Problems with Length Contraction
Special Relativity (SR) tells us that moving objects should shorten in length along their
direction of travel. E.g. assume that an electron has a spherical shape. When at rest
relative to an observer, the electron appears spherical as expected. Then as the electron
moves away from the observer, it appears to shorten. And as it approaches the speed of
light it will appear to flatten out like a discus, approaching zero length as the speed
nears that of light.
This is called Length Contraction (LC). The amount of contraction will be in accordance
with the Lorenz Transform (LT), represented by the symbol γ. The below diagram gives an idea of
what might happen to the electron at different speeds.
The speedy rocket ship
LC is normally not described using single particles but with larger
objects such as a high-speed rocket. The claim is that when the rocket is still it has a
certain length, and when the rocket moves at 0.87c it shrinks to half
that length (relative to an observer).
Given that a rocket is not a single particle but made of many particles
(atoms) the above claim appears to raise an anomaly. What it is saying is that not only do
the atoms shrink but the space between atoms also shrinks, resulting in a shorter rocket.
Somehow the LT has been applied to the space between the atoms. But how can empty space
shrink, especially when space is not a physical object and therefore cannot be considered
to be moving?
The reason for this actually has to do with another aspect of SR called repositioning,
which will be discussed later. For now though we can consider the particles making up the
rocket (atoms) move in unison and thus form a super particle to which we apply
the LT. So lets just take that as being the case and proceed from here.
LC seems easy to picture when a single particle is involved. However a
problem emerges when several moving particles are considered. Lets say we have two
electrons a fixed distance apart and initially at rest. They might look like this:
Now the electrons begin to accelerate at an identical rate until they
reach a speed of 0.87c. At this speed the LT tells us that they should
shrink to about half their length, i.e. γ=2.
Heres a question: according to an external observer, which of the below diagrams
(A or B) now best represents the electrons?
The correct answer must be B. If the distance from the front to the back
of each electron decreases by half, then the distance between the two electrons must also
shrink by half. Both electrons are at rest relative to one another and, like the rocket
above, they form a moving super particle which also needs to reduce its length
Now lets say we have a steady stream of equally-spaced electrons in a circular
particle accelerator. Initially the electrons are at rest and the situation looks like
A small tube sits within the accelerator represented by the red
section at the base. The tube has a length that covers the spacing of two electrons such
that as a new electron enters, an old one exits, and a third electron sits at the mid
point. When the electrons are at rest the tube situation looks like this (for convenience,
the tube is shown flattened out):
Now lets power up the accelerator; applying equal acceleration to
all the electrons until they reach a speed of 0.87c. At this speed the LT
predicts that each electron should shrink to about half its length.
Question: according to an observer in the rest-frame of the laboratory, which of the
following diagrams: A, B or C, best represents the new situation
inside the tube?
(A) shows the electrons shrinking but their spacing staying the same.
(B) shows both electrons and spacing shrinking but the tube staying the same.
(C) shows the electrons, spacing and tube all shrinking.
(A) cannot be correct because the electrons are all moving together as a super
particle so the distance from the front to the end of this particle must also
(B) is correct as far as electrons go, but incorrect as regarding the tube because
electrons must line up with the each end of the tube.
(C) satisfies the requirement for shrinkage of the electrons as a whole while keeping them
matching the tube, but is obviously incorrect because it implies that the accelerator must
also shrink with the tube.
So the correct answer must be: None of the above.
Since the accelerator is circular and the spacing between each electron is equal, it is
not possible for all of the electrons to move closer together without shrinking the
accelerator also. Considering that accelerators are not observed to shrink it seems that
LC could not possibly occur.
A second question regards the direction of shrinkage. LC tells us that
shrinkage should happen along the direction of travel. Consider the following diagram:
Three balls A, B and C are arranged on the points of an equal-sided
triangle and move at equal speeds away from its centre. Suppose we wish to know the
contraction of ball C. First lets consider A and C in isolation. Without having B
available as a reference, A and C are moving directly away from each other. So from the
point of view of A, ball C should look like this:
Now consider B and C in isolation. Without having A available as a
reference, B and C are moving directly away from each other. So from the point of view of
B, ball C should look like this:
This is obviously a problem, since ball C cannot be simultaneously shaped
like both of the above diagrams.
By how much?
A third question relates to the amount of shrinkage relative to an observer.
According to SR all velocities are relative and therefore the amount of shrinkage can be
different for different observers.
For example, suppose we have a rod of fixed length and say it is at rest. Next we take
two observers moving length-wise toward the rod but at different velocities. The first
observer moves at 0.87c (where γ=2)
and the second at 0.94c (where γ=3).
According to the rules of LC, the first observer will see this:
While the second observer sees this:
Again we have a problem because the rod cant have both these
Length Contraction and Repositioning
Well now get to the nuts and bolts of what is said to be causing LC.
Under SR a moving particle will appear to reposition itself according to the formula:
x' = x/γ
Where x is the position the particle would be in if it were not moving, x' is
the position it appears to be in once it starts moving, and γ is the LT. Another way of looking at
it is: x is the position according to Newtonian (classical) mechanics and x'
is the position according to SR .
To see how this leads to LC, consider a rod of length L located a
distance x from an observer as shown:
Initially the rod is not moving, so SR doesnt apply and everything
appears normal. We now set the rod in motion at 0.87c, which makes γ=2. From the view-point of the
observer, the front of the rod repositions itself from x to x/2. Meanwhile
the end of the rod repositions to (x+L)/2, which is x/2+L/2.
The net result is this:
As can be seen, both the distance to the rod and the rod itself has
contracted by a factor of 2. The same applies to all the particles (atoms) within the rod:
they reposition themselves and give the impression that the rod overall has shrunk. For
this reason relativistic length contraction should perhaps more properly be
referred to as relativistic repositioning.
So what about other situations such as when a particle is instead moving toward the
observer or moving at an angle?
In the case where a particle (or rod) is moving toward the observer, it is the treated the
same as if moving away. That is, it moves toward the observer and contracts
along its length by the same amount. This is because the LT function contains a
v2 term, which makes direction unimportant.
In the case where the particle is moving at an angle, i.e. not directly toward or away from
an observer, it is a little more complex. Basically though, you need to trace a line along
the direction of motion until it crosses a line at right angles that passes through the
observer. The length of the first line will be what is used in the calculations. The below
diagram explains this. It shows an objectís position according to classical mechanics
(without SR), and its new position as will be seen according to SR.
Note that the contraction and repositioning occurs along the direction
of travel rather than in the direction of the observer.
Is the Rocket Near or Far?
An interesting story can be told about repositioning.
A rocket starts on Earth and carries an enormous quantity of fuel that allows it to steadily
accelerate at 10m/s2. At this rate it would reach light speed in just under a
year, although we will impose an SR restriction (such as mass-increase) that prevents it
from quite reaching that speed.
After 35 days of travelling it has reached 10% of light speed and is 4.5x1013m
away. After 6 months it is moving at half the speed of light and is 1.2x1015m
away. However due to repositioning it appears closer, at 0.87 times the distance or around
At the 10 month mark it is 87% of light speed and 3.4x1015m away. Yet due to
repositioning, appears to be at half that distance. At the 11.5 month mark the rocket
reaches 99.5% of light speed. It is 4.4x1015m away but appears at only one tenth
that distance. We continue in this manner. When the rocket is at 99.99995%c, the actual
distance is still around the same, but the perceived is 1/1000th of that at
4.4x1012m. We shift to (1-5x10-13)c. The perceived distance is now
one millionth of the original, at 4.4x109m.
Finally we get to (1-2.5x10-30)c. At this point the rocket appears to be only
10 metres away and is extremely compressed along its length. So... build a roof over the
rocket and capture it!
Now obviously this wonít be possible. The people on the ground will see the rocket 10m
above them. Whereas a builder atop the construction would see the rocket 10m above his
head, rather than below the roof. Hence the rocketís apparent location is illusory. But
the above story does serve to call into question the LC hypothesis because it requires the
roof to be obscured by the rocket.
Now lets consider what experiments have to say about LC. Measuring
LC is no easy task because SR tells us that a moving objects length immediately
returns to normal once it stops. And observing a particle in motion is also difficult as
that would normally require bouncing a light beam off the object; and the beam would have
trouble reaching a particle moving at near light speeds.
So what do experiments tell us about LC? Here we draw a blank. Several experiments have
been either tried or proposed, but they failed to measure that a variation in length
occurred , . To date no experiment has been
able to confirm the existence of LC.
Although there is one test which is said to demonstrate LC in terms of magnetism. It
is discussed in a supplemental chapter:
Magnetic Test of Length Contraction (<-- click to read)
The phenomenon of length contraction is said to be responsible for a number
of effects; some of which might be possible from the viewpoint of a single observer, but
would be contradictory from the viewpoint of multiple observers.
There may be a semantic issue here. Namely, is length contraction an
actual contraction or just an observed contraction as seen from a stationary position? If
it is only an observed contraction then it is merely an optical illusion, and is of little
Alternately, if length contraction is purported to be an actual contraction, then based on
the argument that it is not possible to satisfy both the length and spacing requirements
of moving particles relative to a fixed stationary length (see above comments about
circular accelerator), and the lack of experimental evidence to the contrary, I submit
that length contraction does not occur.
 Actually the equation is normally expressed in the
form x'=(x0+vt)/γ where x0
is the initial position, v is velocity and t is the time taken to get to the
 Wood, A.B., Tomlinson, G.A., and Essen, L. 1937.
"The Effect of the Fitzgerald-Lorentz Contraction on the Frequency of Longitudinal
Vibration of a Rod," Proc. Royal Soc., 158, 606-633.
 Renshaw, C. 1999. "Space Interferometry Mission as a Test of
Lorentz Length Contraction," Proc. IEEE Aerospace Conf., 4, 15-24.