Length Contraction, Time Dilation, and the Twin Paradox.

Rolf Muertter

May 12, 1994


In the history of scientific thought, physics and philosophy were not always clearly
distinct disciplines. However, even after they have become established as separate
disciplines, knowledge of the interactions between both fields is important in understanding
their development. A clear illustration of this interdependence is given by Einstein's special
theory of relativity, published 1905. Einstein was deeply impressed by Mach's positivistic
philosophy, which led him to regard only measureable quantities as physically meaningfull.
This allowed him to show that such quantities as time, length, velocity, mass, and energy
cannot be defined absolutely, but depend on the state of motion of the observer. This, in
turn, is philosophically significant since it shows that the ordinary notion of these
fundamental quantities as being absolute is wrong. Meter sticks don't really exist per se,
only gammameter sticks, where v is the velocity of the observer with respect to the stick
(in one dimension), and c is the speed of light. Clocks tick only then at the same rate,
if they are at rest with respect to each other; more generally,

time dilation
 where t0 is the time
of the moving clock. These phenomena are called length contraction and time dilation. Another
interesting consequence of STR is the twin paradox: If a twin leaves her sibling, she will be
younger when she returns.
The basis for STR is the constancy of the speed of light and the assumption that nothing can
move faster than light. This is not an issue in Newtonian mechanics, where anything can have
any speed, in principle (except infinite speed). It was assumed that light was a wave with
the aether as the medium. Newton had assumed that light consisted of particles, but later it
was shown that light is an electro-magnetic wave. Now a wave needs a medium to propagate,
and since it even reaches us from distant stars, it must fill all of space and be otherwise
unnoticeable. Einstein, as a good Machian, never assumed the existence of the aether. As
a consequence, the speed of light must be constant and the same for any observer, regardless
of his state of motion. Based on this assumption, it can then be shown mathematically that
it would take an infinite amount of energy in order to accelerate a body to the speed of light.
In Newtonian mechanics, space and time are independent of each other - there are three spatial
dimensions and one time dimension. Spatial intervals, dr^2 = dx^2+dy^2+dz^2 are measured
idependent of time intervals, dt. However, the fact that length and time are functions of
the state of motion of objects and observers suggests that they cannot be treated independently.
Minkowski was the first to mathematically unify space and time into four-dimensional space-time.
An interval in Minkowski spacetime, ds^2 = dx^2+dy^2+dz^2-c^2dt^2, is a mixed interval, combining
space and time intervals.
Length contraction, time dilation, and the twin paradox can easily be demonstrated
mathematically, but it is very instructive to demonstrate them graphically, using Minkowski
diagrams. A Minkowski diagram is a plot of time versus space. Since it is difficult to
display four dimensions on two-dimensional paper, only one spacial dimension is displayed (fig.1).


Fig.1


The path of a particle in spacetime is called it's world line. If the particle is at rest,
the line is vertical; if it moves with the speed of light, the line makes an angle of 45o
with the x-axis, since the vertical axis is ct instead of just t. A curved line indicates
acceleration.
In order to compare how one observer sees another observer's clock or meter stick,
two coordinate systems are used, one for each observer, in which they are at rest. These
coordinate systems are inertial frames which means that they are not accelerating. Here it
is assumed that observer O' is moving with velocity v in the x-direction. ct' and x' are O''s
coordinates. So how does observer O see O''s coordinates? Clearly, O''s world line (ct'-axis)
makes an angle of less than 45o with the ct-axis. It is less clear how the x'-axis should be
drawn. The Lorenz transformatons can be used to get the exact equations for the primed axes
in terms of the umprimed axes:

equation 2 and 3

where b = v/c. For the ct'-axis, set x'=0. This gives vt = x or ct = x/b. For the x'-axis,
set ct'=0, which gives ct = bx (fig.2).


fig 2

Fig.2


In the appendix it is shown that q = f. The Lorenz transformations can also be used to
figure out how the scale changes as we go from the unprimed to the primed system. For this
the inverse transformations,

equ 3 and 4

 are also needed. To do this, the
hyperbolae c2t2 - x2 = +1 are drawn with the primed and unprimed coordinate systems (fig.3).



fig 3
Fig.3


The hyperbolae c2t2 - x2 = 1 (H1) and x2 - c2t2 = 1 (H2) are used because they give unit
spacetime distance to the origin O (s = 1). This is why the hyperbolae are also called
calibration curves. To see how the calibration works, consider event P1, which is the
intersection of H2 with the x'-axis given by ct = bx. Combining these equations gives the
coordinates of P1,

equ 5 and 6

Comparing equations 5 and 6 with equations 3 and 4 shows that equations 5 and 6 represent
unit length (x' = 1) and zero time (ct' = 0) in the O' frame. In other words, the interval
OP1 gives unit length along the x'-axis. Similarily it can be shown that the interval OP2
represents unit time along the ct'-axis. The result of this calibration is shown in figure 4.



fig 4

Fig.4


In figure 4, b = 0.5, which means that the primed system is moving away from the unprimed
system at half the speed of light. The calibration shows that the unit intervals of the
primed system are longer than the unit intervals of the unprimed system. It is easy to see
how the coordinates of a spacetime event P differ for the two systems. If the coordinates of
P in O are (3, 2.3), they will be (2, 1.2) in O'. Mathematically speaking, the Lorenz
transformations transform an orthogonal system to a nonorthogonal one.
To demonstrate length contraction, a meter stick is placed at rest in O with the
ends of the stick at x=3 and x=4 (fig.5). As time goes on, the ends of the stick trace out
vertical lines parallel to the ct-axis. The stick is parallel to the x-axis because a length
measurement means that the positions of the two ends of the stick are determined simultaneously.
All lines parallel to the x-axis are lines of simultaneity (ct = const.) in O. Lines parallel
to the x'-axis are lines of simultaneity in O'. It is thus clear that events that are
simultaneous in O are not simultaneous in O'. As a matter of fact, length measurements in
O' are different spacetime events from length measurements in O. For example, the events
E1 and E2 constitute a length measurement of the meter stick in O. However, these events
are not simultaneous in O' and cannot, therefore, be used for a length measurement of the
meter stick in O'. A possible pair of events


fig 5

Fig.5


that can be used is E2 and E3. The length measured in O' can then be read off the x'-axis
and is not one meter, but 0.87 meters (for b = 0.5). Therefore, the meter stick appears
to be contracted to an observer in O'.
It has been assumed that O is at rest and O' is moving away from O at half the speed
of light. What this means is that we are pretending to be observers in O. Actually, it is
impossible to tell which frame is the moving one, and which one is at rest. What counts is
the relative motion. The symmetry of the situation suggests that if the meter stick were at
rest in O', it should appear contracted in O. This is indeed the case, as shown in figure 6.
The ends of the meter stick are at x' = 3 and x' = 4. The world lines of the ends of the
stick trace out lines parallel to the ct'-axis, as time moves on. The length as measured in
O is the intersection of the stick's world lines with the x-axis, and again comes out to be
0.87 meters.



fig 6

Fig. 6


Time dilation can also be demonstrated geometrically. A clock at rest in O appears to
tick slower to an observer in O' as shown in figure 7. Also shown is the reciprical nature
of time


fig 7
Fig. 7


dilation. In order for an observer in O' to measure the interval between two ticks of the
clock at rest in O, two clocks are needed, one at x'1 and the other at x'2. The time
interval that they record between ticks t1 and t2 can be read off the ct' axis, and is
larger than one. A clock at rest in O' ticks one unit of time in O', as can be read off
the ct'-axis, but the two clocks in O that record the time of the ticks t'1 and t'2, measure
the interval between the ticks to be more than one unit, as can be read off the ct-axis.
The spacetime geometry of the twin paradox is shown in figure 8. Let's assume that
twin e remains at home while twin d travels with a speed of 0.8c away from e. If d's clock
records 3 years for


fig 8
Fig.8


the journey, e will have aged 5 years. Now e sees d's clock tick at a slower rate, but
this cannot explain the age difference upon reunion since d also sees e's clock tick slower
while she recedes from and approaches e. Time dilation and length contraction, which cause
the age difference, are symmetrical phenomena, whereas the twin paradox is asymmetrical.
This is what makes the twin paradox paradoxical. So what, then, is responsible for breaking
the symmetry? It is simply the fact that one of the twins must change direction in order
to return to the other twin. Actually, the longer the spacetime path taken to get from A to B,
the shorter the proper time, that is the time measured by the traveler, will be. Perhaps this
doesn't make sense intuitively, but can be shown mathematically as follows. Let's compare
proper time for two


fig 9
Fig.9


for two spacetime paths between A and B. The shortest one is Dt1, the longer one is Dt2.

equ 7
 
can also be written as

equ 8

The elapsed proper time along the first world line is then

equ 9
This is clearly longer than Dt2 since dx2 is always positive.
It is often argued that the cause of the asymmetry is the fact that one of the twins is
accelerated. This, however, does not solve the triplet paradox, where no body is actually
accelerated. If triplet 2 recedes from triplet 1 and then encounters triplet 3, who is
heading for triplet 1, triplet 3 will be younger than triplet 1, assuming that triplet 3
synchronized his clock with triplet 2 during their encounter. Thus it seems that all that
needs to be invoked is a change in the direction of motion, irregardless of wether a body
has actually felt an acceleration. However, the triplet paradox can probably be solved
by invoking acceleration since triplet 3 in some point in time had to turn around in order
to head back to triplet 1.
So far the arguments have been fairly abstract, and the question is whether the phenomena
of time dilation, length contraction and the twin paradox are real in the sense that they
could actually be observed. The problem is that these phenomena become important only
at extremely high speeds, close to the speed of light. However, several experiments have
confirmed the reality of these effects. For example, without time dilation, cosmic ray
muons would never make it to the ground, since the half life is too short. In another
experiment, one atomic clock, having circled the earth in an airplane, has been found to
go slower than a second atomic clock that remained on ground, thereby experimentally
confiming the reality of the twin "paradox".
As far as length contraction goes, the question is what is meant by a real physical
phenomenon. Of course, objects are not really sqeezed together in the sense that stress
or strain forces could be measured. But in three dimensions, length contraction might
not even be apparent. According to Terrel, objects sometimes appear to be contracted, but
only because they are rotated. A sphere, for example, would appear to be rotated - not
distorted to an ellipsoid. Even after almost a century of STR, the subject of how moving
bodies would appear if c = 100m/s remains contraversial.







Appendix


proof





References

1. Resnick, Introduction to the Special Theory of Relativity.

2. Arya, P. Atam. Introduction to Classical Mechanics. Allen and Bacon, 1990.

3. Ray, Christopher. Time, Space, and Philosophy. Routledge, 1991.