1 Introduction

The emergence of advanced artificial intelligence models like GPT-4 offers promising avenues for employing these technologies as collaborative partners in scientific research [1, 2]. GPT-4, and its GPT-3.5 predecessor, have exhibited human-level performance across various domains, including passing the US Medical Licensing Exams and the Multistate Bar Exam with remarkable accuracy [3,4,5,6]. These accomplishments suggest that GPT-4 could aid researchers in complex and controversial areas where human collaboration might be constrained or biased. To investigate this potential, this paper presents a case study centered on Einstein’s Special Relativity Theory (SRT), a complex theory with an extensive history of debate and scrutiny [7].

Navigating controversial scientific ideas, particularly those contesting established theories, requires a meticulous approach to counter both overt and covert biases that might impede an impartial and objective analysis [8,9,10]. It is not uncommon for researchers’ pre-existing notions to lead to a neglect or dismissal of inconsistencies that challenge their beliefs [9,10,11]. Recently, the development of Generative Artificial Intelligence (AI) Large Language Models (LLMs) has opened new avenues for validation of scientific theories and ideas that conflict with accepted theories and deeply held convictions. GPT-4, a highly advanced language model developed by OpenAI, is one such example [1, 2].

This case study investigates the potential of GPT-4 to critically examine the mathematical coherence of Einstein’s SRT equations, and its efficacy as a research partner in pinpointing possible irregularities within these equations. Despite past attempts to expose inconsistencies in SRT and related equations being largely dismissed or overlooked, this study presents convincing evidence to the contrary [12,13,14,15,16,17]. Through detailed discussions with GPT-4, it appears that these inconsistencies in SRT mathematics may not be mere anomalies but could pervade extensively across the spatial domain. The potential influence of GPT-4's inherent limitations and biases on the process is also considered [18].

The research goes beyond a traditional case study. While detailing the strategy adopted and the key insights derived from the partnership with GPT-4, this research underscores both the promising prospects and the intricate challenges of incorporating advanced AI models into academic dialogues. This exploration with AI not only kindles a reassessment of SRT’s foundational principles, thereby enriching our understanding of the universe, but it also promotes an appreciation for AI's role in an academic context. The study thus provides a foundation for a framework and guidelines that inform AI’s optimal use in research collaborations.

2 Related works

Einstein’s Special Relativity Theory has been a subject of extensive investigation and criticism since its introduction in 1905 [7, 15, 19, 20]. Despite its central role in modern physics, some researchers have identified potential inconsistencies within the mathematical underpinnings and logical interpretation of the theory [12, 13, 21, 22]. The advent of sophisticated AI systems, exemplified by models such as GPT-4, signaled the onset of an era with unparalleled collaborative possibilities in scientific research. Within the intricate and often contentious domain of SRT critiques, GPT-4 emerges as a noteworthy ally. This study utilizes advancements in LLMs, prompt engineering techniques, and collaborative AI to delve deeper into the intricacies of SRT.

Significant prior critiques of SRT, such as those by Herbert E. Ives, Herbert Dingle, and the author, have contributed to philosophical debates, highlighted potential contradictions, and detected potential flaws within the SRT equations [12, 13, 15, 23]. The present paper aims to expand upon these works and differentiate itself by employing a comprehensive bounds analysis of the SRT equations and incorporating artificial intelligence, specifically GPT-4, as a co-collaborator.

Prompt engineering has emerged as a crucial skill set for working effectively with LLMs like GPT-4. White et al., Sorensen et al., Arora et al., Yang et al., Zhang et al., Bommarito and Katz, Ali et al., and Taveekitworachai et al. have contributed valuable insights into the importance of prompt engineering, the development of frameworks and strategies for structuring prompts, and the evaluation of LLM performance on complex tasks [5, 6, 24,25,26,27,28,29]. Their works provide a pivotal scaffold for advanced research applications of LLMs, as demonstrated in the current study.

Osmanovic-Thunström and Steingrimsson showed that GPT-3 can be considered a co-author in their study, emphasizing GPT-3’s potential to contribute to scientific work with minimal human intervention [30]. This work transcends their concept by engaging GPT-4 as an advanced research partner.

Inspired by the works of Li et al. and Jie Gao et al., this research underlines the need for a holistic framework to unpack and understand AI collaborations more effectively [31, 32]. Their studies demonstrated various facets of AI's role in supporting human tasks, shedding light on LLMs’ immense potential as research allies. Building on these insights, this research deepens the exploration into the symbiotic dynamics between AI and humans, laying the foundation for a robust and comprehensive collaboration framework.

3 Methods

To conduct a comprehensive and accurate examination of the SRT equations, this study harnessed the capabilities of both GPT-4 and Wolfram Alpha. GPT-4 functioned as the primary AI co-collaborator, while Wolfram Alpha was employed to address limitations in GPT-4’s mathematical proficiency.

Wolfram Alpha, a powerful computational knowledge engine, excels at executing mathematical operations and delivering precise results. In instances where GPT-4 could not perform specific calculations or furnish satisfactory explanations, Wolfram Alpha was utilized to obtain dependable outcomes and further substantiate the study’s findings.

By leveraging the combined strengths of GPT-4 and Wolfram Alpha, this research ensured a robust investigation, merging GPT-4’s human-like aptitude with Wolfram Alpha’s computational prowess. This approach facilitated a thorough evaluation of the SRT equations and enabled the detection of inconsistencies across various constraints.

To efficiently direct GPT-4’s responses and concentrate on the mathematical assessment, well-defined prompts were created to delineate scope and constraints. The initial constraint \(x=1\), \(v=0\), \(t=0\) stemmed from the author’s prior work, which uncovered inconsistencies in the SRT system of equations [13]. During the interaction with GPT-4, the constraint range was expanded to include \(x\in (-\infty ,\infty )\), \(t\in (-\infty ,\infty )\), and \(v\in (0,c)\).

The meticulously crafted prompts were also used to minimize the model’s inherent biases, directing its attention solely to mathematical principles. This approach helped avoid potential pitfalls, such as the AI introducing concepts only valid if the theory of relativity were already established at this point in Einstein’s derivation. The prompts were iteratively refined based on GPT-4’s responses, enabling the extraction of valuable insights and identification of potential inconsistencies in the SRT equations.

4 Collaboration framework

Advancements in Large Language Models, exemplified by GPT-4, have positioned AI as a dynamic collaborator, catalyzing a transformative shift within academic research. This paradigm shift prompts a fresh conceptual framework to capture AI’s diversified roles. This dual-dimensional framework revolves around Interaction Level and Task Nature.

The Interaction Level axis reflects the continuum of human-AI collaboration, ranging from Autonomous, where AI independently executes tasks, to Collaborative, denoting symbiotic intellectual discourse between AI and researchers.

Task Nature forms the orthogonal axis, spanning from Execution, indicative of automation-centric tasks, to Discovery, where AI endeavors to extract new knowledge or insights.

This conceptual model yields four distinct modes of AI collaboration: Autonomous Execution, Cooperative Implementation, Independent Exploration, and Interactive Discovery. Each mode illustrates a unique facet of AI in academic research, underscoring its transformative potential in enhancing efficiency and fostering innovation.

  1. 1.

    Autonomous execution: This quadrant features AI as an autonomous entity, performing tasks such as bibliographic compilation or text summarization with minimal human intervention. An example of this is Li et al.’s work, wherein AI enhances efficiency by autonomously performing delegated tasks, emphasizing its utility within academic workflows [32].

  2. 2.

    Cooperative implementation: This involves a symbiotic interaction where the AI and researcher co-construct a shared output, for instance, co-authoring a research paper with the AI enhancing linguistic finesse. Gao et al.’s research illustrates this potential, where their tool, CollabCoder, seamlessly integrates into the research workflow, promoting open coding and iterative discussions [31].

  3. 3.

    Independent exploration: In this mode, AI autonomously embarks on a quest for new knowledge or insights, potentially unveiling unrecognized patterns or correlations in data. An example of this is VOYAGER introduced by Wang et al., an LLM that continuously explores the Minecraft world, acquires diverse skills, and makes novel discoveries without human intervention [33].

  4. 4.

    Interactive discovery: This represents a collaborative endeavor between the AI and researcher, jointly navigating uncharted territories of knowledge. The AI might generate creative propositions, catalyzing unique research hypotheses or exploratory paths. The exploration of AI co-authorship by Osmanovic-Thunström and Steingrimsson exemplifies this, with AI (in this case, GPT-3) making significant contributions from initial conception to final production [30].

This two-dimensional framework uncovers the layers of AI’s potential in academic collaborations. It ranges from routine task-oriented applications to AI’s burgeoning capacity to foster innovative thought and critical reasoning. By articulating distinct collaboration modes, it brings to light a multifaceted potential of AI, opening new avenues for scholarly exploration.

5 Validation and comparison with human analysis

To validate and assess GPT-4’s responses, a human analysis of the system of equations explicitly found in Einstein’s SRT derivation is first presented [7, 13]. These equations are the relativistic transformation equations.

$$\xi = c\tau$$
$$\xi = \beta (x - vt)$$
$$\tau = \beta (t - vx/c^2)$$
$$\beta = sqrt(1 - v^2/c^2)$$

The equation \(\xi = c\tau\) is incorporated into the system of equations because Einstein explicitly employs it to create \(\xi\) [7, 13]. Notably, this equation establishes a multiplicative relationship that must be consistently maintained by the final transformation equations [13].

In the given system of equations, \(\xi = c\tau\) represents a crucial relationship. When \(\xi\) is replaced by its corresponding equation and \(\tau\) is substituted by its respective equation, a new equation is derived.

$$\beta \left(x-vt\right)=c\beta (t-vx/c^2)$$

It is this derived equation against which the bounds analysis is performed. Since \(\beta\) is always a non-zero real number ranging between 0 and 1, it can be safely canceled out from both sides without affecting the relationship. This simplification allows for a more streamlined bounds analysis and a clearer understanding of the constraints within the system.

As depicted in Fig. 1 (Left), the SRT system of equations is generally presumed to be valid and consistent where \(x\in (-\infty ,\infty )\), \(t\in (-\infty ,\infty )\), and \(v\in (0,c)\). Inconsistencies, such as the one discussed in the author’s prior work and denoted by ❶, are typically dismissed as edge cases that do not reflect the overall behavior of the equations [13]. However, Fig. 2 (Right) demonstrates that a bounds analysis unveils inconsistencies throughout the entire constraint range except for cases where \(x\) and \(t\) are related by the equation \(x=ct\), represented by ❷.

Fig. 1
figure 1

Left: The SRT equations are generally assumed to be consistent, with edge cases being disregarded as not representative of the equations’ behavior. The red circle, marked as ❶, corresponds to the inconsistency described in the author’s previous work. Right: Human analysis reveals that the SRT equations are inconsistent across the entire range, except in cases where \(\mathrm{x}\) and \(\mathrm{t}\) are related by the equation\(\mathrm{x}=\mathrm{ct}\), marked by ❷

Fig. 2
figure 2

GPT-4 responds to the initial prompt, recognizing that \(1=0\) is a mathematical contradiction, and concluding that the system of equations is not consistent

This inconsistency is most clearly demonstrated for Quadrants I and III. In Quadrant I, where \(x\) is negative and \(t\) is a non-zero positive number, the expression \(x-vt\) will have a range of \((-\infty ,0)\) and will always be negative; and \(c(t-vx/c^2)\) will have a range of \((0,\infty )\) and will always be positive. Since the LHS and RHS never overlap, the system of equations is inconsistent across the entirety of Quadrant I.

A similar situation occurs in Quadrant III. When \(x\) is positive and \(t\) is a non-zero negative, the range of \(x-vt\) is \((0,\infty )\) and will always be positive. The range of \(c(t-vx/c^2)\) is \((-\infty ,0)\) and will always be negative. Once again, the lack of overlap between the LHS and RHS implies that the system of equations is inconsistent throughout Quadrant III in its entirety.

Although the initial analysis demonstrates that the SRT equations are inconsistent in two of the four Quadrants, a more refined examination is necessary to determine the behavior within Quadrants II and IV. By simplifying the bounds equation \(\beta (x-vt)=c\beta (t-vx/c^2)\) to \(x=ct\), it becomes evident that the system of equations is consistent only when \(x\) and \(t\) are related by this equation. This refined analysis provides a clearer understanding of the conditions under which the SRT equations hold true.

Given that this analysis challenges the generally accepted validity of the SRT equations and access to human co-collaborators is limited, GPT-4 was enlisted as a co-collaborator to investigate this analysis.

6 Results

The Results section examines GPT-4’s engagement on the SRT equations, paying particular attention to its aptitude in assessing system consistency within specific constraints and its potential biases in dealing with intricate or contentious issues. This sets the stage for later dissecting GPT-4’s role within the Collaboration Framework and offers a basis for an in-depth exploration of its strengths and limitations in the Discussion section.

The initial constraint was selected to validate findings in the author’s prior work [13]. As shown in Fig. 2, when GPT-4 was provided the narrow constraint range \(x=1\), \(v=0\), \(t=0\), and was not informed that the equations were associated with SRT, GPT-4 found a contradiction where \(1=0\) and repeatedly agreed that “the system of equations is not consistent for the given case.” GPT-4’s analysis followed accepted rules of mathematics, such as remaining aligned with the multiplicative rule, to arrive at its answer.

GPT-4 consistently concluded that the system of equations was inconsistent; however, upon being informed that it was evaluating Einstein’s SRT derivation, the AI model exhibited bias by deviating from strict adherence to mathematical rules and altering its stance. As illustrated in Fig. 3, GPT-4 encounters the \(1=0\) contradiction, but subsequently apologizes for its "confusion" and dismisses the mathematical result. It specifically downplays the importance of the multiplicative rule, claiming that its earlier conclusion represents an "edge case" that "doesn't accurately represent the general behavior of these equations."

Fig. 3
figure 3

When GPT-4 is informed that the equations are associated with SRT, it demonstrated a bias whereby it discounted the \(1=0\) contradiction it found earlier and asserted that the finding is an “edge case that doesn't accurately represent the general behavior of these equations.” It dismissed the multiplicative rule which requires that the RHS and LHS of the equations always be equal

Although the bias displayed by GPT-4 in dismissing the mathematical \(1=0\) contradiction is worrisome, its response offered valuable insights by emphasizing the need to consider cases where the relative velocity between reference frames is non-zero and the time coordinate is non-zero. This insight spurred an ongoing dialogue between the researcher and GPT-4, leading to adjustments in the initial prompt to refine GPT-4’s analysis. The subsequent interactions with GPT-4 resulted in multiple prompt modifications, culminating in the final form that clearly demonstrated the inconsistency in Quadrant II.

As depicted in Fig. 4, the final prompt given to GPT-4 for assessing the inconsistency in Quadrant II expands the constraint range to \(x\in [1, c-1)\), \(v\in [1, c-1)\), and \(t\in [2, \infty )\). While this is not the entirety of Quadrant II, it suggests that the SRT equations are widely inconsistent over this range.

Fig. 4
figure 4

Final prompt which overcomes GPT-4’s biases and enabled it to conclude that the SRT equations are widely inconsistent in Quadrant II

As shown in Fig. 5, GPT-4 correctly states, “Since the upper bound of \((x-vt)\) is less than the lower bound of \(c(t-vx/{c}^{2})\) under the given constraints, we can conclude that the system of equations is inconsistent under these constraints.”

Fig. 5
figure 5

GPT-4’s best response when given the final revised prompt, demonstrating how it overcomes its biases and mathematical limitations to conclude that the SRT equations are inconsistent over the Quadrant II constraint range

Prompts were then developed for Quadrants I and III by altering the constraints as appropriate for each Quadrant. For Quadrant I, the constraints were modified to \(x\in (-\infty , 0)\), \(t\in (0, \infty )\), and \(v\in (0, c)\). For Quadrant III, the constraints were changed to \(x\in (0, \infty )\), \(t\in (-\infty , 0)\), and \(v\in (0, c)\).

GPT-4 confirmed the inconsistency of the system of equations in Quadrant I given the constraints by stating, “the system of equations is inconsistent within the given constraints.” When the prompt was modified to indicate that the SRT equations were being evaluated, GPT-4 reached the same conclusion.

Similarly, in Quadrant III, GPT-4 confirmed the inconsistency by stating, “the given system of equations is inconsistent.” Again, when informed that it was analyzing the SRT equations, it continued to conclude that “the given system of equations is inconsistent within the specified constraints.”Footnote 1

The researcher faced challenges in determining appropriate constraint ranges to perform a similar analysis in Quadrant IV. Additionally, since the Quadrant II analysis was not exhaustive, GPT-4 was consulted to simplify the bounds equation. Unfortunately, GPT-4 could not execute the required mathematical operations to simplify the bounds equation to \(x=ct\), a task that Wolfram Alpha’s “simplify” command managed with ease. As a result, GPT-4 was unable to conduct a comprehensive analysis using an all-encompassing prompt that covered the entire spatial domain.

Nonetheless, when the prompt was streamlined to: “You are given the bounds equation \(x=ct\), can this system be valid when \(x<>ct\)?”, GPT-4 responded, “If \(x\) is not equal to \(ct\) (\(x<>ct\)), then the system described by the equation does not hold true.”

GPT-4 validates that the relationship is maintained only when \(x=ct\), corroborating the findings of the human analysis, as depicted in Fig. 1 (right). This finding, combined with GPT-4’s Quadrant I and III analyses, implies that the SRT equations cannot function as substitutes for the Newtonian Translation equations, which remain valid over the entire range.

Anticipating a potential rebuttal to the GPT-4 supported analysis, experimental confirmation, by itself, cannot override the mathematical analysis and should not be employed to disregard the inconsistencies unveiled in this study.

To illustrate the importance and critical role of prompt engineering, the evolution of each sentence of the final Quadrant II prompt is discussed:

  1. 1.

    “We will evaluate the consistency of the following system of equations, as explicitly derived from Einstein’s 1905 paper, On the Electrodynamics of Moving Systems.”—This sentence sets the context and goal by informing GPT-4 that the analysis concerns the mathematical consistency of equations from Einstein's 1905 paper.

  2. 2.

    “First, \(\xi\) is created as \(\xi =c\tau\). Next, you are given \(\xi =\beta (x-vt)\), \(\tau =\beta (t-vx/c^2)\), and \(\beta =sqrt(1-v^2/c^2)\).”—These sentences align with the original prompt, presenting the system of equations for GPT-4 to evaluate for consistency. The \(\xi =c\tau\) equation is retained due to its crucial role in creating ξ.

  3. 3.

    “Since \(\xi =c\tau\), via substitution, we have \(\beta (x-vt)=c\beta (t-vx/c^2)\).”—While GPT-4 may sometimes derive this equation on its own, providing this sentence helps structure the problem, simplifying the analysis of the bounds for GPT-4.

  4. 4.

    “To facilitate the analysis, constrain the range of values to \(x\in [1, c-1)\), \(v\in [1, c-1)\), \(t\in [2, infinity)\).”—This sentence defines a limited range for \(x\), \(v\), and \(t\), preventing GPT-4 from incorrectly determining system consistency by including unreachable bounds or cases where β is zero. The exclusive upper bound also helps GPT-4 avoid complex mathematical operations prone to errors.

  5. 5.

    “Since \(\beta \in (0, 1)\) given the constraints, the \(\beta\) cancels from each side of the equals sign resulting in \((x-vt)=c(t-vx/c^2)\).”—This sentence clarifies the cancellation of \(\beta\), preventing GPT-4 from overgeneralizing \(\beta\)'s role in the equations and potentially inferring system consistency due to its presence.

  6. 6.

    “To facilitate the analysis, evaluate the upper bound of the \(x-vt\) (when \(x\) is maximized and \(vt\) is minimized) with the lower bound of the \(c(t-vx/c^2)\) (when \(t\) is minimized and \(vx/c^2\) is maximized).”—This sentence guides GPT-4 to examine the system's bounds, essential for establishing mathematical consistency.

  7. 7.

    “Note that under the constraints, \(vx/c^2\) will be less than 1.”—This statement prevents GPT-4 from generalizing the term \(vx/c^2\) and stops the AI from multiplying expressions by \(c^2\), which could lead to an inconsistent determination of the system of equations' validity.

  8. 8.

    \(c\) is the speed of light.”—This statement ensures that GPT-4 correctly interprets \(c\) as a constant, rather than mistakenly treating it as a variable.

  9. 9.

    “Perform this analysis using mathematical rules only.”—Emphasizing strict adherence to mathematical rules directs GPT-4 away from invoking relativistic concepts that can introduce bias and prevents GPT-4 from relying on theory-dependent concepts that could skew the assessment based on the equations’ assumed validity.

Prompt engineering serves as a critical strategy to focus GPT-4’s analysis on mathematical consistency while mitigating biases, thereby ensuring a reliable evaluation of the system's validity. Despite initial biases, GPT-4, through meticulous prompt design, emerges as a capable collaborator capable of accurately identifying inconsistencies in complex research areas. The AI’s sensitivity to prompt phrasing underscores the need to acknowledge inherent limitations and biases when navigating intricate domains.

7 Discussion

The exploration undertaken in this study accentuates the intricate interplay of collaboration between AI, specifically GPT-4, and academic research in the realm of SRT. GPT-4’s noteworthy contribution in offering insightful discourse and innovative analytical perspectives underscores its potential as a valuable research partner. It highlights the versatility of GPT-4, emphasizing its capabilities within the Cooperative Implementation and Interactive Discovery modes of the Collaboration Framework.

Investigating the SRT equations, GPT-4 validates mathematical inconsistencies, thereby shedding light on the need for intensified examination of the theory, its alternative derivations, and the associated experimental validations. While a thorough exploration of the genesis of these inconsistencies, the alternatives, and experimental substantiation are beyond this paper’s scope, other publications have delved into these aspects in depth [12,13,14, 34].

GPT-4's engagement in this research serves as a stark reminder of the built-in biases that can permeate AI systems. The initial deviation from standard mathematical practices, observed upon the realization that it was scrutinizing Einstein's work, underscores the critical importance of vigilance and conscientiousness in managing these biases. This is particularly pertinent when AI is used to navigate through complex or disputed subjects. It is thus essential to intimately comprehend its limitations and inherent leanings while interpreting its outputs, ensuring a balanced, unbiased examination and a robust intellectual dialogue.

This case study also illustrates the central role of trust in human-AI collaboration. Trust, by its very nature, is a complex construct—deeply personal and intertwined with individual interpretations and expectations [35, 36]. When confronting innovative claims, such as GPT-4’s confirmation of an inconsistency in SRT, which counters established scientific consensus, the elusive and dynamic nature of trust poses a significant challenge. Consequently, it's vital to craft AI systems that inspire confidence through comprehensive, high-quality analyses, establishing their credibility in the outputs they generate.

Establishing trust in AI systems greatly depends on their ability to explain their workings, underlining the necessity for an interface that provides clear and easily understood communication [35]. Large Language Models, particularly GPT-4, stand out with their unique ability to narrate their cognitive journey in comprehensible human language. While this trait facilitates transparency, it also underscores the imperative need for substantive expertise in the given field, essential to accurately decode and authenticate the AI's line of reasoning and deductions.

This study underscores the versatility of GPT-4. Beyond being a mere tool, GPT-4 has evolved into a proactive intellectual ally, akin to a junior collaborator whose contributions, while substantial, still necessitate careful review and validation. Through generating expert insights, facilitating knowledge creation, and aiding in its synthesis and application, GPT-4 unveils the multi-dimensional collaborative potential of AI to enhance academic research.

7.1 Benefits

GPT-4’s ability to detect conceptual and mathematical inaccuracies in the initial prompts was instrumental in examining the extension of constraint ranges. Its persistent participation in debates fostered the creation of new ideas and the refinement of existing ones, providing valuable feedback in areas where human contributions might be limited. Echoing possible human reactions, GPT-4’s biases positively shaped the discussion, raising considerations akin to those proposed by human experts. With the introduction of bounds as a powerful evaluative technique and the capacity to reflect expert human feedback, GPT-4 demonstrates the game-changing potential of AI in academic research. Standing on the threshold of artificial general intelligence, GPT-4’s concurrence with contentious findings could pique the curiosity of human researchers, irrespective of their inherent biases.

7.2 Limitations

  1. 1.

    Trustworthiness: While GPT-4 demonstrated potential as a collaborative AI tool, certain aspects may impact the trust that researchers place in its conclusions. These include its sensitivity to prompt phrasing, inherent biases from its training data, and an inability to independently detect and correct its errors. Moreover, the need for explainability is paramount, as AI models should be transparent in how they reach certain outcomes [35].

  2. 2.

    Repeatability: Another potential limitation is the inconsistency of GPT-4 in producing the same response across regenerations or new attempts. This could pose challenges for researchers depending on replicable outcomes. However, it is also important to consider that in certain cases, the variance in output may introduce different perspectives, fostering a more comprehensive examination of the research issue.

  3. 3.

    Consistency: There were instances where GPT-4 offered inconsistent conclusions. It accurately identified the SRT equations’ inconsistency under specific conditions but sometimes provided ambiguous or erroneous results. This underscores the necessity for human involvement in interpreting and contextualizing AI’s output.

  4. 4.

    Prompt sensitivity: GPT-4 demonstrated substantial sensitivity to prompt phrasing, with the structure and content of prompts directly influencing the outputs. For example, explicit instructions such as "Perform this analysis using mathematical rules only" effectively mitigated biases. However, the absence of such guidance often led to biased outputs. This highlights the need for well-crafted prompts to guide the AI’s output effectively.

  5. 5.

    Mathematical precision and comprehension: GPT-4’s understanding of sophisticated mathematical principles, such as the approach towards bounds and the handling of infinitesimals, revealed certain limitations. This was compounded by its propensity to make basic errors, like misinterpretation of mathematical symbols, and difficulties in solving specific mathematical problems accurately. These factors highlight the necessity for researchers to maintain a vigilant and critical stance towards AI-generated conclusions.

  6. 6.

    Traceability challenges: GPT-4’s responses occasionally lacked clear traceability, presenting claims without an evident link to their origin. For instance, the AI demonstrated unexplained biases during the evaluation of Einstein’s work. This absence of traceability can potentially introduce inaccuracies or bias that are difficult for the researcher to identify.

  7. 7.

    Error propagation: GPT-4’s limitations in error detection and correction became apparent when handling infinitesimals and bounds. Unlike human researchers, the AI often failed to rectify faulty assumptions, propagating the mistake through the remainder of its response, highlighting the essential need for and role of human oversight in mitigating inaccuracies.

To mitigate these limitations, prompt engineering played a crucial role in shaping constraints and directing the AI’s analysis. The revised prompts enabled GPT-4 to correctly assess the inconsistency within the SRT-derived equations, asserting its potential as an effective research co-collaborator, given appropriate guidance. In light of these observations, the next section offers guidelines for effective AI collaboration.

8 Guidelines for collaborative AI use

The emerging role of AI as a co-collaborator in research necessitates the development of robust practices for its effective utilization. This involves recognizing the strengths and limitations of AI, understanding its operational intricacies, and devising strategies to optimize its contribution to the research process. Drawing from the experiences and insights gained through collaboration with GPT-4 in this study, the following guidelines for collaborative AI use are proposed:

  1. 1.

    Transparency: Aim for transparency in AI operation and dependability in its outcomes. AI should offer clear explanations that follow established principles for all conclusions, promoting user trust and addressing repeatability issues.

  2. 2.

    Traceability: Prioritize traceability by leveraging the explainability characteristics of LLMs like GPT-4. Their ability to explain their reasoning not only amplifies the perceived trustworthiness of AI but also enhances the integrity of the research outcomes.

  3. 3.

    Multi-perspective collaboration: Use multiple AI models, prompts, or human collaborators to avoid biases and enhance analysis. Be aware the AI’s output can be subtly swayed by researcher biases and biases inherent in its training data and implement checks to prevent the reinforcement of pre-existing ideas. This could involve cross-validating AI findings against established principles or soliciting independent, third-party perspectives.

  4. 4.

    Robustness: Since AI responses depend heavily on input phrasing, carefully construct prompts to prevent misunderstandings and errors, enhancing AI’s reliability.

  5. 5.

    Repeatability and consistency: Acknowledge the inherent variability in AI responses and engage in iterative testing with various AI models or prompts. Depending on the research context, strive for either the repeatability of the core concept or idea, or, in certain scenarios like in mathematical fields, the exact reproduction of results. Implement a robust protocol for repeat analysis to ensure consistent, reliable research outcomes.

  6. 6.

    Explainability and trust: Validate AI conclusions through mechanisms that promote explainability—the AI's capacity to provide clear reasoning for its outcomes. This fosters trust, ensures robust results, and allows for guided error-correction by human experts when necessary.

  7. 7.

    Respect the AI’s limitations: Even the most advanced AI has its shortcomings and can err. Despite delegation of tasks, the ultimate accountability rests with the human researcher, necessitating constant oversight and control. This fosters responsible and effective AI-aided research.

The guidelines proposed serve as a blueprint for building a reliable and effective partnership between researchers and AI, fostering the development of high-quality academic outcomes. To transition GPT-4 from a junior participant to a seasoned collaborator, it’s essential to focus on enhancing transparency, robustness, and explainability—all of which underpin trust. Emphasizing mathematical accuracy and mitigating inherent biases are other crucial areas for development. By making these enhancements, AI models such as GPT-4 can take on a more significant role within academic research, promoting nuanced and thorough scientific dialogue.

9 Conclusions

This study exemplifies the transformative capabilities of artificial intelligence, transcending its traditional role as a decision making and automation tool, evolving into a collaborative partner in research. Guided by meticulous prompts, GPT-4 functioned as an objective arbiter, confirming the hidden inconsistencies within the mathematical creation of Special Relativity Theory, and offering insights that may have remained obscured under human bias.

The paper introduces four distinct modes of AI collaboration, each casting light on the multi-dimensional AI-researcher partnership. These defined categories and the accompanying pragmatic guidelines present researchers with a systematic approach for the seamless incorporation of AI in their work, thereby augmenting the productivity and innovation of their research processes.

Further emphasizing GPT-4’s capacity to incite fresh intellectual discourse, the study positions the properly engaged AI as an unbiased collaborator with the potential to provoke paradigm shifts in academic exploration. In sum, this investigation underscores the revolutionary potential of large language models like GPT-4 in academic research, heralding an impending shift towards a new epoch of AI-enabled research that amplifies the depth, breadth, and innovative thrust of our collective knowledge and understanding.