Lorentz Covariant Extension of the Geometric Progressive Conservation Law: Unification of Classical and Relativistic Conservation

Author: Zhang Suhang
Affiliation: Luoyang, Henan

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Abstract

The Geometric Progressive Conservation Law (GPCL), within the framework of classical mechanics, relies on the geometric axiom that "linear motion is equivalent to circular motion with a radius of curvature tending to infinity" to construct a chain-like conservation system that progressively transmits energy, kinetic energy, momentum, and angular momentum, with rigorous demonstration provided by Poisson mechanics. A long-standing gap exists between classical mechanics and special relativity in the formulation of conservation laws, where the rules governing low-speed and high-speed scenarios lack a homologous logical foundation. This paper transplants the core geometric idea of GPCL into Minkowski four-dimensional flat spacetime, derives the Lorentz covariant form of the Geometric Progressive Conservation Law, and establishes a conservation transmission chain in four-dimensional spacetime. The results show that the extended covariant GPCL can uniformly describe low-speed classical motion and near-light-speed relativistic motion, achieving an integration of conservation laws across the two major theoretical systems, while further demonstrating the universality of the unified geometric view in spacetime physics.

Keywords: Geometric Progressive Conservation Law; GPCL; Lorentz covariance; Minkowski spacetime; four-momentum; classical-relativistic unification

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I. Introduction

In previous work, this author proposed the Geometric Progressive Conservation Law (GPCL). Based on a unified geometric axiom, the unidirectional recursive chain of "energy conservation → kinetic energy conservation → momentum conservation → angular momentum conservation" was derived, and theoretical self-consistency within the framework of classical mechanics was demonstrated using Poisson brackets and Poissons theorem. This achievement resolved the issue of the four major conservation laws in classical mechanics being discrete from one another, establishing a new homologous and progressive conservation paradigm.

Special relativity established the theory of four-dimensional spacetime and reshaped the physical laws governing high-speed motion. In current theoretical systems, classical mechanical conserved quantities and relativistic covariant conserved quantities employ two independent descriptive systems: classical three-dimensional momentum, energy, and angular momentum cannot be directly adapted to Lorentz transformations, and the conservation logic between the two cannot be connected, forming a clear theoretical gap.

This paper extends the core geometric idea of GPCL, specifically the geometric view of motion that "a straight line is equivalent to the limit of a circular arc," into Minkowski four-dimensional spacetime. Based on Lorentz transformation rules, the expressions of conserved quantities and their transmission relationships are reconstructed, and a geometric recursive conservation chain possessing Lorentz covariance is derived. Ultimately, the same geometric logic and the same recursive structure simultaneously cover the classical low-speed domain and the relativistic high-speed domain, achieving a deep unification of the conservation systems of classical mechanics and special relativity.

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II. Review of Foundational Theories

2.1 Classical GPCL

Core geometric axiom: Any linear motion can be regarded as circular motion with radius of curvature R → ∞, satisfying the kinematic relation v = Rω.

In a free system without external forces or external torques, the classical conservation transmission chain is:

Energy conservation ⇒ Kinetic energy conservation ⇒ Momentum conservation --(L = Rp)→ Angular momentum conservation

This chain relation holds rigorously under conditions of three-dimensional Euclidean space and low-speed motion, with Poisson mechanics providing complete algebraic support.

2.2 Foundational Knowledge of Special Relativity

2.2.1 Minkowski Four-Dimensional Spacetime

Special relativity integrates three-dimensional space and one-dimensional time into four-dimensional flat spacetime, with spacetime coordinates expressed as:

(x₀, x₁, x₂, x₃) = (ct, x, y, z)

where c is the speed of light in vacuum. The four-dimensional spacetime interval satisfies Lorentz invariance:

ds² = -c²dt² + dx² + dy² + dz²

2.2.2 Relativistic Conserved Quantities

1. Four-momentum
The relativistic four-momentum vector is defined as:
Pᵘ = (E/c, p_x, p_y, p_z)
where E is the total energy of the particle, and p is the three-dimensional relativistic momentum:
p = γ m₀ v, γ = 1 / √(1 - v²/c²)
γ is the Lorentz factor, and m₀ is the rest mass.
2. Relativistic energy
The total energy of a particle is given by:
E = γ m₀ c²
In the low-speed approximation, γ ≈ 1, and the total energy reduces to the sum of classical kinetic energy and rest energy, returning to the classical mechanics domain.
3. Lorentz covariance
If a physical equation remains form-invariant under any Lorentz transformation, it is said to possess Lorentz covariance. This is a core requirement of special relativity for physical laws.

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III. Extension of the Geometric Axiom to Four-Dimensional Spacetime

The geometric idea in classical theory that "a straight line is a circular arc with infinite radius of curvature" is essentially the use of limit geometry to unify different forms of motion. This idea does not depend on the assumption of three-dimensional space and can be directly extended to Minkowski four-dimensional spacetime:

1. Uniform linear motion in four-dimensional spacetime is equivalent to timelike circular motion with the four-dimensional radius of curvature tending to infinity;
2. The four-dimensional velocity, radius of curvature, and four-dimensional angular velocity satisfy a geometric relationship homologous to the classical form;
3. Geodesic motion in four-dimensional spacetime is the natural manifestation of this geometric axiom within the relativistic framework.

The above extension ensures that the core of GPCL is fully preserved in four-dimensional spacetime, laying the geometric foundation for the covariant reformulation of the conservation chain.

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IV. Derivation of the Lorentz Covariant Form of GPCL

4.1 Relativistic Energy Conservation (Four-Dimensional Starting Point)

For an isolated free system with no external energy exchange and no interactions, the relativistic total energy is a Lorentz invariant, satisfying:

dE/dτ = 0

where τ is proper time. This equation serves as the starting point of the entire conservation chain in the relativistic domain, corresponding to energy conservation in the classical system.

The relativistic total energy includes rest energy and kinetic energy. Under free motion conditions, the total energy is constant, from which relativistic kinetic energy conservation is directly derived, corresponding to the kinetic energy conservation link in the classical chain.

4.2 Derivation of Four-Momentum Conservation from Kinetic Energy Conservation

From the relativistic energy E = γ m₀ c² and the definition of four-momentum Pᵘ = (E/c, p), the two are strictly bound.

In a free system, energy is constant, so the Lorentz factor γ and the three-dimensional momentum p both remain constant. Therefore:

dPᵘ/dτ = 0

That is, four-momentum conservation holds. In the low-speed limit, the spatial components of the four-momentum reduce to classical three-dimensional momentum, corresponding to the momentum conservation link in the classical chain.

4.3 Derivation of Relativistic Angular Momentum Conservation from Four-Momentum Conservation Combined with the Geometric Relation

Analogous to the classical relation L = Rp, the relativistic angular momentum tensor is defined in four-dimensional spacetime. Combining the four-dimensional radius of curvature Rᵘ and the four-momentum Pᵘ, the generalized geometric relation is:

Lᵘᵛ = Rᵘ Pᵛ - Rᵛ Pᵘ

The four-dimensional radius of curvature Rᵘ of the free motion trajectory is an invariant, and four-momentum has been proven to be conserved. Therefore:

dLᵘᵛ/dτ = 0

Relativistic angular momentum conservation holds. In the low-speed approximation, the relativistic angular momentum reduces to classical angular momentum, completing the final link of transmission.

4.4 Complete Covariant Conservation Chain

Combining the above derivations, the Lorentz covariant geometric recursive conservation chain is obtained:

Relativistic total energy conservation ⇒ Four-momentum conservation ⇒ Relativistic angular momentum conservation

This chain strictly reduces to the classical GPCL form in the low-speed limit, achieving a unification of the classical and relativistic conservation systems.

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V. Conclusion

1. The geometric axiom is successfully extended to four-dimensional Minkowski spacetime, laying the geometric foundation for the covariant conservation chain.
2. The Lorentz covariant GPCL is established: relativistic total energy conservation → four-momentum conservation → relativistic angular momentum conservation.
3. A logical unification of the conservation systems of classical mechanics and special relativity is achieved, with the classical GPCL naturally embedded as the low-speed limit.
4. This chapter maintains strict connection with Chapter One, with consistent structure and consistent recursive logic, providing the covariant foundation for subsequent extensions to general relativity (Chapter Three) and algebraic structure analysis (Chapter Four).

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