Lorentz, Poincaré, and Einstein: Rethinking Doppler, Aberration, and the Fresnel Drag (2509.26389v1)

Abstract: This paper examines Lorentz's 1895 derivations of the classical Doppler formula and Fresnel drag, Einstein's 1905 derivation of the relativistic Doppler effect and aberration, and Einstein's 1907 kinematical route to the exact velocity composition law from which Fresnel drag is obtained as a low-velocity limit. Einstein acknowledged that he had read Lorentz's "Versuch" well before 1905. In 1907, Einstein identified Lorentz's "Versuch" as a crucial precursor to relativity. In that work, Lorentz had already invoked local time to derive Fresnel's drag coefficient from Maxwell's equations. There is a genuine "family resemblance" between Lorentz's and Einstein's treatments in that both preserve the phase of a plane wave under transformation. Yet I demonstrate that this resemblance is only formal. I also discuss the absence of the relativistic Doppler and aberration laws in Poincar\'e's Dynamics of the Electron.

• The paper demonstrates that Einstein's exact relativistic framework resolves anomalies in classical Doppler and Fresnel drag derivations.
• It uses rigorous mathematical analysis of phase invariance and Lorentz transformations to compare differing theoretical approaches.
• The study highlights the paradigm shift from first-order, ether-based approximations to principle-based relativistic kinematics with broad implications.

Lorentz, Poincaré, and Einstein: Rethinking Doppler, Aberration, and the Fresnel Drag

Historical and Conceptual Context

This paper provides a rigorous comparative analysis of the derivations and conceptual underpinnings of the Doppler effect, stellar aberration, and Fresnel drag in the works of Lorentz, Poincaré, and Einstein. The author traces the evolution from Lorentz’s ether-based kinematics and first-order approximations, through Poincaré’s group-theoretic formalism, to Einstein’s principle-based, exact relativistic framework. The analysis is grounded in the mathematical treatment of plane wave phase invariance and the transformation properties of electromagnetic waves under changes of inertial reference frames.

Lorentz’s Ether Theory: Classical Doppler and Fresnel Drag

Lorentz’s 1895 derivations are situated within the context of an immobile ether, employing Galilean transformations and the concept of local time to reconcile Maxwell’s equations with experimental results such as Fizeau’s measurement of light speed in moving water. The classical Doppler effect is derived by considering the observer’s and source’s velocities relative to the ether, resulting in formulas that exhibit a formal symmetry but a physical asymmetry due to the privileged ether frame. The classical Doppler shift for a moving observer is bounded ($0 < \nu' < 2\nu$), while for a moving source, the formulas predict unphysical singularities as $v \to c$.

Lorentz’s treatment of Fresnel drag involves the introduction of local time and a first-order correction to the phase of a plane wave in a moving medium. The resulting formula for the observed light speed in moving water,

$u = \frac{c}{n} + v\left(1 - \frac{1}{n^2}\right),$

matches Fresnel’s empirical drag coefficient but is valid only to first order in $v/c$ and retains the ether as a necessary construct.

Einstein’s Relativistic Kinematics: Exact Doppler, Aberration, and Velocity Addition

Einstein’s 1905 and 1907 works mark a decisive conceptual break. By abolishing the ether and postulating the principle of relativity and the invariance of $c$, Einstein derives the relativistic Doppler and aberration laws from the invariance of the phase of a plane electromagnetic wave under Lorentz transformations. The relativistic Doppler formula,

$$\nu' = \nu \frac{1 - \frac{v}{c}\cos\varphi}{\sqrt{1 - \frac{v^2}{c^2}}},$$

depends only on the relative velocity and angle, eliminating the need for separate observer and source velocities. The aberration law,

$$\cos\varphi' = \frac{\cos\varphi - \frac{v}{c}}{1 - \frac{v}{c}\cos\varphi},$$

follows directly from the transformation of the wave normal.

Einstein’s kinematical approach yields the exact velocity addition law for phase velocity,

$u = \frac{u' + v}{1 + \frac{u'v}{c^2}},$

which, for $u' = c/n$ and $v \ll c$, reduces to Fresnel’s drag as a limiting case. This derivation is valid to all orders in $v/c$ and is grounded in the Lorentz transformation, not in first-order approximations or ether-based constructs.

Poincaré’s Group Theory and Its Limitations

Poincaré’s formalism includes the full Lorentz transformation and recognizes its group properties, but he retains the ether as a physical entity. While the mathematical machinery is present, Poincaré does not derive the relativistic Doppler or aberration laws as Einstein does. The invariance of the phase under Lorentz transformations is not postulated as a physical principle, and the ontological commitment to the ether prevents the full conceptual leap to relativity.

The Doppler Darkness Paradox and Its Resolution

The paper highlights the “Doppler darkness paradox” inherent in Lorentz’s classical theory: the possibility of an observer moving at $v = c$ and experiencing a vanishing frequency, which is physically inadmissible in Maxwellian electrodynamics. Einstein’s relativistic framework resolves this paradox by prohibiting inertial frames with $v = c$ and ensuring that the frequency shift is asymptotic, not attainable by any massive observer.

Methodological and Ontological Implications

The analysis demonstrates that the “family resemblance” between Lorentz’s and Einstein’s treatments is only formal. Lorentz’s derivations are constructive, relying on mathematical devices (local time, Galilean kinematics) to salvage empirical results within an ether theory. Einstein’s approach is principle-based, deriving exact results from the invariance of physical laws and the structure of spacetime. The transition from Lorentz’s constructive theory to Einstein’s principle theory represents a fundamental shift in the ontology of physics, from ether-based dynamics to relativistic kinematics.

Implications and Future Directions

The paper’s comparative methodology clarifies the distinction between first-order approximations and exact relativistic results, emphasizing the necessity of abandoning the ether for a coherent theory of electrodynamics and optics. The analysis suggests that future developments in theoretical physics should prioritize principle-based approaches that unify disparate phenomena under common kinematical frameworks. The treatment of phase invariance and velocity addition in relativity provides a template for addressing analogous issues in quantum field theory and other domains where transformation properties underlie physical law.

Conclusion

This work provides a detailed account of the mathematical and conceptual evolution from Lorentz’s ether-based optics to Einstein’s special relativity, with Poincaré’s group-theoretic formalism as an intermediate step. The author demonstrates that the relativistic Doppler, aberration, and velocity addition laws are not mere technical corrections but reflect a profound rethinking of the foundations of physics. The abolition of the ether and the adoption of Lorentz invariance as a universal principle mark the transition to modern kinematics, with implications for both theoretical understanding and experimental practice.