Prologue
I recently read Anatoli Andrei Vankov's paper "On Controversies in Special Relativity" [Van2006] with great interest. I've always been fascinated by discussions of paradoxes in Relativity (especially, the Ehrenfest Paradox), but I wasn't in the least aware that the Spherical Wave-front example of Einstein was controversial.
Vankov's Analysis
See Section 3 (Shape of light front, and constancy of the speed of light) of Vankov's paper [Van2006] for
I recently read Anatoli Andrei Vankov's paper "On Controversies in Special Relativity" [Van2006] with great interest. I've always been fascinated by discussions of paradoxes in Relativity (especially, the Ehrenfest Paradox), but I wasn't in the least aware that the Spherical Wave-front example of Einstein was controversial.
Vankov's Analysis
See Section 3 (Shape of light front, and constancy of the speed of light) of Vankov's paper [Van2006] for
- Specifics on why the example is controversial.
- An analysis of the physics underlying the example.
- Subsequent interpretation towards the resolution of the controversy.
Specifically, the analysis concludes that the shape of an isotropic (spherical) wave-front in frame S' becomes an ellipsoid in frame S. I do not disagree with the analysis itself. However, I submit that the eventual interpretation is untenable. [There are a few typos in equations (11), (13), and (14). However, the typos don't affect the analysis perse; so I don't dwell on them here.]
Note that we start off with a spherical wave-front in frame S' which is a surface of constant t'. This surface is then parametrized using theta'. Subsequently, using the Lorentz transforms, this spherical surface is determined to be transformed into an ellipsoid in frame S. And the analysis is absolutely correct.
My Comments
However, at this point it is also asserted that an ellipsoid is exactly what is perceived by frame S. But, it must be realized that this ellipsoid surface is not a surface of constant t in frame S (because of relativity of simultaneity). [This is also evident in the analysis: The expression for the time coordinate in frame S is a function of theta/theta'.] As a result, the conclusion that the ellipsoid surface is perceived as-is in frame S, is, philosophically speaking, untenable: If frame S were to specify the shape of the wave-front, it would be based on a surface of constant t, and not based on a surface that corresponds to space-time events belonging to varying, or even arbitrary, t. Thus, the ellipsoid surface has no significance in frame S.
From a physical perspective, frame S would assert that the wave-front is spherical because frame S will make the shape determination at a specific time t which would give an obviously spherical wave-front (just as Einstein had originally remarked). Note that the space-time events that make up this spherical wave-front in frame S similarly carry no significance when transformed to frame S'.
Update [15Sep2009]: I've emailed a one-page write-up to Anatoli Vankov soliciting his comments.
Note that we start off with a spherical wave-front in frame S' which is a surface of constant t'. This surface is then parametrized using theta'. Subsequently, using the Lorentz transforms, this spherical surface is determined to be transformed into an ellipsoid in frame S. And the analysis is absolutely correct.
My Comments
However, at this point it is also asserted that an ellipsoid is exactly what is perceived by frame S. But, it must be realized that this ellipsoid surface is not a surface of constant t in frame S (because of relativity of simultaneity). [This is also evident in the analysis: The expression for the time coordinate in frame S is a function of theta/theta'.] As a result, the conclusion that the ellipsoid surface is perceived as-is in frame S, is, philosophically speaking, untenable: If frame S were to specify the shape of the wave-front, it would be based on a surface of constant t, and not based on a surface that corresponds to space-time events belonging to varying, or even arbitrary, t. Thus, the ellipsoid surface has no significance in frame S.
From a physical perspective, frame S would assert that the wave-front is spherical because frame S will make the shape determination at a specific time t which would give an obviously spherical wave-front (just as Einstein had originally remarked). Note that the space-time events that make up this spherical wave-front in frame S similarly carry no significance when transformed to frame S'.
Update [15Sep2009]: I've emailed a one-page write-up to Anatoli Vankov soliciting his comments.
References
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