Dear Colleague,

The concern you've raised about Einstein's derivation of the stellar aberration formula and the treatment of the angle ϕ' in relation to ϕ touches on the application of the theory of relativity to optical phenomena.

In Einstein's seminal paper, "Zur Elektrodynamik bewegter Körper," he derived the formula for stellar aberration by considering the transformation of direction cosines of a light ray between stationary and moving reference frames. The formula mentioned, cos(ϕ') = (cos(ϕ) - v/V) / (1 - cos(ϕ) * v/V), relates the direction cosines (or equivalently, the angles of incidence) of a light ray in a moving frame (ϕ') to that in a stationary frame (ϕ) with respect to the velocity v of the moving frame and the speed of light V.

This expression, cos(ϕ') = (cos(ϕ) - v/V) / (1 - cos(ϕ) * v/V), is derived from the Lorentz transformation, accounting for the effects of motion on the measurements of space and time, including the direction of light rays. Here, ϕ' and ϕ are not simply related by a direct subtraction (i.e., ϕ' is not replaced by ϕ' - ϕ); rather, ϕ' is defined in a way that inherently considers the relative motion between the source (or wave-normal) and observer.

The confusion might arise from interpreting the transformation as implying a simple geometric relationship between ϕ' and ϕ, whereas it actually represents a relativistic effect of motion on the observed direction of light. The angle ϕ' is derived considering the relativistic composition of velocities, which alters the apparent direction of the incoming light in the frame of the observer moving at velocity v.

Regarding the note about the change in the original text mentioned in the Einstein Papers Project (Note 29, Vol. 2), it reflects Einstein's clarification or correction about the description of the angle ϕ'. The issue with the aberration angle for ϕ = π/2 leading to a second quadrant result contrary to experimental observations highlights the complexity of interpreting relativistic formulas in the context of classical phenomena like stellar aberration. This concern might necessitate a more detailed analysis of the original text and the historical context in which Einstein was writing. Einstein's adjustments or clarifications in later notes or editions of his work often addressed misunderstandings or sought to make the implications of his theories clearer to readers and correspondents.

Thus, the question of replacing ϕ' with the difference between ϕ' and ϕ is not a matter of allowing or disallowing within the derivation; it's more about how the relativistic transformation is applied and interpreted. The relativistic formula for stellar aberration, as derived by Einstein, correctly accounts for the observed phenomena within the framework of special relativity, and any adjustments or clarifications made by Einstein in subsequent notes would aim to address misinterpretations or refine the conceptual framework rather than fundamentally altering the mathematical basis of the derivation.

Regards,

Alessandro Rizzo

May I point out a couple of issues:

Einstein used his brand of Lorentz transformation to derive the expression for the stellar aberration. (Lorentz and Poincare independently derived the so-called Lorentz transformation before Einstein's paper was published. Einstein may not have been aware of these publications.)

I suppose something being 'relativistic' does not allow one to break well-established rules of mathematics.

Dear Joseph Kurusingal ,

Thank you for your insights. Yes Indeed, it is crucial to acknowledge the historical context in which Einstein derived the stellar aberration formula. While it is true that Lorentz and Poincaré had already developed what we now call the Lorentz transformation before Einstein's seminal paper, "Zur Elektrodynamik bewegter Körper," was published, Einstein's contribution was unique in its interpretation and application within the framework of Special Relativity.

Einstein's approach to deriving the stellar aberration formula did not, in any way, break well-established rules of mathematics. Instead, it applied these rules within the novel context of relativistic physics. The Lorentz transformation, as used by Einstein, was employed to reconcile the observed behavior of light with the principles of relativity, thereby providing a deeper understanding of the nature of light and motion.

It's important to emphasize that the development of scientific theories often involves building upon and reinterpreting the work of predecessors. While Einstein may have used the mathematical foundation laid by Lorentz and Poincaré, his interpretation introduced the concept of time dilation and length contraction as natural consequences of the speed of light being constant in all inertial frames of reference. This was a groundbreaking perspective that went beyond the mathematical formalism to include a physical interpretation that was consistent with observations.

The relativistic treatment of stellar aberration by Einstein showcases the application of mathematical principles to explain physical phenomena in a manner that was consistent with the empirical evidence available at the time. It highlights the evolution of scientific thought and the integration of mathematical rigor with physical insight to advance our understanding of the universe.

Best regards,