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# Twin Paradox

The twin paradox is a mind experiment which involves twins that are identical. One twin makes a journey into space in a high-speed rocket and returns home to find that the other twin, who remained on Earth the entire time, has aged more. This is puzzling because each twin sees the other twin as traveling, according to an application of time dilation each should paradoxically find the other to have aged more slowly. This scenario can be looked at within the standard framework of special relativity because the twins are not equivalent; the space twin experiences additional asymmetrical acceleration when switching direction to return home and therefore is not a paradox in the sense of a logical contradiction.

In the theory of special relativity, there is no such concept of ``absolute present``. The present is from the point of view of the observer. It is defined as the set of events that are simultaneous for that observer. Simultaneity depends on the reference frame.

The standard proper time formula to calculate the difference in elapsed times is:

∆τ= ∫_0^∆t〖√(1- ((v(t))/c)^2 ) dt〗

Time dilation is one of the key components in understanding the twin paradox. It is the actual difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. For example, an accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer’s own equally accurate clock. This effect rises from technical aspects neither of the clocks nor from the fact that signals need time to propagate but from the nature of space time.

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### On his 25th birthday, Peter leaves his twin Paul behind on the earth and goes off in a straight line for 7 years of his time at a speed of 0.96c.

On his 25th birthday, Peter leaves his twin Paul behind on the earth and goes off in a straight line for 7 years of his time at a speed of 0.96c. Peter then reverses direction and returns at the same speed. What are the ages of Peter and Paul at the moment of reunion?

### Proper time problem.

See attached file 1. Consider two observers moving in the x direction with respect to a stationary reference frame. The worldline of one observer is given by x1 = (1/3)t while that of the other is given by x2 = (1/18)t^2. (a) Determine the event at which the two observers meet after leaving the origin. (b) Determine the