Is the acceleration of mass A equal to mass B in Born rigid motion?

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Discussion Overview

The discussion centers on the relationship between the accelerations of two point masses A and B in the context of Born rigid motion, particularly when one mass is subjected to a constant proper acceleration. Participants explore the implications of this motion and the conditions under which the accelerations can be compared.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant posits that if mass B has a fixed constant acceleration aB, then mass A's acceleration aA can be expressed in relation to aB, but seeks to determine the exact expression.
  • Another participant argues that in Born rigid motion, both masses must have the same proper acceleration, suggesting that the rear mass (B) experiences a higher proper acceleration than the front mass (A) to maintain the distance L in the original rest frame.
  • A mathematical expression is provided, L = c²/aA - c²/aB, which is claimed to hold under the assumption of proper accelerations.
  • A similar expression is proposed that relates aA to aB, specifically aA = aB/(1 + aBL/c²), and connects this to gravitational fields using Newton's formula for gravity, questioning whether the Equivalence Principle implies that these accelerations should be equal.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between the accelerations of masses A and B. There is no consensus on whether the accelerations are equal or how they should be expressed in the context of Born rigidity and gravitational effects.

Contextual Notes

Participants assume that the accelerations discussed are proper accelerations and not coordinate accelerations, which is crucial for the validity of their arguments. The implications of the Equivalence Principle and the specific conditions of the scenario are also noted but remain unresolved.

Ironhead
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Suppose I am standing in an inertial reference frame and watching two point masses A and B accelerate according to the rules of the Born rigidity. The distance between the masses is L in an inertial frame in which A and B are simultaneously at rest. Mass A is 'in front' of mass B. Mass B has a fixed constant acceleration aB. What is the expression for the acceleration aA of mass A?
 
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If they move in a Born rigid way while one of them is doing constant proper acceleration, the other one is too. The rear has a higher proper acceleration than the front. (Otherwise the distance between them in the original rest frame wouldn't decrease). Check out the Wikipedia page for Rindler coordinates. The world lines of A and B are two different hyperbolic arcs in the first picture. See also this thread, in particular DrGreg's posts.
 
Assuming the accelerations are proper accelerations (not coordinate acceleration, otherwise it wouldn't be Born rigid motion)

[tex]L = \frac{c^2}{a_A} - \frac{c^2}{a_B}[/tex]​
 
DrGreg said:
Assuming the accelerations are proper accelerations (not coordinate acceleration, otherwise it wouldn't be Born rigid motion)

[tex]L = \frac{c^2}{a_A} - \frac{c^2}{a_B}[/tex]​

I prefer to state that as aA=aB/(1+aBL/c2).
Taking Newtons formula for gravity F=GMm/r2 for mA and mB and assuming mass A is above mass B in a planet M gravity field, or
L=rA-rB, you can get the same type of expression for a planet gravity field: aA=aB/(1+L/rB)2

The Equivalence Principle says these two should be equal? Am I wrong?
 

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