Classical Theory (in derivations of SR)

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SPECIAL RELATIVITY

Classical Theory (in derivations of SR)

1905 –

In his explanations of special relativity, Einstein often compared the new theory's predictions to those of a system referred to as "Classical Theory" (CT).

In the context of SR, Classical Theory was a hybrid:

1). For light, the equations concerning motion were supposed to be those associated with a flat aether stationary in the observer's frame (global c-constancy w.r.t. the observer)
2). For matter, the equations concerning motion were supposed to be those of Newtonian mechanics.

Einstein's logic then shows an incompatibility between the Newtonian equations and the assumption of global $c$, which special relativity then resolves by modifying both sets of equations with a Lorentz factor.

Ahistoricity of "Classical Theory"

While Einstein's invocation of "CT" as a starting point provides a good expositional route from CT to SR, one has to remember that this form of "Classical Theory" does not obviously correspond to any form of theory in popular use in the Nineteenth Century. It's an artificial system designed to present a problem, the answer to which is special relativity.

The assumption of a globally fixed c wrt the observer does not seem to belong to any theory from the 1800s, except perhaps that of Fitzgerald –if we assumed that there really was an absolute frame for light-propagation that was fixed with respect to Earthbound laboratories, then since such a laboratory would be circling at high speed due to the Earth's rotation, and also have another varying speed due to the Earth's motion around the Sun, and a third motion w.r.t. the Solar System's orbit around the centre of the Milky Way, the only obvious way for $c$ to be globally constant w.r.t. our experiments would seem to be if we went back to pre-Copernican theory, and assumed with Ptolemy (~100–~170) that the Earth was the fixed centre of the universe, and that all other outside matter performed a crazy sets of "spirograph"-style loop-the-loops around us.

Albert Michelson's experiments (which failed to find evidence of a variation in the Earth's speed wrt an aether) were intended to reveal the existence of an aether wind, not to disprove it. FitzGerald, and then Lorentz, both provided pre-SR explanations of why Fitzgerald's experiment might be unable to show the existence of periodic lightspeed variations wrt the Earth, but these theories don't correspond to Einstein's characterisation of CT.

Incompatibility of CT with Newtonian theory

Neither does Newtonian Mechanics. The extension of NM arguments to light does not generate the shift relationships of (1) (characterised by zero transverse redshift), but those normally associated with ballistic emission theory, characterised by a Lorentz-squared transverse redshift. Where the assumptions of CT for light (1) lead to a recession Doppler effect of $\nu'/\nu = c/{(c+v)}$, those of CT for matter (2) lead to a prediction of $\nu'/\nu = {(c-v)/c}$. The two components of CT are not mathematically compatible.

Utility of CT

What CT does do (if we suppose it to represent pre-SR theory), is provide a clear narrative path to special relativity from the supposed "earlier" system, by allowing us to derive a third set of intermediate equations that lie between the conflicting sets of (1) and (2), eliminating the conflict (see: special relativity considered as an average). However, this path from CT to SR should not be mistaken as a representation of special relativity’s actual relationship with earlier theory.

Notes

In Einstein's earlier writing we sometimes find arguments along the lines of "In pre-SR thinking we arrive at outcome X, which causes problems that can be fixed by applying SR and instead getting outcome Y" These arguments can make SR seem unavoidable, until we check actual C19th textbooks and find that the pre-SR predicted outcome was not X, but Z.