research should focus on aligning penetration
testing practices with evolving regulatory requirements, ensuring compliance
while maintaining robust cybersecurity measures against insider threats [20].

  7. 7.

Cross-sector Collaboration and Information Sharing: Encourage cooperation and
information-sharing programs between businesses, academic institutions, and
governmental entities in order to combat insider threats as a group. To
increase collective resistance against insider threats across many industries
and geographical areas, develop mechanisms for exchanging anonymized threat
data and best practices in penetration testing [25].

  8. 8.

Evaluation of Emerging Technologies: Examine how well-suited and scalable new
technologies are for identifying and reducing insider risks. Examples of these
technologies include AI-driven penetration testing platforms and Internet of
Things (IoT) security frameworks. Research ought to concentrate on the
potential advantages of incorporating these technologies into current
cybersecurity infrastructures as well as the real-world implementation issues
[26].

In this systematic review of the literature, insider threat analysis and
mitigation techniques have been thoroughly examined, with a particular
emphasis on the function of penetration testing in cybersecurity. The results
emphasize the complexity of identifying and reducing insider threats in
organizational settings, underscoring the diverse character of these risks
[22].

Many approaches and technologies have been considered during the evaluation,
exposing both the improvements and ongoing difficulties in effectively
tackling insider threats. One essential tool for finding weak points and
evaluating how resilient organizational defenses are to insider threats is
penetration testing [20].

The literature reviewed emphasizes that while considerable progress has been
made in understanding insider threats, significant gaps and opportunities for
future research remain.

Future Research Directions

  1. 1.

Improved Behavioral Modeling: More study should be done to improve behavioral
modeling methods for identifying insider threats. This entails using cutting-
edge AI and machine learning algorithms to examine intricate user behavior and
identify minute variations that point to malevolent intent [25].

  2. 2.

Psychological and Motivational Factors: It is important to look at the
psychological and motivational elements that contribute to insider threats.
More focused preventative and intervention techniques can be developed by
having a better understanding of the triggers and actions of insiders, whether
they be unhappy workers or unintentional collaborators [26].

  3. 3.

Integration of Emerging Technologies: investigating how to incorporate
cutting-edge technologies like blockchain for safe data authentication and
Internet of Things security frameworks to keep an eye on insider activity.
Assessing the efficiency and expandability of these technologies in practical
situations can improve their use in reducing insider threats [22].

  4. 4.

Regulatory Compliance and Governance: Matching industry norms and changing
regulations with penetration testing procedures. Future studies should
concentrate on creating frameworks that protect cybersecurity against internal
threats and guarantee compliance [23].

  5. 5.

Cross-sector Collaboration: Encouraging cooperation and information exchange
between government agencies, academic institutions, and businesses in order to
combat insider threats as a group. Creating networks for exchanging best
practices and anonymized threat intelligence can strengthen cybersecurity
defenses in a variety of industries [20].

##  X Conclusion

This systematic literature review provides a comprehensive overview of the
role of penetration testing in mitigating insider threats. The findings
underscore the critical importance of penetration testing as part of a broader
insider threat mitigation strategy. By synthesizing existing research,
identifying challenges, and exploring real-world applications, this review
highlights several key insights:

Effectiveness of Penetration Testing: Regular and comprehensive penetration
testing significantly reduces the frequency of insider threat incidents.
Tailored testing scenarios that simulate insider behaviors are essential for
identifying vulnerabilities [25].

Integration with Other Security Practices: Combining penetration testing with
user behavior analytics, access control mechanisms, and employee training
programs enhances the overall effectiveness of insider threat mitigation
strategies [26].

Challenges and Limitations: Organizations face several challenges, including
resource constraints, modeling insider scenarios in the real world, and
ensuring regulatory compliance. Addressing these challenges requires a
multidisciplinary approach and continuous adaptation to evolving threats [20].

Future Research Directions: Advancing the field of insider threat mitigation
necessitates further research in areas such as advanced threat modeling,
behavioral analytics, real-time monitoring, and the integration of emerging
technologies like blockchain and AI-driven systems [22].

In conclusion, penetration testing plays a pivotal role in strengthening an
organization’s defenses against insider threats. By continuously evolving and
integrating with other security practices, organizations can proactively
identify and mitigate potential risks, ensuring a robust security posture.

##  XI Declarations of Interest

None.

##  XII Acknowledgment

I would also like to extend my deepest gratitude to Professor Abbas Yazdinejad
for their assistance in completing the project.

## References

  * [1] M. Rana and Z. Baig, “A survey of recent advances, challenges, and opportunities in insider threat detection,” _IEEE Access_ , 2019.
  * [2] X. Yao and Y. Qiao, “Auth: An adversarial autoencoder based unsupervised insider threat detection system,” _IEEE Access_ , 2024.
  * [3] V. Patel and R. Arora, “Detecting insider threats in cybersecurity using machine learning and data analytics,” _IEEE Access_ , 2022.
  * [4] S. Alshehri, S. Mishra, and R. Raj, “Insider threat mitigation and access control in healthcare systems,” _Technical Report, Dochester Institute of Technology_ , 2013.
  * [5] A. Yazdinejad, R. Parizi, A. Dehghantanha, Q. Zhang, and K.-K. Choo, “An energy-efficient sdn controller architecture for iot networks with blockchain-based security,” _IEEE Transactions on Services Computing_ , vol. 13, no. 4, pp. 625–638, 2020.
  * [6] A. Yazdinejad, R. Parizi, A. Dehghantanha, and K.-K. Choo, “Blockchain-enabled authentication handover with efficient privacy protection in sdn-based 5g networks,” _IEEE Transactions on Network Science and Engineering_ , vol. 8, no. 2, pp. 1120–1132, 2019.
  * [7] J. Schwartz and H. Kurniawati, “Autonomous penetration testing using reinforcement learning,” _arXiv preprint_ , 2019.
  * [8] S. Yuan and X. Wu, “Deep learning for insider threat detection: Review, challenges, and opportunities,” _arXiv preprint_ , 2020.
  * [9] L. Giddens, D. Leidner, and E. Gonzalez, “Gender bias and the impact on managerial evaluation of insider security threats,” _Computers & Security_, vol. 99, p. 102066, 2020.
  * [10] R. Zhou and R. Gupta, “Improving insider threat detection with user and role-based behavior,” _IEEE Transactions on Information Forensics and Security_ , 2020.
  * [11] F. Greitzer and D. Frincke, “Combining traditional cyber security audit data with psychosocial data: Towards predictive modeling for insider threat mitigation,” _Insider Threats in Cyber Security_ , 2010.
  * [12] J. Wang and Z. Yang, “Testing individual differences in phishing vulnerability within an integrated, information processing model,” _Journal of Business Research_ , vol. 68, no. 3, pp. 516–525, 2015.
  * [13] P. Dutta and A. Sengupta, “A structured control selection methodology for insider threat mitigation,” _Procedia Computer Science_ , vol. 181, pp. 1003–1012, 2022.
  * [14] O. Balogun and D. Takabi, “An insider threat mitigation framework using attribute based access control,” _arXiv preprint_ , 2023.
  * [15] U. Oparaji and S. Phithakkitnukoon, “Towards a new insider threat mitigation framework,” _IEEE Access_ , 2022.
  * [16] L. Huang and Q. Zhu, “A dynamic games approach to proactive defense strategies against advanced persistent threats in cyber-physical systems,” _Computers & Security_, vol. 89, p. 101660, 2020.
  * [17] S. Ahmed and J. Shamsi, “Performance analysis of an insider threat mitigation model,” _IEEE Access_ , 2018.
  * [18] S. Temara, “Maximizing penetration testing success with effective reconnaissance techniques using chatgpt,” _arXiv preprint_ , 2023.
  * [19] S. Parnitha and P. Acharya, “Combating insider threats using behavioral analytics,” _IEEE Transactions on Information Forensics and Security_ , 2021.
  * [20] A. Ghorbani and M. Tavallaee, “Information security insider threats in organizations and mitigation strategies,” _IEEE Transactions on Information Forensics and Security_ , 2019.
  * [21] A. Sengupta, C. Mazumdar, and A. Bagchi, “A formal methodology for detecting managerial vulnerabilities and threats in an enterprise information system,” _Journal of Network and Systems Management_ , vol. 19, no. 3, pp. 319–342, 2019.
  * [22] A. Yazdinejad, A. Dehghantanha, R. Parizi, G. Srivastava, and H. Karimipour, “Secure intelligent fuzzy blockchain framework: Effective threat detection in iot networks,” _Computers in Industry_ , vol. 144, p. 103801, 2023.
  * [23] A. Yazdinejad, B. Zolfaghari, A. Azmoodeh, A. Dehghantanha, H. Karimipour, E. Fraser, A. Green, C. Russell, and E. Duncan, “A review on security of smart farming and precision agriculture: Security aspects, attacks, threats and countermeasures,” _Applied Sciences_ , vol. 11, no. 16, p. 7518, 2021.
  * [24] Q. Yaseen and B. Panda, “Insider threat mitigation: Preventing unauthorized knowledge acquisition,” _International Journal of Information Security_ , vol. 11, no. 4, pp. 269–280, 2012.
  * [25] B. Ampel, S. Samtani, H. Chen, and J. J. Nunamaker, “Improving threat mitigation through a cybersecurity risk management framework: A computational design science approach,” _Journal of Management Information Systems_ , vol. 41, no. 1, pp. 236–265, 2021.
  * [26] M. Ferrag and et al., “Generative ai and large language models for cyber security: All insights you need,” _arXiv preprint_ , 2024.
  * [27] A. Yazdinejad, M. Kazemi, R. Parizi, A. Dehghantanha, and H. Karimipour, “An ensemble deep learning model for cyber threat hunting in industrial internet of things,” _Digital Communications and Networks_ , 2022.


# Robinson–Trautman solution with scalar hair

T. Tahamtan<EMAIL_ADDRESS>O. Svítek<EMAIL_ADDRESS>Institute of
Theoretical Physics, Faculty of Mathematics and Physics, Charles University in
Prague, V Holešovičkách 2, 180 00 Prague 8, Czech Republic

(August 27, 2024)

###### Abstract

Explicit Robinson–Trautman solution with minimally coupled free scalar field
is derived and analyzed. It is shown that this solution contains curvature
singularity which is initially naked but later the horizon envelopes it. We
use quasilocal horizon definition and prove its existence in later retarded
times using sub- and supersolution method combined with growth estimates. We
show that the solution is generally of algebraic type II but reduces to type D
in spherical symmetry.

exact solution, black hole, scalar field

###### pacs:

04.20.Jb, 04.70.Bw

##  I Introduction

Solutions to Einstein equations with scalar field source provide very useful
tool for understanding relativity due to the simplicity of the source.
Recently, it becomes progressively plausible that such fields might really
exist (LHC) and potentially play a fundamental role in physics. In classical
General Relativity they were used to study counterexamples to black hole no-
hair theorems and in many other areas. These results were mostly based on
highly symmetric solutions and it is therefore important to provide solutions
with less or no symmetries to subsequently analyze if those results hold in
more generic situations and are not tied to a specific symmetry.

Robinson–Trautman spacetimes represent an important class of expanding
nontwisting and nonshearing solutions RobinsonTrautman:1960 ;
RobinsonTrautman:1962 ; Stephanietal:book  describing non-spherical
generalizations of black holes. In general, they do not posses any Killing
vectors thus providing important solutions devoid of symmetry. Many properties
of this family in four dimensions have been studied, especially in the last 25
years. In particular, the existence, asymptotic behaviour and global structure
of vacuum Robinson–Trautman spacetimes of type II with spherical topology were
investigated by Chruściel and Singleton Chru1 ; Chru2 ; ChruSin .
Robinson–Trautman solutions were shown to exist for generic, arbitrarily
strong smooth initial data for all positive retarded times, and to converge
asymptotically to corresponding Schwarzschild metric. Extensions across the
“Schwarzschild-like” future event horizon can only be made with a finite order
of smoothness. These results were generalized in podbic95 ; podbic97  to
Robinson–Trautman vacuum spacetimes with cosmological constant. These
cosmological solutions settle down to a Schwarzschild–(anti-)de Sitter
solution at large times $u$. Finally, the Chruściel–Singleton analysis was
extended to Robinson-Trautman spacetimes including matter, namely pure
radiation BicakPerjes ; PodSvi:2005 , showing that they approach the
spherically symmetric Vaidya–(anti-)de Sitter metric. Generally, the solutions
of this family settle down to physically important solutions. The location of
the horizon together with its general existence and uniqueness for the vacuum
Robinson–Trautman solutions has been studied by Tod tod . Later, Chow and Lun
chow-lun  analyzed some other useful properties of this horizon and made
numerical study of both the horizon equation and Robinson–Trautman equation.
These results were later extended to nonvanishing cosmological constant
PodSvi:2009 . The anisotropy of Robinson-Trautman horizon and its associated
asymptotic momentum was also used in the analytic explanation of an "antikick"
appearing in numerical studies of binary black hole mergers rezzolla .

Robinson–Trautman spacetimes (containing aligned pure radiation and a
cosmological constant) were also generalized to any dimension podolsky-
ortaggio . Existence of horizons was subsequently analyzed in Svitek2011 .
Finally, Robinson–Trautman solutions with p-form fields in arbitrary dimension
were derived recently ortaggio . One of the results mentioned therein rules
out the existence of aligned scalar field (where alignment refers to the
gradient of the field) for generic Robinson–Trautman case.

The solutions for "stringy" Robinson–Trautman spacetime corresponding to
Einstein–Maxwell–dilaton system were obtained in Guven . Recently, scalar
field solutions for Einstein–Maxwell–Lambda system with a conformally coupled
scalar field belonging to Plebański–Demiański family (containing type D
solutions of Robinson–Trautman class) were derived in Maeda .

##  II Vacuum Robinson–Trautman metric and field equations

The general form of a vacuum Robinson–Trautman spacetime can be given by the
following line element RobinsonTrautman:1960 ; RobinsonTrautman:1962 ;
Stephanietal:book ; GriffithsPodolsky:book

${\rm d}s^{2}=-2H\,{\rm d}u^{2}-\,2\,{\rm d}u\,{\rm
d}r+\frac{r^{2}}{\tilde{P}^{2}}\,({\rm d}y^{2}+{\rm d}x^{2}),$ (1)

where
${2H=\Delta(\,\ln\tilde{P})-2r(\,\ln\tilde{P})_{,u}-{2m/r}-(\Lambda/3)r^{2}}$,

$\Delta\equiv\tilde{P}^{2}(\partial_{xx}+\partial_{yy}),$ (2)

and $\Lambda$ is the cosmological constant. The metric depends on two
functions, ${\,\tilde{P}(u,x,y)\,}$ and ${\,m(u)\,}$, which satisfy the
nonlinear Robinson–Trautman equation

$\Delta\Delta(\,\ln\tilde{P})+12\,m(\,\ln\tilde{P})_{,u}-4\,m_{,u}=0\,.$ (3)

The function $m(u)$ might be set to a constant by suitable coordinate
transformation for vacuum solution.

The spacetime admits a geodesic, shearfree, twistfree and expanding null
congruence generated by ${\mbox{\boldmath$k$}=\partial_{r}}$. The coordinate
$r$ is an affine parameter along this congruence, $u$ is a retarded time
coordinate, and $x,y$ are spatial coordinates spanning transversal 2-space
with their Gaussian curvature (for ${r=1}$) being given by

$\mathcal{K}(x,y,u)\equiv\Delta(\,\ln\tilde{P})\,.$ (4)

For general fixed values of $r$ and $u$, the Gaussian curvature is
${\mathcal{K}/r^{2}}$ so that, as ${r\to\infty}$, they become locally flat.

##  III Solution coupled to a scalar field

We consider the following action, describing a scalar field minimally coupled
to gravity,

$S=\int d^{4}x\sqrt{-g}[\mathcal{R}+\nabla_{\mu}\varphi\nabla^{\mu}\varphi]$
(5)

where $\mathcal{R}$ is the Ricci scalar for the metric $g_{\mu\nu}$. The
massless scalar field $\varphi$ is supposed to be real and we use units in
which $c=\hbar=8\pi G=1$. By applying the variation with respect to the metric
for the action (5), we get Einstein equations

$\mathcal{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathcal{R}=T_{\mu\nu}.$ (6)

The energy momentum tensor generated by the scalar field is given by

$T_{\mu\nu}=\nabla_{\mu}\varphi\nabla_{\nu}\varphi-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\nabla_{\alpha}\varphi\nabla_{\beta}\varphi$
(7)

and the scalar field must satisfy corresponding field equation

$\Box\varphi(u,r)=0$ (8)

where $\Box$ is a standard d’Alembert operator for our metric (9).

For the matter of convenience we will be looking for the metric in the
following form

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-2(H(u,r)+K(u,x,y))\,{\rm d}u^{2}-2\,{\rm d}u{\rm d}r$ (9)
$\displaystyle+\frac{R(u,r)^{2}}{P(x,y)^{2}}({\rm d}x^{2}+{\rm d}y^{2})$

The scalar field is assumed to be function of $u$ and $r$ only
($\varphi(u,r)$). The dependence on $r$ means that the scalar field is not
aligned and thus is not ruled out by the results of ortaggio . The nontrivial
components of the Ricci tensor corresponding to the metric (9) are

$\displaystyle\mathcal{R}_{uu}$ $\displaystyle=$ $\displaystyle
2\left(2\frac{R_{,r}}{R}H_{,r}+H_{,rr}\right)(H+K)+2\frac{R_{,r}}{R}(H+K)_{,u}$
$\displaystyle-\frac{2}{R}(R_{,u}H_{,r}+R_{,uu})+\frac{P^{2}}{R^{2}}(K_{,xx}+K_{,yy})$
$\displaystyle\mathcal{R}_{rr}$ $\displaystyle=$
$\displaystyle-2\frac{R_{,rr}}{R}$ (10) $\displaystyle\mathcal{R}_{ru}$
$\displaystyle=$
$\displaystyle\mathcal{R}_{ur}=2\frac{R_{,r}H_{,r}-R_{,ru}}{R}+H_{,rr}$
$\displaystyle\mathcal{R}_{xx}$ $\displaystyle=$
$\displaystyle\mathcal{R}_{yy}=-\frac{1}{P^{2}}\left\\{k(x,y)+2(H+K)(RR_{,r})_{,r}+\right.$
$\displaystyle\left.+2RR_{,r}H_{,r}-2(RR_{,u})_{,r}\right\\}$

where as usual $()_{,x^{i}}=\frac{\partial}{\partial x^{i}}()$ and

$k(x,y)=\Delta(\,\ln P(x,y))$ (11)

where $\Delta$ is still given by expression (2) with $\tilde{P}$ replaced by
$P$.

We will use the following form of equations equivalent to Einstein equations
(6) coupled to energy momentum tensor (7)

$\mathcal{R}_{\mu\nu}=\varphi_{,\mu}\varphi_{,\nu}=\left(\begin{array}[]{cccc}\varphi_{,u}^{2}&\varphi_{,u}\varphi_{,r}&0&0\\\
\varphi_{,u}\varphi_{,r}&\varphi_{,r}^{2}&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right)$ (12)

From the above equations describing gravitational field and field equation for
the scalar field (8) we obtain the following expressions for unknown metric
functions and scalar field

$\displaystyle H(u,r)$ $\displaystyle=$
$\displaystyle\frac{r}{2U(u)}\frac{\partial U(u)}{\partial u}$ $\displaystyle
R(u,r)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{U(u)^{2}r^{2}-C_{0}^{2}}{U(u)}}$ $\displaystyle
K(u,x,y)$ $\displaystyle=$ $\displaystyle\frac{k(x,y)}{2U(u)}$ (13)
$\displaystyle\varphi(u,r)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\ln{\left\\{\frac{U(u)r-C_{0}}{U(u)r+C_{0}}\right\\}}$
$\displaystyle\Delta k(x,y)$ $\displaystyle=$ $\displaystyle{\alpha^{2}}$
$\displaystyle U(u)$ $\displaystyle=$ $\displaystyle\gamma
e^{\omega^{2}u^{2}+\eta u},$

in which $C_{0}\neq 0,\alpha,\eta,\gamma,\omega$ are constants and
$\omega=\frac{\alpha}{2C_{0}}$. In the following we will assume
$C_{0}>0,\alpha>0,\eta>0,\gamma>0$ for simplicity of discussion.

##  IV Properties of the solution

First, we should ensure that our solution really belongs to the Robinson-
Trautman family. This is simply confirmed by studying the properties of a null
congruence generated by vector $\mathbf{l}=\partial_{r}$. Such congruence is
geodesic, nontwisting, nonshearing and its expansion is given by

$\Theta_{\mathbf{l}}=2\frac{R_{,r}}{R}=\frac{2\,U(u)^{2}r}{U(u)^{2}r^{2}-C_{0}^{2}}.$
(14)

Evidently, the above expression is positive only for $r>\frac{C_{0}}{U(u)}$
which may seem not satisfactory. However by inspecting the Kretschmann scalar

$\kappa\sim\frac{1}{R(u,r)^{8}}$ (15)

and using (III) we immediately see that the geometry has singularities for
$r=\pm r_{0}=\pm\frac{C_{0}}{U(u)}$. Naturally, we are led to constrain the
range of coordinates to $r\in\left(\frac{C_{0}}{U(u)},\infty\right)$. In this
range the expansion (14) is everywhere positive, diverges at the singularity
and approaches zero at infinity (as $r\to\infty$). Also, one can check from
the line element that the singularity is a standard pointlike one. Due to the
asymptotic behaviour of function $U(u)$ (see (III)) the singularity tends to
$r_{0}=0$ as $u\to\infty$. The singularity appears due to the divergence of
the scalar field and its energy momentum tensor.

Asymptotically ($u\to\infty$), the scalar field itself is vanishing everywhere
outside the singularity (see (III)) while it diverges at $r=0$. So there would
be no scalar hair left outside when the spacetime settles down to the final
state. Indeed, our geometry approaches the original Robinson–Trautman form (1)
for $u\to\infty$ when we define $\tilde{P}(u,x,y)=P(x,y)/U(u)$. In this case
one can apply the Chruściel–Singleton analysis Chru1 ; Chru2 ; ChruSin  of
asymptotic behaviour to recover the spherical symmetry of the final state
which neccesarrily points to Schwarzschild solution.

When the singularity is present in our solution we will investigate if it is
covered by a horizon. Due to dynamical nature of the spacetime it is
preferable to use the quasilocal definitions of horizon — apparent horizon
hawking-ellis , trapping horizon hayward  or dynamical horizon krishnan . The
basic local condition is shared by all the standard horizon definitions: these
horizons are sliced by marginally trapped surfaces with vanishing expansion of
outgoing (ingoing) null congruence orthogonal to the surface. We will be
looking for the horizon hypersurface in the following form

$r=\mathcal{M}(u,x,y)$ (16)

and study the expansion of compact slices of such hypersurface given by
$u=u_{0}=const.$ (with $\mathcal{M}(u_{0},x,y)=M(x,y)$). The requirement of
compactness necessarily means that the two-spaces spanned by $x$ and $y$ are
compact as well. We construct null vector fields orthogonal to surface
$r=M(x,y)$

$\displaystyle\mathbf{l}$ $\displaystyle=$ $\displaystyle\partial_{r}$ (17)
$\displaystyle\mathbf{k}$ $\displaystyle=$
$\displaystyle\partial_{u}+\left[\frac{P^{2}}{2R^{2}}(M_{x}^{2}+M_{y}^{2})-(H+K)\right]\partial_{r}+$
(18) $\displaystyle+\frac{P^{2}}{R^{2}}(M_{x}\partial_{x}+M_{y}\partial_{y})$

that satisfy normalization condition $\mathbf{l}\cdot\mathbf{k}=-1$. From the
geometry of the situation one can deduce that congruence $\mathbf{l}$ is
outgoing while $\mathbf{k}$ is ingoing. The expansion of the congruence
generated by $\mathbf{l}$ is always positive (see (14) and the discussion
beneath) so we are looking for vanishing of expansion related to the other
congruence $\mathbf{k}$. These two conditions ($\Theta_{\mathbf{l}}>0$ and
$\Theta_{\mathbf{l}}=0$) mean that we are looking for the past horizon
according to the definition given by hayward . The second expansion is given
by

$\displaystyle\Theta_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}\left[\Delta M-(\ln{R})_{,r}(\nabla M\cdot\nabla
M)-\right.$ (19) $\displaystyle\left.-(K+H)(R^{2})_{,r}+(R^{2})_{,u}\right],$

where Laplace operator and scalar product denoted by dot correspond to metric
$h_{ij}dx^{i}dx^{j}=\frac{1}{P(x,y)^{2}}(dx^{2}+dy^{2})$ on the space $\Sigma$
spanned by $x,y$. So the horizon is given by the solution of the following
quasilinear elliptic partial differential equation

$\displaystyle\left\\{\Delta M-(\ln{R})_{,r}(\nabla M\cdot\nabla
M)-(K+H)(R^{2})_{,r}\right.$
$\displaystyle\left.+(R^{2})_{,u}\right\\}|_{r=M(x,y)\&u=u_{0}}=0$ (20)

where all dependence on $r$ is replaced by the function $M(x,y)$ and $u$ is
evaluated to arbitrary constant value $u_{0}$.

It is impossible to solve this equation generally but fortunately we can get
some useful information about the existence of solution using the technique
developed for the case of Robinson–Trautman spacetime in higher dimensions
Svitek2011 . The proof of existence of the solution to the same type of
quasilinear equation ($\Delta u=F(x,u,\nabla u)$) was given there by combining
several steps motivated by Kuo  and using results from Besse ; Boccardo ;
Gilbarg-Trudinger . The main issues were to provide an estimate for the
function $F$ of the form $|F|\leq B(u)(1+|{\nabla u}|^{2})$ (where $B(u)$ is
increasing function on ${\mathbb{R}}^{+}$), and to show the existence of a
sub- and a super-solution footnote  $u^{-}\leq u^{+}$, $u^{\pm}\in
C^{1,\beta}(\Sigma)\cap L^{\infty}(\Sigma)$ (here $C^{1,\beta}(\Sigma)$ are
Hölder continuous functions of some suitable index $\beta$). Then we know
there is a solution $u\in C^{2,\iota}(\Sigma)$ (for some $\iota$) satisfying
$u^{-}\leq u\leq u^{+}$.

In our case, to provide an estimate of the form $|M|\leq B(M)(1+|{\nabla
M}|^{2})$ (the norm is taken with respect to the two-dimensional metric
$h_{ij}$) for the horizon equation (IV) when considered in the form $\Delta
M=F(x,y,M,\nabla M)$ where

$F=(\ln{R(u_{0},M)})_{,r}|\nabla
M|^{2}+k(x,y)M-\frac{C_{0}^{2}U_{,u}(u_{0})}{U^{2}(u_{0})}$ (21)

one has to deal with the singular behaviour of $(\ln{R})_{,r}$ at
$r=\frac{C_{0}}{U(u)}$. We can do this either by removing the vicinity of
singularity from our domain
$r\in\mathbf{R}^{+}\setminus\left(\frac{C_{0}}{U(u)}(1-\delta),\frac{C_{0}}{U(u)}(1+\delta)\right)$
or by continuing (with some appropriate smoothing) the divergent function on
the problematic interval
$\left(\frac{C_{0}}{U(u)}(1-\delta),\frac{C_{0}}{U(u)}(1+\delta)\right)$ with
a constant value it attains on the boundary of the interval. Now, with all the
coefficients of the equation finite one can construct the bounding function
$B(u)$ easily and thus we can proceed to the construction of sub- and
supersolutions $M^{\pm}$.

First, we note that due to the selection of sign for the free constants made
at the end of previous section ($C_{0}>0,\alpha>0,\eta>0,\gamma>0$) we obtain
$U_{,u}>0$ if we restrict our attention to retarded time region
$u\in(-\frac{\eta}{2\omega^{2}},\infty)$. We can then understand our solution
as being given by initial conditions specified at
$u_{in}=-\frac{\eta}{2\omega^{2}}$ which corresponds to usual understanding of
Robinson–Trautman solution. As usual, we are looking for constant sub- and
supersolutions but we are unable to provide them independently of the value of
$u_{0}$. Generally, we can find the sub- and supersolutions in the following
cases:

  * •

$u_{0}<u_{1}=\frac{\min({k(x,y)})-\max({k(x,y)})\delta-
C_{0}\eta}{2C_{0}\omega^{2}}$

$\displaystyle M^{-}$ $\displaystyle=$ $\displaystyle 0$ (22) $\displaystyle
M^{+}$ $\displaystyle=$ $\displaystyle\frac{C_{0}}{U(u_{0})}(1-\delta)$ (23)

  * •

$u_{0}>u_{2}=\frac{\max({k(x,y)})(1+\delta)-C_{0}\eta}{2C_{0}\omega^{2}}$

$\displaystyle M^{-}$ $\displaystyle=$
$\displaystyle\frac{C_{0}}{U(u_{0})}(1+\delta)$ (24) $\displaystyle M^{+}$
$\displaystyle=$
$\displaystyle\frac{C_{0}^{2}U_{,u}(u_{0})}{\min({k(x,y)})U^{2}(u_{0})}$ (25)

Both bounds $u_{1}$ and $u_{2}$ are in the restricted range of coordinate $u$.
Evidently, the first case would provide existence of solution only beneath the
position of singularity (or, in other words, inside the singularity) which is
irrelevant and moreover we have already restricted the range of
$r\in\left(\frac{C_{0}}{U(u)},\infty\right)$. In the second case, one can
easily check that the necessary condition $M^{-}\leq M^{+}$ is indeed
satisfied for $u_{0}>u_{2}$ and we certainly have a horizon given by
$r=M(x,y)$ where $M^{-}\leq M(x,y)\leq M^{+}$. Note that we suppose that
$\min({k(x,y)})>0$ for the last estimate $M^{+}$ to be valid.

If we allow $k_{min}\equiv\min({k(x,y)})\leq 0$ (we define accordingly
$k_{max}\equiv\max({k(x,y)})$) we are unable to provide constant supersolution
in the case $u_{0}>u_{2}$. Instead we can use the knowledge of how Laplace
operator acts on $k(x,y)$ (III) and the observation that the first term of the
definition of function $F$ (21) is always positive to provide the following
non-constant supersolution

$M^{+}=c[k_{max}-k(x,y)]+\frac{C_{0}}{U(u_{0})}(1+\delta)$ (26)

where

$c=\frac{C_{0}\left[C_{0}U_{,u}(u_{0})-k_{min}U(u_{0})(1+\delta)\right]}{U^{2}(u_{0})(\alpha^{2}+k_{max}k_{min}-k_{min}^{2})}.$
(27)

This estimate works if $(\alpha^{2}+k_{max}k_{min}-k_{min}^{2})>0$.

Even-though our solution possesses singularity at any retarded time $u$ this
singularity appears to be initially naked and the horizon develops only in
later time.

##  V Algebraic type of the solution

Now, we would like to see if the geometry of our spacetime is sufficiently
general. Since vacuum Robinson–Trautman spacetime is generally of algebraic
type II we would like our solution to be at least of the same type and not
more special. Our preferred tetrad for determining the Weyl scalars of our
solution is given by different null vectors compared to (17)

$\displaystyle\mathbf{\tilde{l}}$ $\displaystyle=$ $\displaystyle\partial_{r}$
$\displaystyle\mathbf{\tilde{k}}$ $\displaystyle=$
$\displaystyle\partial_{u}-(H+K)\partial_{r}$ (28) $\displaystyle\mathbf{m}$
$\displaystyle=$ $\displaystyle\frac{P}{\sqrt{2}R}(\partial_{x}+I\partial{y})$

where $I$ is a complex unit. The Weyl spinor computed from this tetrad has
only the following nonzero components

$\displaystyle\Psi_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4UR^{2}}\left[\frac{1}{2}P(k_{,yy}-k_{,xx}+Ik_{,xy})-\right.$
$\displaystyle\left.-(k_{,x}-Ik_{,y})(P_{,x}-IP_{,y})\right]$
$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}PR_{,r}}{UR^{2}}(k_{,x}-Ik_{,y})$ (29)
$\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6UR^{2}}\left[Uk-(U_{,u}r+k)(RR_{,rr}R_{,r}{}^{2})-\right.$
$\displaystyle\left.-2URR_{,ru}+(RU_{,u}+2UR_{,r})R_{,r}\right]$

Now, we can easily determine the type irrespective of possible non-optimal
choice of tetrad by using the review of explicit methods for determining the
algebraic type in Zakhary  that are based on Penrose . Namely, when we use
invariants

$I=\Psi_{0}\Psi_{4}-4\Psi_{1}\Psi_{3}+3\Psi_{2}^{2},\ J={\rm
det}\left(\begin{array}[]{ccc}\Psi_{4}&\Psi_{3}&\Psi_{2}\\\
\Psi_{3}&\Psi_{2}&\Psi_{1}\\\ \Psi_{2}&\Psi_{1}&\Psi_{0}\end{array}\right)$

we can immediately confirm that $I^{3}=27J^{2}$ is satisfied so that we are
dealing with type II or more special. At the same time generally $IJ\neq 0$ so
it cannot be just type III. Moreover, the spinor covariant $R_{ABCDEF}$ has
nonzero components

$\displaystyle R_{000000}$ $\displaystyle=$
$\displaystyle\Psi_{1}(3\Psi_{0}\Psi_{2}-2\Psi_{1}^{2})$ (30) $\displaystyle
R_{000001}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Psi_{2}(3\Psi_{0}\Psi_{2}-2\Psi_{1}^{2})$ (31)

which means that generally the spacetime cannot be of type D. So indeed our
scalar field solution is of the most general type possible for the Robinson-
Trautman vacuum class. Which does not mean that there cannot be a scalar field
solution of type I. Moreover, inspecting the components of the Weyl spinor (V)
one concludes that in the special case of $k(x,y)=const>0$ (constant positive
Gaussian curvature of compact two-space spanned by $x,y$) the algebraic type
becomes D consistent with spherical symmetry. Finally, since $\Psi_{1}=0$
implies $\Psi_{0}=0$ we cannot have all components of spinor covariant
$Q_{ABCD}$ (see Penrose and Zakhary ) vanishing while having nonvanishing Weyl
spinor. This means that our family of solutions does not contain type N
geometries.

##  VI Conclusion and final remarks

We have derived a Robinson–Trautman spacetime with minimally coupled free
scalar field. We have shown that it has a singularity for all retarded times
created by the divergence of the scalar field therein. This singularity is
initially (with respect to retarded time) naked and only later becomes covered
by the quasilocal horizon. Note that the energy momentum tensor of the free
minimally coupled scalar field trivially satisfies null energy condition (as
well as weak and strong ones) and the naked singularity at the beginning of
the evolution is probably caused by a slow buildup of effective energy density
caused by the scalar field at the singularity position which is enough to form
the singularity but not enough to envelop it in horizon initially. This
behaviour suggests similarity with the appearance of a naked curvature
singularity in Vaidya spacetime with linear mass function. The naked
singularity appears there initially depending on the speed of growth of mass
WaughLake:1986  and later becomes covered by horizon as well. From the
properties of both null congruences orthogonal to the horizon we deduced that
we are dealing with past horizon which is natural for standard (retarded) form
of Robinson–Trautman spacetime.

Our solution is asymptotically flat, contains a black hole (at least in the
later stage of development) and has a scalar field, so one is naturally
interested in its connection with the no-hair theorems (see Herdeiro  for
current review). As recently shown Graham , for stationary black hole
spacetimes there are no scalar hairs (even for time-dependent scalar field)
which means that the dynamical nature of Robinson–Trautman family is truly
needed for our solution to be feasible. Also, we have shown that the scalar
field vanishes outside the black hole in infinite retarded time limit when the
geometry settles down to the final state — Schwarzschild black hole.

Finally, we have proved that our geometry is of algebraic type II (the most
general type for vacuum Robinson–Trautman spacetimes) and if we restrict to
spherically symmetric case it is of type D. However, the type N subcase is not
possible for our solution.

###### Acknowledgements.

This work was supported by grant GAČR 14-37086G.

## References

  * (1) I. Robinson and A. Trautman, Phys. Rev. Lett. 4, 431 (1960).
  * (2) I. Robinson and A. Trautman, Proc. Roy. Soc. Lond. A265 , 463 (1962).
  * (3) H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Herlt, Exact Solutions of the Einstein’s Field Equations, 2nd edn (CUPress Cambridge, 2002).
  * (4) P.T. Chruściel, Commun. Math. Phys. 137, 289 (1991).
  * (5) P.T. Chruściel, Proc. Roy. Soc. Lond. A436, 299 (1992).
  * (6) P.T. Chruściel and D.B. Singleton, Commun. Math. Phys. 147, 137 (1992).
  * (7) J. Bičák and J. Podolský, Phys. Rev. D 52, 887 (1995).
  * (8) J. Bičák and J. Podolský, Phys. Rev. D 55, 1985 (1997).
  * (9) J. Bičák, Z. Perjés, Class. Quantum Grav. 4, 595 (1987).
  * (10) J. Podolský and O. Svítek, Phys. Rev. D 71, 124001 (2005).
  * (11) J. Podolský and M. Ortaggio, Class. Quant. Grav. 23, 5785 (2006).
  * (12) K.P. Tod, Class. Quantum Grav. 6, 1159 (1989).
  * (13) E.W.M. Chow and A.W.C. Lun, J. Austr. Math. Soc. B 41, 217 (1999).
  * (14) J. Podolský and O. Svítek, Phys. Rev. D 80, 124042 (2009).
  * (15) L. Rezzolla, R. P. Macedo and J. L. Jaramillo, Phys. Rev. Lett. 104, 221101 (2010).
  * (16) O. Svítek, Phys. Rev. D 84 044027 (2011).
  * (17) M. Ortaggio, J. Podolský, M. Žofka, JHEP 1502 045 (2015).
  * (18) R. Güven, E. Yörük, Phys. Rev. D 54 6413 (1996).
  * (19) A. Anabalon, H. Maeda, Phys.Rev. D 81 041501 (2010).
  * (20) J. B. Griffiths and J. Podolský, Exact Space-Times in Einstein’s General Relativity, (Cambridge University Press, Cambridge, England, 2009).
  * (21) S.W. Hawking and G.F.R. Ellis, _The large scale structure of space-time_ (CUPress Cambridge, 1975).
  * (22) S.A. Hayward, Phys. Rev. D 49, 6467 (1994).
  * (23) A. Ashtekar and B. Krishnan, Phys. Rev. D 68, 104030 (2003).
  * (24) T-H. Kuo, Nonlinear Anal. 63, e427 (2005).
  * (25) A. Besse, Einstein Manifolds (Springer Berlin, 1987).
  * (26) L. Boccardo, F. Murat and J.P. Puel, Annali della Scuola Normale Superiore di Pisa (Classe di Scienze) Sér. 4 vol. 11 no. 2, 213 (1984).
  * (27) D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer Berlin, 2001).
  * (28) E.g. supersolution satisfies $\Delta u^{+}\leq F(x,u^{+},\nabla u^{+})$.
  * (29) E. Zakhary, K. T. Vu, J. Carminati, Gen. Rel. Grav. 35, 1223 (2003).
  * (30) R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 2 (Cambridge University Press, 1990).
  * (31) B. Waugh, K. Lake, Phys. Rev. D 34 2978 (1986).
  * (32) C.A.R. Herdeiro, E. Radu [arXiv:1504.08209].
  * (33) A.A.H. Graham, R. Jha, Phys. Rev. D 90 041501 041501(R).