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Why isn't the symmetric twin paradox a paradox?
Suppose there are two identical rockets, each carrying one of two identical clocks and one of two identical observers. The rockets lie on a straight line through space, face away from each other, and are completely motionless with respect to each other. The two rockets will use exactly the same flight plans except in different directions, like so:
<**********************************************************---\
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/----------------------------------A B----------------------------------/
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\---**********************************************************>
The direction, power, and duration of the various uses of thrust will be identical. The actions of the observers are also completely identical. It seems that the experience of the observer in either rocket should be identical to the experience of the observer in the other rocket.
Also, the rocket do not accelerate at all while on the portions of their paths that are composed of asterisks.
Consider the portions of the paths composed of asterisks and the effect of time dilation. Because of the difference in velocities, the observer in either rocket will "perceive" the clock in the other rocket showing a time earlier than the clock in his own rocket. However, because the situation is completely symmetrical, the reality in either rocket should be the same; the clocks at any given moment read exactly the same thing. For arbitrarily long "asterisk paths", the difference between real time in either rocket and the time observed in that rocket by the the observer in the other rocket becomes arbitrarily large. (I'm not entirely certain about the conclusions in this paragraph.)
Now suppose that both rockets also carry pairs of missiles such that the four missiles are all identical. The passenger in rocket A hates the passenger in rocket B, so he does some calculations to figure out when to fire his missiles in order to kill the passenger in rocket B. (The passenger in rocket A is a nasty fellow.) He has decided to fire his two missiles in exactly opposite directions in order to prevent his rocket from being accelerated.
So, when he does this, what happens?
Will the observer in rocket B, who is being fired upon, perceive the two missiles leaving rocket A but no longer in rocket A? Will he perceive them being in rocket A but not leaving rocket A? Will he perceive them both still being in rocket A but also leaving it?
If he does not perceive them leaving rocket A, and if we assume that there is a sufficiently large "time lag" between rockets A and B, will the observer in rocket A observe a collision when the observer in rocket B does not observe a collision (and does not get hurt or killed)? Will the observer in rocket B have no physical possibility of seeing the missile missile coming but get killed by it anyway?
What happens?
