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Is there an inertial frame ($x'$, $t'$) in which two objects undergoing proper acceleration in the positive x-direction (e.g. starting from rest at $t=0$ from $x=0$ and $x=L$) where the two objects end up at rest?

In this frame, where and when did each object begin moving and how far apart do they end up? I know that in the stationary frame the two objects are always a distance $L$ apart. How far apart are they from the perspective of the objects themselves?

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The short answer is no. There is no other inertial frame in which both can even momentarily be at rest while they are accelerating.

While they are both accelerating, observers on both objects agree that the clocks on both objects are running at different speeds, with the clock on the leading object apparently running faster. They are also moving apart.

For them both to share a new inertial frame each must stop accelerating when it's clock reaches an agreed time. For the accelerating observers this will happen first for the leading object.

When they have both stopped accelerating the leading object's clock will show the moving observers a later time than the trailing one, and the observers will measure the two objects farther apart than when they started, but they will now be stationary in a new inertial frame.

The rest frame always sees them the original distance apart with their clocks showing the same time.

Had they been rigidly connected - think train - they would now be Lorenz contracted and if they synchronised their clocks the lead clock would show an earlier time to the rest frame than the following clock.

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