Insights from the Field: A Comprehensive Analysis of Industrial Accidents in
Plants and Strategies for Enhanced Workplace Safety.

Hasanika Samarasinghe 111 Department of Engineering, Sony UK Technology
Centre, Pencoed Technology Park, Bridgend, United Kindom, CF35 5HZ. E-mail:
<EMAIL_ADDRESS> and Shadi Heenatigala222 Department of Mathematics, Texas
State University, San Marcos, TX, USA, 78666. E-mail<EMAIL_ADDRESS>
Abstract. The study delves into 425 industrial incidents documented on Kaggle
[1], all of which occurred in 12 separate plants in the South American region.
By meticulously examining this extensive dataset, we aim to uncover valuable
insights into the occurrence of accidents, identify recurring trends, and
illuminate underlying causes. The implications of this analysis extend beyond
mere statistical observation, offering organizations an opportunity to enhance
safety and health management practices. Our findings underscore the importance
of addressing specific areas for improvement, empowering organizations to
fortify safety measures, mitigate risks, and cultivate a secure working
environment. We advocate for strategically applying statistical analysis and
data visualization techniques to leverage this wealth of information
effectively. This approach facilitates the extraction of meaningful insights
and empowers decision-makers to implement targeted improvements, fostering a
preventive mindset, and promoting a safety culture within organizations. This
research is a crucial resource for organizations committed to transforming
data into actionable strategies for accident prevention and creating a safer
workplace.

Key Words: Safety regulations, Industry-specific hazards, Auto-electronic
safety management system, Occupational Safety and Health Practice.

##  1 Introduction

Workplace health and safety practices play a crucial role in ensuring the
well-being of both the organization and its employees. Ineffective policies in
this area can adversely affect the firm and its workforce. While safety and
health concepts are globally applicable, the specific actions required may
vary based on the organization’s size, the hazards associated with its
operations, its overall culture, products, or services, and the adequacy of
existing arrangements.
The twentieth century witnessed a significant increase in the risks associated
with technology, prompting companies to prioritize the reduction of workplace
injuries and illnesses. Consequently, safety management has emerged as a
prominent discipline in the business world. Extensive research has been
conducted on safety management practices in various industries globally, but
there is a lack of studies specifically focused on real-world manufacturing
plants.
The Industrial Revolution in the 18th century indeed brought about significant
changes in manufacturing processes, including the introduction of machines to
replace human labor. This led to the establishment of large-scale factories
where workers were closely supervised and tasks were divided among them.
As Felton [12] points out, the rapid growth of the Industrial Revolution was
accompanied by an increase in work-related accidents, resulting in injuries
and fatalities. It is important to note that the Industrial Revolution marked
a pivotal moment in history, where the focus on productivity and efficiency
often overshadowed concerns for worker safety. The lack of proper safety
regulations and protective measures during this period contributed to the rise
in work-related accidents. Over time, however, societies recognized the
importance of workplace safety and began implementing regulations and
practices to protect workers. This led to the development of occupational
health and safety standards that continue to evolve and improve today.
According to Booth and Lee [8], safety management refers to a process driven
by top management aimed at reducing risks to the health and safety of workers.
The primary objective of safety management is to identify and address the
causes of accidents and incidents. This involves actively identifying both
observable and latent hazards. Hazard detection is just one aspect of safety
management, encompassing a comprehensive system for planning, implementing,
and following up on safety operations. Safety management typically involves
risk analysis, coordination of safety training, investigation of accidents and
near-misses, safety promotion, and assessment of human reliability. Grimaldi
and Simmonds [13] emphasized that a successful safety management system
assigns these tasks to all hierarchical levels within the organization. By
implementing a robust safety management system, companies can effectively
mitigate risks and ensure the well-being of their employees.
This paper is organized as follows. In Section 2, we present the literature
review to demonstrate the background of the project. In Section 3, we provide
a comprehensive overview of Brazil’s diverse industries, workforce, and
socioeconomic landscape. Subsection 3.1 focuses on Occupational Safety and
Health Practices in Brazil. In Section 4, we analyze perspectives on
industrial accidents. Finally, in Section 5, we conclude by summarizing the
main contributions of the paper and offering comments on future research
directions.

##  2 Literature Review

The establishment of the International Labour Organization (ILO) in 1919 [3]
marked a significant step towards preventing occupational hazards and creating
safe work environments. The ILO, as a specialized agency of the United
Nations, is responsible for developing policies, promoting programs, and
setting work standards for its 187 member countries. Its primary objective is
to ensure the well-being and safety of workers worldwide. In addition to the
ILO, the World Health Organization (WHO) [6] also plays a role in addressing
occupational health and safety. The WHO focuses on various aspects of health,
including occupational health, and works towards promoting and protecting the
health of individuals globally. At the European level, the European Agency for
Safety and Health at Work (EU-OSHA) [2] takes the lead in addressing
occupational safety and health issues among its member states. It works
towards ensuring safe and healthy working conditions for European workers. It
is important to note that each country has its own set of laws and regulations
for occupational health and safety. These laws and rules are implemented to
ensure compliance and protect workers within their respective jurisdictions.
According to Cox and Tait [10], safety management involves making informed
decisions to meet accepted safety standards and achieve a state of freedom
from emerging risks or harm. Kennedy and Kirwan [15] further explain that
organizations employ various safety management strategies to monitor and
control work activities and methods within the framework of their safety
management system. In developing nations, management voluntarily adopts
different safety management strategies to ensure the health and safety of
their workers. These strategies are implemented to mitigate risks and create a
safe working environment.
The studies conducted by Cohen et al. [9] and Shannon et al. [18] provide
valuable insights into the factors that contribute to lower accident rates and
improved workplace safety. In the study by Cohen et al. [9], several factors
were identified as influential in reducing accidents, including the presence
of high-ranking safety officers, the involvement of leaders in safety
activities, regular training for new and current employees, and increased
interpersonal interaction between managers and employees. Similarly, the study
by Shannon et al.[18] focused on manufacturing enterprises and found that
organizations with lower rates of lost time injuries had managers who believed
in giving the workforce a greater say in the decision-making process and
fostered a harmonious management-worker relationship. These findings highlight
the importance of proactive safety measures, leadership involvement, ongoing
training, and effective communication between managers and employees in
creating a safer work environment. Implementing these practices can help
organizations reduce accidents and promote a culture of safety.
Researchers have conducted investigations into how safety management
techniques impact the safety behaviours of individuals within organizations.
One study by Smith-Crowe et al. [19] found that safety management methods,
particularly safety training, played a moderating role in the relationship
between safety awareness and overall safety performance. Another study by Neal
et al. [17] utilized structural equation procedures to examine the connection
between antecedent factors, determinants, and components of safety performance
among hospital workers in Australia. The study revealed that safety awareness
and safety motivation individually predicted participation in safety
management practices. Additionally, it was found that safety management
techniques were predictive of both safety awareness and safety motivation.
These findings highlight the importance of implementing effective safety
management practices to enhance safety performance within organizations.
In industrialized countries, government agencies have indeed guided the
creation and operation of effective health and safety management systems. This
has led to safety management becoming an established practice in many
industries. In several cases, these guidelines have been integrated with
existing quality management and system standards, creating a comprehensive
framework for organizations to ensure both safety and quality in their
operations. These standards and guidelines serve as valuable resources for
companies seeking to implement robust health and safety management systems.
Ensuring effective safety and health management practices is crucial in
hazardous areas like chemical or process industries. According to the study
mentioned by Hurst et al. [14], the frequent occurrence of accidents due to
safety management failures highlights the need for audit tools. These tools
can help evaluate an organization’s safety climate and complement the
assessment of safety management methods. By conducting audits of both safety
climate and safety management practices, organizations can identify areas for
improvement and enhance overall safety measures.
Indeed, there are similarities between good safety and health management
practices and other management approaches such as quality management,
environmental protection, and business excellence. Various management
techniques can be applied across different areas, including safety and health
management, quality management, environmental protection, and business
excellence. Commercially successful companies often excel in occupational
health and safety because they apply efficient business skills to safety and
health management, just as they do for other aspects of their operations. This
includes implementing effective risk assessment and mitigation strategies,
establishing clear policies and procedures, providing adequate training and
resources, promoting a culture of safety, and continuously improving safety
performance through monitoring and evaluation. By integrating safety and
health management into their overall business practices, these companies
prioritize the well-being of their employees and create a safer working
environment.

##  3 A Comprehensive Overview of Brazil’s Diverse Industries, Workforce, and
Socioeconomic Landscape

Brazil is indeed the world’s fifth largest nation, covering an area of 8.5
million km2. It is also the sixth most populous country, with approximately
150 million inhabitants [5]. The Brazilian economy is diverse and includes
well-developed sectors such as mining, agriculture, manufacturing, and
services. Mining plays a significant role in Brazil’s economy, with rich
mineral deposits supporting an extensive mining industry. Various resources
are mined in different regions of the country. For example, there are major
deposits of iron ore, bauxite, hematite, oil, and gas in many areas. According
to De Medeiros Delgado et al.[11], the Amazon region is known for its aluminum
and manganese mining, while Bahia has chrome, magnesium, and quartz deposits.
Copper and lead can be found in Bahia and Rio Grande do Sul, asbestos in
Goias, and nickel in Goias and Minas Gerais. It’s worth noting that Brazil is
self-sufficient in tin, zinc, and tungsten. Additionally, the Amazon region is
known for its gold deposits, which are mined by approximately 600,000 gold
seekers, also known as ganimpeiros, who often work outside the formal sector.
Overall, the mining industry in Brazil employs about 4 million people
nationwide.
Brazil’s manufacturing sector is indeed a rapidly growing and vibrant part of
the economy. It encompasses a wide range of products, including steel,
chemicals, petrochemicals, textiles, automobiles, pulp and paper, aircraft,
electronics, and footwear. The petrochemical industry in Brazil is the largest
in Latin America. The manufacturing sector has become a significant employer,
currently employing about 25% of the workforce and showing signs of continued
expansion. Civil construction also employs a large number of workers. On the
other hand, the service sector is the fastest-growing part of Brazil’s economy
and now employs over one-third of the workforce.
In terms of workforce participation, approximately 45% of Brazil’s population,
which is nearly 70 million people, is considered economically active, with
around 60 million people being employed. Unfortunately, child labour remains a
concern, with about 20% of children aged 10 to 14 being part of the workforce
(Population by situation in the labor market in Brazil [7]). Regarding gender
distribution, women constitute about 35% of Brazil’s workforce. Female workers
tend to have completed more schooling than their male counterparts. However,
women are relatively more concentrated in the service sector and less
concentrated in manufacturing, mining, agriculture, and construction compared
to men.

###  3.1 Occupational Safety and Health Practice in Brazil

Under the legal framework for occupational health regulation in Brazil, there
are various components and entities involved. The federal constitution, issued
in 1988, plays a significant role in this framework [16]. The Ministry of
Labour has traditionally been responsible for most occupational health
functions, but with the reorganisation of the health sector in 1988, some
responsibilities were transferred to the Unified Health System (SUS).
According to the new constitution, SUS has the authority to protect worker
health, as stated in Article 205, Sections II and VII. However, the
constitution does not provide specific guidelines on how this protection
should be implemented. Additionally, no specific legislation defines the
process for setting occupational health standards. The constitution does grant
the federal government the power to conduct workplace inspections, as
mentioned in Article 21, Section XXIV. The exact manner in which SUS will
carry out regulation and workplace inspections under the new arrangement is
yet to be determined. It is important to note that this information is based
on the provided context and may not encompass all aspects of the legal
framework for occupational health regulation in Brazil.
Currently, the Ministry of Labour in Brazil is responsible for occupational
health regulation following the Consolidated Labour Law of 1943 [4]. This law
serves as the foundation for occupational health regulations in the country.
In addition to federal regulations, state and municipal governments also have
the authority to issue their own laws and regulations about occupational
health. The Ministry of Labour has traditionally relied on threshold limit
values (TLVs) established by the American Conference of Governmental
Industrial Hygienists (ACGIH) as the basis for its regulations. These TLVs
guide exposure limits to various substances in the workplace. It is important
to note that this information is based on the provided context and may not
encompass all aspects of occupational health regulation in Brazil.

##  4 Analytical Perspectives on Industrial Accidents

The database contains records of 425 accidents from 12 different plants
located in Brazil. By studying this data, organizations can gain insights into
the occurrence of accidents, identify trends, and potentially uncover
underlying causes. This analysis can help organizations identify areas for
improvement in their safety and health management practices. By addressing
these areas, organizations can enhance their safety measures, mitigate risks,
and create a safer working environment for their employees. Organizations need
to utilize this data effectively, employing appropriate statistical analysis
and data visualization techniques to extract meaningful insights. This can aid
in making informed decisions and implementing targeted improvements to prevent
accidents and promote a culture of safety.

Figure 4.1: The sector of industry and the frequency of accidents.

According to the depiction in Figure 4.1, the mining industry accounted for
241 reported accidents, whereas the metals industry reported 134 accidents.
Remarkably, a mere 50 accidents were reported across various other industries.
There can be several reasons for this discrepancy:

  1. 1.

Inherent risks: The mining industry is known to have inherent risks due to the
nature of its operations, such as working in underground or hazardous
environments, handling heavy machinery, or dealing with explosives. These
factors can contribute to a higher likelihood of accidents compared to other
industries.

  2. 2.

Safety regulations: The mining industry often operates under strict safety
regulations and protocols due to the potential for serious accidents and
injuries. The reporting and documentation of accidents may be more
comprehensive in this industry, leading to a higher number of reported
incidents compared to other industries with less stringent regulations.

  3. 3.

Workforce size: The mining industry and the metals industry may have larger
workforces compared to other industries, resulting in a higher number of
reported accidents simply due to the larger number of workers.

  4. 4.

Industry-specific hazards: Each industry has its own unique set of hazards and
risks. The specific hazards associated with mining and metals industries, such
as cave-ins, equipment malfunctions, or exposure to toxic substances, may
contribute to a higher number of accidents compared to other industries.

Figure 4.2: The analysis of major industrial accidents by gender.

In Figure 4.2, we demonstrate that accidents within the industry predominantly
involve males, accounting for 94.82% of reported incidents, while females make
up only 5.18%. This article aims to investigate and compare the involvement of
males and females in these accidents, to understand the underlying causes, and
to develop strategies to mitigate risks for both genders.
Possible factors that contribute to this observation are:

  1. 1.

Occupational distribution: Men and women often have different job roles and
occupational distributions within industries. Some industries, such as
construction or mining, have a higher proportion of male workers, which may
expose them to higher-risk tasks and environments.

  2. 2.

Physical differences: Men and women may have different physical capabilities
and strengths, which can influence the types of tasks they perform. Certain
physically demanding jobs may have a higher risk of accidents, and if these
jobs are predominantly performed by men, it can contribute to the observed
difference.

  3. 3.

Cultural and societal factors: Societal norms and expectations can influence
the career choices and preferences of men and women. Men may be more likely to
pursue jobs in industries with higher accident rates, while women may
gravitate towards other sectors. This can contribute to the observed
difference in accident rates.

  4. 4.

Safety culture and training: Workplace safety culture and training programs
may not always address the specific needs and challenges faced by both men and
women equally. This can result in a higher risk of accidents for men if safety
measures are not effectively implemented or followed.

It is important to emphasize that these factors are observations derived from
analyzing databases gathered from mining, metal, and other industries, and may
not have universal applicability. Efforts to promote gender equality in the
workplace, improve safety measures, and provide comprehensive training can
help reduce the disparity in accident rates between men and women in
industries.

Figure 4.3: Analysis of the number of accidents by accident level.

Accidents in various industries are classified on a severity scale ranging
from Category I to Category VI. Category I denotes milder accidents, while
Category VI signifies extremely severe incidents. As depicted in Figure 4.3 in
the metal industry, 107 accidents were recorded at the Category I level, along
with 12 at the Category II level, 7 at the Category III level, 7 at the
Category IV level, and 1 at the Category V level. Additionally, 167 accidents
occurred in the mining industry at the Category I level, with 26 at Category
II, 20 at Category III, 21 at Category IV, and 7 at Category V. Lastly,
accidents in other industries accounted for 42 at Category I, 2 at Category
II, 4 at Category III, and no accidents at Category IV or V. In total, there
were 134 accidents in the metal industry, 241 in the mining industry, and 50
in other industries. This amounts to a total of 316 accidents at the Category
I level, 40 at Category II, 31 at Category III, 30 at Category IV, and 8 at
Category V.
The mining industry is known to have a higher occurrence of dangerous
accidents compared to other industries. Several reasons contribute to this:

  1. 1.

Hazardous working conditions: Mining operations often involve working in
challenging environments such as underground mines, open pits, or remote
locations. These conditions can expose workers to various hazards, including
cave-ins, explosions, toxic gases, and the risk of being trapped or injured by
heavy machinery.

  2. 2.

Complex machinery and equipment: The mining industry relies heavily on complex
machinery and equipment, such as drills, crushers, and loaders. Operating and
maintaining these machines require specialized skills and knowledge. Any
malfunction or improper use of equipment can lead to severe accidents.

  3. 3.

Exposure to harmful substances: Miners may be exposed to harmful substances
like dust, gases, and chemicals, depending on the type of mining being
conducted. Prolonged exposure to these substances can lead to respiratory
problems, occupational diseases, and other health issues.

  4. 4.

Geotechnical risks: Mining operations involve working with the earth’s natural
resources, which can pose geotechnical risks such as rockfalls, landslides,
and collapses. These risks can be unpredictable and pose a significant danger
to workers.

Regarding the comparison between the mining and metal industries, it is worth
noting that the mining industry involves extracting raw materials, while the
metal industry involves processing and refining those materials. The metal
industry may have a higher occurrence of accidents compared to other
industries due to factors such as:

  1. 1.

High-temperature processes: Metal industries often involve high-temperature
operations like smelting, casting, and welding. These processes can expose
workers to burn hazards, fire risks, and the potential release of toxic fumes.

  2. 2.

Heavy machinery and equipment: Similar to the mining industry, the metal
industry utilizes heavy machinery and equipment, which can pose risks if not
operated or maintained properly.

  3. 3.

Handling of hazardous materials: Metal industries may involve working with
hazardous materials such as acids, solvents, and heavy metals. Improper
handling or exposure to these substances can lead to chemical burns,
poisoning, or long-term health effects.

Both the mining and metal industries must prioritize safety measures, provide
comprehensive training, enforce strict regulations, and regularly assess and
mitigate risks to ensure the well-being of workers and minimize accidents.

Figure 4.4: Analysis of injured individuals by employment type.

Within this database, it was discovered that a significant number of accidents
have affected both employees and third-party workers. Figure 4.4 stated that
178 employees and 257 third-party workers have experienced accidents in the
database used to analyze accidents in mining, metals, and other industries.
Possible reasons for a higher occurrence of accidents among third-party
workers compared to direct employees include:

  1. 1.

Lack of familiarity: Third-party workers may be less familiar with the
specific work environment, equipment, and safety protocols compared to direct
employees. This lack of familiarity can increase the risk of accidents.

  2. 2.

Limited training: Third-party workers may receive limited or insufficient
training on safety procedures and protocols compared to direct employees. This
can result in a higher likelihood of accidents due to a lack of knowledge or
awareness.

  3. 3.

Time pressure: Third-party workers may face time constraints or deadlines
imposed by their employers, which can lead to rushing or taking shortcuts.
This can compromise safety measures and increase the risk of accidents.

  4. 4.

Communication challenges: Communication between third-party workers and the
host company’s employees may not always be as effective or clear.
Misunderstandings or miscommunication regarding safety instructions or
procedures can contribute to accidents.

  5. 5.

Lack of supervision: Third-party workers may have less direct supervision
compared to direct employees. This reduced oversight can result in a lower
adherence to safety protocols and an increased risk of accidents.

##  5 Conclusion

A staggering number of industrial accidents in Brazil, particularly within the
mining, metal, and other related industries. A total of 425 cases were
observed, necessitating a detailed analysis to understand the root causes and
potential preventive measures. This article aims to provide an in-depth
examination of these accidents, shedding light on the key factors, the
consequences, and the lessons learned. Ensuring safety within an organization
is of paramount importance in today’s fast-paced and ever-evolving industrial
landscape. Enterprises across all sectors must continuously strive to enhance
their safety management practices and mitigate potential risks. This can be
achieved through various means, including the activation of a safety culture
within the organization, strengthening the determination of business owners,
and providing education and training to workers.
It is evident that managing safety operating procedures and diligently
addressing facility maintenance are critical to fostering a safe work
environment. By placing a greater emphasis on these areas, organizations can
enhance their preventive measures, reduce accidents, and safeguard the well-
being of their workforce. Regularly reviewing and modifying safety procedures,
along with providing comprehensive worker education, are paramount in ensuring
the ongoing effectiveness of safety protocols. Additionally, daily inspections
and effective equipment maintenance play pivotal roles in preventing accidents
and creating a proactive safety culture within organizations.
The safety report submission process is a crucial aspect of preventing and
managing chemical accidents. By maintaining and expanding this process to meet
real-time demands and focusing on substances with high risk, we can
significantly reduce the occurrence of accidents. Additionally, establishing a
newly introduced system through the revision of the Occupational Safety and
Health Act, along with efforts to establish a national safety culture, will
further enhance overall safety. Let us prioritize safety and work towards
creating a safer future for all.
Industrial safety measures have been successful in preventing major accidents
caused by hazardous substance leakage, fires, and explosions. However, the
persistent potential for accidents resulting from equipment defects or
inadequate safety measures necessitates continuous vigilance. By adopting
comprehensive preventive measures, conducting regular safety evaluations, and
investing in ongoing training, industries can further ensure the safety and
well-being of their workers and the broader community.
In modern industrial sites, safety is of utmost importance. Accidents not only
cause harm to individuals but also lead to financial losses and damage a
company’s reputation. It is crucial to prioritize safety measures to ensure
the well-being of employees and the smooth operation of industrial sites. The
integration of auto-electronic systems has revolutionized safety practices,
providing real-time monitoring and proactive action in hazardous situations.
These systems utilize advanced technologies to enhance safety measures and
minimize risks. implementing an auto-electronic safety management system
offers numerous benefits to organizations. From improved safety measures and
real-time monitoring to data-driven decision-making and cost savings, this
advanced system revolutionizes safety practices. By embracing innovation and
technology, companies can create a safer and more secure environment for their
employees and stakeholders. With the advancement of auto-electronic safety
management systems, the goal of zero accidents in industrial sites is within
reach. By prioritizing safety and investing in proactive measures, companies
can create a secure work environment that safeguards the well-being of
employees and facilitates uninterrupted operations. Embracing these
technologies and adopting a safety-first approach is crucial for future
success in the industrial sector.

## References

  * [1] Find open datasets and machine learning projects.  https://www.kaggle.com/datasets.
  * [2] Information, statistics, legislation and risk assessment tools.  https://doi.org/10.1080/00206819409465474.
  * [3] International labour organization.  https://www.ilo.org/global/lang--en/index.htm.
  * [4] Labor and welfare.  https://www.gov.br/en/categories/labor-and-welfare.
  * [5] Site homepage.  https://www.oecd-ilibrary.org/.
  * [6] World health statistics.  https://www.who.int/data/gho/publications/world-health-statistics.
  * [7] Population by situation in the labor market in Brazil Q2 2023.  https://www.statista.com/statistics/1343642/population-situation-labor-market-brazil/, 2003.
  * [8] R. Booth and Lee, The role of human factors and safety culture in safety management, Journal of Engineering Manufacture, 209 (1995), pp. 393–400.
  * [9] A. Cohen, M. Smith, and H. Cohen, Safety program practices in high versus low accident rate companies, Cincinnati, OH, National Institute of Occupational Health and Safety, (1975).
  * [10] S. Cox and N. Tait, Reliability, safety and risk management: An integrated approach, Oxford: Butterworth Heinemann, (1991).
  * [11] I. De Medeiros Delgado, A. Pedreira, and C. Thorman, Geology and mineral resources of brazil: A review, International Geology Review, 36(6) (1994), pp. 503––544.
  * [12] J. Felton, Histry of occupational health and safety., Itasca, IL: National Safty Council, (1986).
  * [13] J. Grimaldi and R. Simmonds, Safety management. illinois, Richard D.Irwin Inc., (1975).
  * [14] N. Hurst, S. Young, I. Donald, H. Gibson, and A. Muyselaar, Measures of safety management performance and attitudes to safety at major hazard sites, Journal of Loss Prevention in the Process Industries, 9 (1996), pp. 161–172.
  * [15] R. Kennedy and B. Kirwan, Development of a hazard and operability based method for identifying safety management vulnerabilities in high risk systems, Safety Science, 30 (1998), pp. 249–274.
  * [16] T. Menicucci, The brazilian unified health system: Thirty years of advances and resistance, (2019).
  * [17] A. Neal, M. Griffin, and P. Hart, The impact of organizational climate on safety climate and individual behavior, Safety Science, 34 (2000), pp. 99–109.
  * [18] H. Shannon, V. Walters, W. Lewchuk, J. Richardson, L. Moran, T. Haines, and D. Verma, Workplace organizational correlates of lost-time accident rates in manufacturing, American Journal of Industrial Medicine, 29 (1996), pp. 258––268.
  * [19] K. Smith-Crowe and R. Burke, M.J.and Landis, Organizational climate as a moderator of safety knowledge-safety performance relationships, Journal of Organizational Behavior, 24 (2003), pp. 861–876.


# Black Hole Zeroth Law in Horndeski Gravity

Aofei Sang<EMAIL_ADDRESS>College of Education for the Future,
Beijing Normal University, Zhuhai, 519087, China  Department of Physics,
Beijing Normal University, Beijing 100875, China  Jie Jiang  Corresponding
author<EMAIL_ADDRESS>College of Education for the Future, Beijing
Normal University, Zhuhai, 519087, China  Department of Physics, Beijing
Normal University, Beijing 100875, China

###### Abstract

The four laws of black hole mechanics have been put forward for a long time.
However, the zeroth law, which states that the surface gravity of a stationary
black hole is a constant on the event horizon, still lacks universal proof in
various modified gravitational theories. In this paper, we study the zeroth
law in a special Horndeski gravity, which is an interesting gravitational
theory with a nonminimally coupled scalar field. After assuming that the
nonminimally coupled scalar field has the same symmetries with the spacetime,
the minimally coupled matter fields satisfy the dominant energy condition and
the Horndeski gravity has a smooth limit to Einstein gravity when the coupling
constant approaches zero, we prove the zeroth law based on the gravitational
equation in Horndeski gravity without any assumption to the spacetime
symmetries.

##  I Introduction

Black hole is an object predicted by general relativity and its existence is
ensured by the singularity theoremPenrose:1969pc . In the 1970s, the four laws
of black hole mechanics are proposedBardeen:1973gs . Later, Hawking radiation,
using the method of the quantum field theory in curved spacetime, gives a
relationship between the surface gravity $\kappa$ and the temperature of a
stationary black horizon and therefore indicates that black hole can be
regarded as a thermodynamic systemHawking:1975vcx . Since then, the four laws
of black hole mechanics and other consequent thermodynamic properties of the
black hole arose wide attention.

The zeroth law of black hole mechanics, which states that the surface gravity
$\kappa$ (temperature) of a stationary black hole is a constant on the event
horizon, is proved in Einstein gravity in 1970sBardeen:1973gs . In Einstein
gravity, guaranteed by the rigidity theoremHawking:1971vc , the event horizon
must be a Killing horizon of a black hole in asymptotic flat stationary
spacetime. Then, using the Einstein equation and the dominant energy
condition, one can easily find that the derivative of $\kappa$ in all
directions along the horizon is zeroBardeen:1973gs . However, this proof
failed in other gravitational theories since the Einstein equation is used in
the process of proof. Later, a proof without the equation of motion for the
gravitational theory is given by Racz and WaldRacz:1995nh . They assume that
the black hole is static or stationary-axisymmetric with ‘$t-\phi$’ reflection
isometry and find that in this case the zeroth law always holds. This proof is
independent of the concrete gravitational theories, however, in contrast to
the argument in the previous proof, this argument makes quite strong
assumptions about the symmetry of black hole spacetime. In short, there is
still a lacks general proof of the black hole zeroth lawSarkar:2012wy ;
Ghosh:2020dkk .

General relativity has been a great success in describing gravity, especially
in explaining phenomena in the solar system, but many observations of the
universe indicate that there is still a lot to be discussed for this theory.
For example, general relativity tells us that the universe is slowing down,
which contradicts the observational result that the universe is accelerating.
Many other cosmological problems, such as dark matter, dark energy, and the
early origins and the evolution of the universe, also remain to be
solveddePireySaintAlby:2017lwc ; Planck:2015fie ; Bertone:2004pz ;
Peebles:2002gy ; Alpher:1949sef . One way to solve these problems is to
introduce the scalar field. The Horndeski gravity considered in this paper,
first proposed in the 1970s, is a general second-order scalar-tensor
theoryHorndeski:1974wa . The full action of this kind of theory is given in
Deffayet:2011gz ; Kobayashi:2011nu . Recently, this gravitational theory
received renewed attention for the possibility to solve the cosmological
constant problemCharmousis:2011bf  and other interesting
propertiesNicolis:2008in ; Saridakis:2010mf ; Maselli:2016gxk . Many
thermodynamic properties of stationary black holes in Horndeski gravity are
also discussed recently such as the black hole first law, second law and
etc.Feng:2015oea ; Miao:2016aol ; Feng:2015wvb ; Hu:2018qsy ; Hajian:2020dcq ;
Wang:2020svl . However, there is no general proof for the black hole zeroth
law in Horndeski gravity and a natural question is whether the zeroth law
holds generally in Horndeski gravity. In this paper, we explored this question
in the stationary black holes of a special Horndeski gravity. Then, with
natural assumptions that the nonminimally coupled scalar field shares the same
symmetries with the spacetime, the Horndeski gravity approach to general
relativity smoothly when the coupling constant approach to zero and the matter
field satisfies the dominant energy condition, we would like to prove that the
surface gravity is constant in the whole event horizon, i.e. the zeroth law of
black hole mechanics established in Horndeski gravity. This result is general
in Horndeski gravity since there is no extra symmetry is assumed in the
process of proof.

This paper is organized as follows. In sec. II, we introduce some properties
of the Killing horizon, which will be used for further proof and review the
proof of the zeroth law in Einstein gravity. Then, we introduce the Horndeski
gravity and gives the equation of motion of this theory in sec. III. The main
calculation of the proof is shown in sec. IV. Finally, in sec. V, we give a
conclusion and prospects the later work.

##  II Geometry of stationary Killing horizon

In this section, we would like to introduce some related properties of the
stationary event horizon which is also the Killing horizon in the
$n$-dimensional stationary black holes.

A null hypersurface $\mathcal{H}$ is called a Killing horizon if there exists
a Killing vector field $\xi^{a}$ which is orthogonal to the null hypersurface
$\mathcal{H}$. The surface gravity $\kappa$ of the Killing horizon is defined
by

$\displaystyle\xi^{a}\nabla_{a}\xi^{b}=\kappa\xi^{b}\,.$ (1)

To prove the zeroth law, we need to check that whether the surface gravity
$\kappa$ is a constant on the hypersurface $\mathcal{H}$. Choose $C$ is a
cross-section of the event horizon and another cross-section in $\mathcal{H}$
is generated by the Killing vector field $\xi^{a}$. For calculation
convenient, we can choose a basis $\\{\xi^{a},l^{a},(e_{i})^{a}\\}$ satisfying

$\displaystyle\xi^{a}l_{a}=-1\,,\quad l^{a}l_{a}=0\,,$ (2)
$\displaystyle\xi_{a}(e_{i})^{a}=$ $\displaystyle 0\,,\quad
l_{a}(e_{i})^{a}=0\,,\quad(e_{i})^{a}(e_{j})_{a}=\delta_{ij}\,.$

on $\mathcal{H}$ and $(e_{i})^{a}$ is tangent vectors of the cross-section
$C$. Then, we can express the metric on the horizon as

$\displaystyle g_{ab}=-\xi_{a}l_{b}-\xi_{b}l_{a}+\gamma_{ab}\,,$ (3)

where $\gamma_{ab}=\delta_{ij}(e^{i})_{a}(e^{j})_{b}$ is the induced metric of
the cross-section $C$. Using this basis and the definition of $\kappa$, we can
get

$\displaystyle\kappa=l_{b}\xi^{a}\nabla_{a}\xi^{b}\,.$ (4)

The Killing horizon of a stationary black hole can be characterized by the
vanishing shear $\sigma_{ab}$ and expansion $\vartheta$ of the horizon, which
are defined by

$\displaystyle\begin{aligned}
\vartheta=\frac{1}{2}\gamma^{ab}\mathcal{L}_{\xi}\gamma_{ab}\,,\quad\quad\sigma_{ab}=\frac{1}{2}\mathcal{L}_{\xi}\gamma_{ab}-\frac{\vartheta}{n-2}\gamma_{ab}\,.\end{aligned}$
(5)

Using the fact that $\xi^{a}$ is a Killing vector field, it is easy to see

$\displaystyle\xi^{a}\nabla_{a}\kappa=0\,.$ (6)

We can note that the derivation of Eq. (6) did not use the equation of motion
and therefore it is true for any gravitational theories. To prove the
constancy of surface gravity, we only need to show $\kappa$ vanishes along the
transverse directions, i.e., $D_{a}\kappa=\gamma_{a}{}^{b}\nabla_{b}\kappa=0$.
Here $D_{a}$ is the induced covariant derivative operator on the cross-section
$C$ which is defined by

$\displaystyle\begin{aligned}
D_{a}T_{bcd\cdots}:=\gamma_{a}{}^{a_{1}}\gamma_{b}{}^{b_{1}}\gamma_{c}{}^{c_{1}}\cdots\nabla_{a_{1}}T_{b_{1}c_{1}d_{1}\cdots}\end{aligned}$
(7)

for any tensor $T_{bcd\cdots}$ on $C$.

Using the Gauss-Codazzi equation Poisson2004 ; Vega:2011ue  and considering
the fact that expansion and shear vanish on the entire horizon, we can get

$\displaystyle R_{abcd}\xi^{a}(e_{i})^{b}\xi^{c}(e_{j})^{d}=0\,,$ (8)
$\displaystyle R_{abcd}\xi^{a}(e_{i})^{b}(e_{j})^{c}(e_{k})^{d}=0\,,$
$\displaystyle R_{ab}\xi^{a}\xi^{b}=0$

on the horizon $\mathcal{H}$. Using these conditions, it is straightforward to
get

$\displaystyle
D_{b}\kappa=R_{aecd}\xi^{a}l^{e}\gamma_{b}{}^{c}\xi^{d}=-R_{ac}\xi^{a}\gamma_{b}{}^{c}\,.$
(9)

For the Einstein gravity, the equation of motion gives

$\displaystyle G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R=8\pi T_{ab}\,.$ (10)

Thus, the derivative of $\kappa$ in the spatial direction becomes

$\displaystyle D_{b}\kappa=-8\pi T_{ac}\xi^{a}\gamma_{b}{}^{c}\,.$ (11)

Moreover, combining (LABEL:Rabcd) and (10) also gives
$T_{ab}\xi^{a}\xi^{b}|_{\mathcal{H}}=0$, which means $T_{ab}\xi^{b}$ can be
written as linear combination of $\xi_{a}$ and $(e_{i})_{a}$ on $\mathcal{H}$.
If the matter field satisfies the dominant energy condition, $T_{ab}\xi^{b}$
must be timelike or null. Therefore, $T_{ab}\xi^{b}\propto\xi_{a}$ on
$\mathcal{H}$, which implies $T_{ac}\xi^{a}\gamma_{b}{}^{c}=0$. These imply
that $D_{a}\kappa=0$ in Einstein gravity, and therefore $\kappa$ is a constant
on the horizon.

##  III The Horndeski Gravitational theory

In this paper, we would like to consider the zeroth law in the Horndeski
gravity with non-minimally coupled scalar field $\chi$ and linear curvature
term. A full Horndeski action can be found in Refs. Deffayet:2011gz ;
Kobayashi:2011nu . In this paper, we would like to consider a special case of
the full theory for simplification. We take $K=\beta X$, $G_{3}=0$,
$G_{4}=G_{5}=1$ in Ref. Kobayashi:2011nu . Then, the action of a simplified
$n$-dimension Horndeski gravitational theory can be written asWang:2020svl

$\displaystyle I=I_{\text{Horn}}+I_{\text{matter}}\,,$ (12)

where $I_{\text{matter}}$ is the action of the minimally coupled matter
fields, and $I_{\text{Horn}}$ is given by

$\displaystyle I_{\text{Horn}}=\frac{1}{16\pi}\int
d^{n}x\sqrt{-g}\left[R-\frac{1}{2}(\beta g^{ab}-\alpha
G^{ab})\nabla_{a}\chi\nabla_{b}\chi\right]\,,$ (13)

where $G_{ab}$ is the Einstein tensor and $\alpha$, $\beta$ are some coupling
constants. The equation of motion is given byWang:2020svl

$\displaystyle H_{ab}=8\pi T_{ab}\,,$ (14)

in which

$\displaystyle\begin{aligned}
H_{ab}=&G_{ab}-\frac{\alpha}{4}G_{ab}(\nabla\chi)^{2}-\frac{\alpha}{4}g_{ab}A-\frac{\alpha}{2}K_{ab}-\beta
H^{(\beta)}_{ab}\end{aligned}$ (15)

with

$\displaystyle
A=(\nabla^{c}\chi\nabla^{d}\chi)(\nabla_{c}\chi\nabla_{d}\chi)-(\nabla^{2}\chi)^{2}+2R^{cd}\nabla_{c}\chi\nabla_{d}\chi\,,$
(16) $\displaystyle
K_{ab}=\frac{R}{2}\nabla_{a}\chi\nabla_{b}\chi-2\nabla_{c}\chi\nabla_{(a}\chi
R^{c}_{b)}+\nabla_{a}\nabla_{b}\chi\nabla^{2}\chi$ $\displaystyle\quad\quad-
R_{acbd}\nabla^{c}\chi\nabla^{d}\chi-\nabla_{a}\nabla^{c}\chi\nabla_{b}\nabla_{c}\chi\,,$
$\displaystyle
H^{(\beta)}_{ab}=\frac{1}{2}\nabla_{a}\chi\nabla_{b}\chi-\frac{1}{4}g_{ab}\nabla^{c}\chi\nabla_{c}\chi\,.$

Here $T_{ab}$ represents the stress-energy tensor of the minimally coupled
matter fields. In this paper, we only assume these minimally coupled matter
fields satisfy the dominant energy condition, i.e., $\xi^{a}T_{ab}$ must be
non-spatial.

##  IV the black hole zeroth law in horndeski gravity

In this section, we would like to investigate the zeroth law in Horndeski
gravity. For this purpose, we divide the main calculation into three parts. In
the first part, we calculate the contraction of the equation of motion and
$\xi^{a}\xi^{b}$. In the second part, we calculate the contraction of the
equation of motion and $\xi^{a}(e_{i})^{b}$. In the third part, we analyze the
identity we have found and gives a discussion.

###  IV.1 Contract with $\xi^{a}\xi^{b}$

We would like to contract $\xi^{a}\xi^{b}$ both side of Eq. (14) in this
subsection. Considering that $\xi^{a}$ is a null vector on the horizon, the
contraction of $H_{ab}$ and $\xi^{a}\xi^{b}$ gives

$\displaystyle H_{ab}\xi^{a}\xi^{b}$
$\displaystyle=\left[1-\frac{\alpha}{4}(\nabla\chi)^{2}\right]R_{ab}\xi^{a}\xi^{b}$
(17)
$\displaystyle-\frac{\alpha}{2}K_{ab}\xi^{a}\xi^{b}-\frac{\beta}{2}\xi^{a}\nabla_{a}\chi\xi^{b}\nabla_{b}\chi\,.$

Note that $\xi^{a}$ is a Killing vector field. In this paper, we would like to
consider a special solution of the Horndeski gravity where the non-minimally
coupled scalar field $\chi$ shares the same symmetries with the spacetime,
i.e., we have $\mathcal{L}_{\xi}\chi=\xi^{a}\nabla_{a}\chi=0$ and therefore
the last term of the above expression vanishes. Together with the third
equation in Eq. (LABEL:Rabcd), the first term of Eq. (17) also vanishes. Then,
Eq. (17) further simplifies to

$\displaystyle H_{ab}\xi^{a}\xi^{b}=-\frac{\alpha}{2}K_{ab}\xi^{a}\xi^{b}\,.$
(18)

Again using the Killing condition $\xi^{a}\nabla_{a}\chi=0$, the lhs of Eq.
(18) becomes

$\displaystyle K_{ab}\xi^{a}\xi^{b}=$
$\displaystyle\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}\chi\nabla^{2}\chi-\xi^{a}\xi^{b}R_{acbd}\nabla^{c}\chi\nabla^{d}\chi$
(19)
$\displaystyle-\xi^{a}\xi^{b}\nabla_{a}\nabla^{c}\chi\nabla_{b}\nabla_{c}\chi\,.$

The first term vanishes on the horizon since

$\displaystyle\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}\chi$
$\displaystyle=\xi^{a}\nabla_{a}(\xi^{b}\nabla_{b}\chi)-(\xi^{a}\nabla_{a}\xi^{b})\nabla_{b}\chi$
(20)
$\displaystyle=\mathcal{L}_{\xi}^{2}\chi-\kappa\xi^{b}\nabla_{b}\chi=0\,,$

where we used the definition of the surface gravity and the fact that
$\xi^{a}$ is the Killing vector field. Then, using the decomposition of the
metric (3) and considering the symmetries of the Riemann tensor, the second
term induces to

$\displaystyle\xi^{a}\xi^{b}R_{acbd}\nabla^{c}\chi\nabla^{d}\chi=\xi^{a}\xi^{b}R_{acbd}g^{ce}g^{df}\nabla_{e}\chi\nabla_{f}\chi$
(21) $\displaystyle=R_{acbd}\xi^{a}\xi^{b}D^{c}\chi
D^{d}\chi-2\xi^{a}\xi^{b}R_{acbd}l^{c}\xi^{e}\nabla_{e}\chi D^{d}\chi$
$\displaystyle\quad+\xi^{a}\xi^{b}l^{c}l^{d}R_{acbd}\xi^{e}\nabla_{e}\chi\xi^{f}\nabla_{f}\chi\,.$

According to (LABEL:Rabcd),
$R_{aebf}\gamma_{c}{}^{e}\gamma_{d}{}^{f}\xi^{a}\xi^{b}$ vanishes on the
horizon. Together with the Killing condition $\xi^{a}\nabla_{a}\chi=0$, it is
easy to get

$\displaystyle\xi^{a}\xi^{b}R_{acbd}\nabla^{c}\chi\nabla^{d}\chi=0$ (22)

on the horizon.

Similarly, using Eq. (3), we can expand the third term of Eq. (19) as

$\displaystyle\xi^{a}\xi^{b}\nabla_{a}\nabla^{c}\chi\nabla_{b}\nabla_{c}\chi=\xi^{a}\xi^{b}g^{cd}\nabla_{a}\nabla_{d}\chi\nabla_{b}\nabla_{c}\chi$
(23)
$\displaystyle=\xi^{a}\xi^{b}\gamma^{cd}\nabla_{a}\nabla_{d}\chi\nabla_{b}\nabla_{c}\chi-2\xi^{a}\xi^{b}l^{c}\xi^{d}\nabla_{a}\nabla_{d}\chi\nabla_{b}\nabla_{c}\chi$
$\displaystyle=\xi^{a}\xi^{b}\gamma^{cd}\nabla_{a}\nabla_{d}\chi\nabla_{b}\nabla_{c}\chi\,,$

where we have used Eq. (20) in the last step. As for the remaining part, we
consider

$\displaystyle\begin{aligned}
\xi^{a}\gamma_{c}{}^{d}\nabla_{d}\nabla_{a}\chi&=\gamma_{c}{}^{d}\nabla_{d}(\xi^{a}\nabla_{a}\chi)-(\gamma_{c}{}^{d}\nabla_{d}\xi^{a})\nabla_{a}\chi\\\
&=-(\gamma_{c}{}^{d}\nabla_{d}\xi^{a})\nabla_{a}\chi\,.\end{aligned}$ (24)

Then, it can be expanded as

$\displaystyle\xi^{a}\gamma_{c}{}^{d}\nabla_{d}\nabla_{a}\chi$
$\displaystyle=-(\delta_{c}{}^{d}+\xi_{c}l^{d}+\xi^{d}l_{c})\nabla_{d}\xi^{a}\nabla_{a}\chi$
(25) $\displaystyle=-\nabla^{a}\chi(\nabla_{c}\xi_{a}+\kappa
l_{c}\xi_{a}+l^{b}\xi_{c}\nabla_{b}\xi_{a})\,.$

For the Killing horizon generated by $\xi^{a}$, we have $\xi^{a}$ is
hypersurface orthogonal, i.e., $\xi^{a}$ satisfies

$\displaystyle\xi_{c}\nabla_{b}\xi_{a}=\xi_{b}\nabla_{c}\xi_{a}+\xi_{a}\nabla_{b}\xi_{c}$
(26)

on the horizon. After taking this condition into account, we can find

$\displaystyle\xi^{a}\gamma_{c}{}^{d}\nabla_{d}\nabla_{a}\chi$
$\displaystyle=-\nabla^{a}\chi(\nabla_{c}\xi_{a}+\kappa
l_{c}\xi_{a}+l^{b}\xi_{c}\nabla_{b}\xi_{a})$ (27)
$\displaystyle=-\nabla^{a}\chi(\nabla_{c}\xi_{a}+\kappa
l_{c}\xi_{a}+l^{b}\xi_{b}\nabla_{c}\xi_{a}+l^{b}\xi_{a}\nabla_{b}\xi_{c})$
$\displaystyle=-\nabla^{a}\chi[\nabla_{c}\xi_{a}-\nabla_{c}\xi_{a}+\xi_{a}(\kappa
l_{c}+l^{b}\nabla_{b}\xi_{c})]$ $\displaystyle=0\,.$

Finally, we find the three terms in Eq. (19) all vanish on the horizon and
therefore we have

$\displaystyle 8\pi T_{ab}\xi^{a}\xi^{b}=H_{ab}\xi^{a}\xi^{b}=0$ (28)

on the horizon $\mathcal{H}$. With a similar discussion in sec. II, we assume
the minimally coupled matter field satisfies the dominant energy condition.
The dominant energy condition gives that for any future-directed, timelike
vector $Z^{a}$, $-T^{a}_{b}Z^{b}$ is further directed timelike or null vector.
Therefore, for the future-directed null vector $\xi^{a}$, there must be
$-T_{ab}\xi^{a}Z^{b}\leq 0$ for any future-directed timelike vector $Z^{a}$,
which means $T_{ab}\xi^{a}$ cannot be spacelike. In addition, considering the
result (28), we can written $T_{ab}\xi^{b}$ as linear combination of $\xi_{a}$
and $(e_{i})_{a}$ on $\mathcal{H}$. Combining with $T_{ab}\xi^{b}$ must be
timelike or null, it is easy to find $T_{ab}\xi^{b}\propto\xi_{a}$ on
$\mathcal{H}$, which implies

$\displaystyle T_{ab}\xi^{a}\gamma_{c}{}^{b}=0$ (29)

on the horizon $\mathcal{H}$ in the Horndeski gravity.

###  IV.2 Contract with $\xi^{a}\gamma_{c}{}^{b}$

In this subsection, we would like to find the corresponding relation of
$D_{b}\kappa$ using the equation of motion in Horndeski gravity to prove the
zeroth law. To do this, we contract $\xi^{a}\gamma_{c}{}^{b}$ both side of
(14). Since $\xi^{a}$ is orthogonal to the cross-section and the Killing
condition $\xi^{a}\nabla_{a}\chi=0$, the contraction
$H_{ac}\xi^{a}\gamma_{b}{}^{c}$ gives

$\displaystyle H_{ab}\xi^{a}\gamma_{c}{}^{b}$
$\displaystyle=\left[1-\frac{\alpha}{4}(\nabla\chi)^{2}\right]R_{ab}\xi^{a}\gamma_{c}{}^{b}-\frac{\alpha}{2}K_{ab}\xi^{a}\gamma_{c}{}^{b}$
(30)
$\displaystyle=-\left[1-\frac{\alpha}{4}(D\chi)^{2}\right]D_{c}\kappa-\frac{\alpha}{2}K_{ab}\xi^{a}\gamma_{c}{}^{b}\,.$

Then, we would like to calculate the remaining five terms in
$K_{ab}\xi^{a}\gamma_{c}{}^{b}$. The first term is vanishing since
$\xi^{a}\nabla_{a}\chi=0$. For the second term, it can be written as

$\displaystyle-2\xi^{a}\gamma_{c}{}^{b}\nabla_{d}\chi\nabla_{(a}\chi
R^{d}_{b)}=-\xi^{a}\gamma_{c}{}^{b}R_{ad}\nabla^{d}\chi\nabla_{b}\chi$ (31)
$\displaystyle=-(-l^{(a}\xi^{e)}+\gamma^{ae})(-l^{(d}\xi^{f)}+\gamma^{df})\xi_{e}\gamma_{c}{}^{b}R_{ad}\nabla_{f}\chi\nabla_{b}\chi\,.$

After expanding Eq. (LABEL:second2) and considering the Killing condition, the
orthogonality as well as $R_{ab}\xi^{a}\xi^{b}=0$, we can find the only term
left is

$\displaystyle-2\xi^{a}\gamma_{c}{}^{b}\nabla_{d}\chi\nabla_{(a}\chi
R^{d}_{b)}$
$\displaystyle=-R_{ad}\gamma_{c}{}^{b}\gamma^{df}\xi^{a}\nabla_{b}\chi\nabla_{f}\chi$
(32) $\displaystyle=D^{a}\chi D_{c}\chi D_{a}\kappa\,.$

For the third term, using what we have found in Eq. (27), it is easy to find
that this term vanishes. The fourth term can be calculated using the same
method. It is not hard to find

$\displaystyle-\xi^{a}\gamma_{c}{}^{b}R_{aebd}\nabla^{e}\chi\nabla^{d}\chi$
(33)
$\displaystyle=-(-l^{(e}\xi^{f)}+\gamma^{ef})(-l^{(d}\xi^{g)}+\gamma^{dg})\xi^{a}\gamma_{c}{}^{b}R_{aebd}\nabla_{f}\chi\nabla_{g}\chi$
$\displaystyle=R_{aedf}\gamma_{g}{}^{e}\gamma_{b}{}^{f}\gamma_{c}{}^{d}\xi^{a}\nabla^{g}\nabla^{b}\chi+R_{gebf}l^{a}\gamma_{c}{}^{e}\gamma_{d}{}^{f}\xi^{g}\xi^{b}\nabla_{a}\chi\nabla^{d}\chi$

after using that $\xi^{a}$ is the Killing vector field. Then, considering Eq.
(LABEL:Rabcd), we can find the above two terms all vanish on the horizon,
i.e., we have

$\displaystyle-\xi^{a}\gamma_{c}{}^{b}R_{aebd}\nabla^{e}\chi\nabla^{d}\chi=0\,.$
(34)

For the fifth term, we do the same as before. Using the expression of the
metric and considering the orthogonal relation between the basis, we can get

$\displaystyle-\xi^{a}\gamma_{c}{}^{b}\nabla_{a}\nabla^{d}\chi\nabla_{b}\nabla_{d}\chi=l_{d}\gamma_{bf}\gamma_{c}{}^{b}\xi_{a}\xi_{e}\nabla^{e}\nabla^{d}\chi\nabla^{f}\nabla^{a}\chi$
(35)
$\displaystyle+l_{a}\gamma_{bf}\gamma_{c}{}^{b}\xi_{d}\xi_{e}\nabla^{e}\nabla^{d}\chi\nabla^{f}\nabla^{a}\chi-\gamma_{bf}\gamma_{c}{}^{b}\gamma_{da}\xi_{e}\nabla^{e}\nabla^{d}\chi\nabla^{f}\nabla^{a}\chi\,.$

Then, using Eqs. (20) and (27), it is obvious that

$\displaystyle-\xi^{a}\gamma_{c}{}^{b}\nabla_{a}\nabla^{d}\chi\nabla_{b}\nabla_{d}\chi=0\,.$
(36)

Combining the above results together, Eq. (30) gives the following identity:

$\displaystyle\begin{aligned}
H_{ab}\xi^{a}\gamma_{c}{}^{b}=-\left[1-\frac{\alpha}{4}(D\chi)^{2}\right]D_{c}\kappa-\frac{\alpha}{2}D^{a}\chi
D_{c}\chi D_{a}\kappa\,.\end{aligned}$ (37)

Considering the condition (29) which is given by the dominant energy
condition, we have

$\displaystyle\left[1-\frac{\alpha}{4}(D\chi)^{2}\right]D_{c}\kappa+\frac{\alpha}{2}D^{a}\chi
D_{c}\chi D_{a}\kappa=0\,.$ (38)

###  IV.3 Generalized zeroth law

In this subsection, we would like to give a discussion to the above results.
As suggested by Ghosh and SarkarGhosh:2020dkk , we rewrite the identity (37)
to a different form

$\displaystyle X_{c}=\alpha M^{a}_{c}X_{a}\,,$ (39)

where

$\displaystyle X_{c}\equiv D_{c}\kappa\,,$ (40)

and

$\displaystyle
M_{c}{}^{a}\equiv\frac{1}{4}(D\chi)^{2}\delta_{c}{}^{a}-\frac{1}{2}D_{c}\chi
D^{a}\chi\,.$ (41)

Since we assume that all quantities have a smooth limit to general relativity
when $\alpha\rightarrow 0$ in Horndeski gravity. With a similar argument to
Ref. Ghosh:2020dkk  will arise, $X_{a}$ can be expanded in terms of $\alpha$
and can be written as

$\displaystyle
X_{a}=(X_{0})_{a}+\alpha(X_{1})_{a}+\alpha^{2}(X_{2})_{a}+\dots\,,$ (42)

where $(X_{0})_{a}$ is the corresponding $X_{a}$ in Einstein gravity and
therefore it is zero. Similarly, $M_{c}^{a}$ can also be written as

$\displaystyle
M_{c}^{a}=(M_{0})_{c}^{a}+\alpha(M_{1})_{c}^{a}+\alpha^{2}(M_{2})_{c}^{a}+\dots\,.$
(43)

According to the assumption that all quantities have a smooth limit to general
relativity, the scalar field should vanish when $\alpha\rightarrow 0$. Then, a
closer look at (41) will reveal that $M_{a}^{c}$ is a quantity of $\alpha^{2}$
order, i.e. $(M_{0})_{c}^{a}=(M_{1})_{c}^{a}=0$. However, as we will discuss
below, we can expand $M_{c}^{a}$ as (43) since the actual order of $M_{c}^{a}$
does not affect the final result. Eq. (39) can be expanded in terms of
$\alpha$

$\displaystyle(X_{0})_{c}+\alpha(X_{1})_{c}+\dots=$
$\displaystyle\alpha(M_{0})_{c}^{a}(X_{0})_{a}+\alpha^{2}[(M_{0})_{c}^{a}(X_{1})_{a}$
(44) $\displaystyle+(M_{1})_{c}^{a}(X_{0})_{a}]+\dots\,.$

This identity is established for any $\alpha$, and every order of $\alpha$
should be equal on both sides. At first, we have $(X_{0})_{a}=0$. Then, we
equal the first order of $\alpha$. With the face that both $(M_{0})_{c}^{a}$
and $(X_{0})_{a}$ are zero, we have
$(X_{1})_{c}=(M_{0})_{c}^{a}(X_{0})_{a}=0$. For the second order, we have
$(X_{2})_{c}=(M_{0})_{c}^{a}(X_{1})_{a}+(M_{1})_{c}^{a}(X_{0})_{a}$, since
$(X_{0})_{a}$ and $(X_{1})_{a}$ are zero, we can find $(X_{2})_{a}=0$. For the
$k$th order of $\alpha$, the identity gives

$\displaystyle(X_{k})_{c}=\sum_{i=0}^{k-1}(M_{i})_{c}^{a}(X_{k-i-1})_{a}\,.$
(45)

By a simple mathematical induction, we can prove that
$(X_{2})_{c}=(X_{3})_{c}=\dots=0$ one by one. And finally, we can get
$X_{c}=0$. As the discussion above, the zeroth law always holds as long as
$X_{c}$ satisfies the equation with the same form as (39) once we throw the
smooth condition to the gravitational theory. And the zeroth law holds in
Horndeski gravity.

##  V conclusion

In this paper, we studied the zeroth law in Horndeski gravity. We first
reviewed the proof of the black hole zeroth law in Einstein gravity and gave
some basic setups on the Killing horizon. Then, according to a similar
calculation as Ref. Ghosh:2020dkk , we calculated the derivative of the
surface gravity $\kappa$ in each direction along the event horizon in the
stationary black holes based on the equation of motion in Horndeski gravity
with the assumption that the scalar field $\chi$ shares the same symmetries
with the spacetime. We found that the derivative of $\kappa$ along the
transverse directions satisfies such an identity

$\displaystyle D_{c}\kappa=\alpha M_{c}^{a}D_{a}\kappa$ (46)

once we assume the minimally coupled matter fields satisfy the dominant energy
condition. With a similar discussion to Ref. Ghosh:2020dkk , according to the
above identity, we can get $D_{a}\kappa=0$ as long as all the quantities have
a smooth limit to Einstein gravity. Then, combining with the universal
relation $\xi^{a}\nabla_{a}\kappa=0$, this result means that the surface
gravity $\kappa$ is a constant on the whole stationary horizon in Horndeski
gravity, and therefore the zeroth law is satisfied.

It is worth noting that we only studied a part of the Horndeski gravity
theory. The zeroth law of the full Horndeski theory is also worth studying.
The tedious equation of motion of the full Horndeski theory is calculated in
Ref. Kobayashi:2011nu  and we can discover the terms appearing in the equation
are quite similar with (14). The further extension of the zeroth law proof to
general Horndeski gravity theory will be complex but not hard. Besides, in our
paper, we assumed that the scalar field $\chi$ also shares the Killing
symmetries. However, in Horndeski gravity, there exists the solution where
$\mathcal{L}_{\xi}\chi\neq 0$. Solutions like this is found in Refs.
Babichev:2013cya ; Babichev:2016rlq . The spacetime in this solution is
spherically symmetric while the scalar field is given by
$\chi(t,r)=qt+\psi(r)$. It is obvious that for this solution, the scalar field
$\chi$ does not satisfy $\mathcal{L}_{\xi}\chi\neq 0$. It is also very
interesting to further consider how to prove the zeroth law in this kind of
solution.

## Acknowledgement

This research was supported by National Natural Science Foundation of China
(NSFC) with Grants No. 11775022 and 11873044.

## References

  * (1) R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cim. 1 , 252-276 (1969).
  * (2) J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161-170 (1973) doi:10.1007/BF01645742
  * (3) S. W. Hawking, Commun. Math. Phys. 43, 199-220 (1975) [erratum: Commun. Math. Phys. 46, 206 (1976)] doi:10.1007/BF02345020
  * (4) S. W. Hawking, Commun. Math. Phys. 25, 152-166 (1972) doi:10.1007/BF01877517
  * (5) I. Racz and R. M. Wald, Class. Quant. Grav. 13, 539-553 (1996) doi:10.1088/0264-9381/13/3/017 [arXiv:gr-qc/9507055 [gr-qc]].
  * (6) S. Sarkar and S. Bhattacharya, Phys. Rev. D 87, no.4, 044023 (2013) doi:10.1103/PhysRevD.87.044023 [arXiv:1205.2042 [gr-qc]].
  * (7) R. Ghosh and S. Sarkar, Phys. Rev. D 102, no.10, 101503 (2020) doi:10.1103/PhysRevD.102.101503 [arXiv:2009.01543 [gr-qc]].
  * (8) T. A. de Pirey Saint Alby and N. Yunes, Phys. Rev. D 96, no.6, 064040 (2017) doi:10.1103/PhysRevD.96.064040 [arXiv:1703.06341 [gr-qc]].
  * (9) P. A. R. Ade et al. [Planck], Astron. Astrophys. 594, A13 (2016) doi:10.1051/0004-6361/201525830 [arXiv:1502.01589 [astro-ph.CO]].
  * (10) G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405, 279-390 (2005) doi:10.1016/j.physrep.2004.08.031 [arXiv:hep-ph/0404175 [hep-ph]].
  * (11) P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559-606 (2003) doi:10.1103/RevModPhys.75.559 [arXiv:astro-ph/0207347 [astro-ph]].
  * (12) R. A. Alpher and R. C. Herman, Phys. Rev. 75, no.7, 1089-1095 (1949) doi:10.1103/physrev.75.1089
  * (13) G. W. Horndeski, Int. J. Theor. Phys. 10, 363-384 (1974) doi:10.1007/BF01807638
  * (14) C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, Phys. Rev. D 84, 064039 (2011) doi:10.1103/PhysRevD.84.064039 [arXiv:1103.3260 [hep-th]].
  * (15) T. Kobayashi, M. Yamaguchi and J. Yokoyama, Prog. Theor. Phys. 126, 511-529 (2011) doi:10.1143/PTP.126.511 [arXiv:1105.5723 [hep-th]].
  * (16) C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saffin, Phys. Rev. Lett. 108, 051101 (2012) doi:10.1103/PhysRevLett.108.051101 [arXiv:1106.2000 [hep-th]].
  * (17) A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D 79, 064036 (2009) doi:10.1103/PhysRevD.79.064036 [arXiv:0811.2197 [hep-th]].
  * (18) E. N. Saridakis and S. V. Sushkov, Phys. Rev. D 81, 083510 (2010) doi:10.1103/PhysRevD.81.083510 [arXiv:1002.3478 [gr-qc]].
  * (19) A. Maselli, H. O. Silva, M. Minamitsuji and E. Berti, Phys. Rev. D 93, no.12, 124056 (2016) doi:10.1103/PhysRevD.93.124056 [arXiv:1603.04876 [gr-qc]].
  * (20) X. H. Feng, H. S. Liu, H. Lü and C. N. Pope, JHEP 11, 176 (2015) doi:10.1007/JHEP11(2015)176 [arXiv:1509.07142 [hep-th]].
  * (21) Y. G. Miao and Z. M. Xu, Eur. Phys. J. C 76, no.11, 638 (2016) doi:10.1140/epjc/s10052-016-4482-1 [arXiv:1607.06629 [hep-th]].
  * (22) X. H. Feng, H. S. Liu, H. Lü and C. N. Pope, Phys. Rev. D 93, no.4, 044030 (2016) doi:10.1103/PhysRevD.93.044030 [arXiv:1512.02659 [hep-th]].
  * (23) Y. P. Hu, H. A. Zeng, Z. M. Jiang and H. Zhang, Phys. Rev. D 100, no.8, 084004 (2019) doi:10.1103/PhysRevD.100.084004 [arXiv:1812.09938 [gr-qc]].
  * (24) K. Hajian, S. Liberati, M. M. Sheikh-Jabbari and M. H. Vahidinia, Phys. Lett. B 812, 136002 (2021) doi:10.1016/j.physletb.2020.136002 [arXiv:2005.12985 [gr-qc]].
  * (25) X. Y. Wang and J. Jiang, Phys. Rev. D 102, no.8, 084020 (2020) doi:10.1103/PhysRevD.102.084020 [arXiv:2008.09774 [gr-qc]].
  * (26) Poisson, E. (2004). A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511606601
  * (27) I. Vega, E. Poisson and R. Massey, Class. Quant. Grav. 28, 175006 (2011) doi:10.1088/0264-9381/28/17/175006 [arXiv:1106.0510 [gr-qc]].
  * (28) E. Babichev and C. Charmousis, JHEP 08, 106 (2014) doi:10.1007/JHEP08(2014)106 [arXiv:1312.3204 [gr-qc]].
  * (29) E. Babichev, C. Charmousis and A. Lehébel, Class. Quant. Grav. 33, no.15, 154002 (2016) doi:10.1088/0264-9381/33/15/154002 [arXiv:1604.06402 [gr-qc]].