[Paper Review] An Invariant Formulation of Special Relativity, or the "True Transformation Relativity," and its Comparison with Experiments

Tomislav Ivezić|ArXiv.org|Mar 9, 2001

Relativity and Gravitational Theory3 citations

TL;DR

This paper proposes an invariant formulation of special relativity—'true transformations (TT) relativity'—where physical quantities are represented as complete 4D tensors, ensuring frame- and coordinatization-invariant descriptions. Unlike conventional 'apparent transformations (AT) relativity' that treat components like time dilation and length contraction as separate effects, TT relativity unifies them via full tensor transformations, showing that all experiments—muon decay, Michelson-Morley, Ives-Stilwell, and Kennedy-Thorndike—agree with TT relativity but only appear to agree with AT relativity due to incomplete measurements of 4D tensor quantities.

ABSTRACT

Different formulations of special relativity are theoretically discussed. First an invariant formulation, i.e., the ''true transformations (TT) relativity,'' is exposed. There a physical quantity is represented by a true tensor which comprises both components and a basis. Also the usual covariant formulation and the ''apparent transformations (AT) relativity'' are considered. It is shown that all the experiments are in agreement with the ''TT relativity'' but not always with the ''AT relativity.''

Motivation & Objective

• To establish a fully invariant formulation of special relativity based on true tensor fields in 4D spacetime, independent of coordinate systems or observers.
• To resolve inconsistencies in conventional special relativity formulations by distinguishing between 'true' tensor quantities and 'apparent' component-based transformations.
• To demonstrate that experimental results traditionally attributed to 'time dilation' and 'length contraction' are actually consistent only with the incomplete 'apparent transformations' (AT) relativity, not the complete 'true transformations' (TT) relativity.
• To show that the apparent agreement between experiments and Einstein's formulation arises from measuring only parts of 4D tensor quantities, not the full geometric object.
• To validate the TT relativity formulation by comparing it with key experiments (muon decay, Ives-Stilwell, Michelson-Morley, Kennedy-Thorndike) across different inertial frames and coordinatizations.

Proposed method

• Formalizes special relativity using true tensor fields on a 4D pseudo-Riemannian spacetime with Lorentzian metric, ensuring invariance under isometries (Lorentz transformations).
• Introduces coordinate-based geometric quantities (CBGQs) that combine tensor components with basis one-forms and vectors, enabling explicit calculations in any inertial frame.
• Compares two coordinatizations: Einstein (e) and 'radio' (r), showing that physical predictions must be independent of the choice of coordinatization in TT relativity.
• Applies the TT formalism to key experiments by treating physical observables (e.g., phase shift, Doppler shift) as full 4D tensor quantities, such as the wave 4-vector $k^a$ or phase $\phi$.
• Derives phase increments and fringe shifts using the full tensor product $k_{e}^{0'}l_{0'e}$ in the moving frame, ensuring consistency across frames and coordinatizations.
• Contrasts TT relativity with AT relativity by showing that AT formulations incorrectly equate only parts of 4D tensors (e.g., $\omega$ or $\ell$) across frames, neglecting full geometric structure.

Experimental results

Research questions

• RQ1 How does a fully invariant tensor formulation of special relativity differ from conventional 'apparent transformation' formulations in describing physical quantities?
• RQ2 Why do experiments like muon decay and Ives-Stilwell appear to confirm Einstein's special relativity when the underlying theory is actually incomplete?
• RQ3 Can the same experimental results be consistently explained using a 4D tensor-based approach that treats all components of physical quantities as unified geometric objects?
• RQ4 What is the role of coordinatization (e.g., Einstein vs. radio) in determining the validity and consistency of relativistic predictions?
• RQ5 Why does the conventional treatment of the Doppler effect in Ives-Stilwell experiments fail at non-zero observation angles, and how does TT relativity resolve this?

Key findings

• The 'true transformations (TT) relativity' formulation, based on complete 4D tensor quantities, is fully consistent with all tested experiments—muon decay, Ives-Stilwell, Michelson-Morley, and Kennedy-Thorndike—across all inertial frames and coordinatizations.
• In contrast, 'apparent transformations (AT) relativity'—the standard Einstein formulation—only appears to agree with experiments because it measures only parts of 4D tensor quantities (e.g., $\omega$ or $\ell$), not the full geometric object.
• The non-null fringe shift predicted in the Michelson-Morley experiment using AT relativity is shown to be an artifact of incomplete tensor treatment; TT relativity yields consistent, frame-independent results.
• For Ives-Stilwell experiments, the AT relativity approach fails at non-zero observation angles ($\theta \neq 0^\circ$ or $180^\circ$) because it neglects the transformation of the spatial wave vector $\mathbf{k}$, while TT relativity treats $k^a$ as a unified 4D object, correctly predicting the Doppler shift.
• The TT relativity calculation correctly predicts a positive transverse Doppler shift $\Delta\lambda = \lambda_0 \beta^2$ in experiments like [47] and [48] when the aberration condition $\Delta\theta$ is satisfied, resolving discrepancies in AT relativity.
• The TT relativity framework is fully independent of coordinatization: results derived in the Einstein (e) or 'radio' (r) coordinatizations are equivalent, proving its geometric consistency and superiority over AT formulations.