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

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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)

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

L = \frac{c^2}{a_A} - \frac{c^2}{a_B}​

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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