[Paper Review] Frequent Errors in Special Relativity

Diego Saa|ArXiv.org|Jun 28, 2005

Relativity and Gravitational Theory7 references3 citations

TL;DR

This paper argues that Special Relativity's Lorentz transformations are fundamentally flawed because they treat finite coordinates instead of infinitesimal intervals, leading to inconsistencies and paradoxes. By reinterpreting the transformations using differentials and redefining the Lorentz factor as a quotient of time differentials, the author shows that the standard form of the Lorentz contraction factor is mathematically inconsistent, challenging core assumptions of Einstein's theory and suggesting a need for theoretical revision.

ABSTRACT

Some reasons are given to suggest that the interpretation of the Lorentz' transformations as if they referred to coordinates instead of to intervals could be incorrect. Besides, the usual form of such transformations, by using variables that represent finite values instead of differentials, could be another error. Later it is shown that the Lorentz contraction factor must not have the form currently accepted for it if the Lorentz contraction factor is assumed to be equal to the quotient between time differentials.

Motivation & Objective

• To identify a foundational error in Special Relativity: the treatment of Lorentz transformations as acting on finite coordinates rather than infinitesimal intervals.
• To demonstrate that the standard form of the Lorentz contraction factor is inconsistent when derived from time differential quotients.
• To challenge the validity of Einstein’s original derivation of the Lorentz transformations, particularly the use of finite variables instead of differentials.
• To argue that many so-called 'paradoxes' in relativity stem from this persistent mathematical error.
• To propose a reformulation of the Lorentz transformations using infinitesimal intervals to restore internal consistency.

Proposed method

• Reinterprets the Lorentz transformations as operating on infinitesimal intervals rather than finite coordinates, arguing that finite variables introduce logical inconsistencies.
• Uses a Gedanken experiment with a small relative velocity (e.g., 1 m/day) to expose the dependency of time coordinates on frame origin positions, showing time is not an independent variable.
• Analyzes the transformation equations under the assumption that x = 0 (event at origin of S frame), simplifying x′ = −Vt to reveal time dependence on frame displacement.
• Derives the Lorentz factor γ from the quotient of time differentials between observers, showing it does not match the standard expression if γ is defined this way.
• Compares the standard γ = 1/√(1−v²/c²) with an alternative form derived from differential time ratios, revealing an internal contradiction in Special Relativity.
• Argues that the standard coordinate-based formulation leads to unphysical results (e.g., negative time for small velocities), indicating the need for a differential formulation.

Experimental results

Research questions

• RQ1 Why do the standard Lorentz transformations produce unphysical results when applied to finite coordinates, especially at low relative velocities?
• RQ2 What happens to the time coordinate t′ when the relative velocity V approaches zero and the frame origins are offset?
• RQ3 Is the Lorentz factor γ truly equal to the ratio of time differentials between two observers, and does this lead to a consistent expression?
• RQ4 Can the paradoxes in Special Relativity be traced back to the use of finite coordinates instead of infinitesimal intervals in the Lorentz transformations?
• RQ5 What would a consistent formulation of the Lorentz transformations look like if based on differentials rather than finite coordinates?

Key findings

• The standard Lorentz transformation equations fail when applied to finite coordinates, as they produce unphysical results such as negative time values for small relative velocities.
• The time coordinate t′ is not independent but depends on the relative position of the frame origins, contradicting the assumption of independence in Einstein’s formulation.
• When the event occurs at the origin of the unprimed frame (x = 0), the transformation simplifies to x′ = −Vt, revealing that t is not an absolute time but depends on frame displacement.
• The Lorentz factor γ cannot be consistently defined as the ratio of time differentials between observers if it is also required to have the standard form 1/√(1−v²/c²), indicating an internal contradiction.
• The correct form of the Lorentz contraction factor must be derived from differential intervals, not finite coordinates, leading to a different expression than the classical γ.
• The paper concludes that the entire foundation of Special Relativity, based on finite coordinate transformations, is flawed and requires reformulation using infinitesimal intervals to eliminate paradoxes.