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Case Study – Methods, Examples and Guide

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A case study is a research method that involves an in-depth examination and analysis of a particular phenomenon or case, such as an individual, organization, community, event, or situation.

It is a qualitative research approach that aims to provide a detailed and comprehensive understanding of the case being studied. Case studies typically involve multiple sources of data, including interviews, observations, documents, and artifacts, which are analyzed using various techniques, such as content analysis, thematic analysis, and grounded theory. The findings of a case study are often used to develop theories, inform policy or practice, or generate new research questions.

Types of Case Study

Types and Methods of Case Study are as follows:

Single-Case Study

A single-case study is an in-depth analysis of a single case. This type of case study is useful when the researcher wants to understand a specific phenomenon in detail.

For Example , A researcher might conduct a single-case study on a particular individual to understand their experiences with a particular health condition or a specific organization to explore their management practices. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as content analysis or thematic analysis. The findings of a single-case study are often used to generate new research questions, develop theories, or inform policy or practice.

Multiple-Case Study

A multiple-case study involves the analysis of several cases that are similar in nature. This type of case study is useful when the researcher wants to identify similarities and differences between the cases.

For Example, a researcher might conduct a multiple-case study on several companies to explore the factors that contribute to their success or failure. The researcher collects data from each case, compares and contrasts the findings, and uses various techniques to analyze the data, such as comparative analysis or pattern-matching. The findings of a multiple-case study can be used to develop theories, inform policy or practice, or generate new research questions.

Exploratory Case Study

An exploratory case study is used to explore a new or understudied phenomenon. This type of case study is useful when the researcher wants to generate hypotheses or theories about the phenomenon.

For Example, a researcher might conduct an exploratory case study on a new technology to understand its potential impact on society. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as grounded theory or content analysis. The findings of an exploratory case study can be used to generate new research questions, develop theories, or inform policy or practice.

Descriptive Case Study

A descriptive case study is used to describe a particular phenomenon in detail. This type of case study is useful when the researcher wants to provide a comprehensive account of the phenomenon.

For Example, a researcher might conduct a descriptive case study on a particular community to understand its social and economic characteristics. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as content analysis or thematic analysis. The findings of a descriptive case study can be used to inform policy or practice or generate new research questions.

Instrumental Case Study

An instrumental case study is used to understand a particular phenomenon that is instrumental in achieving a particular goal. This type of case study is useful when the researcher wants to understand the role of the phenomenon in achieving the goal.

For Example, a researcher might conduct an instrumental case study on a particular policy to understand its impact on achieving a particular goal, such as reducing poverty. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as content analysis or thematic analysis. The findings of an instrumental case study can be used to inform policy or practice or generate new research questions.

Case Study Data Collection Methods

Here are some common data collection methods for case studies:

Interviews involve asking questions to individuals who have knowledge or experience relevant to the case study. Interviews can be structured (where the same questions are asked to all participants) or unstructured (where the interviewer follows up on the responses with further questions). Interviews can be conducted in person, over the phone, or through video conferencing.

Observations

Observations involve watching and recording the behavior and activities of individuals or groups relevant to the case study. Observations can be participant (where the researcher actively participates in the activities) or non-participant (where the researcher observes from a distance). Observations can be recorded using notes, audio or video recordings, or photographs.

Documents can be used as a source of information for case studies. Documents can include reports, memos, emails, letters, and other written materials related to the case study. Documents can be collected from the case study participants or from public sources.

Surveys involve asking a set of questions to a sample of individuals relevant to the case study. Surveys can be administered in person, over the phone, through mail or email, or online. Surveys can be used to gather information on attitudes, opinions, or behaviors related to the case study.

Artifacts are physical objects relevant to the case study. Artifacts can include tools, equipment, products, or other objects that provide insights into the case study phenomenon.

How to conduct Case Study Research

Conducting a case study research involves several steps that need to be followed to ensure the quality and rigor of the study. Here are the steps to conduct case study research:

• Define the research questions: The first step in conducting a case study research is to define the research questions. The research questions should be specific, measurable, and relevant to the case study phenomenon under investigation.
• Select the case: The next step is to select the case or cases to be studied. The case should be relevant to the research questions and should provide rich and diverse data that can be used to answer the research questions.
• Collect data: Data can be collected using various methods, such as interviews, observations, documents, surveys, and artifacts. The data collection method should be selected based on the research questions and the nature of the case study phenomenon.
• Analyze the data: The data collected from the case study should be analyzed using various techniques, such as content analysis, thematic analysis, or grounded theory. The analysis should be guided by the research questions and should aim to provide insights and conclusions relevant to the research questions.
• Draw conclusions: The conclusions drawn from the case study should be based on the data analysis and should be relevant to the research questions. The conclusions should be supported by evidence and should be clearly stated.
• Validate the findings: The findings of the case study should be validated by reviewing the data and the analysis with participants or other experts in the field. This helps to ensure the validity and reliability of the findings.
• Write the report: The final step is to write the report of the case study research. The report should provide a clear description of the case study phenomenon, the research questions, the data collection methods, the data analysis, the findings, and the conclusions. The report should be written in a clear and concise manner and should follow the guidelines for academic writing.

Examples of Case Study

Here are some examples of case study research:

• The Hawthorne Studies : Conducted between 1924 and 1932, the Hawthorne Studies were a series of case studies conducted by Elton Mayo and his colleagues to examine the impact of work environment on employee productivity. The studies were conducted at the Hawthorne Works plant of the Western Electric Company in Chicago and included interviews, observations, and experiments.
• The Stanford Prison Experiment: Conducted in 1971, the Stanford Prison Experiment was a case study conducted by Philip Zimbardo to examine the psychological effects of power and authority. The study involved simulating a prison environment and assigning participants to the role of guards or prisoners. The study was controversial due to the ethical issues it raised.
• The Challenger Disaster: The Challenger Disaster was a case study conducted to examine the causes of the Space Shuttle Challenger explosion in 1986. The study included interviews, observations, and analysis of data to identify the technical, organizational, and cultural factors that contributed to the disaster.
• The Enron Scandal: The Enron Scandal was a case study conducted to examine the causes of the Enron Corporation’s bankruptcy in 2001. The study included interviews, analysis of financial data, and review of documents to identify the accounting practices, corporate culture, and ethical issues that led to the company’s downfall.
• The Fukushima Nuclear Disaster : The Fukushima Nuclear Disaster was a case study conducted to examine the causes of the nuclear accident that occurred at the Fukushima Daiichi Nuclear Power Plant in Japan in 2011. The study included interviews, analysis of data, and review of documents to identify the technical, organizational, and cultural factors that contributed to the disaster.

Application of Case Study

Case studies have a wide range of applications across various fields and industries. Here are some examples:

Business and Management

Case studies are widely used in business and management to examine real-life situations and develop problem-solving skills. Case studies can help students and professionals to develop a deep understanding of business concepts, theories, and best practices.

Case studies are used in healthcare to examine patient care, treatment options, and outcomes. Case studies can help healthcare professionals to develop critical thinking skills, diagnose complex medical conditions, and develop effective treatment plans.

Case studies are used in education to examine teaching and learning practices. Case studies can help educators to develop effective teaching strategies, evaluate student progress, and identify areas for improvement.

Social Sciences

Case studies are widely used in social sciences to examine human behavior, social phenomena, and cultural practices. Case studies can help researchers to develop theories, test hypotheses, and gain insights into complex social issues.

Law and Ethics

Case studies are used in law and ethics to examine legal and ethical dilemmas. Case studies can help lawyers, policymakers, and ethical professionals to develop critical thinking skills, analyze complex cases, and make informed decisions.

Purpose of Case Study

The purpose of a case study is to provide a detailed analysis of a specific phenomenon, issue, or problem in its real-life context. A case study is a qualitative research method that involves the in-depth exploration and analysis of a particular case, which can be an individual, group, organization, event, or community.

The primary purpose of a case study is to generate a comprehensive and nuanced understanding of the case, including its history, context, and dynamics. Case studies can help researchers to identify and examine the underlying factors, processes, and mechanisms that contribute to the case and its outcomes. This can help to develop a more accurate and detailed understanding of the case, which can inform future research, practice, or policy.

Case studies can also serve other purposes, including:

• Illustrating a theory or concept: Case studies can be used to illustrate and explain theoretical concepts and frameworks, providing concrete examples of how they can be applied in real-life situations.
• Developing hypotheses: Case studies can help to generate hypotheses about the causal relationships between different factors and outcomes, which can be tested through further research.
• Providing insight into complex issues: Case studies can provide insights into complex and multifaceted issues, which may be difficult to understand through other research methods.
• Informing practice or policy: Case studies can be used to inform practice or policy by identifying best practices, lessons learned, or areas for improvement.

Advantages of Case Study Research

There are several advantages of case study research, including:

• In-depth exploration: Case study research allows for a detailed exploration and analysis of a specific phenomenon, issue, or problem in its real-life context. This can provide a comprehensive understanding of the case and its dynamics, which may not be possible through other research methods.
• Rich data: Case study research can generate rich and detailed data, including qualitative data such as interviews, observations, and documents. This can provide a nuanced understanding of the case and its complexity.
• Holistic perspective: Case study research allows for a holistic perspective of the case, taking into account the various factors, processes, and mechanisms that contribute to the case and its outcomes. This can help to develop a more accurate and comprehensive understanding of the case.
• Theory development: Case study research can help to develop and refine theories and concepts by providing empirical evidence and concrete examples of how they can be applied in real-life situations.
• Practical application: Case study research can inform practice or policy by identifying best practices, lessons learned, or areas for improvement.
• Contextualization: Case study research takes into account the specific context in which the case is situated, which can help to understand how the case is influenced by the social, cultural, and historical factors of its environment.

Limitations of Case Study Research

There are several limitations of case study research, including:

• Limited generalizability : Case studies are typically focused on a single case or a small number of cases, which limits the generalizability of the findings. The unique characteristics of the case may not be applicable to other contexts or populations, which may limit the external validity of the research.
• Biased sampling: Case studies may rely on purposive or convenience sampling, which can introduce bias into the sample selection process. This may limit the representativeness of the sample and the generalizability of the findings.
• Subjectivity: Case studies rely on the interpretation of the researcher, which can introduce subjectivity into the analysis. The researcher’s own biases, assumptions, and perspectives may influence the findings, which may limit the objectivity of the research.
• Limited control: Case studies are typically conducted in naturalistic settings, which limits the control that the researcher has over the environment and the variables being studied. This may limit the ability to establish causal relationships between variables.
• Time-consuming: Case studies can be time-consuming to conduct, as they typically involve a detailed exploration and analysis of a specific case. This may limit the feasibility of conducting multiple case studies or conducting case studies in a timely manner.
• Resource-intensive: Case studies may require significant resources, including time, funding, and expertise. This may limit the ability of researchers to conduct case studies in resource-constrained settings.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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The Ultimate Guide to Qualitative Research - Part 1: The Basics

• Introduction and overview
• What is qualitative research?
• What is qualitative data?
• Examples of qualitative data
• Qualitative vs. quantitative research
• Mixed methods
• Qualitative research preparation
• Theoretical perspective
• Theoretical framework
• Literature reviews

Research question

• Conceptual framework
• Conceptual vs. theoretical framework

Data collection

• Qualitative research methods
• Focus groups
• Observational research

What is a case study?

Applications for case study research, what is a good case study, process of case study design, benefits and limitations of case studies.

• Ethnographical research
• Ethical considerations
• Confidentiality and privacy
• Power dynamics
• Reflexivity

Case studies

Case studies are essential to qualitative research , offering a lens through which researchers can investigate complex phenomena within their real-life contexts. This chapter explores the concept, purpose, applications, examples, and types of case studies and provides guidance on how to conduct case study research effectively.

Whereas quantitative methods look at phenomena at scale, case study research looks at a concept or phenomenon in considerable detail. While analyzing a single case can help understand one perspective regarding the object of research inquiry, analyzing multiple cases can help obtain a more holistic sense of the topic or issue. Let's provide a basic definition of a case study, then explore its characteristics and role in the qualitative research process.

Definition of a case study

A case study in qualitative research is a strategy of inquiry that involves an in-depth investigation of a phenomenon within its real-world context. It provides researchers with the opportunity to acquire an in-depth understanding of intricate details that might not be as apparent or accessible through other methods of research. The specific case or cases being studied can be a single person, group, or organization – demarcating what constitutes a relevant case worth studying depends on the researcher and their research question .

Among qualitative research methods , a case study relies on multiple sources of evidence, such as documents, artifacts, interviews , or observations , to present a complete and nuanced understanding of the phenomenon under investigation. The objective is to illuminate the readers' understanding of the phenomenon beyond its abstract statistical or theoretical explanations.

Characteristics of case studies

Case studies typically possess a number of distinct characteristics that set them apart from other research methods. These characteristics include a focus on holistic description and explanation, flexibility in the design and data collection methods, reliance on multiple sources of evidence, and emphasis on the context in which the phenomenon occurs.

Furthermore, case studies can often involve a longitudinal examination of the case, meaning they study the case over a period of time. These characteristics allow case studies to yield comprehensive, in-depth, and richly contextualized insights about the phenomenon of interest.

The role of case studies in research

Case studies hold a unique position in the broader landscape of research methods aimed at theory development. They are instrumental when the primary research interest is to gain an intensive, detailed understanding of a phenomenon in its real-life context.

In addition, case studies can serve different purposes within research - they can be used for exploratory, descriptive, or explanatory purposes, depending on the research question and objectives. This flexibility and depth make case studies a valuable tool in the toolkit of qualitative researchers.

Remember, a well-conducted case study can offer a rich, insightful contribution to both academic and practical knowledge through theory development or theory verification, thus enhancing our understanding of complex phenomena in their real-world contexts.

What is the purpose of a case study?

Case study research aims for a more comprehensive understanding of phenomena, requiring various research methods to gather information for qualitative analysis . Ultimately, a case study can allow the researcher to gain insight into a particular object of inquiry and develop a theoretical framework relevant to the research inquiry.

Why use case studies in qualitative research?

Using case studies as a research strategy depends mainly on the nature of the research question and the researcher's access to the data.

Conducting case study research provides a level of detail and contextual richness that other research methods might not offer. They are beneficial when there's a need to understand complex social phenomena within their natural contexts.

The explanatory, exploratory, and descriptive roles of case studies

Case studies can take on various roles depending on the research objectives. They can be exploratory when the research aims to discover new phenomena or define new research questions; they are descriptive when the objective is to depict a phenomenon within its context in a detailed manner; and they can be explanatory if the goal is to understand specific relationships within the studied context. Thus, the versatility of case studies allows researchers to approach their topic from different angles, offering multiple ways to uncover and interpret the data .

The impact of case studies on knowledge development

Case studies play a significant role in knowledge development across various disciplines. Analysis of cases provides an avenue for researchers to explore phenomena within their context based on the collected data.

This can result in the production of rich, practical insights that can be instrumental in both theory-building and practice. Case studies allow researchers to delve into the intricacies and complexities of real-life situations, uncovering insights that might otherwise remain hidden.

Types of case studies

In qualitative research , a case study is not a one-size-fits-all approach. Depending on the nature of the research question and the specific objectives of the study, researchers might choose to use different types of case studies. These types differ in their focus, methodology, and the level of detail they provide about the phenomenon under investigation.

Understanding these types is crucial for selecting the most appropriate approach for your research project and effectively achieving your research goals. Let's briefly look at the main types of case studies.

Exploratory case studies

Exploratory case studies are typically conducted to develop a theory or framework around an understudied phenomenon. They can also serve as a precursor to a larger-scale research project. Exploratory case studies are useful when a researcher wants to identify the key issues or questions which can spur more extensive study or be used to develop propositions for further research. These case studies are characterized by flexibility, allowing researchers to explore various aspects of a phenomenon as they emerge, which can also form the foundation for subsequent studies.

Descriptive case studies

Descriptive case studies aim to provide a complete and accurate representation of a phenomenon or event within its context. These case studies are often based on an established theoretical framework, which guides how data is collected and analyzed. The researcher is concerned with describing the phenomenon in detail, as it occurs naturally, without trying to influence or manipulate it.

Explanatory case studies

Explanatory case studies are focused on explanation - they seek to clarify how or why certain phenomena occur. Often used in complex, real-life situations, they can be particularly valuable in clarifying causal relationships among concepts and understanding the interplay between different factors within a specific context.

Intrinsic, instrumental, and collective case studies

These three categories of case studies focus on the nature and purpose of the study. An intrinsic case study is conducted when a researcher has an inherent interest in the case itself. Instrumental case studies are employed when the case is used to provide insight into a particular issue or phenomenon. A collective case study, on the other hand, involves studying multiple cases simultaneously to investigate some general phenomena.

Each type of case study serves a different purpose and has its own strengths and challenges. The selection of the type should be guided by the research question and objectives, as well as the context and constraints of the research.

The flexibility, depth, and contextual richness offered by case studies make this approach an excellent research method for various fields of study. They enable researchers to investigate real-world phenomena within their specific contexts, capturing nuances that other research methods might miss. Across numerous fields, case studies provide valuable insights into complex issues.

Critical information systems research

Case studies provide a detailed understanding of the role and impact of information systems in different contexts. They offer a platform to explore how information systems are designed, implemented, and used and how they interact with various social, economic, and political factors. Case studies in this field often focus on examining the intricate relationship between technology, organizational processes, and user behavior, helping to uncover insights that can inform better system design and implementation.

Health research

Health research is another field where case studies are highly valuable. They offer a way to explore patient experiences, healthcare delivery processes, and the impact of various interventions in a real-world context.

Case studies can provide a deep understanding of a patient's journey, giving insights into the intricacies of disease progression, treatment effects, and the psychosocial aspects of health and illness.

Asthma research studies

Specifically within medical research, studies on asthma often employ case studies to explore the individual and environmental factors that influence asthma development, management, and outcomes. A case study can provide rich, detailed data about individual patients' experiences, from the triggers and symptoms they experience to the effectiveness of various management strategies. This can be crucial for developing patient-centered asthma care approaches.

Other fields

Apart from the fields mentioned, case studies are also extensively used in business and management research, education research, and political sciences, among many others. They provide an opportunity to delve into the intricacies of real-world situations, allowing for a comprehensive understanding of various phenomena.

Case studies, with their depth and contextual focus, offer unique insights across these varied fields. They allow researchers to illuminate the complexities of real-life situations, contributing to both theory and practice.

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Understanding the key elements of case study design is crucial for conducting rigorous and impactful case study research. A well-structured design guides the researcher through the process, ensuring that the study is methodologically sound and its findings are reliable and valid. The main elements of case study design include the research question , propositions, units of analysis, and the logic linking the data to the propositions.

The research question is the foundation of any research study. A good research question guides the direction of the study and informs the selection of the case, the methods of collecting data, and the analysis techniques. A well-formulated research question in case study research is typically clear, focused, and complex enough to merit further detailed examination of the relevant case(s).

Propositions

Propositions, though not necessary in every case study, provide a direction by stating what we might expect to find in the data collected. They guide how data is collected and analyzed by helping researchers focus on specific aspects of the case. They are particularly important in explanatory case studies, which seek to understand the relationships among concepts within the studied phenomenon.

Units of analysis

The unit of analysis refers to the case, or the main entity or entities that are being analyzed in the study. In case study research, the unit of analysis can be an individual, a group, an organization, a decision, an event, or even a time period. It's crucial to clearly define the unit of analysis, as it shapes the qualitative data analysis process by allowing the researcher to analyze a particular case and synthesize analysis across multiple case studies to draw conclusions.

Argumentation

This refers to the inferential model that allows researchers to draw conclusions from the data. The researcher needs to ensure that there is a clear link between the data, the propositions (if any), and the conclusions drawn. This argumentation is what enables the researcher to make valid and credible inferences about the phenomenon under study.

Understanding and carefully considering these elements in the design phase of a case study can significantly enhance the quality of the research. It can help ensure that the study is methodologically sound and its findings contribute meaningful insights about the case.

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Conducting a case study involves several steps, from defining the research question and selecting the case to collecting and analyzing data . This section outlines these key stages, providing a practical guide on how to conduct case study research.

Defining the research question

The first step in case study research is defining a clear, focused research question. This question should guide the entire research process, from case selection to analysis. It's crucial to ensure that the research question is suitable for a case study approach. Typically, such questions are exploratory or descriptive in nature and focus on understanding a phenomenon within its real-life context.

Selecting and defining the case

The selection of the case should be based on the research question and the objectives of the study. It involves choosing a unique example or a set of examples that provide rich, in-depth data about the phenomenon under investigation. After selecting the case, it's crucial to define it clearly, setting the boundaries of the case, including the time period and the specific context.

Previous research can help guide the case study design. When considering a case study, an example of a case could be taken from previous case study research and used to define cases in a new research inquiry. Considering recently published examples can help understand how to select and define cases effectively.

Developing a detailed case study protocol

A case study protocol outlines the procedures and general rules to be followed during the case study. This includes the data collection methods to be used, the sources of data, and the procedures for analysis. Having a detailed case study protocol ensures consistency and reliability in the study.

The protocol should also consider how to work with the people involved in the research context to grant the research team access to collecting data. As mentioned in previous sections of this guide, establishing rapport is an essential component of qualitative research as it shapes the overall potential for collecting and analyzing data.

Collecting data

Gathering data in case study research often involves multiple sources of evidence, including documents, archival records, interviews, observations, and physical artifacts. This allows for a comprehensive understanding of the case. The process for gathering data should be systematic and carefully documented to ensure the reliability and validity of the study.

Analyzing and interpreting data

The next step is analyzing the data. This involves organizing the data , categorizing it into themes or patterns , and interpreting these patterns to answer the research question. The analysis might also involve comparing the findings with prior research or theoretical propositions.

Writing the case study report

The final step is writing the case study report . This should provide a detailed description of the case, the data, the analysis process, and the findings. The report should be clear, organized, and carefully written to ensure that the reader can understand the case and the conclusions drawn from it.

Each of these steps is crucial in ensuring that the case study research is rigorous, reliable, and provides valuable insights about the case.

The type, depth, and quality of data in your study can significantly influence the validity and utility of the study. In case study research, data is usually collected from multiple sources to provide a comprehensive and nuanced understanding of the case. This section will outline the various methods of collecting data used in case study research and discuss considerations for ensuring the quality of the data.

Interviews are a common method of gathering data in case study research. They can provide rich, in-depth data about the perspectives, experiences, and interpretations of the individuals involved in the case. Interviews can be structured , semi-structured , or unstructured , depending on the research question and the degree of flexibility needed.

Observations

Observations involve the researcher observing the case in its natural setting, providing first-hand information about the case and its context. Observations can provide data that might not be revealed in interviews or documents, such as non-verbal cues or contextual information.

Documents and artifacts

Documents and archival records provide a valuable source of data in case study research. They can include reports, letters, memos, meeting minutes, email correspondence, and various public and private documents related to the case.

These records can provide historical context, corroborate evidence from other sources, and offer insights into the case that might not be apparent from interviews or observations.

Physical artifacts refer to any physical evidence related to the case, such as tools, products, or physical environments. These artifacts can provide tangible insights into the case, complementing the data gathered from other sources.

Ensuring the quality of data collection

Determining the quality of data in case study research requires careful planning and execution. It's crucial to ensure that the data is reliable, accurate, and relevant to the research question. This involves selecting appropriate methods of collecting data, properly training interviewers or observers, and systematically recording and storing the data. It also includes considering ethical issues related to collecting and handling data, such as obtaining informed consent and ensuring the privacy and confidentiality of the participants.

Data analysis

Analyzing case study research involves making sense of the rich, detailed data to answer the research question. This process can be challenging due to the volume and complexity of case study data. However, a systematic and rigorous approach to analysis can ensure that the findings are credible and meaningful. This section outlines the main steps and considerations in analyzing data in case study research.

Organizing the data

The first step in the analysis is organizing the data. This involves sorting the data into manageable sections, often according to the data source or the theme. This step can also involve transcribing interviews, digitizing physical artifacts, or organizing observational data.

Categorizing and coding the data

Once the data is organized, the next step is to categorize or code the data. This involves identifying common themes, patterns, or concepts in the data and assigning codes to relevant data segments. Coding can be done manually or with the help of software tools, and in either case, qualitative analysis software can greatly facilitate the entire coding process. Coding helps to reduce the data to a set of themes or categories that can be more easily analyzed.

Identifying patterns and themes

After coding the data, the researcher looks for patterns or themes in the coded data. This involves comparing and contrasting the codes and looking for relationships or patterns among them. The identified patterns and themes should help answer the research question.

Interpreting the data

Once patterns and themes have been identified, the next step is to interpret these findings. This involves explaining what the patterns or themes mean in the context of the research question and the case. This interpretation should be grounded in the data, but it can also involve drawing on theoretical concepts or prior research.

Verification of the data

The last step in the analysis is verification. This involves checking the accuracy and consistency of the analysis process and confirming that the findings are supported by the data. This can involve re-checking the original data, checking the consistency of codes, or seeking feedback from research participants or peers.

Like any research method , case study research has its strengths and limitations. Researchers must be aware of these, as they can influence the design, conduct, and interpretation of the study.

Understanding the strengths and limitations of case study research can also guide researchers in deciding whether this approach is suitable for their research question . This section outlines some of the key strengths and limitations of case study research.

Benefits include the following:

• Rich, detailed data: One of the main strengths of case study research is that it can generate rich, detailed data about the case. This can provide a deep understanding of the case and its context, which can be valuable in exploring complex phenomena.
• Flexibility: Case study research is flexible in terms of design , data collection , and analysis . A sufficient degree of flexibility allows the researcher to adapt the study according to the case and the emerging findings.
• Real-world context: Case study research involves studying the case in its real-world context, which can provide valuable insights into the interplay between the case and its context.
• Multiple sources of evidence: Case study research often involves collecting data from multiple sources , which can enhance the robustness and validity of the findings.

On the other hand, researchers should consider the following limitations:

• Generalizability: A common criticism of case study research is that its findings might not be generalizable to other cases due to the specificity and uniqueness of each case.
• Time and resource intensive: Case study research can be time and resource intensive due to the depth of the investigation and the amount of collected data.
• Complexity of analysis: The rich, detailed data generated in case study research can make analyzing the data challenging.
• Subjectivity: Given the nature of case study research, there may be a higher degree of subjectivity in interpreting the data , so researchers need to reflect on this and transparently convey to audiences how the research was conducted.

Being aware of these strengths and limitations can help researchers design and conduct case study research effectively and interpret and report the findings appropriately.

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noun as in case history

Strongest match

• medical history

Weak matches

• medical record
• psychiatric history

Example Sentences

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A case study would be your Twilight director Catherine Hardwicke.

A good case study for the minority superhero problem is Luke Cage.

He was asked to review a case study out of Lebanon that had cited his work.

Instead, now we have a political science case-study proving how political fortunes can shift and change at warp speed.

One interesting case study is Sir Arthur Evans, the original excavator and “restorer” of the Minoan palace of Knossos on Crete.

As this is a case study, it should be said that my first mistake was in discrediting my early religious experience.

The author of a recent case study of democracy in a frontier county commented on the need for this kind of investigation.

How could a case study of Virginia during this period illustrate these developments?

Related Words

Words related to case study are not direct synonyms, but are associated with the word case study . Browse related words to learn more about word associations.

noun as in record of what happened

From Roget's 21st Century Thesaurus, Third Edition Copyright © 2013 by the Philip Lief Group.

Case Study vs. Survey

What's the difference.

Case studies and surveys are both research methods used in various fields to gather information and insights. However, they differ in their approach and purpose. A case study involves an in-depth analysis of a specific individual, group, or situation, aiming to understand the complexities and unique aspects of the subject. It often involves collecting qualitative data through interviews, observations, and document analysis. On the other hand, a survey is a structured data collection method that involves gathering information from a larger sample size through standardized questionnaires. Surveys are typically used to collect quantitative data and provide a broader perspective on a particular topic or population. While case studies provide rich and detailed information, surveys offer a more generalizable and statistical overview.

Further Detail

Introduction.

When conducting research, there are various methods available to gather data and analyze it. Two commonly used methods are case study and survey. Both approaches have their own unique attributes and can be valuable in different research contexts. In this article, we will explore the characteristics of case study and survey, highlighting their strengths and limitations.

A case study is an in-depth investigation of a particular individual, group, or phenomenon. It involves collecting detailed information about the subject of study through various sources such as interviews, observations, and document analysis. Case studies are often used in social sciences, psychology, and business research to gain a deep understanding of complex issues.

One of the key attributes of a case study is its ability to provide rich and detailed data. Researchers can gather extensive information about the subject, including their background, experiences, and perspectives. This depth of data allows for a comprehensive analysis and interpretation of the case, providing valuable insights into the phenomenon under investigation.

Furthermore, case studies are particularly useful when studying rare or unique cases. Since case studies focus on specific individuals or groups, they can shed light on situations that are not easily replicated or observed in larger populations. This makes case studies valuable in exploring complex and nuanced phenomena that may not be easily captured through other research methods.

However, it is important to note that case studies have certain limitations. Due to their in-depth nature, case studies are often time-consuming and resource-intensive. Researchers need to invest significant effort in data collection, analysis, and interpretation. Additionally, the findings of a case study may not be easily generalized to larger populations, as the focus is on a specific case rather than a representative sample.

Despite these limitations, case studies offer a unique opportunity to explore complex issues in real-life contexts. They provide a detailed understanding of individual experiences and can generate hypotheses for further research.

A survey is a research method that involves collecting data from a sample of individuals through a structured questionnaire or interview. Surveys are widely used in social sciences, market research, and public opinion studies to gather information about a larger population. They aim to provide a snapshot of people's opinions, attitudes, behaviors, or characteristics.

One of the main advantages of surveys is their ability to collect data from a large number of respondents. By reaching out to a representative sample, researchers can generalize the findings to a larger population. Surveys also allow for efficient data collection, as questionnaires can be distributed electronically or in person, making it easier to gather a wide range of responses in a relatively short period.

Moreover, surveys offer a structured approach to data collection, ensuring consistency in the questions asked and the response options provided. This allows for easy comparison and analysis of the data, making surveys suitable for quantitative research. Surveys can also be conducted anonymously, which can encourage respondents to provide honest and unbiased answers, particularly when sensitive topics are being explored.

However, surveys also have their limitations. One of the challenges is the potential for response bias. Respondents may provide inaccurate or socially desirable answers, leading to biased results. Additionally, surveys often rely on self-reported data, which may be subject to memory recall errors or misinterpretation of questions. Researchers need to carefully design the survey instrument and consider potential biases to ensure the validity and reliability of the data collected.

Furthermore, surveys may not capture the complexity and depth of individual experiences. They provide a snapshot of people's opinions or behaviors at a specific point in time, but may not uncover the underlying reasons or motivations behind those responses. Surveys also rely on predetermined response options, limiting the range of possible answers and potentially overlooking important nuances.

Case studies and surveys are both valuable research methods, each with its own strengths and limitations. Case studies offer in-depth insights into specific cases, providing rich and detailed data. They are particularly useful for exploring complex and unique phenomena. On the other hand, surveys allow for efficient data collection from a large number of respondents, enabling generalization to larger populations. They provide structured and quantifiable data, making them suitable for statistical analysis.

Ultimately, the choice between case study and survey depends on the research objectives, the nature of the research question, and the available resources. Researchers need to carefully consider the attributes of each method and select the most appropriate approach to gather and analyze data effectively.

Comparisons may contain inaccurate information about people, places, or facts. Please report any issues.

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Thesaurus for Case study

Related terms for case study - synonyms, antonyms and sentences with case study, similar meaning.

• case history
• medical history
• medical record
• psychiatric history
• pilot study
• examination

Opposite meaning

• creative writing
• form a theory
• form theories
• formulate a theory
• hypothesize
• let's say
• make guesses
• philosophize
• take as a hypothesis
• traditional practice

Common usage

• child study
• field study
• follow up study
• panel study
• study group
• study leave
• clinical study
• market study
• blind study
• feasibility study
• literary study
• quick study
• case in point
• extreme case

Sentence Examples

Proper usage in context.

• A case study was presented on best practices in Guatemala
• Begin your case study story with the ending
• For a more detailed case study of SSCA, see annex 1.3
• More information about this customer case study is available on
• NATO would be a useful case study in this respect

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Blog Beginner Guides 6 Types of Case Studies to Inspire Your Research and Analysis

6 Types of Case Studies to Inspire Your Research and Analysis

Written by: Ronita Mohan Sep 20, 2021

Case studies have become powerful business tools. But what is a case study? What are the benefits of creating one? Are there limitations to the format?

If you’ve asked yourself these questions, our helpful guide will clear things up. Learn how to use a case study for business. Find out how cases analysis works in psychology and research.

We’ve also got examples of case studies to inspire you.

Haven’t made a case study before? You can easily  create a case study  with Venngage’s customizable case study templates .

Click to jump ahead:

What is a case study?

6 types of case studies, what is a business case study, what is a case study in research, what is a case study in psychology, what is the case study method, benefits of case studies, limitations of case studies, faqs about case studies.

A case study is a research process aimed at learning about a subject, an event or an organization. Case studies are use in business, the social sciences and healthcare.

A case study may focus on one observation or many. It can also examine a series of events or a single case. An effective case study tells a story and provides a conclusion.

Healthcare industries write reports on patients and diagnoses. Marketing case study examples , like the one below, highlight the benefits of a business product.

Now that you know what a case study is, let’s look at the six different types of case studies next.

There are six common types of case reports. Depending on your industry, you might use one of these types.

Descriptive case studies

Explanatory case studies, exploratory case reports, intrinsic case studies, instrumental case studies, collective case reports.

We go into more detail about each type of study in the guide below.

Related:  15+ Professional Case Study Examples [Design Tips + Templates]

When you have an existing hypothesis, you can design a descriptive study. This type of report starts with a description. The aim is to find connections between the subject being studied and a theory.

Once these connections are found, the study can conclude. The results of this type of study will usually suggest how to develop a theory further.

A study like the one below has concrete results. A descriptive report would use the quantitative data as a suggestion for researching the subject deeply.

When an incident occurs in a field, an explanation is required. An explanatory report investigates the cause of the event. It will include explanations for that cause.

The study will also share details about the impact of the event. In most cases, this report will use evidence to predict future occurrences. The results of explanatory reports are definitive.

Note that there is no room for interpretation here. The results are absolute.

The study below is a good example. It explains how one brand used the services of another. It concludes by showing definitive proof that the collaboration was successful.

Another example of this study would be in the automotive industry. If a vehicle fails a test, an explanatory study will examine why. The results could show that the failure was because of a particular part.

Related: How to Write a Case Study [+ Design Tips]

An explanatory report is a self-contained document. An exploratory one is only the beginning of an investigation.

Exploratory cases act as the starting point of studies. This is usually conducted as a precursor to large-scale investigations. The research is used to suggest why further investigations are needed.

An exploratory study can also be used to suggest methods for further examination.

For example, the below analysis could have found inconclusive results. In that situation, it would be the basis for an in-depth study.

Intrinsic studies are more common in the field of psychology. These reports can also be conducted in healthcare or social work.

These types of studies focus on a unique subject, such as a patient. They can sometimes study groups close to the researcher.

The aim of such studies is to understand the subject better. This requires learning their history. The researcher will also examine how they interact with their environment.

For instance, if the case study below was about a unique brand, it could be an intrinsic study.

Once the study is complete, the researcher will have developed a better understanding of a phenomenon. This phenomenon will likely not have been studied or theorized about before.

Examples of intrinsic case analysis can be found across psychology. For example, Jean Piaget’s theories on cognitive development. He established the theory from intrinsic studies into his own children.

Related: What Disney Villains Can Tell Us About Color Psychology [Infographic]

This is another type of study seen in medical and psychology fields. Instrumental reports are created to examine more than just the primary subject.

When research is conducted for an instrumental study, it is to provide the basis for a larger phenomenon. The subject matter is usually the best example of the phenomenon. This is why it is being studied.

Take the example of the fictional brand below.

Assume it’s examining lead generation strategies. It may want to show that visual marketing is the definitive lead generation tool. The brand can conduct an instrumental case study to examine this phenomenon.

Collective studies are based on instrumental case reports. These types of studies examine multiple reports.

There are a number of reasons why collective reports are created:

• To provide evidence for starting a new study
• To find pattens between multiple instrumental reports
• To find differences in similar types of cases
• Gain a deeper understanding of a complex phenomenon
• Understand a phenomenon from diverse contexts

A researcher could use multiple reports, like the one below, to build a collective case report.

Related: 10+ Case Study Infographic Templates That Convert

A business or marketing case study aims at showcasing a successful partnership. This can be between a brand and a client. Or the case study can examine a brand’s project.

There is a perception that case studies are used to advertise a brand. But effective reports, like the one below, can show clients how a brand can support them.

Hubspot created a case study on a customer that successfully scaled its business. The report outlines the various Hubspot tools used to achieve these results.

Hubspot also added a video with testimonials from the client company’s employees.

So, what is the purpose of a case study for businesses? There is a lot of competition in the corporate world. Companies are run by people. They can be on the fence about which brand to work with.

Business reports  stand out aesthetically, as well. They use  brand colors  and brand fonts . Usually, a combination of the client’s and the brand’s.

With the Venngage  My Brand Kit  feature, businesses can automatically apply their brand to designs.

A business case study, like the one below, acts as social proof. This helps customers decide between your brand and your competitors.

Don’t know how to design a report? You can learn  how to write a case study  with Venngage’s guide. We also share design tips and examples that will help you convert.

Related: 55+ Annual Report Design Templates, Inspirational Examples & Tips [Updated]

Research is a necessary part of every case study. But specific research fields are required to create studies. These fields include user research, healthcare, education, or social work.

For example, this UX Design  report examined the public perception of a client. The brand researched and implemented new visuals to improve it. The study breaks down this research through lessons learned.

Clinical reports are a necessity in the medical field. These documents are used to share knowledge with other professionals. They also help examine new or unusual diseases or symptoms.

The pandemic has led to a significant increase in research. For example,  Spectrum Health  studied the value of health systems in the pandemic. They created the study by examining community outreach.

The pandemic has significantly impacted the field of education. This has led to numerous examinations on remote studying. There have also been studies on how students react to decreased peer communication.

Social work case reports often have a community focus. They can also examine public health responses. In certain regions, social workers study disaster responses.

You now know what case studies in various fields are. In the next step of our guide, we explain the case study method.

In the field of psychology, case studies focus on a particular subject. Psychology case histories also examine human behaviors.

Case reports search for commonalities between humans. They are also used to prescribe further research. Or these studies can elaborate on a solution for a behavioral ailment.

The American Psychology Association  has a number of case studies on real-life clients. Note how the reports are more text-heavy than a business case study.

Famous psychologists such as Sigmund Freud and Anna O popularised the use of case studies in the field. They did so by regularly interviewing subjects. Their detailed observations build the field of psychology.

It is important to note that psychological studies must be conducted by professionals. Psychologists, psychiatrists and therapists should be the researchers in these cases.

Related: What Netflix’s Top 50 Shows Can Teach Us About Font Psychology [Infographic]

The case study method, or case method, is a learning technique where you’re presented with a real-world business challenge and asked how you’d solve it.

After working through it independently and with peers, you learn how the actual scenario unfolded. This approach helps develop problem-solving skills and practical knowledge.

This method often uses various data sources like interviews, observations, and documents to provide comprehensive insights. The below example would have been created after numerous interviews.

Case studies are largely qualitative. They analyze and describe phenomena. While some data is included, a case analysis is not quantitative.

There are a few steps in the case method. You have to start by identifying the subject of your study. Then determine what kind of research is required.

In natural sciences, case studies can take years to complete. Business reports, like this one, don’t take that long. A few weeks of interviews should be enough.

The case method will vary depending on the industry. Reports will also look different once produced.

As you will have seen, business reports are more colorful. The design is also more accessible . Healthcare and psychology reports are more text-heavy.

Designing case reports takes time and energy. So, is it worth taking the time to write them? Here are the benefits of creating case studies.

• Collects large amounts of information
• Helps formulate hypotheses
• Builds the case for further research
• Discovers new insights into a subject
• Builds brand trust and loyalty
• Engages customers through stories

For example, the business study below creates a story around a brand partnership. It makes for engaging reading. The study also shows evidence backing up the information.

We’ve shared the benefits of why studies are needed. We will also look at the limitations of creating them.

Related: How to Present a Case Study like a Pro (With Examples)

There are a few disadvantages to conducting a case analysis. The limitations will vary according to the industry.

• Responses from interviews are subjective
• Subjects may tailor responses to the researcher
• Studies can’t always be replicated
• In certain industries, analyses can take time and be expensive
• Risk of generalizing the results among a larger population

These are some of the common weaknesses of creating case reports. If you’re on the fence, look at the competition in your industry.

Other brands or professionals are building reports, like this example. In that case, you may want to do the same.

What makes a case study a case study?

A case study has a very particular research methodology. They are an in-depth study of a person or a group of individuals. They can also study a community or an organization. Case reports examine real-world phenomena within a set context.

How long should a case study be?

The length of studies depends on the industry. It also depends on the story you’re telling. Most case studies should be at least 500-1500 words long. But you can increase the length if you have more details to share.

What should you ask in a case study?

The one thing you shouldn’t ask is ‘yes’ or ‘no’ questions. Case studies are qualitative. These questions won’t give you the information you need.

Ask your client about the problems they faced. Ask them about solutions they found. Or what they think is the ideal solution. Leave room to ask them follow-up questions. This will help build out the study.

How to present a case study?

When you’re ready to present a case study, begin by providing a summary of the problem or challenge you were addressing. Follow this with an outline of the solution you implemented, and support this with the results you achieved, backed by relevant data. Incorporate visual aids like slides, graphs, and images to make your case study presentation more engaging and impactful.

Now you know what a case study means, you can begin creating one. These reports are a great tool for analyzing brands. They are also useful in a variety of other fields.

Use a visual communication platform like Venngage to design case studies. With Venngage’s templates, you can design easily. Create branded, engaging reports, all without design experience.

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• Introduction

The case study creation process

Types of case studies, benefits and limitations.

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• Table Of Contents

case study , detailed description and assessment of a specific situation in the real world created for the purpose of deriving generalizations and other insights from it. A case study can be about an individual, a group of people, an organization, or an event, among other subjects.

By focusing on a specific subject in its natural setting, a case study can help improve understanding of the broader features and processes at work. Case studies are a research method used in multiple fields, including business, criminology , education , medicine and other forms of health care, anthropology , political science , psychology , and social work . Data in case studies can be both qualitative and quantitative. Unlike experiments, where researchers control and manipulate situations, case studies are considered to be “naturalistic” because subjects are studied in their natural context . ( See also natural experiment .)

The creation of a case study typically involves the following steps:

• The research question to be studied is defined, informed by existing literature and previous research. Researchers should clearly define the scope of the case, and they should compile a list of evidence to be collected as well as identify the nature of insights that they expect to gain from the case study.
• Once the case is identified, the research team is given access to the individual, organization, or situation being studied. Individuals are informed of risks associated with participation and must provide their consent , which may involve signing confidentiality or anonymity agreements.
• Researchers then collect evidence using multiple methods, which may include qualitative techniques, such as interviews, focus groups , and direct observations, as well as quantitative methods, such as surveys, questionnaires, and data audits. The collection procedures need to be well defined to ensure the relevance and accuracy of the evidence.
• The collected evidence is analyzed to come up with insights. Each data source must be reviewed carefully by itself and in the larger context of the case study so as to ensure continued relevance. At the same time, care must be taken not to force the analysis to fit (potentially preconceived) conclusions. While the eventual case study may serve as the basis for generalizations, these generalizations must be made cautiously to ensure that specific nuances are not lost in the averages.
• Finally, the case study is packaged for larger groups and publication. At this stage some information may be withheld, as in business case studies, to allow readers to draw their own conclusions. In scientific fields, the completed case study needs to be a coherent whole, with all findings and statistical relationships clearly documented.

Case studies have been used as a research method across multiple fields. They are particularly popular in the fields of law, business, and employee training; they typically focus on a problem that an individual or organization is facing. The situation is presented in considerable detail, often with supporting data, to discussion participants, who are asked to make recommendations that will solve the stated problem. The business case study as a method of instruction was made popular in the 1920s by instructors at Harvard Business School who adapted an approach used at Harvard Law School in which real-world cases were used in classroom discussions. Other business and law schools started compiling case studies as teaching aids for students. In a business school case study, students are not provided with the complete list of facts pertaining to the topic and are thus forced to discuss and compare their perspectives with those of their peers to recommend solutions.

In criminology , case studies typically focus on the lives of an individual or a group of individuals. These studies can provide particularly valuable insight into the personalities and motives of individual criminals, but they may suffer from a lack of objectivity on the part of the researchers (typically because of the researchers’ biases when working with people with a criminal history), and their findings may be difficult to generalize.

In sociology , the case-study method was developed by Frédéric Le Play in France during the 19th century. This approach involves a field worker staying with a family for a period of time, gathering data on the family members’ attitudes and interactions and on their income, expenditures, and physical possessions. Similar approaches have been used in anthropology . Such studies can sometimes continue for many years.

Case studies provide insight into situations that involve a specific entity or set of circumstances. They can be beneficial in helping to explain the causal relationships between quantitative indicators in a field of study, such as what drives a company’s market share. By introducing real-world examples, they also plunge the reader into an actual, concrete situation and make the concepts real rather than theoretical. They also help people study rare situations that they might not otherwise experience.

Because case studies are in a “naturalistic” environment , they are limited in terms of research design: researchers lack control over what they are studying, which means that the results often cannot be reproduced. Also, care must be taken to stay within the bounds of the research question on which the case study is focusing. Other limitations to case studies revolve around the data collected. It may be difficult, for instance, for researchers to organize the large volume of data that can emerge from the study, and their analysis of the data must be carefully thought through to produce scientifically valid insights. The research methodology used to generate these insights is as important as the insights themselves, for the latter need to be seen in the proper context. Taken out of context, they may lead to erroneous conclusions. Like all scientific studies, case studies need to be approached objectively; personal bias or opinion may skew the research methods as well as the results. ( See also confirmation bias .)

Business case studies in particular have been criticized for approaching a problem or situation from a narrow perspective. Students are expected to come up with solutions for a problem based on the data provided. However, in real life, the situation is typically reversed: business managers face a problem and must then look for data to help them solve it.

10 Case Study Advantages and Disadvantages

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Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

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A case study in academic research is a detailed and in-depth examination of a specific instance or event, generally conducted through a qualitative approach to data.

The most common case study definition that I come across is is Robert K. Yin’s (2003, p. 13) quote provided below:

“An empirical inquiry that investigates a contemporary phenomenon within its real-life context, especially when the boundaries between phenomenon and context are not clearly evident.”

Researchers conduct case studies for a number of reasons, such as to explore complex phenomena within their real-life context, to look at a particularly interesting instance of a situation, or to dig deeper into something of interest identified in a wider-scale project.

While case studies render extremely interesting data, they have many limitations and are not suitable for all studies. One key limitation is that a case study’s findings are not usually generalizable to broader populations because one instance cannot be used to infer trends across populations.

Case Study Advantages and Disadvantages

1. in-depth analysis of complex phenomena.

Case study design allows researchers to delve deeply into intricate issues and situations.

By focusing on a specific instance or event, researchers can uncover nuanced details and layers of understanding that might be missed with other research methods, especially large-scale survey studies.

As Lee and Saunders (2017) argue,

“It allows that particular event to be studies in detail so that its unique qualities may be identified.”

This depth of analysis can provide rich insights into the underlying factors and dynamics of the studied phenomenon.

2. Holistic Understanding

Building on the above point, case studies can help us to understand a topic holistically and from multiple angles.

This means the researcher isn’t restricted to just examining a topic by using a pre-determined set of questions, as with questionnaires. Instead, researchers can use qualitative methods to delve into the many different angles, perspectives, and contextual factors related to the case study.

We can turn to Lee and Saunders (2017) again, who notes that case study researchers “develop a deep, holistic understanding of a particular phenomenon” with the intent of deeply understanding the phenomenon.

3. Examination of rare and Unusual Phenomena

We need to use case study methods when we stumble upon “rare and unusual” (Lee & Saunders, 2017) phenomena that would tend to be seen as mere outliers in population studies.

Take, for example, a child genius. A population study of all children of that child’s age would merely see this child as an outlier in the dataset, and this child may even be removed in order to predict overall trends.

So, to truly come to an understanding of this child and get insights into the environmental conditions that led to this child’s remarkable cognitive development, we need to do an in-depth study of this child specifically – so, we’d use a case study.

4. Helps Reveal the Experiences of Marginalzied Groups

Just as rare and unsual cases can be overlooked in population studies, so too can the experiences, beliefs, and perspectives of marginalized groups.

As Lee and Saunders (2017) argue, “case studies are also extremely useful in helping the expression of the voices of people whose interests are often ignored.”

Take, for example, the experiences of minority populations as they navigate healthcare systems. This was for many years a “hidden” phenomenon, not examined by researchers. It took case study designs to truly reveal this phenomenon, which helped to raise practitioners’ awareness of the importance of cultural sensitivity in medicine.

5. Ideal in Situations where Researchers cannot Control the Variables

Experimental designs – where a study takes place in a lab or controlled environment – are excellent for determining cause and effect . But not all studies can take place in controlled environments (Tetnowski, 2015).

When we’re out in the field doing observational studies or similar fieldwork, we don’t have the freedom to isolate dependent and independent variables. We need to use alternate methods.

Case studies are ideal in such situations.

A case study design will allow researchers to deeply immerse themselves in a setting (potentially combining it with methods such as ethnography or researcher observation) in order to see how phenomena take place in real-life settings.

6. Supports the generation of new theories or hypotheses

While large-scale quantitative studies such as cross-sectional designs and population surveys are excellent at testing theories and hypotheses on a large scale, they need a hypothesis to start off with!

This is where case studies – in the form of grounded research – come in. Often, a case study doesn’t start with a hypothesis. Instead, it ends with a hypothesis based upon the findings within a singular setting.

The deep analysis allows for hypotheses to emerge, which can then be taken to larger-scale studies in order to conduct further, more generalizable, testing of the hypothesis or theory.

7. Reveals the Unexpected

When a largescale quantitative research project has a clear hypothesis that it will test, it often becomes very rigid and has tunnel-vision on just exploring the hypothesis.

Of course, a structured scientific examination of the effects of specific interventions targeted at specific variables is extermely valuable.

But narrowly-focused studies often fail to shine a spotlight on unexpected and emergent data. Here, case studies come in very useful. Oftentimes, researchers set their eyes on a phenomenon and, when examining it closely with case studies, identify data and come to conclusions that are unprecedented, unforeseen, and outright surprising.

As Lars Meier (2009, p. 975) marvels, “where else can we become a part of foreign social worlds and have the chance to become aware of the unexpected?”

Disadvantages

1. not usually generalizable.

Case studies are not generalizable because they tend not to look at a broad enough corpus of data to be able to infer that there is a trend across a population.

As Yang (2022) argues, “by definition, case studies can make no claims to be typical.”

Case studies focus on one specific instance of a phenomenon. They explore the context, nuances, and situational factors that have come to bear on the case study. This is really useful for bringing to light important, new, and surprising information, as I’ve already covered.

But , it’s not often useful for generating data that has validity beyond the specific case study being examined.

2. Subjectivity in interpretation

Case studies usually (but not always) use qualitative data which helps to get deep into a topic and explain it in human terms, finding insights unattainable by quantitative data.

But qualitative data in case studies relies heavily on researcher interpretation. While researchers can be trained and work hard to focus on minimizing subjectivity (through methods like triangulation), it often emerges – some might argue it’s innevitable in qualitative studies.

So, a criticism of case studies could be that they’re more prone to subjectivity – and researchers need to take strides to address this in their studies.

3. Difficulty in replicating results

Case study research is often non-replicable because the study takes place in complex real-world settings where variables are not controlled.

So, when returning to a setting to re-do or attempt to replicate a study, we often find that the variables have changed to such an extent that replication is difficult. Furthermore, new researchers (with new subjective eyes) may catch things that the other readers overlooked.

Replication is even harder when researchers attempt to replicate a case study design in a new setting or with different participants.

Comprehension Quiz for Students

Question 1: What benefit do case studies offer when exploring the experiences of marginalized groups?

a) They provide generalizable data. b) They help express the voices of often-ignored individuals. c) They control all variables for the study. d) They always start with a clear hypothesis.

Question 2: Why might case studies be considered ideal for situations where researchers cannot control all variables?

a) They provide a structured scientific examination. b) They allow for generalizability across populations. c) They focus on one specific instance of a phenomenon. d) They allow for deep immersion in real-life settings.

Question 3: What is a primary disadvantage of case studies in terms of data applicability?

a) They always focus on the unexpected. b) They are not usually generalizable. c) They support the generation of new theories. d) They provide a holistic understanding.

Question 4: Why might case studies be considered more prone to subjectivity?

a) They always use quantitative data. b) They heavily rely on researcher interpretation, especially with qualitative data. c) They are always replicable. d) They look at a broad corpus of data.

Question 5: In what situations are experimental designs, such as those conducted in labs, most valuable?

a) When there’s a need to study rare and unusual phenomena. b) When a holistic understanding is required. c) When determining cause-and-effect relationships. d) When the study focuses on marginalized groups.

Question 6: Why is replication challenging in case study research?

a) Because they always use qualitative data. b) Because they tend to focus on a broad corpus of data. c) Due to the changing variables in complex real-world settings. d) Because they always start with a hypothesis.

Lee, B., & Saunders, M. N. K. (2017). Conducting Case Study Research for Business and Management Students. SAGE Publications.

Meir, L. (2009). Feasting on the Benefits of Case Study Research. In Mills, A. J., Wiebe, E., & Durepos, G. (Eds.). Encyclopedia of Case Study Research (Vol. 2). London: SAGE Publications.

Tetnowski, J. (2015). Qualitative case study research design.  Perspectives on fluency and fluency disorders ,  25 (1), 39-45. ( Source )

Yang, S. L. (2022). The War on Corruption in China: Local Reform and Innovation . Taylor & Francis.

Yin, R. (2003). Case Study research. Thousand Oaks, CA: Sage.

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What Is a Case Study and Why You Should Use Them

Case studies can provide more insights into your business while helping you conduct further research with robust qualitative data analysis to learn more.

If you're in charge of running a company, then you're likely always looking for new ways to run your business more efficiently and increase your customer base while streamlining as many processes as possible.

Unfortunately, it can sometimes be difficult to determine how to go about implementing the proper program in order to be successful. This is why many business owners opt to conduct a case study, which can help significantly. Whether you've been struggling with brand consistency or some other problem, the right case study can identify why your problem exists as well as provide a way to rectify it.

A case study is a great tool that many businesses aren't even aware exists, and there are marketing experts like Mailchimp who can provide you with step-by-step assistance with implementing a plan with a case study. Many companies discover that not only do they need to start a blog in order to improve business, but they also need to create specific and relevant blog titles.

If your company already has a blog, then optimizing your blog posts may be helpful. Regardless of the obstacles that are preventing you from achieving all your professional goals, a case study can work wonders in helping you reverse this issue.

What is a case study?

A case study is a comprehensive report of the results of theory testing or examining emerging themes of a business in real life context. Case studies are also often used in the healthcare industry, conducting health services research with primary research interest around routinely collected healthcare data.

However, for businesses, the purpose of a case study is to help small business owners or company leaders identify the issues and conduct further research into what may be preventing success through information collection, client or customer interviews, and in-depth data analysis.

Knowing the case study definition is crucial for any business owner. By identifying the issues that are hindering a company from achieving all its goals, it's easier to make the necessary corrections to promote success through influenced data collection.

Why are case studies important?

Now that we've answered the questions, "what is a case study?" Why are case studies important? Some of the top reasons why case studies are important include:

• Understand complex issues: Even after you conduct a significant amount of market research , you might have a difficult time understanding exactly what it means. While you might have the basics down, conducting a case study can help you see how that information is applied. Then, when you see how the information can make a difference in business decisions, it could make it easier to understand complex issues.
• Collect data: A case study can also help with data tracking . A case study is a data collection method that can help you describe the information that you have available to you. Then, you can present that information in a way the reader can understand.
• Conduct evaluations: As you learn more about how to write a case study, remember that you can also use a case study to conduct evaluations of a specific situation. A case study is a great way to learn more about complex situations, and you can evaluate how various people responded in that situation. By conducting a case study evaluation, you can learn more about what has worked well, what has not, and what you might want to change in the future.
• Identify potential solutions: A case study can also help you identify solutions to potential problems. If you have an issue in your business that you are trying to solve, you may be able to take a look at a case study where someone has dealt with a similar situation in the past. For example, you may uncover data bias in a specific solution that you would like to address when you tackle the issue on your own. If you need help solving a difficult problem, a case study may be able to help you.

Remember that you can also use case studies to target your audience . If you want to show your audience that you have a significant level of expertise in a field, you may want to publish some case studies that you have handled in the past. Then, when your audience sees that you have had success in a specific area, they may be more likely to provide you with their business. In essence, case studies can be looked at as the original method of social proof, showcasing exactly how you can help someone solve their problems.

What are the benefits of writing a business case study?

Although writing a case study can seem like a tedious task, there are many benefits to conducting one through an in depth qualitative research process.

• Industry understanding: First of all, a case study can give you an in-depth understanding of your industry through a particular conceptual framework and help you identify hidden problems that are preventing you from transcending into the business world.
• Develop theories: If you decide to write a business case study, it provides you with an opportunity to develop new theories. You might have a theory about how to solve a specific problem, but you need to write a business case study to see exactly how that theory has unfolded in the past. Then, you can figure out if you want to apply your theory to a similar issue in the future.
• Evaluate interventions: When you write a business case study that focuses on a specific situation you have been through in the past, you can uncover whether that intervention was truly helpful. This can make it easier to figure out whether you want to use the same intervention in a similar situation in the future.
• Identify best practices: If you want to stay on top of the best practices in your field, conducting case studies can help by allowing you to identify patterns and trends and develop a new list of best practices that you can follow in the future.
• Versatility: Writing a case study also provides you with more versatility. If you want to expand your business applications, you need to figure out how you respond to various problems. When you run a business case study, you open the door to new opportunities, new applications, and new techniques that could help you make a difference in your business down the road.
• Solve problems: Writing a great case study can dramatically improve your chances of reversing your problem and improving your business.
• These are just a few of the biggest benefits you might experience if you decide to publish your case studies. They can be an effective tool for learning, showcasing your talents, and teaching some of your other employees. If you want to grow your audience , you may want to consider publishing some case studies.

What are the limitations of case studies?

Case studies can be a wonderful tool for any business of any size to use to gain an in-depth understanding of their clients, products, customers, or services, but there are limitations.

One limitation of case studies is the fact that, unless there are other recently published examples, there is nothing to compare them to since, most of the time, you are conducting a single, not multiple, case studies.

Another limitation is the fact that most case studies can lack scientific evidence.

Types of case studies

There are specific types of case studies to choose from, and each specific type will yield different results. Some case study types even overlap, which is sometimes more favorable, as they provide even more pertinent data.

Here are overviews of the different types of case studies, each with its own theoretical framework, so you can determine which type would be most effective for helping you meet your goals.

Explanatory case studies

Explanatory case studies are pretty straightforward, as they're not difficult to interpret. This type of case study is best if there aren't many variables involved because explanatory case studies can easily answer questions like "how" and "why" through theory development.

Exploratory case studies

An exploratory case study does exactly what its name implies: it goes into specific detail about the topic at hand in a natural, real-life context with qualitative research.

The benefits of exploratory case studies are limitless, with the main one being that it offers a great deal of flexibility. Having flexibility when writing a case study is important because you can't always predict what obstacles might arise during the qualitative research process.

Collective case studies

Collective case studies require you to study many different individuals in order to obtain usable data.

Case studies that involve an investigation of people will involve many different variables, all of which can't be predicted. Despite this fact, there are many benefits of collective case studies, including the fact that it allows an ongoing analysis of the data collected.

Intrinsic case studies

This type of study differs from the others as it focuses on the inquiry of one specific instance among many possibilities.

Many people prefer these types of case studies because it allows them to learn about the particular instance that they wish to investigate further.

Instrumental case studies

An instrumental case study is similar to an intrinsic one, as it focuses on a particular instance, whether it's a person, organization, or something different.

One thing that differentiates instrumental case studies from intrinsic ones is the fact that instrumental case studies aren't chosen merely because a person is interested in learning about a specific instance.

Tips for writing a case study

If you have decided to write case studies for your company, then you may be unsure of where to start or which type to conduct.

However, it doesn't have to be difficult or confusing to begin conducting a case study that will help you identify ways to improve your business.

Here are some helpful tips for writing your case studies:

1. Your case study must be written in the proper format

When writing a case study, the format that you should be similar to this:

Administrative summary

The executive summary is an overview of what your report will contain, written in a concise manner while providing real-life context.

Despite the fact that the executive summary should appear at the beginning of your case studies, it shouldn't be written until you've completed the entire report because if you write it before you finish the report, this summary may not be completely accurate.

Key problem statement

In this section of your case study, you will briefly describe the problem that you hope to solve by conducting the study. You will have the opportunity to elaborate on the problem that you're focusing on as you get into the breadth of the report.

Problem exploration

This part of the case study isn't as brief as the other two, and it goes into more detail about the problem at hand. Your problem exploration must include why the identified problem needs to be solved as well as the urgency of solving it.

Additionally, it must include justification for conducting the problem-solving, as the benefits must outweigh the efforts and costs.

Proposed resolution

This case study section will also be lengthier than the first two. It must include how you propose going about rectifying the problem. The "recommended solution" section must also include potential obstacles that you might experience, as well as how these will be managed.

Furthermore, you will need to list alternative solutions and explain the reason the chosen solution is best. Charts can enhance your report and make it easier to read, and provide as much proof to substantiate your claim as possible.

Overview of monetary consideration

An overview of monetary consideration is essential for all case studies, as it will be used to convince all involved parties why your project should be funded. You must successfully convince them that the cost is worth the investment it will require. It's important that you stress the necessity for this particular case study and explain the expected outcome.

Execution timeline

In the execution times of case studies, you explain how long you predict it will take to implement your study. The shorter the time it will take to implement your plan, the more apt it is to be approved. However, be sure to provide a reasonable timeline, taking into consideration any additional time that might be needed due to obstacles.

Always include a conclusion in your case study. This is where you will briefly wrap up your entire proposal, stressing the benefits of completing the data collection and data analysis in order to rectify your problem.

2. Make it clear and comprehensive

You want to write your case studies with as much clarity as possible so that every aspect of the report is understood. Be sure to double-check your grammar, spelling, punctuation, and more, as you don't want to submit a poorly-written document.

Not only would a poorly-written case study fail to prove that what you are trying to achieve is important, but it would also increase the chances that your report will be tossed aside and not taken seriously.

3. Don't rush through the process

Writing the perfect case study takes time and patience. Rushing could result in your forgetting to include information that is crucial to your entire study. Don't waste your time creating a study that simply isn't ready. Take the necessary time to perform all the research necessary to write the best case study possible.

Depending on the case study, conducting case study research could mean using qualitative methods, quantitative methods, or both. Qualitative research questions focus on non-numerical data, such as how people feel, their beliefs, their experiences, and so on.

Meanwhile, quantitative research questions focus on numerical or statistical data collection to explain causal links or get an in-depth picture.

It is also important to collect insightful and constructive feedback. This will help you better understand the outcome as well as any changes you need to make to future case studies. Consider using formal and informal ways to collect feedback to ensure that you get a range of opinions and perspectives.

4. Be confident in your theory development

While writing your case study or conducting your formal experimental investigation, you should have confidence in yourself and what you're proposing in your report. If you took the time to gather all the pertinent data collected to complete the report, don't second-guess yourself or doubt your abilities. If you believe your report will be amazing, then it likely will be.

5. Case studies and all qualitative research are long

It's expected that multiple case studies are going to be incredibly boring, and there is no way around this. However, it doesn't mean you can choose your language carefully in order to keep your audience as engaged as possible.

If your audience loses interest in your case study at the beginning, for whatever reason, then this increases the likelihood that your case study will not be funded.

Case study examples

If you want to learn more about how to write a case study, it might be beneficial to take a look at a few case study examples. Below are a few interesting case study examples you may want to take a closer look at.

• Phineas Gage by John Martin Marlow : One of the most famous case studies comes from the medical field, and it is about the story of Phineas Gage, a man who had a railroad spike driven through his head in 1848. As he was working on a railroad, an explosive charge went off prematurely, sending a railroad rod through his head. Even though he survived this incident, he lost his left eye. However, Phineas Gage was studied extensively over the years because his experiences had a significant, lasting impact on his personality. This served as a case study because his injury showed different parts of the brain have different functions.
• Kitty Genovese and the bystander effect : This is a tragic case study that discusses the murder of Kitty Genovese, a woman attacked and murdered in Queens, New York City. Shockingly, while numerous neighbors watched the scene, nobody called for help because they assumed someone else would. This case study helped to define the bystander effect, which is when a person fails to intervene during an emergency because other people are around.
• Henry Molaison and the study of memory : Henry Molaison lost his memory and suffered from debilitating amnesia. He suffered from childhood epilepsy, and medical professionals attempted to remove the part of his brain that was causing his seizures. He had a portion of his brain removed, but it completely took away his ability to hold memories. Even though he went on to live until the age of 82, he was always forced to live in the present moment, as he was completely unable to form new memories.

Case study FAQs

When should you do a case study.

There are several scenarios when conducting a case study can be beneficial. Case studies are often used when there's a "why" or "how" question that needs to be answered. Case studies are also beneficial when trying to understand a complex phenomenon, there's limited research on a topic, or when you're looking for practical solutions to a problem.

How can case study results be used to make business decisions?

You can use the results from a case study to make future business decisions if you find yourself in a similar situation. As you assess the results of a case study, you can identify best practices, evaluate the effectiveness of an intervention, generate new and creative ideas, or get a better understanding of customer needs.

How are case studies different from other research methodologies?

When compared to other research methodologies, such as experimental or qualitative research methodology, a case study does not require a representative sample. For example, if you are performing quantitative research, you have a lot of subjects that expand your sample size. If you are performing experimental research, you may have a random sample in front of you. A case study is usually designed to deliberately focus on unusual situations, which allows it to shed new light on a specific business research problem.

Writing multiple case studies for your business

If you're feeling overwhelmed by the idea of writing a case study and it seems completely foreign, then you aren't alone. Writing a case study for a business is a very big deal, but fortunately, there is help available because an example of a case study doesn't always help.

Mailchimp, a well-known marketing company that provides comprehensive marketing support for all sorts of businesses, can assist you with your case study, or you can review one of their own recently published examples.

Mailchimp can assist you with developing the most effective content strategy to increase your chances of being as successful as possible. Mailchimp's content studio is a great tool that can help your business immensely.

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What Is a Case Study? Definition, Examples, Types & Methods

What is the definition of a case study.

A case study is typically a research paper to generate an in-depth and multi-faced understanding of any complicated issue in a life scenario. It is a well-written research design that is very commonly used in a wide range of disciplines.

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Case Study Examples

• Marketing case study examples: Case studies in marketing are written to show your success, and you must always prominently showcase your buoyant suits. You can use bright, bold colours with many contesting fonts, shapes, and simple icons to highlight your case study.
• You need to highlight your big win on the 2nd page with a bright orange colour with highlighted circles.
• Make the essential data stand out exceptionally to track your prospective customers.
• Marketing all the critical data is very important in your marketing case study.

Use a straightforward and crystal clear layout of the case study.

• Using a straightforward layout in any case study is very effective. For example, keeping a spotless white background and drawing slim lines helps to separate these sections in a specific way for formatting the case study.
• Making the information clear helps draw attention to the results and helps to improve the accessibility of the design.
• The case study examples must sit nicely with more extended reports and a consistent layout.

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What Are the Types of Case Studies?

Case studies can be categorized into several types based on their focus and purpose. Here are some common types of case studies:

• Collective Case Studies : These types of case studies involve investigating any group of individuals. Here, the researchers need to study a group of people in a specific setting or any community. Ex: Psychologists must explore how access to the resources in any society can affect people’s mental wellness.
• Descriptive Case Studies: These involve starting with any descriptive theory. The subjects are then observed, and the gathered information is compared to the preexisting approaches.
• Explanatory Case Studies: These types of case studies are primarily used to conduct any casual investigation. Here, the researchers are more interested in looking for the factors that caused specific results.
• Exploratory Case Studies : These case studies are conducted when researchers want to explore a new or relatively unexplored topic. They are more open-ended and aim to generate hypotheses and ideas for further research.
• Instrumental Case Studies : These case studies are selected because they provide insights into a broader issue or theory. The case is used as a means to investigate a more general phenomenon.
• Intrinsic Case Studies : In these case studies, the case itself is of particular interest due to its uniqueness or rarity. The goal is not to generalize findings to a larger population but to understand the specific case deeply.
• Pilot Case Studies : Pilot case studies are conducted as a preliminary investigation before launching a larger study. They help researchers refine their research questions, methods, and procedures.
• Problem-Oriented Case Studies : These case studies focus on solving a specific problem or addressing a particular issue. Researchers aim to provide practical solutions based on their analysis of the case.
• Ethnographic Case Studies : Ethnographic case studies involve immersing the researcher in the subject’s environment to gain an in-depth cultural understanding. This is often used in anthropology and sociology.
• Longitudinal Case Studies : Longitudinal studies involve observing and analyzing a case over an extended period of time. This allows researchers to track changes, developments, and trends that occur over time.
• Comparative Case Studies : Comparative case studies involve comparing two or more cases to draw similarities, differences, and patterns between them. This type of study is often used to test hypotheses or theories.
• Critical Instance Case Studies : Critical instance cases are chosen because they represent a crucial or pivotal event that can provide insights into a larger issue or theory.

Each type of case study serves a different purpose and is designed to answer specific research questions. Researchers choose the type of case study that best aligns with their objectives and the nature of the phenomenon they are investigating.

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What Are the Methods of a Case Study?

A   case study research   is a qualitative research design. It is often used in the social sciences since it involves observing the cases or subjects in their settings with the most minor interference from the researcher.

In the case study method, the researchers pose a definite question raging any individual or group for testing their hypotheses or theories. This is done by gathering data from the interviews with the essential data.

Case study research is a perfect way to understand the nuances of any matter often neglected in quantitative research methods. A case study is distinct from any other qualitative study in the following ways:

• Focused on the effect of any set of circumstances in any group or any individual
• It mostly begins with any specific question regarding one or more cases
• It usually focuses on the individual accounts and its experiences

The primary features of case study research methods are as follows:

• The case study methods   must involve the researcher asking a few questions of one person or a small group of people who are known as the respondents for testing the survey.
• The case study in the research mythology might apply triangulation to collect data. It is then analyzed and interpreted to form a hypothesis to be tested through further research or validated by other researchers.
• Concepts are defined using objective language with references to the Preconceived Notions. These individuals may have about them. A researcher sets out to discover by asking any specific question on how people think about their findings.
• The case study method needs a clear concept and theory to guide the processes. A well-organized research question is fundamental while conducting any case study since its results depend on it. The best approach for answering the research questions is challenging the preexisting theories, assumptions or hypotheses.

Benefits and Limitations of Case Studies

The benefits of case studies are as follows:

• Case studies give many details to be collected and will be easily obtained by the other research designs. The collected data is mostly richer than that can be funded via different experimental methods.
• Case studies are primarily conducted on the rare cases where more extensive samples of similar participants are unavailable.
• Within certain case studies, scientific experiments can also be conducted.
• The case studies can also help the experimenters adapt the ideas and produce novel hypotheses for later testing.

Disadvantages of Case Studies

• One of the main criticisms in case studies is that the collected data cannot necessarily be generated for any broader population. This can lead to data being collected over any case study that is only sometimes relevant or useful.
• Some of the case studies still need to be scientific. Many scientists used case studies for conducting several experiments, the results of which were only sometimes very successful.
• Case studies are primarily based on one person, so it can be only one experimenter who is collecting the data. This can lead to a bias in data collection that can influence the results in frequent designs.
• Drawing any definite cause or effect from many case studies is sometimes challenging.

Importance of Case Study

• A case study is a particular research h method involving an up-close and in-depth investigation of any subject, and it is related to a contextual position. These are produced by following a research form. The case study helps in bringing the understanding of any complex issue. This can extend experience or add strength to the already existing knowledge via the previous research. The contextual analysis revolves around a small number of events or situations.
• Researchers have used case studies for an extended period, and they have been successfully applied in various disciplines like social sciences.

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Types of Case Studies

There are several different types of case studies, as well as several types of subjects of case studies. We will investigate each type in this article.

Different Types of Case Studies

There are several types of case studies, each differing from each other based on the hypothesis and/or thesis to be proved. It is also possible for types of case studies to overlap each other.

Each of the following types of cases can be used in any field or discipline. Whether it is psychology, business or the arts, the type of case study can apply to any field.

Explanatory

The explanatory case study focuses on an explanation for a question or a phenomenon. Basically put, an explanatory case study is 1 + 1 = 2. The results are not up for interpretation.

A case study with a person or group would not be explanatory, as with humans, there will always be variables. There are always small variances that cannot be explained.

However, event case studies can be explanatory. For example, let's say a certain automobile has a series of crashes that are caused by faulty brakes. All of the crashes are a result of brakes not being effective on icy roads.

What kind of case study is explanatory? Think of an example of an explanatory case study that could be done today

When developing the case study, the researcher will explain the crash, and the detailed causes of the brake failure. They will investigate what actions caused the brakes to fail, and what actions could have been taken to prevent the failure.

Other car companies could then use this case study to better understand what makes brakes fail. When designing safer products, looking to past failures is an excellent way to ensure similar mistakes are not made.

The same can be said for other safety issues in cars. There was a time when cars did not have seatbelts. The process to get seatbelts required in all cars started with a case study! The same can be said about airbags and collapsible steering columns. They all began with a case study that lead to larger research, and eventual change.

Exploratory

An exploratory case study is usually the precursor to a formal, large-scale research project. The case study's goal is to prove that further investigation is necessary.

For example, an exploratory case study could be done on veterans coming home from active combat. Researchers are aware that these vets have PTSD, and are aware that the actions of war are what cause PTSD. Beyond that, they do not know if certain wartime activities are more likely to contribute to PTSD than others.

For an exploratory case study, the researcher could develop a study that certain war events are more likely to cause PTSD. Once that is demonstrated, a large-scale research project could be done to determine which events are most likely to cause PTSD.

Exploratory case studies are very popular in psychology and the social sciences. Psychologists are always looking for better ways to treat their patients, and exploratory studies allow them to research new ideas or theories.

Multiple-Case Studies or Collective Studies

Multiple case or collective studies use information from different studies to formulate the case for a new study. The use of past studies allows additional information without needing to spend more time and money on additional studies.

Using the PTSD issue again is an excellent example of a collective study. When studying what contributes most to wartime PTSD, a researcher could use case studies from different war. For instance, studies about PTSD in WW2 vets, Persian Gulf War vets, and Vietnam vets could provide an excellent sampling of which wartime activities are most likely to cause PTSD.

If a multiple case study on vets was done with vets from the Vietnam War, the Persian Gulf War, and the Iraq War, and it was determined the vets from Vietnam had much less PTSD, what could be inferred?

Furthermore, this type of study could uncover differences as well. For example, a researcher might find that veterans who serve in the Middle East are more likely to suffer a certain type of ailment. Or perhaps, that veterans who served with large platoons were more likely to suffer from PTSD than veterans who served in smaller platoons.

An intrinsic case study is the study of a case wherein the subject itself is the primary interest. The "Genie" case is an example of this. The study wasn't so much about psychology, but about Genie herself, and how her experiences shaped who she was.

Genie is the topic. Genie is what the researchers are interested in, and what their readers will be most interested in. When the researchers started the study, they didn't know what they would find.

They asked the question…"If a child is never introduced to language during the crucial first years of life, can they acquire language skills when they are older?" When they met Genie, they didn't know the answer to that question.

Instrumental

An instrumental case study uses a case to gain insights into a phenomenon. For example, a researcher interested in child obesity rates might set up a study with middle school students and an exercise program. In this case, the children and the exercise program are not the focus. The focus is learning the relationship between children and exercise, and why certain children become obese.

What is an example of an instrumental case study?

Focus on the results, not the topic!

Types of Subjects of Case Studies

There are generally five different types of case studies, and the subjects that they address. Every case study, whether explanatory or exploratory, or intrinsic or instrumental, fits into one of these five groups. These are:

Person – This type of study focuses on one particular individual. This case study would use several types of research to determine an outcome.

The best example of a person case is the "Genie" case study. Again, "Genie" was a 13-year-old girl who was discovered by social services in Los Angeles in 1970. Her father believed her to be mentally retarded, and therefore locked her in a room without any kind of stimulation. She was never nourished or cared for in any way. If she made a noise, she was beaten.

When "Genie" was discovered, child development specialists wanted to learn as much as possible about how her experiences contributed to her physical, emotional and mental health. They also wanted to learn about her language skills. She had no form of language when she was found, she only grunted. The study would determine whether or not she could learn language skills at the age of 13.

Since Genie was placed in a children's hospital, many different clinicians could observe her. In addition, researchers were able to interview the few people who did have contact with Genie and would be able to gather whatever background information was available.

This case study is still one of the most valuable in all of child development. Since it would be impossible to conduct this type of research with a healthy child, the information garnered from Genie's case is invaluable.

Group – This type of study focuses on a group of people. This could be a family, a group or friends, or even coworkers.

An example of this type of case study would be the uncontacted tribes of Indians in the Peruvian and Brazilian rainforest. These tribes have never had any modern contact. Therefore, there is a great interest to study them.

Scientists would be interested in just about every facet of their lives. How do they cook, how do they make clothing, how do they make tools and weapons. Also, doing psychological and emotional research would be interesting. However, because so few of these tribes exist, no one is contacting them for research. For now, all research is done observationally.

If a researcher wanted to study uncontacted Indian tribes, and could only observe the subjects, what type of observations should be made?

Location – This type of study focuses on a place, and how and why people use the place.

For example, many case studies have been done about Siberia, and the people who live there. Siberia is a cold and barren place in northern Russia, and it is considered the most difficult place to live in the world. Studying the location, and it's weather and people can help other people learn how to live with extreme weather and isolation.

Location studies can also be done on locations that are facing some kind of change. For example, a case study could be done on Alaska, and whether the state is seeing the effects of climate change.

Another type of study that could be done in Alaska is how the environment changes as population increases. Geographers and those interested in population growth often do these case studies.

Organization/Company – This type of study focuses on a business or an organization. This could include the people who work for the company, or an event that occurred at the organization.

An excellent example of this type of case study is Enron. Enron was one of the largest energy company's in the United States, when it was discovered that executives at the company were fraudulently reporting the company's accounting numbers.

Once the fraud was uncovered, investigators discovered willful and systematic corruption that caused the collapse of Enron, as well as their financial auditors, Arthur Andersen. The fraud was so severe that the top executives of the company were sentenced to prison.

This type of case study is used by accountants, auditors, financiers, as well as business students, in order to learn how such a large company could get away with committing such a serious case of corporate fraud for as long as they did. It can also be looked at from a psychological standpoint, as it is interesting to learn why the executives took the large risks that they took.

Most company or organization case studies are done for business purposes. In fact, in many business schools, such as Harvard Business School, students learn by the case method, which is the study of case studies. They learn how to solve business problems by studying the cases of businesses that either survived the same problem, or one that didn't survive the problem.

Event – This type of study focuses on an event, whether cultural or societal, and how it affects those that are affected by it. An example would be the Tylenol cyanide scandal. This event affected Johnson & Johnson, the parent company, as well as the public at large.

The case study would detail the events of the scandal, and more specifically, what management at Johnson & Johnson did to correct the problem. To this day, when a company experiences a large public relations scandal, they look to the Tylenol case study to learn how they managed to survive the scandal.

A very popular topic for case studies was the events of September 11 th . There were studies in almost all of the different types of research studies.

Obviously the event itself was a very popular topic. It was important to learn what lead up to the event, and how best to proven it from happening in the future. These studies are not only important to the U.S. government, but to other governments hoping to prevent terrorism in their countries.

Planning A Case Study

You have decided that you want to research and write a case study. Now what? In this section you will learn how to plan and organize a research case study.

Selecting a Case

The first step is to choose the subject, topic or case. You will want to choose a topic that is interesting to you, and a topic that would be of interest to your potential audience. Ideally you have a passion for the topic, as then you will better understand the issues surrounding the topic, and which resources would be most successful in the study.

You also must choose a topic that would be of interest to a large number of people. You want your case study to reach as large an audience as possible, and a topic that is of interest to just a few people will not have a very large reach. One of the goals of a case study is to reach as many people as possible.

Who is your audience?

Are you trying to reach the layperson? Or are you trying to reach other professionals in your field? Your audience will help determine the topic you choose.

If you are writing a case study that is looking for ways to lower rates of child obesity, who is your audience?

If you are writing a psychology case study, you must consider whether your audience will have the intellectual skills to understand the information in the case. Does your audience know the vocabulary of psychology? Do they understand the processes and structure of the field?

You want your audience to have as much general knowledge as possible. When it comes time to write the case study, you may have to spend some time defining and explaining terms that might be unfamiliar to the audience.

Lastly, when selecting a topic you do not want to choose a topic that is very old. Current topics are always the most interesting, so if your topic is more than 5-10 years old, you might want to consider a newer topic. If you choose an older topic, you must ask yourself what new and valuable information do you bring to the older topic, and is it relevant and necessary.

Determine Research Goals

What type of case study do you plan to do?

An illustrative case study will examine an unfamiliar case in order to help others understand it. For example, a case study of a veteran with PTSD can be used to help new therapists better understand what veterans experience.

An exploratory case study is a preliminary project that will be the precursor to a larger study in the future. For example, a case study could be done challenging the efficacy of different therapy methods for vets with PTSD. Once the study is complete, a larger study could be done on whichever method was most effective.

A critical instance case focuses on a unique case that doesn't have a predetermined purpose. For example, a vet with an incredibly severe case of PTSD could be studied to find ways to treat his condition.

Ethics are a large part of the case study process, and most case studies require ethical approval. This approval usually comes from the institution or department the researcher works for. Many universities and research institutions have ethics oversight departments. They will require you to prove that you will not harm your study subjects or participants.

This should be done even if the case study is on an older subject. Sometimes publishing new studies can cause harm to the original participants. Regardless of your personal feelings, it is essential the project is brought to the ethics department to ensure your project can proceed safely.

Developing the Case Study

Once you have your topic, it is time to start planning and developing the study. This process will be different depending on what type of case study you are planning to do. For thissection, we will assume a psychological case study, as most case studies are based on the psychological model.

Once you have the topic, it is time to ask yourself some questions. What question do you want to answer with the study?

For example, a researcher is considering a case study about PTSD in veterans. The topic is PTSD in veterans. What questions could be asked?

Do veterans from Middle Eastern wars suffer greater instances of PTSD?

Do younger soldiers have higher instances of PTSD?

Does the length of the tour effect the severity of PTSD?

Each of these questions is a viable question, and finding the answers, or the possible answers, would be helpful for both psychologists and veterans who suffer from PTSD.

Research Notebook

1. What is the background of the case study? Who requested the study to be done and why? What industry is the study in, and where will the study take place?

2. What is the problem that needs a solution? What is the situation, and what are the risks?

3. What questions are required to analyze the problem? What questions might the reader of the study have? What questions might colleagues have?

4. What tools are required to analyze the problem? Is data analysis necessary?

5. What is your current knowledge about the problem or situation? How much background information do you need to procure? How will you obtain this background info?

6. What other information do you need to know to successfully complete the study?

7. How do you plan to present the report? Will it be a simple written report, or will you add PowerPoint presentations or images or videos? When is the report due? Are you giving yourself enough time to complete the project?

The research notebook is the heart of the study. Other organizational methods can be utilized, such as Microsoft Excel, but a physical notebook should always be kept as well.

Planning the Research

The most important parts of the case study are:

1. The case study's questions

2. The study's propositions

3. How information and data will be analyzed

4. The logic behind the propositions

5. How the findings will be interpreted

The study's questions should be either a "how" or "why" question, and their definition is the researchers first job. These questions will help determine the study's goals.

Not every case study has a proposition. If you are doing an exploratory study, you will not have propositions. Instead, you will have a stated purpose, which will determine whether your study is successful, or not.

How the information will be analyzed will depend on what the topic is. This would vary depending on whether it was a person, group, or organization.

When setting up your research, you will want to follow case study protocol. The protocol should have the following sections:

1. An overview of the case study, including the objectives, topic and issues.

2. Procedures for gathering information and conducting interviews.

3. Questions that will be asked during interviews and data collection.

4. A guide for the final case study report.

When deciding upon which research methods to use, these are the most important:

1. Documents and archival records

2. Interviews

3. Direct observations

4. Indirect observations, or observations of subjects

5. Physical artifacts and tools

Documents could include almost anything, including letters, memos, newspaper articles, Internet articles, other case studies, or any other document germane to the study.

Archival records can include military and service records, company or business records, survey data or census information.

Research Strategy

Before beginning the study you want a clear research strategy. Your best chance at success will be if you use an outline that describes how you will gather your data and how you will answer your research questions.

The researcher should create a list with four or five bullet points that need answers. Consider the approaches for these questions, and the different perspectives you could take.

The researcher should then choose at least two data sources (ideally more). These sources could include interviews, Internet research, and fieldwork or report collection. The more data sources used, the better the quality of the final data.

The researcher then must formulate interview questions that will result in detailed and in-depth answers that will help meet the research goals. A list of 15-20 questions is a good start, but these can and will change as the process flows.

Planning Interviews

The interview process is one of the most important parts of the case study process. But before this can begin, it is imperative the researcher gets informed consent from the subjects.

The process of informed consent means the subject understands their role in the study, and that their story will be used in the case study. You will want to have each subject complete a consent form.

The researcher must explain what the study is trying to achieve, and how their contribution will help the study. If necessary, assure the subject that their information will remain private if requested, and they do not need to use their real name if they are not comfortable with that. Pseudonyms are commonly used in case studies.

Informed Consent

The process by which permission is granted before beginning medical or psychological research

A fictitious name used to hide ones identity

It is important the researcher is clear regarding the expectations of the study participation. For example, are they comfortable on camera? Do they mind if their photo is used in the final written study.

Interviews are one of the most important sources of information for case studies. There are several types of interviews. They are:

Open-ended – This type of interview has the interviewer and subject talking to each other about the subject. The interviewer asks questions, and the subject answers them. But the subject can elaborate and add information whenever they see fit.

A researcher might meet with a subject multiple times, and use the open-ended method. This can be a great way to gain insight into events. However, the researcher mustn't rely solely on the information from the one subject, and be sure to have multiple sources.

Focused – This type of interview is used when the subject is interviewed for a short period of time, and answers a set of questions. This type of interview could be used to verify information learned in an open-ended interview with another subject. Focused interviews are normally done to confirm information, not to gain new information.

Structured – Structured interviews are similar to surveys. These are usually used when collecting data for large groups, like neighborhoods. The questions are decided before hand, and the expected answers are usually simple.

When conducting interviews, the answers are obviously important. But just as important are the observations that can be made. This is one of the reasons in-person interviews are preferable over phone interviews, or Internet or mail surveys.

Ideally, when conducing in-person interviews, more than one researcher should be present. This allows one researcher to focus on observing while the other is interviewing. This is particularly important when interviewing large groups of people.

The researcher must understand going into the case study that the information gained from the interviews might not be valuable. It is possible that once the interviews are completed, the information gained is not relevant.

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• Volume 21, Issue 1
• What is a case study?
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• Roberta Heale 1 ,
• Alison Twycross 2
• 1 School of Nursing , Laurentian University , Sudbury , Ontario , Canada
• 2 School of Health and Social Care , London South Bank University , London , UK
• Correspondence to Dr Roberta Heale, School of Nursing, Laurentian University, Sudbury, ON P3E2C6, Canada; rheale{at}laurentian.ca

https://doi.org/10.1136/eb-2017-102845

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What is it?

Case study is a research methodology, typically seen in social and life sciences. There is no one definition of case study research. 1 However, very simply… ‘a case study can be defined as an intensive study about a person, a group of people or a unit, which is aimed to generalize over several units’. 1 A case study has also been described as an intensive, systematic investigation of a single individual, group, community or some other unit in which the researcher examines in-depth data relating to several variables. 2

Often there are several similar cases to consider such as educational or social service programmes that are delivered from a number of locations. Although similar, they are complex and have unique features. In these circumstances, the evaluation of several, similar cases will provide a better answer to a research question than if only one case is examined, hence the multiple-case study. Stake asserts that the cases are grouped and viewed as one entity, called the quintain . 6  ‘We study what is similar and different about the cases to understand the quintain better’. 6

The steps when using case study methodology are the same as for other types of research. 6 The first step is defining the single case or identifying a group of similar cases that can then be incorporated into a multiple-case study. A search to determine what is known about the case(s) is typically conducted. This may include a review of the literature, grey literature, media, reports and more, which serves to establish a basic understanding of the cases and informs the development of research questions. Data in case studies are often, but not exclusively, qualitative in nature. In multiple-case studies, analysis within cases and across cases is conducted. Themes arise from the analyses and assertions about the cases as a whole, or the quintain, emerge. 6

Benefits and limitations of case studies

If a researcher wants to study a specific phenomenon arising from a particular entity, then a single-case study is warranted and will allow for a in-depth understanding of the single phenomenon and, as discussed above, would involve collecting several different types of data. This is illustrated in example 1 below.

Using a multiple-case research study allows for a more in-depth understanding of the cases as a unit, through comparison of similarities and differences of the individual cases embedded within the quintain. Evidence arising from multiple-case studies is often stronger and more reliable than from single-case research. Multiple-case studies allow for more comprehensive exploration of research questions and theory development. 6

Despite the advantages of case studies, there are limitations. The sheer volume of data is difficult to organise and data analysis and integration strategies need to be carefully thought through. There is also sometimes a temptation to veer away from the research focus. 2 Reporting of findings from multiple-case research studies is also challenging at times, 1 particularly in relation to the word limits for some journal papers.

Examples of case studies

Example 1: nurses’ paediatric pain management practices.

One of the authors of this paper (AT) has used a case study approach to explore nurses’ paediatric pain management practices. This involved collecting several datasets:

Observational data to gain a picture about actual pain management practices.

Questionnaire data about nurses’ knowledge about paediatric pain management practices and how well they felt they managed pain in children.

Questionnaire data about how critical nurses perceived pain management tasks to be.

These datasets were analysed separately and then compared 7–9 and demonstrated that nurses’ level of theoretical did not impact on the quality of their pain management practices. 7 Nor did individual nurse’s perceptions of how critical a task was effect the likelihood of them carrying out this task in practice. 8 There was also a difference in self-reported and observed practices 9 ; actual (observed) practices did not confirm to best practice guidelines, whereas self-reported practices tended to.

Example 2: quality of care for complex patients at Nurse Practitioner-Led Clinics (NPLCs)

The other author of this paper (RH) has conducted a multiple-case study to determine the quality of care for patients with complex clinical presentations in NPLCs in Ontario, Canada. 10 Five NPLCs served as individual cases that, together, represented the quatrain. Three types of data were collected including:

Review of documentation related to the NPLC model (media, annual reports, research articles, grey literature and regulatory legislation).

Interviews with nurse practitioners (NPs) practising at the five NPLCs to determine their perceptions of the impact of the NPLC model on the quality of care provided to patients with multimorbidity.

Chart audits conducted at the five NPLCs to determine the extent to which evidence-based guidelines were followed for patients with diabetes and at least one other chronic condition.

The three sources of data collected from the five NPLCs were analysed and themes arose related to the quality of care for complex patients at NPLCs. The multiple-case study confirmed that nurse practitioners are the primary care providers at the NPLCs, and this positively impacts the quality of care for patients with multimorbidity. Healthcare policy, such as lack of an increase in salary for NPs for 10 years, has resulted in issues in recruitment and retention of NPs at NPLCs. This, along with insufficient resources in the communities where NPLCs are located and high patient vulnerability at NPLCs, have a negative impact on the quality of care. 10

These examples illustrate how collecting data about a single case or multiple cases helps us to better understand the phenomenon in question. Case study methodology serves to provide a framework for evaluation and analysis of complex issues. It shines a light on the holistic nature of nursing practice and offers a perspective that informs improved patient care.

• Gustafsson J
• Calanzaro M
• Sandelowski M

Competing interests None declared.

Provenance and peer review Commissioned; internally peer reviewed.

Read the full text or download the PDF:

• DOI: 10.1109/CCECE59415.2024.10667327
• Corpus ID: 272604113

Exploring Long-term Memory in Evolutionary Multi-objective Algorithms: A Case Study with NSGA-III

• Masoud Kermani Poor , Shahryar Rahnamayan , +1 author Mehran Ebrahimi
• Published in Canadian Conference on… 6 August 2024
• Computer Science, Engineering

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24 References

Machine learning-based framework to cover optimal pareto-front in many-objective optimization, an improved multi-objective workflow scheduling using f-nspso with fuzzy rules, multiobjective resnet pruning by means of emoas for remote sensing scene classification, interactive multiobjective optimization: a review of the state-of-the-art, application and benchmarking of multi-objective evolutionary algorithms on high-dose-rate brachytherapy planning for prostate cancer treatment, data mining methods for knowledge discovery in multi-objective optimization: part a - survey, elitensga-iii: an improved evolutionary many-objective optimization algorithm, multi-objective pigeon-inspired optimization for brushless direct current motor parameter design, dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, an efficient approach to nondominated sorting for evolutionary multiobjective optimization, related papers.

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On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes

• Open access
• Published: 16 September 2024
• Volume 66 , article number  10 , ( 2024 )

Cite this article

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• Markus Wolff 1  

We study the effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes, both in the spacelike and the timelike case, respectively. In the spacelike case, we study totally umbilic warped product graphs and give a full characterization of embedded surfaces with constant spacetime mean curvature using an Alexandrov Theorem by Brendle and Borghini–Fogagnolo–Pinamonti. In the timelike case, we achieve a characterization of photon surfaces with constant umbilicity factor similar to a result by Cederbaum–Galloway.

Avoid common mistakes on your manuscript.

1 Introduction

The null energy condition, also known as null convergence condition, states that for any null vector field L the Einstein–Tensor \({\mathfrak {G}}\) satisfies

In cosmology, this assumption on the spacetime takes a critical role in the formation of singularities (cf. [ 1 ] Section 8.2). Moreover, in a class of static, warped product spacetimes and on their respective time-symmetric slices, it was used to establish versions of a (Spacetime) Alexandrov Theorem [ 2 , 3 , 4 ]. In this paper, we study the effects of the (NEC) on both spacelike and timelike totally umbilic hypersurfaces in this class of static spacetimes.

In the spacelike case, we study initial data sets given as warped product graphs and show that in the totally umbilic case an Alexandrov Theorem by Brendle becomes applicable if the null energy condition is satisfied. The classical Alexandrov Theorem [ 5 ] in Euclidean space states that any compact surface of constant mean curvature is a round sphere. In [ 2 ], Brendle generalized this to a large class of Riemannian warped product manifolds. Brendle proved that if certain conditions (H1)–(H3) are satisfied on the manifold, then any orientable, embedded, closed surface of constant mean curvature is necessarily totally umbilic [ 2 , Theorem 1.1]. If an additional assumption (H4) is met Brendle showed that the surface is a leaf of the canonical foliation. After a suitable coordinate change using condition (H2), one can embed the Riemannian manifolds under consideration as time-symmetric \(\{t={\text {const.}}\}\) -slices into a certain Class \({\mathfrak {H}}\) of static spacetimes of the form

where h is a smooth, positive function on some open intervall I , and \(({\mathcal {N}},g_{{\mathcal {N}}})\) is a compact Riemannian manifold. Then, the conditions (H1) and (H3) translate into physically reasonable assumptions on the spacetime, more precisely, that the inner boundary is a non-degenerate Killing horizon and that the spacetime satisfies the null energy condition, see [ 3 ]. In [ 3 ], Wang–Wang–Zhang used the results of Brendle in this spacetime setting to proof a spacetime Alexandrov theorem which characterizes round null cones if they admit a spacelike cross section satisfying a spacetime CMC condition (see [ 3 ] Section 3.2). Moreover, Wang [ 4 ] noted that in spherical symmetry condition (H4) is already met under (H1)–(H3), and in a recent paper [ 6 ], Borghini–Fogagnolo–Pinamonti completely removed assumption (H4) as an application for a rigidity statement of the Heintze–Karcher inequality in substatic manifolds.

Regarding the results in [ 2 , 6 ], the question whether these conditions on the spacetime allow for a similar characterization of surfaces not only on the time-symmetric slices, but general initial data sets with warped product structure, naturally arises, and one would expect that the respective geometric property of the surfaces also arises out of the structure of the spacetime. Due to its Lorentz invariance, it seems natural to consider a characterization of surfaces that posses constant spacetime mean curvature \({\mathcal {H}}^2\) (STCMC surfaces), where \({\mathcal {H}}^2\) is the Lorentzian length of the mean curvature vector \(\mathbf {\vec {\mathcal {H}}}\) in the spacetime. If the surface is contained within a time-symmetric initial data set this reduces to the Alexandrov Theorem proven in [ 2 , 6 ]. A foliation of such STCMC surfaces at spacelike infinity has recently been used by Cederbaum–Sakovich [ 7 ] in a center of mass formulation in the asymptotically flat setting. To apply the results of Brendle [ 2 ], we utilize a recent construction by Cederbaum and the author [ 8 ] of generalized Kruskal–Szekeres coordinates.

In the timelike case, we study photon surfaces, i.e., timelike, totally umbilic hypersurfaces, within the same Class \({\mathfrak {H}}\) of static spacetimes. Imposing an assumption on the eigenvalues of the Ricci tensor as Brendle [ 2 ] in the Riemannian setting, we obtain a short proof for a characterization of photon surfaces with constant umbilicity factor \(\lambda \) in the spacetime setting. In a recent paper, Cederbaum–Galloway [ 9 ] achieved a similar characterization of photon surfaces in the same class of spacetimes assuming spherical symmetry and a nowhere locally spacetime conformally flat condition. Although they do not need to assume that \(\lambda ={\text {const.}}\) , their arguments rely heavily on the spherical symmetry and can not be extended across a non-degenerate Killing horizon without additional assumptions. We include a detailed discussion and give an illustrative example of a 1-parameter family within this Class \({\mathfrak {H}}\) of spacetimes that are locally spacetime conformally flat but satisfy the assumption on the eigenvalues of the Ricci tensor.

1.1 Main results

Given a non-negative solution x of an ordinary differential inequality ( 13 ) related to the (NEC), we show that the results in [ 2 , 6 ] can also be applied to spacelike warped product graphs with metric coefficient \(h+x\) (Theorem  10 ). For some non-negative constant C , \(x=Cs^2\) is an exact solutions of ( 13 ). We will refer to the resulting graphs as hyperboloids, and note that they are totally umbilic with constant umbilicity factor (cf. Corollary  7 ). In particular, the characterization of embedded CMC surfaces allows for a characterization of STCMC surfaces on hyperboloids. Note that this requires to extend the graphs past the non-degenerate Killing horizon into a suitable spacetime extension. Here, we utilize a recent construction of generalized Kruskal–Szekeres coordinates [ 8 ]. See also [ 10 , 11 ].

Let \(h:(0,\infty )\rightarrow {\mathbb {R}}\) be a smooth function with finitely many, positive simple zeroes \(r_1<\dotsc < r_N\) , \(({\mathcal {N}},g_{{\mathcal {N}}})\) an \((n-1)\) -dimensional Riemannian manifold \((n\ge 3)\) . Let \(({\mathfrak {M}},{\mathfrak {g}})\) be the corresponding spacetime of Class \({\mathfrak {H}}\) with metric coefficient h and fibre \({\mathcal {N}}\) . Assume that the generalized Kruskal–Szekeres extension satisfies the (NEC) condition.Then there exists a constant \(C_0=C_0(h,h')\in (0,\infty ]\) such that for any hyperboloid \((M_T,g^T,K^T)\) with umbilicity factor \(\lambda _T\) satisfying \(\lambda _T^2<C_0\) we have: If \(\Sigma \subset M_T\) is an orientable, closed, embedded hypersurface with constant spacetime mean curvature, then \(\Sigma \) is a slice \(\{s\}\times {\mathcal {N}}\) .

Note that the Kruskal–Szekeres extension of the \((n+1)\) -dimensional Schwarzschild spacetime with positive masss satisfies all of the above assumptions with \(C_0=\infty \) .

Corollary 2

Let \(({\mathfrak {M}},{\mathfrak {g}})\) denote the Schwarzschild spacetime with positive mass. Then any STCMC surface \(\Sigma \) in an hyperboloid \((M_T,g^T,K^T)\) is a slice \(\{s\}\times {\mathbb {S}}^{n-1}\) .

Further, if the (NEC) is strict in some sense (cf. Sect.  5 ), this also allows for a characterization of photon surfaces \(({\mathcal {P}}^n,{\mathfrak {p}})\) , i.e., timelike and totally umbilic hypersurfaces, with constant umbilicity factor.

Let \(({\mathcal {P}}^n,{\mathfrak {p}})\) be a connected photon surface with constant umbilicity factor \(\lambda \) in a spacetime of Class \({\mathfrak {H}}\) or its respective generalized Kruskal–Szekeres extension as constructed in [ 8 ]. Assume further, that h satisfies

for any unit tangent vector X in \(({\mathcal {N}},g_{\mathcal {N}})\) on a dense set of radii in \((0,\infty )\) .Then \({\mathcal {P}}^n\) is either rotationally symmetric, or \({\mathcal {P}}^n\) is totally geodesic with parallel unit normal vector \(\eta \) everywhere tangent to \({\mathcal {N}}\) .

This paper is structured as follows: In Sect.  2 we introduce basic notation and some preliminary Lemmas. In Sect.  3 we study the geometry of spacelike warped product graphs within spacetimes of Class \({\mathfrak {H}}\) . In Sect.  4 we prove the characterization of STCMC surfaces on spacelike totally umbilic warped product graphs. In Sect.  5 we prove the characterization of photon surfaces in the timelike case and compare it to a result by Cederbaum–Galloway [ 9 ].

2 Preliminaries

Let \(({\mathfrak {M}},{\mathfrak {g}})\) be a spacetime. We denote its Ricci curvature tensor, and scalar curvature by \(\mathfrak {Ric}\) and \({\mathfrak {R}}\) , respectively. The Einstein tensor is then given as \({\mathfrak {G}}\,{{:}{=}}\,\mathfrak {Ric}-\frac{1}{2}{\mathfrak {R}}{\mathfrak {g}}\) , and we say that \(({\mathfrak {M}},{\mathfrak {g}})\) satisfies the null energy condition (NEC) (or null convergence condition ) if

for any null vector field \(L\in \Gamma (T{\mathfrak {M}})\) . If we consider an initial data set ( M ,  g ,  K ) in \(({\mathfrak {M}},{\mathfrak {g}})\) , i.e., a spacelike hypersurface M of \({\mathfrak {M}}\) with induced metric g and second fundamental form K with respect to a future timelike unit normal \(\vec {{\textbf {n}}}\) , the Gauß-Codazzi equations imply the well-known constraint equations on ( M ,  g ,  K ):

where \(\mu \,{{:}{=}}\,{\mathfrak {G}}({\textbf {n}},{\textbf {n}})\) , and \(J(\cdot )\,{{:}{=}}\,{\mathfrak {G}}({\textbf {n}},\cdot )\) are called the energy- and momentum density , respectively, and where \({\text {R}}\) denotes the scalar curvature of ( M ,  g ). Hence, for any unit vector field V on ( M ,  g ), the (NEC) implies

Note that we use the following conventions for the Riemann curvature tensor \({\text {Rm}}\) , Ricci curvature tensor \({\text {Ric}}\) and scalar curvature \({\text {R}}\) , respectively:

Troughout this paper, we consider the following class \({\mathfrak {H}}\) of spacetimes: We say an \((n+1)\) -dimensional spacetimes \(({\mathfrak {M}},{\mathfrak {g}})\) ( \(n\ge 3\) ) is of class \({\mathfrak {H}}\) with metric coefficient h and fibre \({\mathcal {N}}\) if \({\mathfrak {M}}={\mathbb {R}}\times I\times {\mathcal {N}}\) for an open, non-negative interval \(I=(r_H,\infty )\) , \(r_H\ge 0\) , and some compact \((n-1)\) -dimensional Riemannian manifold \(({\mathcal {N}},g_{{\mathcal {N}}})\) such that

for a smooth function \(h:(0,\infty )\rightarrow {\mathbb {R}}\) that is strictly positive on I and satisfies \(h(r_H)=0\) if \(r_H>0\) . Hence, the inner boundary \(\{r=r_H\}\) is a Killing horizon of \(({\mathfrak {M}},{\mathfrak {g}})\) . We will denote coordinates on \({\mathcal {N}}\) with capital Roman letters. In spherical symmetry, i.e., \({({\mathcal {N}},g_{{\mathcal {N}}})=({\mathbb {S}}^{n-1},\,\text {d}\Omega ^2)}\) , where \(\,\text {d}\Omega ^2\) denotes the round metric on \({\mathbb {S}}^{n-1}\) , these spacetimes compose a large class of physically significant models both in the context of isolated gravitating systems as well as in the context of cosmology, such as the Schwarzschild and Reissner-Nordström spacetime, and the de Sitter and anti-de Sitter spacetime. The properties of this class of spherically symmetric spacetimes are well understood and have been subject to extensive research, see e.g. Cederbaum–Galloway [ 9 ], Schindler–Aguirre [ 11 ]. Note that the more general form has also been considered in other works, see e.g. Wang–Wang–Zhang [ 3 ], Brill–Hayward [ 10 ], and a joint work of Cederbaum and the author [ 8 ]. If we assume that \(({\mathcal {N}},g^{{\mathcal {N}}})\) has constant sectional curvature, then a spacetime of class \({\mathfrak {H}}\) is equipped with a Birmingham–Kottler metric, see e.g. [ 12 , 13 , 14 ]. In spherical symmetry we will adopt the notion of Cederbaum–Galloway and call them spacetimes of class \({\mathcal {S}}\) .

Let \((M_0, g^0)\) denote the time-symmetric ( \(K\equiv 0\) ) time slice \(\{t=0\}\) with induced metric

Note that the warped product manifolds considered by Brendle [ 2 ] are precisely of the form \((M_0,g^0)\) after a change of coordinates if condition (H2) imposed by Brendle is satisfied. Then condition (H1) implies that \(\{r=r_H\}\) is a non-degenerate Killing horizon in \(({\mathfrak {M}},{\mathfrak {g}})\) in the sense that \(h'(r_H)\not =0\) , cf. [ 15 , Equation (12.5.16)]. Moreover, Wang–Wang–Zhang [ 3 ] pointed out that condition (H3) of Brendle precisely translates to \(({\mathfrak {M}},{\mathfrak {g}})\) satisfying the null energy condition. For the convenience of the reader, we collect the respective equivalences in the notation of this paper in the following lemma:

Let \(({\mathfrak {M}},{\mathfrak {g}})\) be a spacetime of class \({\mathfrak {H}}\) with metric coefficient h , and let \(f\,{{:}{=}}\,\sqrt{h}\) on \((r_H,\infty )\) . Then the following are equivalent:

\(({\mathfrak {M}},{\mathfrak {g}})\) satisfies the (NEC).

\(\Delta _0fg^0-\text {Hess}_0f+f\text {Ric}^0\ge 0\) on \(M_0\) , where \(\Delta _0\) , \(\text {Hess}_0\) , and \(\text {Ric}^0\) denote the Laplacian, the Hessian and Ricci curvature with respect to \(g^0\) , respectively.

on \(M_0\) for all unit tangent vector fields X in \(({\mathcal {N}},g_{{\mathcal {N}}})\) , where \(\text {Ric}_{g_{\mathcal {N}}}\) denotes the Ricci curvature on \({\mathcal {N}}\) .

The function \(x=h-\alpha :(r_H,\infty )\rightarrow {\mathbb {R}}\) is a solution of the ordinary differential inequality

where \((n-2)\alpha \) is the minimum of the smallest eigenvalue of \(\text {Ric}_{g_{\mathcal {N}}}\) on \({\mathcal {N}}\) .

For the equivalences (i) to (iii), we refer to the respective results of Brendle and Wang–Wang–Zhang, see Proposition 2.1. in [ 2 ] and Lemma 3.8 in [ 3 ], where \(\alpha \) is the same as the constant \(\rho \) considered by Brendle [ 2 ] as above. However, note the different conventions for the functions f , h used here compared to both [ 2 ] and [ 3 ]. The fourth equivalence is immediate, since we assume \({\mathcal {N}}\) to be compact. We moreover observe that

where \(L^*_g\) denotes the formal \(L^2\) adjoint of the linearization of the scalar curvature operator, cf. [ 16 ] Lemma 2.2. Thus, the (NEC) implies the existence of a non-trivial supersolution \(f>0\) of the formal \(L^2\) adjoint of the linearization of the scalar curvature operator. As equality for the (NEC) implies a non-trivial kernel, we can conclude from [ 16 ] Lemma 2.3 that the scalar curvature \(R^0\) of the time-symmetric slices must necessarily be constant in this case. See also Remark  5 below.

Wang [ 4 ] further noticed that the (NEC) is in fact also related to an eigenvalue analysis of the Ricci curvature tensor \(\text {Ric}^0\) of the time-symmetric time slices. Namely, the (NEC) implies monotonocity for the difference between the eigenvalue \(h\text {Ric}^0_{rr}\) and any eigenvalue of \(\text {Ric}^0\vert _{T{\mathcal {N}}\times T{\mathcal {N}}}\) , cf. [ 4 , Lemma 5.1]. Analogous to spherical symmetry, we can establish this monotonicity in the general case by direct computation:

Let \(({\mathfrak {M}},{\mathfrak {g}})\) be a spacetime of class \({\mathfrak {H}}\) satisfying the (NEC), and let X be a unit tangent vector in \(({\mathcal {N}},g_{{\mathcal {N}}})\) . Then

is monotone non-decreasing in r .

Thus, this holds true for any unit eigenvector \(X\in \Gamma (T{\mathcal {N}})\) of \(\text {Ric}_{g_{\mathcal {N}}}\) , in particular for the minimum \((n-2)\alpha \) , and condition (H4) of Brendle [ 2 ] is equivalent to the monotone quantity ( 2 ) being strictly positive everywhere. Due to the monotonicity, it suffices to check this at the inner boundary, so condition (H4) is in particular implied by the boundary condition

and hence immediate for \(\alpha \ge 0\) , cf. [ 4 , Remark 5.2]. Note that it suffices to assume that ( 2 ) is non-vanishing, cf. (H4’) [ 2 , page 3]. In a recent paper [ 6 ], Borghini–Fogagnolo–Pinamonti obtain a similar rigidity statement in Class \({\mathfrak {H}}\) for strictly mean convex surfaces that satisfy equality for the substatic Heintze–Karcher inequality assuming (H1)–(H3), cf. [ 6 , Theorem 1.2]. As already remarked by Brendle [ 2 , Section 6], CMC surfaces satisfy equality for the substatic Heintze–Karcher inequality which yields the desired Alexandrov Theorem even without imposing any version of condition (H4), cf. [ 6 , Corollary 1.3]. As they need to verify an additional technical assumption in the case that \(\Sigma \) is homologous to \(\partial M\) to apply a splitting theorem [ 6 , Theorem 3.1], they state their results only in this case, whereas the nullhomologous case follows directly from [ 6 , Theorem 3.1].

We close this section by briefly introducing our notation for spacelike codimension-2 surfaces \((\Sigma ,\gamma )\) in a spacetime \(({\mathfrak {M}},{\mathfrak {g}})\) . Recall that the vector-valued second fundamental form \(\mathbf {\text{ II }}\) of \(\Sigma \) in \(({\mathfrak {M}},{\mathfrak {g}})\) is defined as

for all tangent vector fields \(V, W\in \Gamma (T\Sigma )\) , where \({\overline{\nabla }}\) denotes the Levi-Civita connection of \(({\mathfrak {M}},{\mathfrak {g}})\) . Then, the codimension-2 mean curvature vector \(\mathbf {\vec {\mathcal {H}}}\) of \(\Sigma \) is given by the trace of \(\vec {\text {II}}\) with respect to \(\gamma \) , i.e, \(\mathbf {\vec {\mathcal {H}}}=\text{ tr}_\gamma \mathbf {\text{ II }}\) . In the following, we will always assume that \((\Sigma ,\gamma )\) is closed, and embedded as a hypersurface in an initial data set ( M ,  g ,  K ), with second fundamental form K with respect to a future timelike unit normel \(\vec {{\textbf {n}}}\) . Let \(\nu \) denote a unit normal of \((\Sigma ,\gamma )\) in ( M ,  g ). Then \(\mathbf {\vec {\mathcal {H}}}\) admits the decomposition

where H denotes the mean curvature of \(\Sigma \) in ( M ,  g ), and \(P\,{{:}{=}}\,\text {tr}_\Sigma K=\text {tr}_MK-K(\nu ,\nu )\) . Moreover, the expansions \(\theta _\pm \) with respect to the null directions \(l_\pm =\nu \pm \vec {{\textbf {n}}}\) are given by

Here, we define the spacetime mean curvature of \(\Sigma \) as

Assuming that \({\mathcal {H}}^2>0\) , this agrees with the notion of spacetime mean curvature by Cederbaum–Sakovich [ 7 ] upon taking a square root. However, in general \({\mathcal {H}}^2\) might be at least locally negative. Indeed, surfaces where \({\mathcal {H}}^2<0\) everywhere along \(\Sigma \) naturally occur in our setting, cf. Remark  6 , and are called trapped in the context of General Relativity. Note that the expansions \(\theta _\pm \) are the change of area of \(\Sigma \) in the null directions \(l_\pm \) , and the spacetime mean curvature is the Lorentzian length of \(\mathbf {\vec {\mathcal {H}}}\) , i.e.,

We say a surface \((\Sigma ,\gamma )\) in an initial data set ( M ,  g ,  K ) has constant expansion (CE), and constant spacetime mean curvature (STCMC) if \(\theta _\pm ={\text {const.}}\) , and \({\mathcal {H}}^2={\text {const.}}\) respectively. In the special case when \(\theta _\pm =0\) , and \({\mathcal {H}}^2=0\) , we call \(\Sigma \) a marginally outer/inner trapped surfaces (MOTS/MITS) , and a generalized apparent horizon , respectively. Any MOTS/MITS is always a generalized apparent horizon, but the converse is not true in general. See Carrasco–Mars [ 17 ] for an explicit counterexample.

3 Warped product graphs in Class \({\mathfrak {H}}\)

We consider spacelike warped product graphs over the canonical \(\{t=0\}\) time slice \(M_0\) in spacetimes of Class \({\mathfrak {H}}\) . More precisely, we look at hypersurfaces \(M_T\) of the form

for some smooth function \(T:(r_1,r_2)\rightarrow {\mathbb {R}}\) with \(r_H\le r_1<r_2\le \infty \) . We will refer to T as the radial height function of \(M_T\) . We further denote the induced metric and second fundamental form of \(M_T\) as \(g^T\) and \(K^T\) , respectively.

Note that any spacelike slice in a static spacetime can always be written as a graph, so the above assumption is only restrictive in the sense that we assume that T is only depending on r . For general graphical initial data sets \((M_T,g^T,K^T)\) given as \(M_T=\,\text {graph}_{M_0}T\) , the spacelike condition yields a restriction on the gradient of T , i.e., that \(1-h\left| \nabla _0T\right| ^2>0\) , where \(\nabla _0\) denotes the gradient on \(M_0\) . Using the computations of Cederbaum–Nerz [ 18 ] for graphs in general static spacetimes with coordinates \(\{x^i\}\) on \(M_0\) , we get

and the future timelike unit normal \(\vec {{\textbf {n}}}\) is given by \(\vec {{\textbf {n}}}=\frac{\partial _t+h\nabla _0T}{f\sqrt{1-h\left| \nabla _0T\right| ^2}}\) with \(f\,{{:}{=}}\,\sqrt{h}\) as above. Note that there is a slight mistake in the formula of the second fundamental form \(K^T\) in [ 18 ]. As corrected in [ 7 ], \(K^T\) is given by

If \(M_T=\{(T(s),s,x^I)\}\) embeds in Class \({\mathfrak {H}}\) with coordinates \(\{s,x^I\}\) , where \(x^I\) denote (local) coordinates on \({\mathcal {N}}\) , this yields

with \(s\equiv r\) along \(M_T\) , and where

with the tangent vector fields \(\partial _s=\partial _r+T'\partial _t\) , \(\partial _I\) and the future unit normal

In particular, we see from ( 3 ) that \((M_T,g^T)\) is again a warped product manifold as considered in [ 2 , 6 ] with \(h_T\ge h\) , and note that K also satisfies a similar block diagonal form. Thus, we will refer to spacelike graphs \((M_T,g^T,K^T)\) such that \(T=T(s)\) as (spacelike) warped product graphs. The main observation in this subsection is to see that both the intrinsic and extrinsic curvature for such spacelike warped product graphs are fully determined by the difference \(h_T-h\) in Class \({\mathfrak {H}}\) . This essentially follows from the following lemma:

By ( 4 ) and using that \(\left| \nabla _0T\right| ^2=h\cdot (T')^2\) , we see that

Thus, the first identity follows by taking a square of ( 6 ). Taking a derivative of ( 7 ) the second identity follows from straightforward computation. \(\square \)

Further, the difference \(h_T-h\) also uniquely determines the radial height function T up to a choice of sign of the derivative and a constant of integration, as

More precisely \((M_T,g^T,K^T)\) is fully determined by the choice of function \(b_T\) with \({h_T=h+r^2b_T^2}\) and

up to a constant of integration.

As a consequence, this rigid structure yields a full characterization of totally umbilic spacelike warped product graphs in Class \({\mathfrak {H}}\) :

Corollary 7

Let \((M_T,g^T,K^T)\) be a spacelike warped product graph as above, and we further assume that \(K^T=\lambda _T g^T\) for some smooth function \(\lambda _T\) . Then \(\lambda _T\) is constant, and \((M_T,g^T,K^T)\) is fully determined by the choice \(b_T=\lambda _T\) up to a shift in t -direction.

In the Minkowski spacetime, where \(h=1\) , we precisely recover the connected components of an hyperboloid centered around the origin with \(h_T=1+\lambda _T^2s^2\) , where the sign of \(\lambda _T\) determines the choice of connected component. As a totally umbilic, spacelike warped product graph now corresponds to \(h_T=h+\lambda _T^2s^2\) in the general case, we will similarly refer to such a graphs as a hyperboloid in Class \({\mathfrak {H}}\) .

Since \(K^T=\lambda _T g^T\) , we have

in particular

Using Lemma  6 , we see that

and solving the ODE gives \(h_T-h=Cs^2\) , where necessarily \(C\ge 0\) since \(h_T\ge h\) . By Lemma  6

so \(\lambda _T\) is constant. As T is uniquely determined by \(b_T=\lambda _T\) up to a constant of integration, the claim is proven. \(\square \)

Similarly, we can characterize all spacelike warped product graphs with \({\text {tr}_{M_T}K^T\equiv C}\) via

for some real constant \(c_1\in {\mathbb {R}}\) . The choice \(c_1=0\) corresponds to the totally umbilic case and we recover the hyperboloids. For \(C=0\) , we obtain a 1-parameter family of maximal hypersurfaces. These CMC graphs have been considered by Bartnik–Simon [ 19 ] as barries in the Minkowski spacetime. See also [ 20 , 21 , 22 ].

Although the assumption of being a graph in the above sense is much more restrictive in the case of a timelike hypersurface, we can establish a similar warped product structure for timelike graphs with radial height function \(T=T(s)\) where we now require that \({h\left| \nabla _0T\right| ^2-1>0}\) . More precisely, we find that

and find the relations

In the totally umbilic case, this leads to the same ODE system characterizing rotationally symmetric photon surfaces in class \({\mathcal {S}}\) derived by Cederbaum–Galloway [ 9 ]. Note that this ODE system has been extended to Class \({\mathfrak {H}}\) by Cederbaum and the author in [ 8 ]. In [ 23 ], Cederbaum–Jahns–Vičánek Martínez fully characterize the behavior of solutions to this ODE, in particular showing that rotationally symmetric photon surfaces are either photon spheres or warped product graphs of the above sense away from singular radii.

In particular we find \(h_T=\lambda _T^2s^2-h\) in the umbilic case with \(\lambda _T\not =0\) constant, so up to dividing \(h_T\) by \(s^2\) , the function determining the induced metric is the effective potential studied by Cederbaum–Jahns–Vičánek Martínez [ 23 ] in order the characterize solutions of the ODE system (away from singular radii where these coordinates break down).

4 Characterization of STCMC surfaces in spacelike totally umbilic warped product graphs

In this section, we will always assume that \((M_T,g^T,K^T)\) is a spacelike warped product graph in Class \({\mathfrak {H}}\) . Assuming the (NEC), we prove a characterization of STCMC surfaces in \((M_T,g^T,K^T)\) . Note that in time symmetry we have that \({\mathcal {H}}^2=H^2\) , so STCMC surfaces are CMC surfaces and the characterization directly follows from [ 2 , 6 ]. In this sense, we extend the characterization of STCMC surfaces in the time-symmetric \(\{t={\text {const.}}\}\) -slices to totally umbilic warped product graphs. In fact, we prove this by directly applying the Alexandrov Theorem [ 2 , 6 ].

4.1 The (NEC) on spacelike warped product graphs

We first show that we can rewrite the (NEC) along any spacelike warped product graph \((M_T,g^T,K^T)\) as a tensor inequality adapted to the slice. Recall that the (NEC) along \((M_T,g^T,K^T)\) equivalently implies that for any unit vector \(V_T\) on \((M_T,g^T,K^T)\)

where we used the contracted Gauss equation in the last line. Here \(\mu _T\) , \(J_T\) denote the energy and momentum density of \((M_T,g^T,K^T)\) , respectively, and \(\text {tr}_T\) denotes the trace with respect to \(g^T\) . We will similarly denote the respective quantities on the \(\{t=0\}\) time slice \(M_0\) with a subscript 0. We refer to Appendix A , where we collect and derive the well-known curvature quantities for warped product graphs and spacetimes of Class \({\mathfrak {H}}\) for the convenience of the reader.

We further define an isomorphism between the tangent bundles of \(M_T\) and \(M_0\) in the following way: Let \(V_T=c_1f_T(s)\partial _s+\frac{c_2}{s}X\) be a tangent vector field along \(M_T\) , where X is a unit vector field tangent in \(({\mathcal {N}},g_{{\mathcal {N}}})\) . We define the vector field \(V_0\) tangent to \(M_0\) as \(V_0\,{{:}{=}}\,c_1f(s)\partial _r+\frac{c_2}{s}X\) . Note that this isomorphism is not induced by an isometry between \(M_T\) and \(M_0\) unless \(T={\text {const.}}\) . Together with Lemma  6 , we establish the following:

For \(f_0\,{{:}{=}}\,f=\sqrt{h}\) as above, we find

\(\mu _T=\mu _0=\frac{1}{2}R_0=\frac{1}{2}{\mathfrak {R}}+\frac{\Delta _0f_0}{f_0}\) ,

\(J_T\equiv 0\) ,

\(\text {Ric}^T(V_T,V_T)+(\text {tr}_TK^TK^T-(K^T)^2)(V_T,V_T)=\text {Ric}^0(V_0,V_0)\) ,

\(\mathfrak {Rm}(V_T,\vec {{\textbf {n}}}^T,V_T,\vec {{\textbf {n}}}^T)=\mathfrak {Rm}(V_0,\partial _t,V_0,\partial _t)=\frac{\text{ Hess}_0f_0(V_0,V_0)}{f_0}\)

As the spacetime is static, it is unsurprising that with the above identities at hand Equation ( 8 ) directly reduces to Lemma  4 (ii). However, as we aim to employ the result of Brendle [ 2 ] directly on the slice, we will instead rewrite ( 8 ) as a tensor inequality involving \(f_T:=\sqrt{h_T}\) .

We also want to emphasize the vanishing momentum constraint \(J_T\equiv 0\) , as the converse is also true in the following sense: If \(({\widetilde{M}},{\widetilde{g}},{\widetilde{K}})\) is an initial data set with warped product structure as above determined by the functions \({\widetilde{h}}\) , \({\widetilde{a}}\) , \({\widetilde{b}}\) , then \({\widetilde{J}}\equiv 0\) if and only if \(({\widetilde{M}},{\widetilde{g}},{\widetilde{K}})\) embeds as a warped product graph into a spacetime of Class \({\mathfrak {H}}\) with \(h:={\widetilde{h}}-s^2{\widetilde{b}}^2\) .

In [ 24 ], Cabrera Pacheco and the author construct non-time symmetric initial data sets from the constraint equations with similar warped product structure. In spherical symmetry, the above observation then allows one to further construct the maximal future development under the additional assumption \(J\equiv 0\) .

Proof of Lemma 8

In view of the Remark, we only prove (ii) and refer to the proof of (i), (iii), (iv) to Appendix  A . In coordinates, \(J_T\) is given as

for indices \(i,j,k,l\in \{s,I,J,K,L\}\) . Using the block diagonal structure of K and the well-known identities for the Christoffel symbols in class \({\mathfrak {H}}\) , a direct computation shows that

Thus, it remains to show that \(h_Ta_T-b_T-sb_T'\) vanishes along \(M_T\) . As \(b_T\) is in particular continuous, there exists a closed set \({\mathcal {X}}\subset I_T:=(r_1,r_2)\) of measure zero, such that \( I_T\setminus {\mathcal {X}}\) lies dense in \(I_T\) , and for all \(s\in I_T\setminus {\mathcal {X}}\) there exists an open neighborhood \(U_s\) of s , such that either \(b_T\not =0\) or \(b_T\) vanishes identically on \(U_s\) . In the first case, multiplying the equation by \(b_T\) yields

which vanishes by the identities of Lemma  6 and using

Since we assumed \(b_T\not =0\) , we have \(h_Ta_T-b_T-sb_T'=0\) . On the other hand, if \(b_T\) vanishes identically on a neighborhood, then \(T={\text {const.}}\) by Remark  1 , so \(h_T=h\) and \(a=0\) . In particular \(h_Ta_T-b_T-sb_T'=0\) . Therefore \(h_Ta_T-b_T-sb_T'\) vanishes on \(I_T\setminus {\mathcal {X}}\) . By continuity, it has to vanish on all of \(I_T\) , which concludes the proof of (ii). \(\square \)

We now establish the relevant tensor inequality adapted to the warped product graphs \((M_T,g^T,K^T)\) .

Proposition 9

Let \(({\mathfrak {M}},{\mathfrak {g}})\) be a spacetime of Class \({\mathfrak {H}}\) that satisfies the (NEC), and let \((M_T,g^T,K^T)\) be a warped product graph in \(({\mathfrak {M}},{\mathfrak {g}})\) . Then, for all unit vector fields \(V_T\) on \((M_T,g^T,K^T)\) we have

and using Lemma  6 , a direct computation shows

For a unit vector \(V_T=c_1f_T\partial _s+\frac{c_2}{s}X\) on \(M_T\) , the (NEC) gives

by Lemma  8 . Note that a direct computation yields

Inserting this into the tensor inequality above yields the claim. \(\square \)

Provided that \(B^T\) is positive semi-definite, the (NEC) on \(({\mathfrak {M}},{\mathfrak {g}})\) in particular implies the desired tensor inequality

on general warped product initial data sets \((M_T,g^T,K^T)\) . Moreover, \(B^T\ge 0\) if and only if \(x:=h_T-h\) is a non-negative function satisfying the linear ordinary differential inequality

Note that this is the same differential inequality as in Lemma  4 (iv) for the function \(h-\alpha \) , which is equivalent to the (NEC) on \(({\mathfrak {M}},{\mathfrak {g}})\) . By linearity, we have that \(h_T-\alpha \) solves the above differential inequality ( 13 ), which by Lemma  4 implies that the spacetime of Class \({\mathfrak {H}}\) with metric coefficient \(h_T\) (and the same fibre \({\mathcal {N}}\) ) satisfies the (NEC).

The exact solutions of ( 13 ) as an ODE are given by a 2-parameter family of solutions of the form

In spherical symmetry, \(h=1+x\) correspond to the Schwarzschild de Sitter and Schwarzschild anti-de Sitter family depending on the sign of \(C_2\) , which describe the static, spherically symmetric Vacuum solutions of the Einstein Equations (with cosmological constant depending on \(C_2\) ). These are precisely the spacetimes of class \({\mathcal {S}}\) such that the time-symmetric slices have constant scalar curvature. Compare [ 16 , Lemma 2.3].

In this sense, if \(B^T\ge 0\) , the spacetime of Class \({\mathfrak {H}}\) with metric coefficient \(h_T\) inherits the (NEC) from the spacetime of Class \({\mathfrak {H}}\) with metric coefficient h . This makes apparent that we can use the (NEC) on \(({\mathfrak {M}},{\mathfrak {g}})\) to classify CMC surfaces on a large class of general warped product graphs \((M_T,g^T,K^T)\) provided we can extend the tensor inequality in an appropriate way until the first zero \(r_T\) of \(h_T\) to verify Brendle’s condition (H1) in [ 2 ]. As \(h_T\ge h\) on I , a minimal, inner boundary of \((M_T,g^T)\) will in general be hidden behind a Killing horizon of \(({\mathfrak {M}},{\mathfrak {g}})\) .

4.2 Extending the graphs past the Killing horizon

We recall that Killing horizons arise in spacetimes of Class \({\mathfrak {H}}\) as zeros of h . We will always assume from now on that h has finitely many positive, simple zeroes

such that all arising Killing horizons are non-degenerate, i.e., \(h'(r_l)\not =0\) for all \(1\le l\le N\) . As (H2) is equivalent to the fact that the outermost Killing horizon \(\{r=r_H\}\) is non-degenerate, this is rather a necessary than a restrictive assumption in lieu of applying the results of Brendle [ 2 ] and Borghini–Fogagnolo–Pinamonti [ 6 ].

However, as \(r\rightarrow r_N\) the ( t ,  r )-coordinate system breaks down, and in general \(M_T\) can no longer be described as a graph of a radial function T , since by Remark  1

We want to argue that we can extend the graph of T past any Killing horizon \(\{r=r_l\}\) for all \(1\le l\le N\) with \(h_T(r_l)>0\) in different coordinates and extend the notion of the (NEC) as an ordinary differential inequality on a suitable spacetime extension. As we only need this up to the inner boundary of \(M_T\) , i.e., \(\{s=r_T\}\) for the largest zero \(r_T\) of \(h_T\) , we always have that \({h_T(r_l)>h(r_l)=0}\) for all \(1\le l\le N\) with \(r_l>r_T\) .

Using the above assumptions that each Killing horizon is non-degenerate, Brill–Hayward [ 10 ], Schindler–Aguirre [ 11 ], and Cederbaum and the author [ 8 ] showed independently that a spacetime \(({\mathfrak {M}},{\mathfrak {g}})\) of Class \({\mathfrak {H}}\) admits a spacetime extension called the generalized Kruskal–Szekeres extension under the above assumptions, that extends the radial coordinate r to \((0,\infty )\) across the zeros of h . Throughout this section we will use similar conventions as in [ 8 ]. In each coordinate chart, the spacetime extension is given by a warped product manifold \(({\mathbb {P}}_l\times {\mathcal {N}},{\widetilde{g}}_l)\) , where

with \(\rho =\Phi _l^{-1}(uv)\) , \(F_l=\frac{2C_l}{\Phi _l'}\) , where \(\Phi _l\) is the unique strictly increasing solution of

on \((r_{l-1},r_{l+ 1})\) with \(\Phi '(r_l)=1\) , \(C_l\,{{:}{=}}\,\frac{1}{h'(r_l)}\) , \(1\le l\le N\) . Note that if \(h>0\) on \((r_N,\infty )\) , the original spacetime \(({\mathfrak {M}},{\mathfrak {g}})\) corresponds to \(\{u,v>0\}\) in \({\mathbb {P}}_N\times {\mathcal {N}}\) . Moreover, we have the explicit coordinate transformations between ( u ,  v ) and ( t ,  r ) coordinates

on each coordinate patch \({\mathbb {R}}\times (r_j,r_{j+1})\times {\mathcal {N}}\) ( \(j\in \{l-1,l\}\) ), where ( t ,  r )-coordinates are defined. A direct computation in the ( u ,  v )-coordinates using Eq. ( 14 ) gives

cf. [ 8 , Proposition B.1]. Let \(L=a\partial _u+b\partial _v+\frac{c}{\rho }X\) be a null vector field, where X is a unit vector in \(({\mathcal {N}},g_{{\mathcal {N}}})\) . Then

Using this identity, we see that

which is again equivalent to the linear ODI ( 13 ) for \(x=h-\alpha \) by the same arguments as in the proof of Lemma  4 . Therefore the spacetime extension satisfies the (NEC) if and only if \(x=h-\alpha \) satisfies ( 13 ) on \((0,\infty )\) .

A crucial observation in [ 8 ] is that any solution \(\Phi _l\) of ( 14 ) is of the form

where \(R_l\) is a smooth function on \((r_{l-1},r_{l+1})\) uniquely determined up to a constant. Although the construction circumvents the need to do so, this yields a tortoise function \(R^*\) , i.e., a primitive of \(\frac{1}{h}\) , a-posteriori on each of the intervals \((r_{l-1},r_l)\) and \((r_l,r_{l+1})\) by defining \(R^*\,{{:}{=}}\,C_l\ln (\left| \Phi _l\right| )\) .

Now, we notice that by Remark  1 T is a primitive of the function \(\pm \frac{1}{\eta _T}\) , where \({\eta _T:=h\sqrt{\frac{h_T}{h_T-h}}}\) is well-defined and \(\eta _T'(r_l)=h'(r_l)\not =0\) for any \(r_l>r_T\) . In particular, the construction of Cederbaum and the author [ 8 ] yields that T satisfies

for some smooth function \({\widetilde{R}}_T\) on \((r_{l-1},r_{l+1})\) . Using this explicit behavior of T , we see that in the \(+\) -case

so \(M_T\) extends smoothly across the horizon and crosses the horizon at \(v=0\) , \(u=u(r_l)\) , and similarly at \(v=v(r_l)\) , \(u=0\) in the case of "-". Therefore we can extend any warped product graph across any non-degenerate Killing horizon up to its minimal, inner boundary.

4.3 STCMC characterization on totally umbilic warped product graphs

Combining the result of the previous subsections with the results of Brendle [ 2 ] and Borghini–Fogagnolo–Pinamonti [ 6 ], and of Cederbaum and the author [ 8 ], we acquire the following:

Let \(h:(0,\infty )\rightarrow {\mathbb {R}}\) be a smooth function with finitely many, positive simple zeroes \(r_1<\dotsc < r_N\) , \(({\mathcal {N}},g_{{\mathcal {N}}})\) an \((n-1)\) -dimensional Riemannian manifold ( \(n\ge 3\) ). Let \(({\mathfrak {M}},{\mathfrak {g}})\) be the corresponding spacetime of Class \({\mathfrak {H}}\) with metric coefficient h and fibre \({\mathcal {N}}\) , and assume that the generalized Kruskal–Szekeres extension of \(({\mathfrak {M}},{\mathfrak {g}})\) satisfies the (NEC) condition. Let x be some non-negative function satisfying ( 13 ), and consider the warped product graph \(M_T\) , where T is such that \(h_T=h+x\) . Then \(M_T\) extends into the generalized Kruskal–Szekeres extension until its minimal inner boundary, corresponding to the first zero \(r_T\) of \(h_T\) .Additionally, if \(r_T>0\) with \(h_T'(r_T)>0\) , then all compact CMC surfaces in \(M_T\) are leaves \(\{s\}\times {\mathcal {N}}\) .

By the results in [ 8 ] as summarized in Sect.  4.2 , \(({\mathfrak {M}},{\mathfrak {g}})\) extends onto all positive radii, and the generalized Kruskal–Szekeres extension is covered by a countable, smooth atlas. As observed in the previous subsection the (NEC) implies that the ODI ( 13 ) holds for \(h-\alpha \) on all of \((0,\infty )\) , and the graph \(M_T\) of T with \(h_T=h+x\) is well defined across any non-degenerate Killing horizon up until the first zero \(r_T\) of \(h_T\) .

Since x is a non-negative solution of ( 13 ), the tensor inequality ( 12 ) holds, and by Remark  1 there exists a graph T such that \(h_T=h+x\) , where T is uniquely determined by x up to a choice of sign of \(T'\) and a constant of integration. Now assume that the first zero \(r_T\) of \(h_T\) satisfies \(h_T'(r_T)>0\) . In particular, conditions (H1)–(H3) in [ 2 , Theorem 1.1] are satisfied. Thus, any compact CMC surface in \(M_T\) is totally umbilic. Moreover, by [ 6 , Corollary 1.3] any compact CMC surface is in fact a leaf of the canonical foliation \(\{s\}\times {\mathcal {N}}\) . \(\square \)

Note that this result is independent of the extrinsic curvature K . Therefore it also suffices to apply the results in [ 2 , 6 ] directly to the totally geodesic slices in the spacetime of Class \({\mathfrak {H}}\) with metric coefficient \(h_T\) , as this spacetime will satisfy the (NEC) by Remark  5 . Note that this observation is consistent with the duality of constant, positive mean curvature slices in spacetimes with zero cosmological constant and maximal slices in spacetimes with negative cosmological constant, cf. [ 25 ].

We now want to incorporate the extrinsic curvature K into our result. Due to the difficulty in adapting the methods in [ 2 , 6 ] in the presence of \(P=\text {tr}_\Sigma K\) and its evolution, we restrict ourselves to the special case of totally umbilic warped product graphs, which we have fully characterized in Corollary  7 . Note that on a hyperboloid as defined in Sect.  3 , \(P=\text {tr}_\Sigma K=(n-1)\lambda \) is constant and the same for any embedded surface \(\Sigma \) , so the evolution of P along any deformation is trivial. Hence, any surface \(\Sigma \) in \((M_T,g^T,K^T)\) has constant spacetime mean curvature \({\mathcal {H}}^2\) , and constant expansion \(\theta _\pm \) if and only if it is a CMC surface. Moreover, by Remark  5 \(Cs^2\) is an exact solution of ( 13 ), so it in fact solves ( 13 ) as an ODE. Using Theorem  10 , we acquire our main result for totally umbilic warped product graphs in Class \({\mathfrak {H}}\) .

Let \(h:(0,\infty )\rightarrow {\mathbb {R}}\) be a smooth function with finitely many, positive simple zeroes \(r_1<\dotsc < r_N\) , \(({\mathcal {N}},g_{{\mathcal {N}}})\) an \((n-1)\) -dimensional Riemannian manifold ( \(n\ge 3\) ). Let \(({\mathfrak {M}},{\mathfrak {g}})\) be the corresponding spacetime of Class \({\mathfrak {H}}\) with metric coefficient h and fibre \({\mathcal {N}}\) . Assume that the generalized Kruskal–Szekeres extension satisfies the (NEC) condition.Then there exists a constant \(C_0=C_0(h,h')\in (0,\infty ]\) such that for any hyperboloid \((M_T,g^T,K^T)\) with umbilicity factor \(\lambda _T\) satisfying \(\lambda _T^2<C_0\) we have: If \(\Sigma \subset M_T\) is an orientable, closed, embedded hypersurface with constant spacetime mean curvature, then \(\Sigma \) is a leaf \(\{s\}\times {\mathcal {N}}\) .

For \(C=\lambda _T^2\) small enough, \(h_T=h+Cs^2\) has at least one positive zero \(r_T\) with \(r_T\rightarrow r_N\) as \(C\rightarrow 0\) . By continuity, we have \(h'(q_T)>0\) for small enough C . We define \(C_0\) as the supremum over all C , such that these conditions are still satisfied. Thus, Theorem  10 applies to a hyperboloid \((M_T,g^T,K^T)\) satisfying \(\lambda _T^2<C_0\) , and any CMC surface \(\Sigma \) is a leaf \(\{s\}\times {\mathcal {N}}\) . Since \((M_T,g^T,K^T)\) has constant umbilicity factor, any STCMC surface is a leaf \(\{s\}\times {\mathcal {N}}\) . \(\square \)

Note that all assumptions are in particular satisfied for any constant \(C\ge 0\) in the Kruskal–Szekeres extension of the Schwarzschild spacetime with positive mass \(m>0\) corresponding to

in spherical symmetry.

Corollary 12

Let \(({\mathfrak {M}},{\mathfrak {g}})\) be the Schwarzschild spacetime with positive mass. Then any closed, embedded STCMC surface \(\Sigma \) in an hyperboloid \((M_T,g^T,K^T)\) is a slice \(\{s\}\times {\mathbb {S}}^{n-1}\) .

Note that a direct computation yields that

for a leaf \(\{s\}\times {\mathcal {N}}\) for any warped product graph \((M_T,g^T,K^T)\) . As we have to extend any hyperboloid with \(\lambda _T\not =0\) across the horizon where \(h(r_H)=0\) into a region where \(h<0\) , both the case of generalized apparent horizons \({\mathcal {H}}^2=0\) , and trapped STCMC surfaces with \({\mathcal {H}}^2<0\) naturally occur in hyperboloids.

5 Characterizing photon surfaces with constant umbilicity factor

Arguing in analogy to Brendle [ 2 ] that umbilic CMC surfaces are canonical leaves in the Riemannian setting under condition (H4), we obtain a short proof for a characterization of photon surfaces with constant umbilicity factor \(\lambda \) imposing a condition on the eigenvalues of the Ricci tensor in the spacetime setting in Class \({\mathfrak {H}}\) (and the respective Kruskal–Szekeres extension). Although utilizing completely different methods, the statement is similar to a result by Cederbaum–Galloway [ 9 , Theorem 3.8] in spherical symmetry under the additional assumption that the umbilicity factor is constant.

5.1 A spacetime Ricci eigenvalue argument

Recall that a photon surface \({\mathcal {P}}^n\) is a smooth, timelike, totally umbilic hypersurface in an \((n+1)\) -dimensional spacetime \(({\mathfrak {M}},{\mathfrak {g}})\) , so denoting the induced metric on \({\mathcal {P}}^n\) by \({\mathfrak {p}}\) and its second fundamental form by \({\mathfrak {h}}\) , we have

for some smooth function \(\lambda \) on \({\mathcal {P}}^n\) . Observe that photons on \({\mathcal {P}}^n\) remain trapped in the following sense: A timelike hypersurface is totally umbilic if and only if null geodesics starting tangent to the hypersurface must remain tangent to \({\mathcal {P}}^n\) [ 26 , 27 ]. We refer the interested reader to [ 9 ] for a more complete introduction and list of references. In this section, we will consider photon surfaces \({\mathcal {P}}^n\) in spacetimes of Class \({\mathfrak {H}}\) and their respective generalized Kruskal–Szekeres extension as defined in Sect.  4.2 . For photon surfaces, the Codazzi equation implies

for any tangent vector field \(Y\in \Gamma (T{\mathcal {P}}^n)\) , where \(\nabla ^n\) denotes the exterior derivative on \({\mathcal {P}}^n\) and \(\eta \) the spacelike unit normal to \({\mathcal {P}}^n\) , respectively. In spacetimes of Class \({\mathfrak {H}}\) , the Ricci curvature tensor has an eigenvalue \(\beta \) with a corresponding eigenspace that is at least 2-dimensional and contains \(\{\partial _t,\partial _r\}\) . More precisely

Since \(\mathfrak {Ric}_{IJ}=\left( \text {Ric}_{g_{\mathcal {N}}}\right) _{IJ}-\left( (n-2)h+rh'\right) \left( g_{{\mathcal {N}}}\right) _{IJ}\) , all other eigenvalues of \(\mathfrak {Ric}\) are characterized by the eigenvalues of \(\text {Ric}_{g_{\mathcal {N}}}\) . To ensure that the corresponding eigenspace of \(\beta \) is exactly 2-dimensional and thus spanned by \(\{\partial _t,\partial _r\}\) , we require that the difference between \(\beta \) and any eigenvalue of \(\mathfrak {Ric}\vert _{T{\mathcal {N}}\times T{\mathcal {N}}}\) is non-trivial. Equivalently,

for any unit tangent vector field X in \(({\mathcal {N}},g_{{\mathcal {N}}})\) . Here, it suffices to assume this for any unit tangent eigenvector X of \(\text {Ric}_{g_{\mathcal {N}}}\) . Note that if ( 16 ) is strictly positive, then \(\beta \) is the smallest eigenvalue of \(\mathfrak {Ric}\) and \(({\mathfrak {M}},{\mathfrak {g}})\) satisfies the (NEC) by Lemma  4 . Conversely, the (NEC) implies that \(\beta \) is the smallest eigenvalue of \(\mathfrak {Ric}\) . However, we need the additional assumption that the (NEC) is strict for any null vector field L that is not perpendicular to \({\mathcal {N}}\) to conclude that ( 16 ) is indeed satisfied. Recall from Remark  5 that in spherical symmetry equality implies \(h=1+\frac{C_1}{r^{n-2}}+C_2r^2\) , so ( 16 ) in Class \({\mathfrak {H}}\) is generally satisfied in spherical symmetry outside of a dense subset unless h locally corresponds to the Schwarzschild de Sitter/Schwarzschild anti de Sitter family. All of the above observations naturally extend to the generalized Kruskal–Szekeres extension, and in this case ( 16 ) on \((0,\infty )\) is equivalent to the fact, that the corresponding eigenspace of \(\beta \) is exactly 2-dimensional and spanned by \(\{\partial _u,\partial _v\}\) .Using the Codazzi equation as the main tool we now prove a characterization of photon surfaces with constant umbilicity factor in a spacetime of Class \({\mathfrak {H}}\) or its generalized Kruskal–Szekeres extension satisfying ( 16 ).

Let \({\mathcal {P}}^n\) be a connected photon surface with constant umbilicity factor \(\lambda \) in a spacetime of Class \({\mathfrak {H}}\) or its respective generalized Kruskal–Szekeres extension. Assume further that ( 16 ) is satisfied on a dense set of radii in \((0,\infty )\) . Then \({\mathcal {P}}^n\) is either symmetric or \({\mathcal {P}}^n\) is totally geodesic with parallel unit normal vector \(\eta \) everywhere tangent to \({\mathcal {N}}\) .

In this context, we understand a photon surface to be symmetric in a spacetime of Class \({\mathfrak {H}}\) or its generalized Kruskal–Szekeres extension if for every point \(p\in {\mathcal {P}}^n\) the canonical lift of the tangent space of \({\mathcal {N}}\) at p is a subspace of the tangent space \(T_p{\mathcal {P}}^n\) of \({\mathcal {P}}^n\) , and the unit normal \(\eta \) is everywhere spanned by \(\{\partial _t,\partial _r\}\) or \(\{\partial _u,\partial _v\}\) respectively. This notion agrees with the definition of symmetric photon surfaces in [ 8 ], and Cederbaum–Galloway [ 9 ], Cederbaum–Jahns–Vičánek Martínez [ 23 ] in spherical symmetry via a profile curve \(\gamma \) mapping into the ( t ,  r )- or ( u ,  v )-coordinate plane, respectively.

Assuming to be in the exact setting of Cederbaum–Galloway in [ 9 ], we note that a non-empty intersection between a \(\{t={\text {const.}}\}\) slice and a totally geodesic photon surface with parallel unit normal vector \(\eta \) everywhere tangent to \({\mathcal {N}}\) is (a piece of) a centered hyperplane in isotropic coordinates as defined in [ 9 ]. In particular, Theorem  13 draws the same conclusion in this setting as the result by Cederbaum–Galloway [ 9 , Theorem 3.8] under the additional assumption of \(\lambda ={\text {const.}}\) . We refer to the next subsection for a comparison of the two statements.

Let \({\mathcal {P}}^n\) be a photon surface with constant umbilicity factor \(\lambda \) . Then the Codazzi equation implies

for any tangent vector Y . Therefore \(\eta \) is an eigenvector of \(\mathfrak {Ric}\) along \({\mathcal {P}}^n\) . Since ( 16 ) is satisfied on a dense set of radii in \((0,\infty )\) , continuity yields that ( 16 ) is in fact satisfied on \((0,\infty )\setminus {\mathcal {X}}\) , where \({\mathcal {X}}\) is a closed set that has measure zero. By assumption the eigenspace of the eigenvalue \(\beta \) of \(\mathfrak {Ric}\) is spanned by \(\{\partial _t,\partial _r\}\) ( \(\{\partial _u,\partial _v\})\) away from \({\mathcal {X}}\) . If \({\mathcal {P}}^n\subset {\mathbb {R}}\times {\mathcal {X}}\times {\mathcal {N}}\) , as \({\mathcal {P}}^n\) is connected, \({\mathcal {P}}^n\subset {\mathbb {R}}\times \{r=r_0\}\times {\mathcal {N}}\) for some \(r_0\in {\mathcal {X}}\) , in particular \({\mathcal {P}}^n\) is rotationally symmetric. Now assume \({\mathcal {P}}^n\subset {\mathbb {R}}\times I\times {\mathcal {N}}\) for some open interval I with \(I\cap {\mathcal {X}}=\emptyset \) . Then ( 16 ) holds on I , so we have \(\eta \in {\text {span}}(\partial _t,\partial _r)\) (or \(\eta \in {\text {span}}(\partial _u,\partial _v)\) in the case of the generalized Kruskal–Szekeres extension) or \(\eta \) is everywhere tangent to \({\mathcal {N}}\) . In the first case \(\eta \) is everywhere perpendicular to \({\mathcal {N}}\) and hence everywhere a lift of \(T{\mathcal {N}}\) has to be a subspace of the tangent bundle of \({\mathcal {P}}^n\) , i.e., \({\mathcal {P}}^n\) is rotationally symmetric. If \(\eta \) is everywhere tangent to \({\mathcal {N}}\) , then away from any Killing horizon (which \({\mathcal {P}}^n\) can only cross at a discrete set which we can also exclude from I ) \(h\not =0\) , and \(\partial _t\) is well-defined and tangent to \({\mathcal {P}}^n\) . Staticity then implies

so \(\lambda =0\) . Hence \({\mathfrak {h}}\) vanishes identically and \({\mathcal {P}}^n\) is totally geodesic. In particular, \(\eta \) is parallel along \({\mathcal {P}}^n\) .

Lastly, if \(I\cap {\mathcal {X}}\not =\emptyset \) , then the above argument holds on any connected component of \({\mathcal {P}}^n\cap ({\mathbb {R}}\times (I\setminus {\mathcal {X}})\times {\mathcal {N}})\) , i.e., on any connected component \(\eta \) is either orthogonal or tangential to \({\mathcal {N}}\) . Assume that there is an open component C such that \(\eta \) is everywhere tangent to \({\mathcal {N}}\) . In particular, \({\overline{C}}\subset {\mathcal {P}}^n\) intersects \({\mathbb {R}}\times {\mathcal {X}}\times {\mathcal {N}}\) transversally. As \(\eta \) is continuous and nowhere-vanishing, this implies that \(\eta \) is tangent to \({\mathcal {N}}\) for all open connected components of \({\mathcal {P}}^n\cap ({\mathbb {R}}\times (I\setminus {\mathcal {X}})\times {\mathcal {N}})\) with closure in \({\mathcal {P}}^n\) intersecting \(\partial C\) . As \({\mathcal {P}}^n\) is connected, applying the above argument iteratively eventually covers all of \({\mathcal {P}}^n\) . This concludes the proof. \(\square \)

5.2 Comparison to the characterization of Cederbaum–Galloway

Cederbaum–Galloway characterized photon surfaces in static, spherically symmetric spacetimes in isotropic coordinates that are of the form \(({\mathbb {R}}\times D^n,-{\widetilde{N}}^2\,\text {d}t^2+\Psi ^2\delta )\) , where \({D^n\,{{:}{=}}\,\{x\in R^n:\left| \left| x\right| \right| =s\in I\}}\) for some positive open Interval \(I\in {\mathbb {R}}\) , and \(\Psi ,{\widetilde{N}}\) are smooth, positive functions on I . In their assumption, they exclude spacetimes satisfying

as this implies that these spacetimes are spacetime conformally flat

for some positive constant A , and thus posses the same plethora of “off-center” photon surfaces as the Minkowski spacetime. Assuming that ( 17 ) does not hold, Cederbaum–Galloway showed in [ 9 ], that any photon surface must necessarily be rotationally symmetric or a centered vertical hyperplane in this coordinate system. See [ 9 , Theorem 3.8] for a precise statement. A spacetime of class \({\mathcal {S}}\) can always be locally rewritten in isotropic coordinates by defining s as a primitive of \(\left( r\sqrt{h(r)}\right) ^{-1}\) , and setting \(\Psi (s)\,{{:}{=}}\,\frac{r(s)}{s}\) and \({\widetilde{N}}^2(s)=h(r(s))\) , where we denote the inverse of s ( r ) by r ( s ). On the other hand, a static, spherically symmetric spacetime in isotropic coordinates can be globally rewritten as a spacetime of class \({\mathcal {S}}\) if and only if

with \(r\,{{:}{=}}\,s\Psi \) and \(h(r)={\widetilde{N}}^2(s(r))\) , where s ( r ) denotes the inverse of r ( s ). Thus [ 9 , Theorem 3.8] and Theorem  13 achieve a similar characterization of photon surfaces under different assumptions. Recall that Theorem  13 additionally imposes that \(\lambda \) is constant, however the proof of Cederbaum–Galloway heavily relies on the conformally Euclidean structure of the time–symmetric time slices in class \({\mathcal {S}}\) and that any timelike hypersurface intersects them transversally. Thus, it is not immediate to extend this result to cases \({\mathcal {N}}\not ={\mathbb {S}}^{n-1}\) or past the Killing horizon in the generalized Kruskal–Szekeres extension as the totally geodesic \(\{t={\text {const.}}\}\) -slices are timelike in a region with \(h<0\) , and hence are photon surfaces themselves.

However, even within class \({\mathcal {S}}\) the assumptions of both theorems appear to be distinctly different. On the one hand, the Schwarzschild spacetime satisfies the assumptions of Cederbaum–Galloway, but can not satisfy ( 16 ) as a vacuum solution. Conversely, the following example shows that there is a 1-parameter family of spacetimes within class \({\mathcal {S}}\) that satisfy ( 17 ) locally, but globally allow for a characterization of photon surfaces with constant umbilicity factor using Theorem  13 .

Combining ( 17 ) and ( 18 ), we see that this question reduces to an ODE on \(\Psi \) , which we can explicitly solve for. The solutions belong to a 2-parameter family and are given by

for some constants C , \(C_0\) , where the open interval I is chosen such that \(\Psi \) is well-defined and positive on I . We find \({\widetilde{N}}^2=C_0^2\Psi ^2\) , so we need to impose \(C_0\not =0\) . Changing coordinates as above, using \(h(r)={\widetilde{N}}^2\) and \(r=s\Psi \) , we see that the corresponding metric coefficient h satisfies \(h(r)=(1-Cr)^2\) independent of the choice of \(C_0\not =0\) . For \(C=0\) we recover the Minkowski spacetime. Explicit computation gives

so Theorem  13 applies for \(C\not =0\) ( \(n\ge 3\) ). Thus all photon surfaces with constant umbilicity factor \(\lambda \) must necessarily be centered around the origin. However, in contrast to the result of Cederbaum–Galloway, this illustrates that assumption ( 16 ) does not prevent the formation of “off-center” photon surfaces in general. If \(C>0\) the corresponding spacetime of class \({\mathcal {S}}\) with profile h satisfies the (NEC) and the (DEC) (with negative cosmological constant \(\Lambda =-\frac{n(n-1)}{2}C^2\) ), and the spacetime possesses a degenerate Killing horizon at \(r=\frac{1}{C}\) . As \(s\Psi :(0,\infty )\rightarrow (0,\frac{1}{C})\) , only the interior of the degenerate Killing horizon is spacetime conformally flat, while the exterior is not. These spacetimes are therefore an interesting toy-model to compare the result of Cederbaum–Galloway and Theorem  13 , as there are a wide array of “off-center” photon surfaces in the interior of the degenerate horizon (which have to be conformal images of either pseudospheres or hyperplanes), while in the exterior any photon surface has to be centered in the sense of Theorem 3.8 in [ 9 ]. Althewhile, Theorem  13 implies that any photon sphere with constant umbilicity factor is necessarily centered around the origin. In fact, the centered pseudospheres \(\{s=\sqrt{R^2+t^2}\}\) of radius R in isotropic coordinates in the interior of the degenerate horizon retain \(\lambda =\frac{1}{R}\) .

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I would like to express my sincere gratitude towards my supervisor Carla Cederbaum for her guidance and helpfull discussions.

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Curvature identities

For any warped product graph \((M_T,g^T)\) , we have that

where \(\text {Ric}^T\) , \(R^T\) , \(\text {Hess}_T\) , and \(\Delta _T\) denote the Ricci curvature, scalar curvature, Hessian and Laplacian along \((M_T, g^T)\) respectively, \(\text {Ric}_{g_{\mathcal {N}}}\) , and \(\text {R}_{g_{\mathcal {N}}}\) denote the Ricci curvature and scalar curvature of \(({\mathcal {N}},g_{\mathcal {N}})\) , respectively, and \(f_T\) is defined as \(f_T\,{{:}{=}}\,\sqrt{h_T}\) on I . Furthermore, for a spacetime \(({\mathfrak {M}},{\mathfrak {g}})\) of Class \({\mathfrak {H}}\) , we have

Assuming the warped product structure, all non-trivial Christoffel symbols are given by

where \({ ^\mathcal {N}}\!{\Gamma }_{IJ}^K\) denotes Christoffel symbols of \(({\mathcal {N}},g_{{\mathcal {N}}})\) . By definition, the Ricci curvature components are given by the formula

where i ,  j denote the coordinates \(\{s,x^I\}\) on \(M_T\) . Then, a straightforward computation yields the above identities for the Ricci curvature components. Taking the metric trace with respect to \(g^T\) yields the scalar curvature

The Hessian \(\text {Hess}_T\) of \(f_T\) on \(M_T\) is given by

where \(f_{T,i}\,{{:}{=}}\,\partial _if_T\) . Using the above identities for the Christoffel symbols and taking the metric trace with respect to \(g^T\) , the identities for the Hessian and Laplacian of \(f_T\) are immediate.Computing all non-trivial Christoffel symbols on \(({\mathfrak {M}},{\mathfrak {g}})\) yields the identities for the relevant curvature components of \(\mathfrak {Rm}\) in a similarly straightforward way. Having computed the Ricci curvature on \(M_0\) by putting \(h_T=h\) , we get the expressions for \(\mathfrak {Ric}\) using O’Neill’s formula (Proposition 2.7 [ 16 ]). Taking the metric trace with respect to \({\mathfrak {g}}\) yields the explicit formula for the scalar curvature \({\mathfrak {R}}\) . \(\square \)

We complete the proof of Lemma  8 with the following two lemmas. Recall that for a tangent vector \(V_T=c_1f_T\partial _s+\frac{c_2}{s}X\) , we define \(V_0:=c_1f\partial _r+\frac{c_s}{s}X\) tangent to \(M_0\) .

For a warped product graph \((M_T,g^T,K^T)\) we have

For the first identity, it suffices to show \(\mu _T=\frac{1}{2}R_0\) . The rest is immediate or follows from Lemma  14 . Since

the claims follow from Equations ( 10 ), ( 11 ), Lemma  6 and Lemma  14 . The identity for \(\text {Ric}^T\) follows directly from Lemma  14 and Lemma  6 . \(\square \)

Recall that \(\vec {{\textbf {n}}}=\frac{\partial _t+h\nabla _0T}{f\sqrt{1-h\left| \nabla _0T\right| ^2}}\) . As \(\nabla _0T=hT'\partial _r\) , a direct computation using Lemma  14 gives

where we used Lemma  14 and the explicit form ( 3 ) for \(h_T\) . Translating \(M_T\) in t -direction, we can extend \(\partial _s\) to a vectorfield on \({\mathfrak {M}}\) . It is straightforward to see that

Thus, for \(V=c_1f_T\partial _s+\frac{c_2}{s}X\)

where we used the definition of \(V_0\) . \(\square \)

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Wolff, M. On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes. Ann Glob Anal Geom 66 , 10 (2024). https://doi.org/10.1007/s10455-024-09969-6

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Received : 26 January 2024

Accepted : 30 August 2024

Published : 16 September 2024

DOI : https://doi.org/10.1007/s10455-024-09969-6

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• Prescribed mean curvature
• Alexandrov Theorem
• Umbilic hypersurfaces

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