And now, having finished the rehearsal on simultaneity, let us
continue with the trip to Canopus and its mysteries
.
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
d ( LI )
d ( t 2  s2 )
dt
 d   dt dt  
dt
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.
Advanced EM - Master
in Physics - 2011-2012
1
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
2
10
5
B
P
0
O
In this drawing 3 different worldlines are
shown, which represent 3 different trips
from O to B: OPB, OQB and OLB.
L
Q
4
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.
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
3
In conclusion of all that
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
4