The Adaptation of Laws in Physics: Adaptation of Mathematical Models in Physics

1. Introduction

In science, the laws of mathematics coincide with the laws of physics. When these mathematical laws are applied to physics, they often require adaptations to align with the complex realities of the universe, to accurately reflect the complexities of physical phenomena.

This paper is part of the broader research on "Adapting Laws in Science" and aims to examine how mathematical laws, despite being powerful and valid, can become imprecise or irrelevant without the necessary adaptations. We will explore two key examples, Euler's exponential laws and the concept of space-time dilation derived from Lorentz transformations. Through these examples, the paper will demonstrate how and why these laws require adjustments when used to model complex physical phenomena.

The paper will also argue that despite the broad applicability of mathematics, it is essential to maintain it as a distinct science, separate from physics, to avoid misinterpretations. Mathematics must remain a pure and abstract concept to serve as a stable foundation upon which new theories and models can be developed. In this context, we will emphasize the need to reevaluate how mathematics and physics interact in scientific research to ensure a better understanding of the realities of the universe and to prevent errors that may arise from the improper application of mathematical laws.

2. Methods

This paper employs a theoretical analytical approach to evaluate the applicability of mathematical concepts within the context of physics. The analysis focuses on two key examples: Euler's exponential laws and spacetime dilation according to Lorentz transformations.

• Conceptual Analysis: We examined how mathematical laws are applied in physics, identifying their challenges and limitations. This involved a thorough review of existing literature and integration of previous studies to better understand the context and implications of these concepts. The analysis was conducted without the use of complex mathematical language or formulas, focusing instead on theoretical and conceptual insights.
• Data Sources: Relevant scientific literature, research articles, and academic resources were utilized to gather necessary information and data for the analysis. The sources provided a comprehensive overview of how these mathematical concepts are adapted and applied in various physical contexts.
• Analytical Tools: The analysis was carried out through theoretical examination and comparison techniques. Differences and similarities between the mathematical models and their physical applications were assessed qualitatively, rather than through quantitative mathematical analysis.

This methodology allows for a critical evaluation of the adaptability of mathematical concepts in physics and offers insights into improving their application based on theoretical understanding.

3. Example Analysis

In this section, we will detail the two main examples used to illustrate how mathematical laws have been applied and adapted in physics: Euler's exponential laws and the concept of spacetime dilation derived from Lorentz transformations.

3.1. Euler's Exponential Laws as Mathematical Models Adapted in Physics

3.1.1. The Initial Mathematical Concept

Euler's exponential laws are foundational in mathematics, particularly in the study of complex functions and the modeling of exponential growth. Initially, these laws were developed to describe financial systems, where they provided a mathematical framework for understanding processes such as compound interest and investment growth.

$$\mathrm{A}=\mathrm{P}\text{×}1+r^t$$

Where:

A is the final amount,

P is the initial principal,

r is the interest rate per period (e.g., per year),

t is the number of periods (years).

3.1.2. Adapting the Mathematical Model to Physical Reality

Euler's exponential laws have been adapted to model various phenomena, such as radioactive decay, and in other exponential growth in physics.

Radioactive Decay:

For radioactive decay, the number of undecayed nuclei N(t) at a time t is given by:

N(t)=N0×e−λt

Where:

N0 is the initial quantity of the substance.

λ is the decay constant, which is specific to the substance.

t is the time elapsed.

This adaptation of Euler's laws in physics demonstrates both the flexibility of mathematical models and the critical need to adjust theoretical laws to fit the nuanced realities of physical phenomena. Despite their robustness, mathematical laws must be refined and adapted to account for the inherentvariability and complexity of nature.

3.2. Lorentz Transformations and the Theories of Relativity as Mathematical Models Adapted in Physics

3.2.1. The Initial Mathematical Concept

In the late 19th century, physicists faced a significant problem: how to explain that the speed of light remains constant in all inertial reference frames, as demonstrated by the Michelson-Morley experiment. According to the classical physics of that time, which was based on Galilean transformations, the speed of light should vary depending on the observer's motion relative to the hypothetical ether, a medium through which light was thought to propagate.

Lorentz initially developed a mathematically derived concept to maintain the constancy of the speed of light by adjusting time and space according to motion. Lorentz hypothesized that if time for an observer in motion were slower than for a stationary observer, then the speed of light could remain constant across all reference frames. Essentially, this means that time dilates for the moving observer. Therefore, t′ (dilated time) is greater than ttt (proper time), so t′>t

3.2.2. Adapting the Mathematical Model to Physical Reality

Lorentz adapted the mathematical concept to match physical observations by developing and introducing the Lorentz factor, γ. Through several adjustments to the formulas, he arrived at the simplified time dilation formula:

tΔt′=γΔt

where:

Δt′ is the time interval measured by the moving observer.

Δt is the proper time interval (measured by the stationary observer).

γ is the Lorentz factor.

In addition to time dilation, Lorentz also predicted length contraction, which states that the length of an object moving relative to an observer will appear shorter along the direction of its motion. This is expressed by the formula:

$$\mathrm{L}\text{’}={\displaystyle \frac{L}{\gamma }}$$

where:

• L′ is the length measured by the moving observer.
• L is the proper length measured by an observer at rest relative to the object.

3.2.3. Integrating Lorentz's Principles into Albert Einstein’s Theory of Special Relativity

This adjustment ensured the consistent application of Lorentz transformations in physical theories, particularly in relativity, by correctly predicting time dilation effects. Lorentz’s work laid the foundation for Albert Einstein’s theory of special relativity, published in 1905. Einstein extended Lorentz’s ideas, showing that the Lorentz transformations are not just mathematical tools but fundamental to our understanding of space and time.

Einstein’s theory introduced the concept of space-time and unified space and time into a single framework. This allowed for a new understanding where the speed of light is constant across all inertial frames. The time dilation formula, simplified by Einstein, is given by:​

$$\mathsf{\Delta }\mathrm{t}\text{′}={\displaystyle \frac{\Delta t}{\gamma }}$$

where Δt′ is the time measured by the moving observer, Δt is the proper time, and γ is the Lorentz factor. Thus, Einstein’s special relativity provided a comprehensive explanation of how space and time are interconnected.

3.3. Conclusions from the Example Analysis

The analysis of the two examples; Euler's exponential laws and space-time dilation highlights a critical observation: although mathematics is a separate science, mathematical models are valid for physics, but with adaptations and readaptions for different aspects of physics.

Without these adaptations, there is a significant risk of oversimplifying the physical phenomena or misinterpreting the results, leading to inaccuracies in scientific conclusions. The necessity of adapting mathematical laws to fit the physical context underscores the importance of a deep understanding of both the mathematical framework and the physical realities it aims to describe.

In short, strong correspondence between mathematical laws and physical principles is often achievable, but only through a process of adaptation for each aspect of physics. Therefore, these laws specific to an aspect of physics cannot be taken as general for the deeper study of the universe.

4. Results and Discussions

4.1.Analysis of Results

In our analysis, we examined two essential examples: Euler's exponential laws and spacetime dilation derived from Lorentz transformations. The results obtained show that these laws, although initially developed within a purely theoretical mathematical framework, were successfully adapted to describe complex physical phenomena. Specifically, Euler's laws and the concept of space-time dilation have demonstrated significant applicability in physics, but they required adjustments to align with the realities of the universe. These adjustments allowed the mathematical models to be effectively used in the context of physics, providing an accurate description of natural behavior. At the same time, the laws that have become complex refer only to the depths of physics for which they were adapted and developed, and it is unfair to generalize with them.

4.2. Discussion of Results

The results of this study suggest that despite the power and versatility of mathematics, it is essential to recognize and manage its limitations when applied in physics. It is important for future research to focus on developing more precise and flexible methods for adapting mathematical concepts to physical realities. One solution could be to preserve the initial forms of the laws and re-adapt them to new research, avoiding the use of adapted formulas from certain environments in the study of environments with insufficient information for adaptation.

The adaptation of laws must be modified as long as they delve deeper into the same aspect of nature. For example, the exponential growth model in banking systems was initially straightforward, making banking operations simpler, but was later adjusted for more complex processes.

However, re-adaptation from the original concept occurs when the model finds its place in a completely different context, even though it manifests through similar laws. For instance, applying Euler's law of exponential growth to the reproduction of animals should not involve the adaptations made for other environments, as they are not relevant. The necessity for re-adaptation is underscored by historical examples, such as the uncontrolled proliferation of rabbits in Australia. Initially, mathematical models might have predicted stable growth patterns, but the real-world application revealed that natural systems can change unexpectedly and dramatically. This example illustrates that while mathematical laws can provide valuable insights, their application to natural systems requires careful consideration and adaptation, as the natural world is subject to complex and often unpredictable changes.

4.2.1. Validation of Relativity Theories as Initially Well-Functioning Mathematical Models

Also, the study showed why relativity theories are correct despite the existence of many critics, being as in the case of Euler's theory of exponential growth, a mathematical origin that coincided with the respective laws of nature. Both examples in this work are mathematical concepts, which do not give errors, but adaptations for physics are still being researched.

5. Conclusions

5.1. Summary of Main Findings

This work explored the adaptation of mathematical laws, such as Euler's laws and the concepts of space-time dilation, in the field of physics. We demonstrated that these mathematical laws, although initially developed in a purely theoretical context, have almost entirely coincided and were successfully adjusted and applied to describe complex physical phenomena. The adaptation of these concepts in physics by scientists not only allowed for a deeper understanding of relativistic phenomena and exponential growth but also contributed to the development of new methods of analysis and modeling. It was emphasized that mathematics is a universal science for all aspects of the universe, including potential other universes, but the laws of physics are specific to a particular aspect and require adaptation for others.

5.2. Implications of the Results

The results obtained have important implications for future research in physics and mathematics. They suggest that, to achieve success in new and deeper research in any field, it is necessary to avoid using heavily adapted laws as a basis and instead use the initial concepts, which should later be re-adapted.

5.3. Recommendations for Future Research

For future research, it is recommended to explore mathematical laws from their initial, simplified stages and then re-adapt them to new situations. This approach involves revisiting the original formulations of mathematical concepts as proposed by their authors, avoiding over-reliance on complex, adapted laws that may have evolved through numerous modifications. By starting with the fundamental forms of these laws, researchers can better understand their base principles and how they can be tailored to specific physical contexts.

In cases where physical phenomena are not fully observable or detectable, such as in the study of subatomic structures, intuitive and theoretical approaches may be employed. However, it is crucial to acknowledge that such methods carry a lower probability of accuracy compared to adaptations based on well-documented physical observations. Therefore, the research should be conducted with a clear indication of the limitations and uncertainties involved when working with less detectable aspects of nature.

5.4. Final Conclusion

In conclusion, for the continued advancement of science and to address the challenges faced in contemporary physics, it is imperative to revisit and refine the foundational mathematical concepts. Rather than relying on extensively adapted laws, researchers should focus on the original formulations and their fundamental principles. These initial concepts provide a solid basis for understanding, and their re-adaptation should be carefully aligned with empirical observations and physical contexts.

Revisiting foundational principles allows for a more precise application of mathematical laws to real-world phenomena, reducing the risk of inaccuracies introduced by excessive modifications. This approach helps ensure that mathematical models retain their relevance and accuracy in describing complex physical systems.

Additionally, acknowledging the limitations of current models is crucial. In cases where direct observation is difficult, such as with subatomic structures, theoretical and intuitive approaches can be valuable, though they carry inherent uncertainties. It is important to approach such methods with caution and recognize their limitations.

Ultimately, maintaining a balance between the theoretical purity of mathematical laws and their practical adaptation to physical realities is essential for making meaningful scientific progress and overcoming the current challenges in physics.