From Everyday Phenomena to Physical Understanding: An Exploratory Essay on Electromagnetism and Special Relativity

Author: Guorui Li

Introduction

As an undergraduate majoring in Internet of Things Engineering, I serve as the person in charge of the Fundamental Physics Division on campus, assisting teaching for university general physics courses for lower-grade students, and also work as a part-time physics tutor for middle school students during vacations. Throughout the history of physics, numerous classic theories originated from careful observation of ordinary real-life phenomena: Newton explored the nature of universal gravitation inspired by falling objects, while Einstein established the spacetime framework of special relativity based on observable scenes involving trains, lightning and pocket watches. The research paths of these pioneering physicists fully confirm a core pedagogical idea adopted in this article: construct intuitive physical images from perceptible daily phenomena, then carry out mathematical derivation and theoretical analysis step by step.

Physical knowledge can be expressed through three interrelated symbolic systems that form an integrated cognitive framework. Verbal language qualitatively interprets phenomena and logical relations; intuitive mental images establish visualized thinking models; mathematical formulas realize quantitative characterization of physical laws. The three systems are complementary to each other. If learned separately, students tend to memorize formulas mechanically and fail to apply them under new scenarios. This cognitive framework runs through the whole article. Focusing on two closely connected subjects, electromagnetism and special relativity, the article demonstrates a unified learning workflow: verbal description of phenomena → construction of intuitive mental images → quantitative derivation via mathematical formulas.

Years ago, observing inconsistent readings between a wristwatch and a wall clock triggered thought about the concept of simultaneity in relativity, and gradually formed an intuitive understanding of four-dimensional unified spacetime. Meanwhile, wireless communication, which transmits signals over long distances without physical cables, provides an intuitive real carrier for electromagnetic fields and wave laws. These two daily examples jointly implement the integrated learning logic of the three physical symbolic systems.

This article only takes clock observation and wireless communication as core real-life entry points. It first establishes basic intuitive cognition of flat spacetime and electromagnetism, supplements core quantitative formulas of electromagnetism, then completes standard pedagogical derivation of the Lorentz transformations by adopting the mathematical form of wave equations. It also distinguishes the classical luminiferous aether hypothesis from the mathematical analogy adopted herein, and defines the applicable scope of such analogical methods. The whole text solely centers on electromagnetism and special relativity, without expanding on thermodynamics and geometric optics, aiming to provide a progressive reference for physics classroom teaching and independent learning.

I. Two Core Daily Scenarios: Constructing Basic Physical Images of Electromagnetism and Spacetime

1.1 Mechanics and Kinematics: Auxiliary Perceptual Examples

Falling raindrops and flowing clouds are common cases in classical mechanics, which can be used to explain reference frames and variable motion laws. The phenomenon that lightning is seen far earlier than thunder intuitively reflects that the speed of light greatly exceeds the speed of sound, serving as a simple real-life material for wave propagation rules. This article does not systematically elaborate the complete system of classical mechanics, only treating it as auxiliary perceptual material. The main focus here is on physical images of electromagnetic fields and flat spacetime.

1.2 Electromagnetism: Invisible Electromagnetic Fields and Wireless Communication

Signal receivers at home can stably send and receive information with remote base stations without physical wires, whose core principle lies in the excitation, propagation and reception of electromagnetic waves. A changing electric field excites a changing magnetic field, and a changing magnetic field in turn generates a new electric field. Electromagnetic waves propagate in vacuum and air without relying on physical material media.

Electromagnetic fields are typical physical entities that cannot be observed by naked eyes but exist objectively, which constitute a core difficulty in teaching electromagnetism. Real-life

examples including wireless communication and mobile radio signals help learners break the

inherent stereotype that interactions must rely on physical contact, establish standardized intuitive images of electromagnetic fields, and build a bridge connecting wave rules and spacetime theory.

Electromagnetic laws can be interpreted from the perspective of the three symbolic systems:

1. Verbal Language: Alternating electric fields and alternating magnetic fields couple and excite each other, generating electromagnetic waves capable of independent propagation in vacuum; wireless communication is the most intuitive real embodiment of this law.

2. Intuitive Mental Images: Form visualized mental pictures of outward-spreading, alternating electromagnetic field fluctuations.

3. Mathematical Formulas: The vacuum Maxwell differential equations quantitatively characterize electromagnetic coupling relations, whose simplest form is written as:

   
   

   

This article does not expand on complete solving processes or quantitative calculation of electromagnetic fields based on these equations. It only adopts the mature conclusion that electromagnetic wave equations can be directly derived from Maxwell equations as the mathematical foundation for subsequent covariance deduction. The equations naturally lead to electromagnetic wave equations, which act as the theoretical basis for the later Lorentz transformation derivation, realizing the unification of verbal description, mental images and mathematical formulas for electromagnetism.

1.3 Reflections on Spacetime: From Clock Observation to Special Relativity

Classical physics focuses on interaction laws between matter and fields, while spacetime serves as the fundamental background carrying all material motions and physical events. Spacetime itself is neither physical objects, particles, nor traditional material media.

My early observation of clocks helped me clearly classify three types of events: simultaneous and co-located events, simultaneous events at different positions, and non-simultaneous events at different positions. Classical cognition treats time and space as independent systems, while further study reveals they are inseparable and jointly form a four-dimensional unified background bearing all physical motions.

Historical physical models such as luminiferous aether assume a universe-wide background. This article only absorbs their characteristic of universal distribution for pure mathematical analogy. It merely uses wave equations as calculation tools to complete standard pedagogical derivation of Lorentz transformation under flat spacetime, without introducing any physical hypotheses about material media.

1.4 Overall Research Framework

The complete research flow of this article is listed as follows: observation of real-life phenomena (qualitative verbal description) → interpretation of fundamental electromagnetic wave laws, construction of intuitive electromagnetic field images, quantitative expression via Maxwell equations → recognition that all material motions and physical processes take four-dimensional spacetime as the unified carrying background → clarification that spacetime is not material media, only adopting wave equations for mathematical calculation → mathematical hypotheses and equation derivation → verification of equation covariance → discrimination of confusing concepts → summary of limitations of the adopted model.

The whole article follows the pedagogical logic: perceptual observation → image construction → theoretical derivation, and only covers two modules: electromagnetic waves and special relativity.

II. Distinction of Core Concepts: Classical Luminiferous Aether and the Mathematical Analogy in This Article

Before proceeding to relativistic deduction, it is necessary to clarify the essential differences between the classical luminiferous aether hypothesis and the mathematical tool adopted herein, which is a key cognitive transition for learners from classical physics to modern spacetime theories.

Classical physics put forward the luminiferous aether hypothesis, which mistakenly regards spacetime as a rigid, elastic material medium filling the whole universe, assumes light propagates through such mechanical substance, and presupposes an absolute rest reference frame. The null result of the Michelson-Morley experiment thoroughly falsifies this hypothesis. Its fundamental flaw lies in equating the non-material spacetime background with macroscopic material media such as water and air, which contradicts modern fundamental cognition of spacetime.

This article only uses the mathematical expressions of wave equations for calculation, without

any physical assumption that spacetime possesses fluctuating or mechanical properties. The core distinctions are sorted out as below:

1. Spacetime acts as a four-dimensional geometric background framework for the evolution of matter and events. It does not contain mechanical parameters including mass, density and elastic modulus, being neither matter nor wave-propagating media, which fundamentally differs from luminiferous aether.

2. Wave equations are merely mathematical computing tools in this article. Their forms are only used to complete covariance deduction, and do not represent real mechanical fluctuations or spatial fluctuations of spacetime itself.

3. The whole text strictly abides by the fundamental postulates of special relativity, avoids introducing any hypotheses of universe-wide material media, and only carries out analogical calculation on a mathematical level.

In short, the core error of luminiferous aether lies in treating spacetime as physical propagation media. This article only uses the mathematical form of wave equations for standard pedagogical derivation, without endowing spacetime with any medium fluctuating properties. Though they share superficial similarities, their physical connotations are completely different, which is critical for learners to break inherent misunderstandings.

III. Analogical Deduction based on the Mathematical Form of Wave Equations (Lorentz Transformation)

Based on the electromagnetic wave equations (Maxwell equations) above, this section continues to follow the three symbolic system logic: verbal description of spacetime covariance requirements, construction of reference frame transformation images, and quantitative formula calculation. This article does not trace the complete derivation of wave equations from Maxwell equations, directly adopting standard wave equations as computing carriers to focus on demonstrating the pedagogical connection between wave equation covariance and Lorentz transformation. All calculations in this section are standard pedagogical demonstrations for physics learners, aiming to display the inherent mathematical relation between wave equation covariance and Lorentz transformation, rather than original theoretical derivation.

3.1 Mathematical Form of Wave Equations

The standard one-dimensional wave equation reads:

In the formula, c stands for the propagation speed of disturbances, equal to the speed of light in vacuum.

Extended three-dimensional form for mathematical disturbances at arbitrary spatial positions:

3.2 Inertial Reference Frames and Linear Transformation

Two inertial reference frames S (x, y, z, t) and S' (x', y', z', t') are defined:

1. Frame S' moves uniformly along the positive x-axis of frame S with constant speed v;

2. The origins of the two coordinate systems coincide when t = 0 and t' = 0;

3. The frames are taken in standard configuration, with their spatial axes parallel and the relative motion along the x-axis; the transverse coordinates are therefore unchanged: y' = y, z' = z.

Assume general linear transformation forms:

where γ, a, b are undetermined coefficients.

3.3 Covariance Requirement and Coefficient Solving

Lorentz covariance - the requirement that the laws of physics retain the same form in all inertial frames - is a core feature of special relativity. Variable substitution is performed on partial differential operators:

Substitute the operators into the three-dimensional wave equation. Solve simultaneous equations according to covariance conditions that cross-term coefficients equal zero and derivative coefficients match, yielding:

and the Lorentz factor:

3.4 Final Form of Lorentz Transformation

Substitute solved coefficients into linear transformation formulas to obtain standard Lorentz

transformation:

To reemphasize, the calculations herein only reproduce mature classical conclusions of special relativity relying on wave equation covariance, merely serving as mathematical analogical tools for teaching. This article per does not propose any new spacetime theories or original physical derivation results.

IV. Limitations of the Adopted Research Model

The mathematical analogical calculations adopted in this article are intuitive and helpful for beginners to understand electromagnetism and relativity, yet applicable boundaries must be clearly defined to guarantee academic rigor.

First, strictly distinguish two types of physical images: mechanical waves and electromagnetic waves within material media all belong to the category of matter; spacetime acts as the geometric background carrying all matter and motions, which is neither matter nor medium and possesses no particle or mechanical properties. This article only uses the mathematical forms of wave equations, and the mathematical disturbances inside equations cannot be equated to objective mechanical fluctuations.

Second, there exists an essential distinction in reference frames: classical material media inevitably lead to the concept of absolute rest frame, which is completely inapplicable to spacetime background. Special relativity proposes all inertial frames are equivalent, fundamentally differing from motion laws of material media.

Third, the applicable scope is limited. This article only discusses pedagogical analogies under flat spacetime and inertial frames without gravitational effects, excluding complex frontier contents such as spacetime curvature, cosmic evolution and vacuum quantum effects. Related extended questions are only reserved as open thinking after class and not incorporated into the argument system of this article.

In general, thermal motions and electromagnetic fields belong to matter, while spacetime exists as an independent geometric background. Mature mathematical tools can be adopted to assist teaching understanding, but readers must always distinguish between mathematical analogical forms and objective physical essence to avoid conceptual confusion. If learners only memorize formulas while ignoring verbal definitions and intuitive mental images, they will easily obscure applicable boundaries of theories and generate cognitive misunderstandings.

V. Summary and Reflections

Years ago, reflections triggered by inconsistent clock readings laid the foundation of my cognition of simultaneity; ubiquitous wireless communication in daily life intuitively demonstrates the objective existence of electromagnetic fields and electromagnetic waves. The two cases respectively serve as the entry point of spacetime cognition and the carrier of wave physics, fully presenting a complete learning chain based on the three physical symbolic systems: describe daily phenomena and physical connotations via verbal language, build corresponding intuitive mental images of electromagnetism and spacetime in mind, and finally complete quantitative calculations using Maxwell equations, wave equations and Lorentz transformation.

This article does not expand other classical physics branches including mechanics, thermodynamics and geometric optics, with its core content only focusing on teaching methods of electromagnetic waves and special relativity. Physics learning should not rely on mechanical formula memorization and exercise drilling. Instead, learners should build visualized physical images based on surrounding intuitive phenomena, and integrate the three expression systems: verbal language, intuitive mental images and mathematical formulas. If separated, learners will easily confuse applicable conditions of theories and fail to distinguish mathematical analogical tools from objective physical essence. Though the classical luminiferous aether hypothesis and the pure mathematical calculation method adopted herein share superficial formal similarities, their physical connotations are totally different, which is a confusing point for novice learners.

This article only provides a pedagogical learning framework applicable to electromagnetism and relativity modules, completing standard pedagogical derivation of Lorentz transformation through wave equations. No new physical hypotheses or original theoretical derivations are put forward. It is hoped that this article can offer simple references for physics

classroom teaching and independent physics learning, helping more learners start from daily

phenomena and understand electromagnetism and spacetime theories in a standardized, rigorous way.

References

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[3] Zhao, K., and Luo, W. New Concept Physics: Mechanics. Beijing: Higher Education Press. ISBN 9787040152012.

[4] Liu, L., and Zhao, Z. General Relativity. Beijing: Higher Education Press. ISBN 9787040144307.

[5] Guo, S. Electrodynamics. Beijing: Higher Education Press. ISBN 9787040239249.