# Time-Dependent Retarded Microwave Electromagnetic Motors
## 1 Introduction
Specifically, for a two-point particle system with charges \(q_{1}\) and \(q_{2}\), we obtain the following [20]:  
\[\vec{F}_{T}^{[2]} = \vec{F}_{Ta}^{[2]}+\vec{F}_{Tv}^{[2]}\] \[\vec{F}_{Ta}^{[2]} = -\frac{kq_{1}q_{2}}{2c^{2}R}\big{[}(\overrightarrow{ac}_{1}+ \overrightarrow{ac}_{2})+((\overrightarrow{ac}_{1}+\overrightarrow{ac}_{2}) \cdot\hat{R})\hat{R}\big{]},\qquad\hat{R}=\frac{\vec{R}  
where the above expression is correct for the second order in \(\frac{R}{c}\).  
This means that the total linear momentum is not conserved (which appears to contradict the [PERSON] theorem applied to the symmetry with respect to spatial translations). This also suggests that \(\vec{v}_{cm}\) must not remain null and may acquire a finite value.  
The prevalent vehicles of today (cars, trucks, motorcycles, ships, etc.) rely on reacting material parts. (For example, the car moves forward only by pushing the road backward, and this backward motion is not apparent because the road is attached to the Earth, which has a rather large mass). Each material component obtains momentum which is equal and opposite to the momentum gained by the other. A well-known example for this momentum balance is a rocket ejecting gas to propel itself in the prescribed direction. However, the retardation phenomena suggest a different engine which is not a combination of two material parts but of field and matter. If we ignore the field, then it may seem to us that the material component gains momentum, and thus the total momentum grows, violating the law of momentum conservation. However, it has been shown that the opposite amount of momentum is carried by the field [19]. Thus, the total momentum of the physical system composed of both field and matter is indeed conserved. This can be traced to [PERSON]'s theorem, which implies that a system which possesses vector translational symmetry will conserve vector linear momentum. As previously stated, the total physical system which is a combination of matter and field will not change under translations, while every specific component of the system (be it either matter or field) will indeed change with respect to the other. This result is not new, as it was already pointed out by a few authors [21; 22; 23; 24; 25]. In particular, [PERSON] [26] indicated that two orthogonally moving point charges apparently contradict [PERSON]'s third law, as the forces that the charges induce do not cancel each other out (last part of (26-2)). This is explained in (27-6), in which it is shown that the momentum gained by the two-charge system is subtracted from the field momentum.  
[PERSON]'s first law of motion, often referred to as the law of inertia, is a fundamental concept in physics. It states that an object at rest tends to stay at rest, and an object in motion tends to stay in motion unless acted upon by an external force. In essence, it underscores the idea that objects will maintain their state of motion, be it stationary or moving in a straight line, unless compelled to change by an external force. This law lays the foundation for understanding how objects behave and is crucial for comprehending motion, equilibrium, and the natural tendency of objects to resist changes in their state of rest or motion. However, again, retardation implies that this law is incorrect (in relativistic motion), and the body's center of mass may acquire a finite velocity \(\vec{v}_{cm}\
eq 0\) even if there is no external force and the body's center of mass was initially at rest.  
This naturally leads to the technological question, of whether can one take advantage of retardation in order to propel bodies to move. Thus, a retarded motor is defined as a system where its center of mass moves due to the interactions between its material components. These components can either move relative to one another or remain stationary within a rigid framework; the specifics are irrelevant since the focus is solely on the motion of the center of mass. This type of engine has several notable features:  
1. **Three-dimensional motion**: It supports motion in all directions, including vertically.
2. **No moving parts**: Unlike conventional engines, it operates without any mechanical moving components.
3. **Fuel-free operation**: It does not consume fuel, thereby producing no carbon emissions. Instead, it converts electromagnetic energy into kinetic energy.
4. **Energy efficiency**: The required electromagnetic energy can be sourced from solar panels and stored in batteries. Once the motor reaches its destination, the energy can be reverted back to electromagnetic form, hinting at exceptionally high efficiency.  
Given these properties, the relativistic motor is particularly well suited for space travel, where current technology dedicates a large portion of spacecraft volume to fuel storage. By eliminating this need, it offers a compact, eco-friendly, and efficient solution for propulsion in space exploration.  
A comprehensive review and introduction to the subject of retarded electromagnetic motors in general and microscopic retarded electromagnetic engines in particular with many references is given in [11; 27] and will not be rewritten here; any reader interested in the additional background and history of the subject is welcome to read the original text. However, for the readers' convenience, we provide a short introduction to the current work, which is given below.  
The work on retarded electromagnetic motors is dependent on the retarded magnetic and electric field formulae introduced by [PERSON] [6; 28]. These mathematical expressions were used to evaluate the total force operating on the center of mass of a system of two time-dependent loop currents [29]. This was later generalized to evaluate the total force in a system composed of a current loop and a permanent magnet [30]. If a force is generated for a non-vanishing length of time, then momentum and kinetic energy will gained by the material part of the system. It may superficially look like that the laws of momentum and energy conservation are not respected, but this is so only if one ignores the interchange of linear momentum and energy with the field. It was shown [19] that the linear momentum gained by the material part of the system is lost by the field component of the same system; that is, it was demonstrated that the linear momentum obtained by the material parts of the retarded electromagnetic motor has an equal amount of momentum (but in the opposite direction) deducted by the electromagnetic field. The exchange of energy between the kinetic material part of the relativistic engine and the electromagnetic field energy was discussed in [31]. It was demonstrated that the electromagnetic energy used was six times the kinetic energy provided to the engine. It was also deduced that energy is radiated from the motor if the coils are not aligned.  
Our preliminary analysis assumed bodies that were devoid of electric charge (neutral). In a later paper [11], charged bodies were analyzed. The charged motor allows obtaining non-vanishing linear momentum in cases in which the current is not continuously enlarged as well. This is not the case in electrically neutral engines. Notice, however, that dielectric breakdown dictates a maximal value for the charge density. The current density is also practically limited even for superconducting wires. It was demonstrated that for any practical motor size, the linear momentum which can be obtained in a retarded charged motor is quite limited, as can be seen in Table 1 for the engine described in Figures 1 and 2, which is of the stationary type.  
The limitations of a macroscopic relativistic motor imply that one should use the extremely high charge and current densities which are available in the microscopic realm, such as in ionic crystals. This idea was studied in earlier papers [20; 27], in which we made good use of the high charge densities and high current densities on the atomic scale.  
Figure 1: A relativistic engine. The charged capacitor is painted in blue, the coil is painted in gold.  
The basic equations and notation relevant to the retarded electromagnetic engine were derived in [11; 27], and they are also given here in Appendix A for the reader's convenience.  
In the previous paper, we only considered a relativistic engine at a stationary state while ignoring the possibility of engines which did not reach a stationary state as well as the transient stage of engines which did. The same criticism holds for the second part of the same paper, dealing with a relativistic engine on the atomic scale [27]. Here, we conduct a more general analysis leading to a different kind of charged relativistic engine of a type which does not reach a stationary state.  
We shall quote the two main results of [11] but omit the definition of symbols and derivation, which can be found in the original publication and in Appendix A. The total force (correct to the second order in \(\frac{R}{c}\)) in the center of mass of the retarded electromagnetic engine composed of two subsystems 1 and 2 is  
\[\vec{F}_{T}^{[2]}=\frac{\mu_{0}}{4\pi}\partial_{t}\int\int d^{3}x_{1}d^{3}x_{2 }\left[\frac{1}{2}(\rho_{2}\partial_{t}\rho_{1}-\rho_{1}\partial_{t}\rho_{2}) \hat{R}-(\rho_{1}\vec{J}_{2}+\rho_{2}\vec{J}_{1})R^{-1}\right] \tag{9}\]  
and the momentum gained by the system which is assumed to be at rest (that is, \(\vec{P}(-\infty)=0\)) is  
\[k = \frac{1}{4\pi e_{0}},\quad\vec{R}\equiv\vec{x}_{1}-\vec{x}_{2}, \quad R=|\vec{R}|,\quad t_{ret}\equiv t-\frac{R}{c}. \tag{1}\]0