Answer to problem 5.4.5

 

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These refer to the simplified answer for the transformation between the inertial frame unprimed, and the accelerated primed frame:

The counter rotating frame transformation to the inertial frame will also be

From these relate the two accelerated frame coordinates

Consider what the double primed frame observer's watch reads according to the single primed frame. It is at the origin of the double primed frame, so:

Solve the last equation for y' and insert into the other two and simplify.

Substitute the last into the previous equation and simplify

They cross each at

Each time they cross this yields,

This says that the watch re-synchs with the single primed frame observer's watch at the locations where the rockets meet.

Differentiate

 

Next consider those re-synch points where they met at

Simplify to

Inserting this and simplification results in

This demonstrates that one rocket frame observer will observe the other as time dilated by special relativistic time dilation as they pass each other.

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