Undergrad Einstein's Derivation of Elapsed Time for Remote Comoving Object

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TL;DR

Question on Einsteins derivation of elapsed time for remote comoving object

This is a question on Einstein's 1907 paper first discussing equivalence principle and uniform acceleration.

Picture a rigid accelerating object of length £ with a clock at each end. The rear accelerates for time τ (measured by the clock there) at a proper acceleration γ. The clock at the front of the object advances by time δ relative to the accelerating frame ∑ of the object, which is what Einstein is computing here.

Reference is http://www.relativitycalculator.com/pdfs/Einstein_1907_the_relativity_principle.pdf at the bottom of page 305

If we move the first point event to the coordinate origin, so that rt = r and E1 = 0, we obtain, omitting the subscript for the second point event,

δ=τ[1+γ£/c²] (30)

This equation holds first of all if τ and £ lie below certain limits. It is obvious that it holds for arbitrarily large τ if the acceleration γ is constant with respect to ∑, because the relation between δ and τ must then be linear. Equation (30) does not hold for arbitrarily large £. From the fact that the choice of the coordinate origin must not affect the relation, one must conclude that, strictly speaking, equation (30) should be replaced by the equation

δ=τ exp(γ£/c²)

Nevertheless, we shall maintain formula (30)

Equation 30 seems fine to me. For really hard accelerations, the time to get to an arbitrary change in velocity drops to negligible levels and the 1+ part becomes insignificant. For the same change in speed in half the time, τ halves and γ doubles. The resulting change in the front clock time is nearly identical in both cases, not being much of a function of the acceleration rate. This is as it should be.

First question, not all that important: Why does Einstein say (30) doesn't hold for large £? If the object is twice as long, the clock there advances twice as much for the same action at the rear. I don't see why it falls apart.

Second question, which is why I opened this topic:
How is the 'strictly speaking' equation at the bottom (not numbered) the better equation? It doesn't seem to yield proper results at all. If I double the aggressive acceleration and halve the time, the clock in front advances not the same, but massively move since it replaces a linear relation τγ£ with the non-linear τ exp(γ£). This seems wrong. Einstein says he's not going to use this equation, but rather will maintain (30) for the subsequent discussion, but is the bottom formula correct? Am I just not reading it right?

 

[Sagittarius A-Star]

Second question, which is why I opened this topic:
How is the 'strictly speaking' equation at the bottom (not numbered) the better equation?

According to Wikipedia, this is the time dilation formula of Radar coordinates (Lass coordinates).

Albert Einstein (1907)[H 13] studied the effects within a uniformly accelerated frame, obtaining equations for coordinate dependent time dilation and speed of light equivalent to (2c), and in order to make the formulas independent of the observer's origin, he obtained time dilation (2i) in formal agreement with Radar coordinates.

Source:
https://en.wikipedia.org/wiki/Rindler_coordinates#Overview

The first formula (30) of Einstein is the time dilation formula of Kottler–Møller coordinates. There, the reference clock (observer) must be located at $x=0$:
https://www.physicsforums.com/threa...an-accelerating-elevator.1046071/post-6806971