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

[Ironhead]

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 a_B. What is the expression for the acceleration a_A of mass A?

 

[Fredrik]

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.

 

[DrGreg]

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}​

 

[Ironhead]

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 a_A=a_B/(1+a_BL/c²).
Taking Newtons formula for gravity F=GMm/r² for m_A and m_B and assuming mass A is above mass B in a planet M gravity field, or
L=r_A-r_B, you can get the same type of expression for a planet gravity field: a_A=a_B/(1+L/r_B)²

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