Time Jump

This was a result from studying the twin paradox. The key idea is that one can start a new line of simultaneity by reversing velocity, just like jumping over time points.

A pair of twins, they have a relative speed \(v\) to each other. We want to know which one of the twins is younger or in other words comparing their times. Because of the relative speed \(v\), the twins may not be on a same simultaneous line, when we check their times.

We want to compare people’s ages up to a same date. We do not want to use today’s age compare to future’s age or passed age. So we require the twins come back to earth to meet together, to be on a same point of our spacetime diagram when checking their times.

A pair of twins, one stay on earth, the other one is an astronaut leaving earth traveling into space with speed \(v=c/2\). \(c\) is the light speed. On the 1st second mark, the astronaut twin reverses his velocity to \(v=-c/2\) to back to earth. On the 2nd second mark the astronaut twin arrives on earth, and comparing time with the earth twin.

E has the coordinates x=1, t=2; C has the coordinates x=0.5, t=1

Let’s look at this graph. If the astronaut twin does not reverse velocity. The earth twin should arrive at point M and the astronaut twin should arrive at point E in 2 earth’s seconds. Astronaut twin’s time passes 1.732051 seconds only. The twins cannot compare their times at points E and M. Because the earth twin at M considers himself to be simultaneous with point E. The Astronaut twin at point E considers himself is a passed event of the earth twin, because D and E are simultaneous in the astronaut twin frame.

But if the astronaut twin does reverse velocity at point C \(x=0.5\) \(t=1\), then get back to earth at point M. The astronaut twin is no longer at the pass of the earth twin. Because the twins are simultaneous at point M. Still earth time passed 2 seconds, the astronaut twin time passed 1.732051 seconds. The astronaut twin is younger.

c0=1
v=c0/2

beta=v/c0
gamma=1/sqrt(1-beta^2)
x=1
t=2
x2=gamma*(x-beta*t)
t2=gamma*(t-beta*x)
c(x2,t2)
## [1] 0.000000 1.732051

The astronaut twin was on the pass of the earth twin. But the astronaut twin was able to catch up and meet the earth twin at point M simultaneously. This is because the astronaut twin reversed velocity at point C, and making a new simultaneous line BC instead of simultaneous line AC. Events on earth suddenly jumped from A to B. Any earth events in between A to B did not physically happened according to the astronaut twin.