by Filippos Atlasis
In this article, I will be explaining an effect that occurs from Albert Einstein's famous Special Theory of Relativity: Length Contraction. My knowledge about this topic was primarily acquired while completing the Special theory of Relativity course offered by Standford university.
An introduction on length contraction
Length contraction is a concept which was
firstly introduced by Albert Einstein as a part of his special theory of
relativity. Einstein’s special theory of relativity was published as an
extension paper to one of his other publications, “The Electrodynamics of
Moving Bodies”, during the so-called “Miracle Year” in 1905. Although it might
come as a surprise to many that Einstein never won a Nobel prize for either of
his theories of relativity (special or general), his work can’t be undermined
as it is proven correct time and time again in the years following its
publication. Today, the effects of the special theory of relativity (including
length contraction) are used in many aspects of modern life e.g Satellites
which control the Global Navigation System.
What is Length contraction and When does it occur?
Length contraction is a consequence of time
dilation. Length contraction states that the measurement of length is different
for different observers that are moving with respect to each other. Measuring
the length of a moving object will give a result of a shorter length to when
the length of the object is measured whilst the object is at rest. All the
effects of the special theory of relativity are applied when two objects move
with respect to each other while maintaining a constant velocity/being in an
inertial frame of reference. This means that if one or more of the objects
accelerate/be in a non-inertial frame of reference, the special theory of
relativity can’t be used to describe the effects on time and length. However,
the effects of length contraction cannot be observed in our everyday world,
because objects need to move with velocities close to that of light, and thus their
effects are close to none. In most of our common everyday experiences, we can
use the classical Length=ΔTime*Velocity equation to calculate a length. When an object’s velocity reaches
significantly close to the speed of light, this classical equation won’t give a
correct result as the effect of length contraction is significant enough to
change our observations. As each object/person is in their own frame of
reference, a person would always see the other’s length to contract because both
of them would see themselves as being at rest and the other moving.
Do all measurements of “distance” (e.g width) contract?
One key concept about the special theory of
relativity is the fact that several measurements of distance aren’t affected by
length contraction; they remain invariant. In fact, the original name for the
special theory of relativity was the “Theory of Invariance” because several
quantities used in calculations remained constant. For example, the width of an
object does not contract; there isn’t a width contraction effect. To justify
this, imagine a train on tracks passing by a station platform. If the train
could reach a speed very close to that of light, an observer standing on the
station platform would see the length of the train contract as it moves past.
However, if the observer saw the width of the train contract, suddenly the rails
of the train would be off the tracks and the train would most likely crush. A
person standing inside the train rather than observing it from the station
platform would run into similar problems. For the person inside the train, the
train is at rest and the rails are moving towards him at a speed very close to
that of light. As the rails move towards him, he observes their length to
contract. If he was to see their width contract as well, the rails would
suddenly overextend off the tracks and again the train would most likely crush.
Similarly to a width contraction effect, the person inside the train would
observe an unfortunate event if the height of the object moving towards him contracted.
Now, along with the tracks moving towards the person inside the train, imagine
a tunnel coming towards the train at a speed very close to that of light. If
the height of the tunnel was to contract, the train wouldn’t be able to fit
through. On the other hand, the observer on the station would see the train
easily fit through the tunnel because the height of the train would contract.
In both cases, the observer on the platform and inside the train do not agree
on the photo clock principle. This means that if a photo was to be taken at an
instant moment in time, the two observers wouldn’t agree on the apparent
position of the train.
How do objects contract?
In order to understand how objects contract, we have to imagine a situation where Person A is found inside a rocket travelling close to the speed of light and Person B is an observer outside of the rocket. The rocket of Person A is travelling at a velocity v. As it passes by Person B, the rocket accelerates slightly from v to v + Δv*[1]. To understand how objects contract it is essential to know the relativity of simultaneity and the concept of “leading clocks lag”. In short, if we place two imaginary clocks, one at the rear and one at the front of the rocket, Person B (the outside observer) would see the clock at the front of the ship fall behind the clock at the rear of the ship. Therefore, Person B would observe the rear part of the ship accelerate before the front of the ship. this slight acceleration at the rear of the ship would cause it to move before the rest of the ship and so start to compress towards the front. If the whole ship were to accelerate simultaneously for Person B, then each part of the ship would move together and so there would be no compression as the rear isn’t the part that is moving first. It is helpful to imagine that the particles that make up the ship would maintain fixed distances if the whole ship was seen to accelerate simultaneously, but in reality Person B observes the rear accelerate before every other part and so the particles found there “compress” towards the front.
How do you work out how much length contracts?
As mentioned above, the classical equation for
calculating the length of an object is Length=ΔTime*Velocity where an
observer observes the length, time and velocity of an object. However, as the
object travels very close to the speed of light, the observer wouldn’t see the
time on his/her run in sync with the time on the clock of the object because of
time dilation. In order for the observer to calculate the passage of time on
the object compared to his/her clock, he/she would have to use the following Time
Dilation equation:
Time (object)=1/γ * Time(observer)
where γ
is known as the Lorentz factor. If we substitute this into the
classical equation for length, we can derive the Length contraction equation:
Epilogue
It has to be noted that all the effects of
special theory along with length contraction aren’t simply illusions to our
eyes; they happen for real. For example, going back to the people in the train
and train station, the person inside the train would indeed travel a shorter
distance because the length of the tracks in front of him contract and the person
on the train station would find out that the length of the train has contracted
if he measured its length properly.
References
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