Time (m)
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Time (m)

Every “trajectory in spacetime”, such as the two drawn in the figure, is called a worldline. What is invariant of a worldline is not the distance covered nor the time taken, but the proper time, also called wristwatch time. It is defined as the integral over the line of the infinitesimal proper times:

B

O

X (m)

If we divide the axis of the ordinates in equal segments of length dt, for each infinitesimal segment the length of the proper time will be smaller than dt (or at most equal).

Therefore the integral, which is the proper time of that particular worldline, is maximum for the blue line. But the blue line is equivalent (I mean: has the same LI ) to a straight line, which can be obtained with a LT. Then: given a timelike pair of events, the straight worldline joining them has the maximum proper time (principle of maximal aging).

The straight line between two events is also the worldline followed by a free particle: it actually represents a motion with uniform velocity.

Combining the two statements, we obtain that: between two fixed events a free particle follows the line of maximum aging.

And now, having finished the rehearsal on simultaneity, let us continue with the trip to Canopus and its mysteries

.

Advanced EM - Master in Physics - 2011-2012


  • We can apply the principle of maximal aging to “the two twins”: the two fixed events between which they travel are located on Earth, and are the departure and the arrival of the spacecraft, obviously at different times.

  • The two worldlines are:

  • The twin who stayed at home , obviously did not move, only saw time passing: he followed the blue line of previous page. His worldline was a straight line (in spacetime!). The wristwatch time was 202 years.

  • The other twin travelled to Canopus: a very long distance indeed. The wristwatch time was only 39.6 years.

A worldline with kinks

A non-straight worldline, i.e. a curved line or with sharp corners implies an acceleration, since the line’s slope is the particle’s velocity. An interstellar travel is, as represented in the spacetime plot, a long one-dimensional travel followed by a slow deceleration, a change of direction and a new acceleration near the faraway star, and a long travel again in direction opposite to the first stretch. The acceleration period can be so short as to be undistinguishable in the spacetime plot. The acceleration is of course the larger the shorter is the acceleration period.

We can consider it as the route of an observer who travels on an IRF and, arrived at destination, gets off and catches on the fly another IRF’ which was just passing by, at the same speed but in the opposite direction.

Advanced EM - Master in Physics - 2011-2012


B twins”: the two fixed events between which they travel are located on Earth, and are the departure and the arrival of the spacecraft, obviously at different times.

5

L

Q

P

4

O

In this drawing 3 different worldlines are shown, which represent 3 different trips from O to B: OPB, OQB and OLB.

OPB, the first trip, stays where he was, does not move, and is therefore on a straight line between O and B: the line of maximal aging. He is like the twin who stayed on Earth. His wristwatch indicated 10.

OQB is similar to the trip of the twin who went to Canopus. While time runs to 5 he has travelled a distance of 4. His β takes then the value of 0.8. His proper time in the stretch from O to Q is obviously 3. Then, at Q, he changes IRF’, i.e. reverses velocity to go back to the same point (in space!) from where he left. Since is velocity is still 0.8, when he gets to B another proper time of 3 has added to the total wristwatch: 6.

10

0

The OLB travel covers two stretches, with a distance of 5 in a time of 5 in the first stretch and also in the second. His wristwatch time is ZERO! How can that be? It can if who does that trip is a light flash reflected by a mirror at L.

Advanced EM - Master in Physics - 2011-2012


In conclusion of all that twins”: the two fixed events between which they travel are located on Earth, and are the departure and the arrival of the spacecraft, obviously at different times.

We have seen the Worldlines and their invariant property, call it Aging, Proper time, Wristwatch time.

We know that Worldlines are trajectories in space covered by moving objects, and that they are always timelike, not only between the end events (departure and arrival), but also between any pair of events along them.

The concept of Proper Time that we knew, as the “distance” between two timelike events, is still present, and in the case of a worldline it is the wristwatch time of the line of maximum aging. Because a new concept has emerged, the wristwatch time. It is the integral of the infinitesimal proper times along the trajectory, or along the worldline. For a curved worldline, i.e. a route which implies changes of velocity, the proper time is shorter than that of a straight line (that is, of motion with uniform velocity) owing to displacements in space and in time from the straight line. This of course implies accelerations.

But…. If we shrink the time during which there were accelerations causing changes of direction or speed, in the direction of sharp turns, the difference between the straight line and the “curved line” does not change much: it is due not to the acceleration but to the long stretches of trip away from the straight line.

Advanced EM - Master in Physics - 2011-2012


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