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

  • I
  • Thread starter Thread starter Halc
  • Start date Start date
  • Tags Tags
    Derivation Time
Halc
Gold Member
Messages
453
Reaction score
372
TL;DR Summary
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?
 
Physics news on Phys.org
Halc said:
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).
Wikipedia said:
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
 
Last edited:
Thread 'Can this experiment break Lorentz symmetry?'
1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
According to the General Theory of Relativity, time does not pass on a black hole, which means that processes they don't work either. As the object becomes heavier, the speed of matter falling on it for an observer on Earth will first increase, and then slow down, due to the effect of time dilation. And then it will stop altogether. As a result, we will not get a black hole, since the critical mass will not be reached. Although the object will continue to attract matter, it will not be a...

Similar threads

Back
Top