New Derivation of the Lorentz Transformation and Applications

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The Lorentz transformation, also known as the Lorentz factor or gamma factor, is a quantity that describes how measurements of time, length, and other physical properties change for an object in motion. This expression appears in several equations in special relativity and is derived from the Lorentz transformations. The term originates from its earlier use in Lorentzian electrodynamics, named after the Dutch physicist Hendrik Lorentz.

In most textbooks, the Lorentz transformation is derived from two postulates: the equivalence of all inertial reference frames and the invariance of the speed of light. However, a more general transformation of space and time coordinates can be derived using only the equivalence of all inertial reference frames and the symmetries of space and time. This general transformation relies on one free parameter with the dimensionality of speed, which can be identified with the speed of light (c).

There has yet to be a simple derivation of the transformation equation. Typically, it is derived from a matrix representation of the transformation between two spatial coordinate systems, treating time as a fourth virtual dimension. Time is only relative to the observer and the animate beings observing common physical events, not the events themselves. Below is an alternative derivation of the equations from a quantum science perspective, as this equation applies not only to spacetime but also influences the symmetries in the quantum realm.

General Quantum Theory Basys

The ancient Greek philosopher Plato proposed that the fundamental structure of physical objects is based on material rectangular triangles. He posited that the scalene rectangular triangle is the foundation for the lightest and most movable objects, while the equilateral rectangular triangle underlies the most stable objects. Plato explained the differences between objects through the various combinations of these triangles that can form regular geometric figures. He further suggested that particles could mutually transform, a concept experimentally validated in modern times.

The idea of world unity appears in ancient philosophy, stating that every individual object contains all other objects, effectively embodying the entirety of existence in each manifestation. This concept extends into the philosophical view that variety and continuity represent reality’s surface and visible aspects, while beneath this, the world exists as one inseparable whole. Following this reasoning, one concludes that a universal structure of a unique material particle exists, with every material object in the universe being a manifestation of that particle.

As a result, all phenomena are correlated and mutually determined; they represent a “mutated” state of one or more fundamental correlations within the boundaries of this unique material particle. Each material object can be understood in one aspect as elementary (or singular) and, in another aspect, as complex (or composite). For example, a cell, the basic unit of living matter, is considered elementary, yet viewed from a chemical and physical perspective, it is a complex entity.

From one perspective, every object can be regarded as an individual entity with its internal structure and organization. From another perspective, it is merely one of the many manifestations of the unique material object, which possesses external dependencies (i.e., its external structure and organization). The world’s contradictory nature necessitates the existence

of three independent, mutually perpendicular spatial dimensions. Thus, space is three-dimensional, as nature adheres to the principle of minimizing elements.

The concept of the world’s eternality implies that some elements must be preserved. These elements can safely be identified as the three preservation laws science has unambiguously verified: the law of energy preservation, momentum preservation, and mass preservation.

According to the principle of minimizing elements, a material object comprises two fundamental aspects. The equally fundamental principles of the struggle and unity of opposites and the principle of progress predict an inevitable division of the whole into opposites. This division generates opposing forces, reflecting every object’s simultaneous tendencies toward struggle and conformity. Figure 1 illustrates the model of such an elementary object.

In the course of quantum time , an object is considered identical to itself; it comprises two opposite elements that share the same mass , along the “x” axis. However, it is also not identical to itself as it develops through a different mass, , and , along the ” ” axis. Thus, at any given moment of quantum time , the object can be described as identical and not identical. For this statement to hold true, the two axes, “x” and “y,” must be independent and mutually orthogonal.

According to the philosophical concept of motion, an object at a specific point in time (quantum ) occupies a specific position (identical to itself, ). At the same time, it does not occupy that position because it has changed . This principle serves as a fundamental requirement for modeling a basic material object. As illustrated in Figure 1, we can derive the following equation:

The left side describes the momentum of motion before development, while the right side represents the momentum of motion after development. Here, “c” denotes the maximum quantum velocity of material object interaction (or the interaction between the two opposite elements of the object). “V” represents the development velocity along the -axis, and “u” is the development velocity along the -axis. ” ” is the mass of the object before development (which is identical to itself), and ” ” is the mass of the object after development. This transformation will result in:

This formula is fundamental in the Special Theory of Relativity (STR). It can be derived by assuming that the energy of a material object at rest is , an assumption that may seem arbitrary. The above ratio is derived based on a reasoned choice of the model for the fundamental material object and the law of conservation of momentum.

It’s important to note that the formula has four solutions for , which depend on the values for particles and antiparticles, as well values for the opposing internal momentum of motion (spins).

Now, let’s take a unique look at the triangular model of a quantum object.

From where:

For :

Where is the Lorentz factor

Therefore, the mass change in the individual form within the boundaries of the existing universe changes according to the Lorenz factor.

Hydrogen Atom

Let’s apply the above model to calculate the hydrogen atom dependencies.

If (mass of proton)

and (mass of the electron)

   

Conclusion

This paper shows an alternative derivation of the Lorenz transformation based on matter’s common and individual properties. The triangular dependency model of matter can open new doors for scientific reasoning and explain ordinary quantum events. The elementary stable structural units of matter are the electron and proton. We have shown how this model can be applied to the most fundamental hydrogen atom structure and derive the most common dependencies of that atom.