# Time-Dependent Retarded Microwave Electromagnetic Motors
## 1 Introduction  
The emergence of special relativity in the early 20 th century was a revolutionary event in the world of physics, fundamentally altering our understanding of space and time. [PERSON], a relatively unknown Swiss patent examiner at the time, published this ground-breaking theory in 1905 [1, 2]. Special relativity originated from [PERSON]'s quest to resolve inconsistencies between the principles of electromagnetism and classical mechanics. The key insight came with the recognition that the speed of light is constant for all observers, regardless of their relative motion. This revelation challenged the classical notion of absolute space and time and led to the development of a new framework. Moreover, it was postulated that no particle of matter, field, or signal can move faster than the velocity of light in a vacuum, denoted by the letter \(c\). [PERSON]'s theory introduced several revolutionary concepts. The most famous one is the equation \(E=mc^{2}\), revealing the equivalence of the rest energy \(E\) and rest mass \(m\). Special relativity also introduced time dilation and length contraction, demonstrating that time passes differently for observers in relative motion and objects shrink along their direction of motion at high speeds. Special relativity was initially met with skepticism, but its validity was confirmed through experimental evidence, such as the Michelson-Morley experiment [3]. It laid the foundation for subsequent developments in theoretical physics, including general relativity, which describes gravitational force, and later, it was integrated by [PERSON] and others into quantum mechanics, the other pillar of modern physics. In summary, the emergence of special relativity marked a pivotal moment in scientific history, reshaping our understanding of the universe and laying the groundwork for many scientific advancements to come. One of the important consequences of special relativity, which in fact can be deduced based on Maxwell electromagnetism [4, 5, 6] alone, even for slow (nonrelativistic) matter,is the phenomena of retardation, which is the fact that any change will not affect a distant observer before a time is passed which is equal to the distance between the source and observer over \(c\). This fact is expressed mathematically through retarded scalar and vector potentials:  
\[\Phi(\vec{x}_{2}) = k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R},\] \[k = \frac{1}{4\pi e_{0}},\quad\vec{R}\equiv\vec{x}_{1}-\vec{x}_{2}, \quad R=|\vec{R}|,\quad t_{ret}\equiv t-\frac{R}{c}. \tag{1}\]  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{f}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{2}\]  
The notation and symbols in the above expressions are standard [7; 8; 9; 10] and specifically can also be found in [11]. Retarded electromagnetic potentials are a fundamental concept in the realm of classical electrodynamics, and they play a pivotal role in understanding the behavior of electromagnetic fields in a time-dependent manner. These potentials are intimately connected to the theory of electromagnetism and provide a means to describe the propagation of electromagnetic waves and the interaction of charged particles with their own electromagnetic fields. The term \"retarded\" in this context refers to the notion that the electromagnetic potentials at a given point and time are influenced only by the charge and current distributions in the past, taking into account the finite speed of light. In essence, when a charged particle moves, it generates electromagnetic fields which travel outward at the speed of light. The potentials at a particular location are the result of these waves arriving from all past events. The concept of retarded potentials helps address the issue of causality in electrodynamics, ensuring that the effects of electromagnetic interactions do not propagate faster than the speed of light.