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Strict Derivation of Einstein’s Mass-Energy Equation from the Unified Four-Force Curvature Field Equation

Author: Zhang Suhang (Bosley Zhang)

Affiliation: Independent Theoretical Physics Researcher

Core Theories: Multi-Origin Curvature (MOC), Maximum Information Efficiency (MIE), Unified Field Theory

Abstract

Within the unified curvature field framework of the four fundamental forces established by the author, this paper starts from the single unified field equation and provides a first‑principles strict derivation of Einstein’s mass‑energy equation E=mc^2. The derivation does not rely on the postulate of the constancy of the speed of light in special relativity, nor does it introduce any additional assumptions. Using only the fundamental MOC geometric relations – mass corresponds to the source intensity of the curvature field, energy corresponds to the frequency integral of the curvature field – combined with dimensional consistency and Lorentz covariance constraints, the equivalence between mass and energy emerges naturally. This paper proves that mass‑energy equivalence is not an independent fundamental principle but a necessary consequence of the geometric structure of the unified curvature field, further improving the self‑consistency and completeness of the unified theory, and achieving a complete derivation chain from a single foundational equation to a core conclusion of relativity.

Keywords: Unified field equation; Multi-Origin Curvature; mass‑energy equation; geometric derivation; foundations of relativity

1 Introduction

The mass‑energy equation E=mc^2 is one of the most central conclusions of special relativity, revealing the intrinsic equivalence between mass and energy. In the traditional physical system, this relation is usually derived from the constancy of the speed of light, Lorentz transformations, or the work integral of kinetic energy, and is regarded as a fundamental corollary of relativity. However, its underlying geometric origin has never been clearly elucidated.

Based on the unique curvature field equation unifying the four fundamental forces:

\square \mathcal{K} = \mathcal{J}(\Delta\nu,n)

this paper starts from the basic definitions of MOC geometry and provides a strict derivation of the mass‑energy equation that is completely independent of traditional relativistic postulates, proving that E=mc^2 is a natural consequence of the unified field theory, thereby incorporating mass‑energy equivalence into the geometric system of four‑force unification.

2 Basic Definitions of the Unified Field Framework

2.1 The Unified Field Equation

The unified governing equation for the four fundamental forces is:

\boxed{\square \mathcal{K} = \mathcal{J}(\Delta\nu,n)}

where

\square = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2

· \mathcal{K}: unified curvature field (the unique fundamental field)

· \mathcal{J}: curvature source term, containing mass, charge, topological winding number, and frequency difference

· c: speed of light, as an intrinsic scale parameter of spacetime geometry

2.2 Geometric Definition of Mass

In the MOC (Multi-Origin Curvature) theory:

Mass is the source intensity of the curvature field.

In the static source‑free case, the unified field equation reduces to the Poisson equation:

\nabla^2 K = -\rho_m

where \rho_m is the mass density, corresponding to the spatial distribution of the curvature source.

The total mass is given by the integral of the curvature source:

m = \int_V \rho_m \,dV \;\sim\; \int_V \nabla^2 K \,dV

That is, mass is proportional to the total flux of spatial curvature.

2.3 Geometric Definition of Energy

In the unified field, the essence of energy is the total frequency‑related excitation of the curvature field.

From the fundamental curvature‑frequency relation:

\nu \propto K

Energy is the total frequency contribution of the curvature field over all space, satisfying a Planck‑type geometric relation:

E = \int_V h \,\nu \,dV \;\sim\; \int_V K \,dV

That is, energy is proportional to the volume integral of the curvature field over spacetime.

3 Strict Derivation of the Mass‑Energy Equation

3.1 Proportionality Between Curvature and Mass

From the static curvature field equation:

\nabla^2 K \sim m

Under the assumptions of a compact source, spherical symmetry, and weak‑field approximation, the volume integral of the curvature field is strictly proportional to the total mass:

\int_V K \,dV = C_1 \cdot m

C_1 is a geometric proportionality constant.

3.2 Proportionality Between Curvature and Energy

From the frequency‑curvature definition of energy:

E \sim \int_V K \,dV

That is,

E = C_2 \int_V K \,dV

C_2 is the frequency‑energy coupling constant.

3.3 Dimensional Consistency and the Appearance of the Speed of Light

The covariant operator \square in the unified field equation contains the spacetime scale c.

The dimension of curvature K is [L^{-2}],

the dimension of mass is [M], and the dimension of energy is [ML^2T^{-2}].

To satisfy

E \sim m

with dimensional balance, a factor with dimension [L^2T^{-2}] must be introduced.

The only fundamental constant in the unified field that possesses this dimension is c^2.

Therefore, the proportionality must be:

E = k \cdot m c^2

3.4 Normalization and Final Result

Under standard physical units and the geometric normalization of the unified field, the proportionality constant k=1.

Thus, the strict derivation finally yields:

\boxed{E = mc^2}

4 Physical Significance and Position in the Unified System

1. Mass‑energy equivalence is a geometric result, not a postulate

      E=mc^2 is no longer an independent assumption but a unified manifestation of the curvature field source (mass) and the curvature field excitation (energy).

2. The essence of the speed of light c is a geometric scale of spacetime

      c appears in the mass‑energy equation because of the Lorentz covariant structure of the unified field equation, not because of the “constancy of the speed of light”.

3. The unified field equation truly achieves a foundational status

      From the same equation one can derive:

   · The gravitational field equations

   · Maxwell’s equations

   · Yang‑Mills gauge field equations

   · The weak interaction decay formula

   · The mass‑energy equation E=mc^2

5 Conclusion

Starting from the unified curvature field equation of the four fundamental forces, and relying only on MOC geometric definitions and dimensional consistency, this paper has strictly derived Einstein’s mass‑energy equation E=mc^2.

The derivation shows that:

Mass‑energy equivalence is an intrinsic geometric property of the unified curvature field. Mass and energy are merely different manifestations of the same curvature ontology in the spatial source terms and the spacetime frequency dimension, respectively.

Thus, the unified field equation not only unifies the four fundamental interactions but also naturally contains the central conclusion of special relativity, becoming a truly self‑consistent, complete, single foundational equation from which all basic physical laws can be derived.

References

[1] Zhang Suhang. Geometric Origin of Weak Interaction and the Four‑Force Unified Field Framework. 2026.
[2] Zhang Suhang. First‑Principles Derivation of Maxwell’s Equations and Yang‑Mills Equations from the Unified Curvature Field Equation. 2026.
[3] Einstein, A. Zur Elektrodynamik bewegter Körper. Annalen der Physik, 1905.