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A unifying physically meaningful relativistic action

The paper introduces a new Lorentz-covariant relativistic action that unifies gravitational and electromagnetic interactions into a single framework, yielding equations of motion consistent with established physical laws while reducing to standard classical mechanics in the appropriate limit.

Original authors: Y. Friedman

Published 2026-04-29

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CC BY 4.0

Original authors: Y. Friedman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible stage where everything happens. For a long time, scientists have had two different rulebooks for how things move on this stage: one for gravity (like planets orbiting the sun) and one for electricity and magnetism (like electrons zipping around). Usually, these rulebooks are written in very different languages.

This paper proposes a single, unified rulebook that explains how everything moves—whether it's a heavy planet, a tiny charged particle, or a beam of light—by treating all forces as changes to the "shape" of the stage itself.

Here is the core idea broken down into simple concepts and analogies:

1. The "Influenced" Stage

In normal physics, we think of space as a flat, empty grid. If you throw a ball, it goes in a straight line unless something pushes it.

The Paper's Idea: The author suggests that forces (like gravity or magnetism) and even materials (like water or glass) don't just "push" objects; they actually reshape the stage the object is walking on.
The Analogy: Imagine a hiker walking on a flat field. If the ground is flat, they walk in a straight line. But if the ground suddenly turns into a giant, invisible trampoline that dips down, the hiker will naturally curve toward the dip. They aren't being "pulled" by a rope; they are just following the easiest path on the new, curved surface.
The Twist: In this paper, the "trampoline" isn't just for gravity. It can also be shaped by electric charges or by the material a light beam is traveling through. Every object has its own version of this "influenced stage" depending on what forces are acting on it.

2. The "Least Effort" Rule

Nature loves to be lazy. This is known as the "Principle of Least Action."

The Analogy: Think of a lifeguard running to save a swimmer. The lifeguard doesn't run in a straight line from the beach to the water because they run faster on sand than in water. Instead, they take a slightly curved path that gets them to the swimmer in the shortest possible time.
The Paper's Claim: Objects in the universe do the same thing. They always choose the path that requires the "least action" (the most efficient route) on their specific, reshaped stage. The author created a simple mathematical formula (an "action function") that calculates this path for any situation.

3. The Two "Knobs" of the Universe

The author found that this unified formula only needs two main ingredients (called "potentials") to describe everything:

The Linear Knob (Electromagnetism): This describes forces like electricity and magnetism. It acts like a gentle slope or a wind that pushes things in a specific direction. The paper shows this is exactly how light and charged particles move.
The Quadratic Knob (Gravity & Media): This describes gravity and how light moves through materials like water or glass. It acts more like a deep valley or a heavy distortion in the fabric of space.
- Gravity: This knob explains why planets orbit and why light bends around stars.
- Media: This knob explains why light slows down in water (refraction) and how it behaves when the water itself is moving (like the famous Fizeau experiment).

4. Why This Matters (According to the Paper)

The author claims this single formula solves several puzzles that usually require different, complex theories:

It unifies the rules: You don't need one set of equations for gravity and another for electricity. One equation handles both.
It explains the "why": Instead of just saying "gravity pulls," it explains that gravity is just the geometry of space changing.
It passes the tests: The paper shows that if you plug in the numbers for Mercury's orbit around the Sun, this new formula predicts the exact same weird wobble (precession) that Einstein's General Relativity does. It also correctly predicts how light bends and slows down in moving water.

5. The "No Magic" Approach

A key point of the paper is that you don't need to solve complicated, messy equations to find these forces.

The Analogy: Usually, to know how a river flows, you have to measure the water at every single point. The author suggests you can just look at the source (the mountain spring) and the shape of the land to know exactly how the water will flow.
The Claim: The paper states that you can calculate these "reshaped stages" directly from the sources (like a star or a charged particle) without needing the complex field equations usually required in physics.

Summary

In short, this paper proposes that forces are just geometry. Whether you are a planet, a photon, or a charged particle, you are simply walking the straightest possible line on a stage that has been warped by the objects around you. The author provides a simple, unified mathematical map to describe this warped stage for gravity, electricity, and even light traveling through glass or water.

1. Problem Statement

The paper addresses several fundamental issues in theoretical physics regarding the Principle of Least Action:

Lack of Physical Interpretation: While the action principle is successful in various domains (Newtonian mechanics, General Relativity, Quantum Electrodynamics), the physical meaning of the action function is often obscure.
Fragmentation: Different areas of physics define the action differently, creating difficulties when describing motion influenced by multiple forces (e.g., a charged particle in both electromagnetic and gravitational fields).
Limitations of General Relativity (GR): GR successfully geometrizes gravity (force = geometry) but fails to geometrize other forces like electromagnetism. In GR, the acceleration of a charged particle depends on its charge-to-mass ratio ( $q/m$ ), meaning there is no single spacetime geometry common to all particles.
Goal: The author seeks to extend Bernhard Riemann's program of "force equals geometry" to unify the relativistic dynamics of electromagnetism, gravity, and motion in isotropic media under a single, physically meaningful action function.

2. Methodology

The author proposes a framework based on an Extended Principle of Inertia and a specific geometric construction of spacetime:

Extended Principle of Inertia: An inanimate object cannot change its velocity on its own. Therefore, in the presence of force fields or media, the object follows a "straight line" (geodesic) in its own "influenced spacetime," rather than in the flat Minkowski spacetime of the observer.
Action Function Definition ( $L(x, u)$ ):
- Defined as the infinitesimal distance in influenced spacetime between events $x$ and $x + \epsilon u$ .
- Constraints: To satisfy the Principle of Relativity and parameter independence, $L(x, u)$ $L (x, u)$ must be:
  1. Lorentz invariant (scalar).
  2. Positive homogeneous of degree 1 in the four-velocity $u$ .
  3. Reducible to the Minkowski metric ( $\sqrt{\eta_{\mu\nu}u^\mu u^\nu}$ ) when fields vanish.
The Unifying Action Function:
Applying Occam's razor, the author constructs the simplest function satisfying these constraints using two four-potentials:
$L(x, u) = \sqrt{\eta_{\mu\nu}u^\mu u^\nu - (l_\mu(x)u^\mu)^2} + k A_\mu(x)u^\mu$
- $A_\mu(x)$ : A linear four-potential (rank 1 tensor).
- $l_\mu(x)$ : A quadratic four-potential (rank 1 tensor, but squared in the metric term).
- $k$ : A constant depending on the object (e.g., $q/m$ for charge, 0 for photons).
Derivation of Equations of Motion:
Using the Euler-Lagrange equations, the author derives a unified relativistic dynamics equation (analogous to Newton's Second Law):
$\ddot{x}^\lambda = G^\lambda_{\mu\nu}(l) \dot{x}^\mu \dot{x}^\nu + k F^\lambda_\nu(A) \dot{x}^\nu$
Where:
- $F^\lambda_\nu$ is the field tensor derived from $A_\mu$ (linear in velocity).
- $G^\lambda_{\mu\nu}$ is a rank-3 tensor derived from $l_\mu$ (quadratic in velocity).

3. Key Contributions

A. Geometrization of Electromagnetism

The paper identifies the linear four-potential $A_\mu$ with the electromagnetic field.
For a single moving charge, $A_\mu$ is derived directly from the source's worldline and retarded time, yielding the Liénard–Wiechert potential without invoking Maxwell's field equations.
The resulting spacetime geometry is conformal.
The model correctly predicts the precession of elliptic orbits for charged particles, showing that gravity cannot be described by a linear potential (as it yields incorrect precession values for Mercury).

B. Geometrization of Gravity

Gravity is described by the quadratic four-potential $l_\mu$ .
Static Fields: For a static gravitational field with Newtonian potential $\Phi$ , $l_\mu$ is derived such that the action reproduces the Newtonian limit.
Spherically Symmetric Bodies: The model predicts the anomalous precession of Mercury's orbit ( $3\pi \mu$ per revolution), matching General Relativity (GR) exactly.
Moving Sources: The paper proposes a Liénard–Wiechert analog for gravity (Eq. 34) for moving spherically symmetric bodies.
Conjecture: For systems of multiple moving bodies (e.g., binary stars), the rank-3 tensor $G$ is the sum of individual source tensors. This implies the model predicts gravitational waves similar to GR.
Distinction from Whitehead's Theory: While similar to Whitehead's theory for a single static source, this model differs for extended bodies and multiple sources, aligning with GR predictions where Whitehead's theory fails.

C. Motion in Isotropic Media

The propagation of light in a medium is modeled by a constant quadratic potential $l_\mu$ .
The model successfully derives:
- Fizeau Experiment: The velocity of light in a moving medium ( $v \pm v_m(1 - 1/n^2)$ ).
- Snell's Law: Derived directly from the conservation of four-momentum across the boundary of two media.

4. Results

Unified Equation: The derived equation of motion (Eq. 13) successfully describes massive/massless, charged/uncharged particles in electromagnetic, gravitational, and media environments.
Validation against GR:
- The model passes all standard solar system tests (Mercury's precession, light deflection, Shapiro time delay).
- It correctly describes the geometry of spacetime near a black hole (Fig. 2), showing the light cone "pulled" inward but remaining within the flat spacetime cone (preserving the speed of light limit for inertial observers).
Source-Based Derivation: A major result is that the four-potentials can be defined directly from the sources (position and velocity of charges/masses) without solving differential field equations (like Maxwell's or Einstein's equations).

5. Significance

Conceptual Unification: The paper provides a geometric interpretation for electromagnetism and media interaction, extending Riemann's "force equals geometry" program beyond gravity.
Physical Clarity: The action function is physically meaningful, representing the infinitesimal distance in an object's influenced spacetime.
Simplicity: By relying on potentials derived directly from sources, the theory offers a potentially simpler computational framework for relativistic dynamics than solving complex field equations.
Bridging Gaps: It resolves the issue of object-dependent forces (like $q/m$ ) by incorporating the object's properties into the geometry of its specific influenced spacetime, rather than forcing a universal geometry for all particles.

In summary, Friedman presents a Lorentz-invariant, homogeneous action function that unifies the dynamics of electromagnetism, gravity, and optics in media, successfully reproducing known experimental results and General Relativity predictions while offering a distinct geometric perspective on non-gravitational forces.