Cosmic Adventure 5.5 Relativistic Length Contraction

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RELATIVISTIC LENGTH
CONTRACTION
Cosmic Adventure 5.5

A Length and a Spatial
Coordinate Separation
First we need to distinguish
between a “length” and a
“spatial coordinate separation”.
A “length” is the measurement
of the dimension or coordinate
difference of a solid body like
a rod or a ruler. A “spatial
coordinate separation” is the
measurement of some empty
space. In our discussion, we
concentrate on length.
0’
𝑥′
0’’
𝑥′′
0’
𝑥′
0’’
𝑥′′

Above: Metric system Below: Imperial system
Length
Representation –
The Ruler
A ruler is the best
material to represent a
length on a one
dimensional scale.
There are many types
of rulers. The one we
are going to use is the
universal one. It has
no units. So it can be
of any scale or length.
Normal Ruler
Universal Ruler

Transformation of a
Length
One of the well-known
magic of the relativists is the
ability to squeeze a long rod
or ruler into a shorter one. It
is called ‘length contraction’.
It happens only in the
direction parallel to the
direction in which the
observed body is travelling.
T r a n s f o r m ! ! !

Rest Length or Proper Length
The length of a rod or ruler can be determined by direct measurement
when it is at rest with respect to an observer. The resultant length of the
ruler at rest is called the ‘rest length’ or ‘proper length’ of the body.
proper length 𝐿0

Length in Classical Physics
In classical physics, in addition to direct measurement, another way to
determine the length of the ruler is by calculating the difference between
the spatial coordinates of the endpoints of the ruler.
Here 𝐿0 = 𝑥′′ − 𝑥′.
0’
𝑥′
0’’
𝑥′′𝐿0

Coordinates of the Ends
According to the Theory of Special
Relativity, the coordinates of the
ends in relativity are formed by
Lorentz transformation:
𝑥′ =
𝑥′′
+ 𝑣𝑡
1 −
𝑣2
𝑐2
𝑥′′ =
𝑥′
+ 𝑣𝑡
1 −
𝑣2
𝑐2
0’
𝑥′
0’’
𝑥′′𝐿0
𝑥′′
+ 𝑣𝑡′′
1 −
𝑣2
𝑐2
𝑥′
+ 𝑣𝑡′
1 −
𝑣2
𝑐2

Length in Relativity
Thus the length is:
𝐿0 = 𝑥′′ − 𝑥′
=
𝑥′′
+ 𝑣𝑡′′
1 −
𝑣2
𝑐2
−
𝑥′
+ 𝑣𝑡′
1 −
𝑣2
𝑐2
=
𝑥′′ − 𝑥′ + 𝑣 𝑡′′ − 𝑡′
1 −
𝑣2
𝑐2
0’
𝑥′
0’’
𝑥′′𝐿0
𝑥′′ − 𝑥′ + 𝑣 𝑡′′ − 𝑡′
1 −
𝑣2
𝑐2

Measured by
Different Viewer
𝑥′′
− 𝑥′
= 𝐿0
Is the proper length of the
ruler as measured by 𝑂′.
𝑡′′ − 𝑡′ will be zero if 𝑥′′
and 𝑥′ are measured by O at
the same time.
Then 𝑥′′
− 𝑥′
will be the
length 𝐿 of the ruler
measured by O.
𝐿0 = 𝑥′′
− 𝑥′
𝑥′′
− 𝑥′
+ 𝑣 𝑡′′
− 𝑡′
1 −
𝑣2
𝑐2
𝑡′′ − 𝑡′ = 0
𝐿 = 𝑥′′ − 𝑥′
Measured by O
Measured by O’

Lorentz contraction
With 𝐿0 as the proper length and
𝐿 as the measured length, the
relationship between the two
measurements is:
𝐿 = 𝐿0 1 −
𝑣2
𝑐2
L = changed length
𝐿0 = 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ
𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑐 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡

𝐿 = 𝐿0 1 −
𝑣2
𝑐2

Lorentz contraction
Since 1 −
𝑣2
𝑐2 is smaller
than 1, 𝐿 is greater than 𝐿0.
This means that there is a
phenomenon of length
contraction.
Historically, it is called the
Lorentz-Fitzgerald
contraction or simply:
Lorentz contraction
Hendrik Lorentz (1853-1928)

Velocity as a fraction of the speed of light
𝐿 = 𝐿 𝑜 1 −
𝑣2
𝑐2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.0
0.8
0.6
0.4
0.2
0.0
LengthContraction
Graph of Length Contraction

This effect is negligible at everyday
speeds, and can be ignored for all
regular purposes. Only at greater speeds
does it become dominant.
Velocity as a fraction of the speed of light
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.0
0.8
0.6
0.4
0.2
0.0
LengthContraction
𝐿 = 𝐿 𝑜
𝐿 = 0

Example 1
Example 1: At a speed of
15,000 km/s or 0.05c,
contracted length is 86.6% of
the length at rest;
𝐿 = 𝐿 𝑜 1 −
15,0002
30,0002
= 0.866 𝐿 𝑜
𝑣 = 0.0 𝑐
𝑣 = 0.4 𝑐
𝑣 = 0.6 𝑐
𝑣 = 0.7 𝑐
𝑣 = 0.99 𝑐

Example 2
At a speed of 29,500 km/s or
0.983c, contracted length is
1% of the length at rest;
𝐿 = 𝐿 𝑜 1 −
29.5002
30,0002
= 0.01 𝐿 𝑜

Is Length Contraction Real?
But is this contraction real?
Some say yes and some say no.
Even the great scientists could
not make up their minds.
In 1911, the Serbian physicist
Vladimir Varićak (1865-1942)
had to assert that length
contraction is "real" according to
Lorentz, while it is "apparent or
subjective" according to Einstein.
Wikipedia.

Einstein was not that certain either,
he replied:
. . . The question as to whether
length contraction really exists or
not is misleading. It doesn't "really"
exist, in so far as it doesn't exist for
a comoving observer; though it
"really" exists, i.e. in such a way that
it could be demonstrated in
principle by physical means by a
non-comoving observer.
— Albert Einstein, 1911 - Wikipedia

Proper length 𝐿 0
Contracted length 𝐿
The phenomenon of
Length contraction is
Allegedly Real
The body of the ruler or
any length will be
shortened due to the
transformation of the
equations. The higher
the velocity, the more
shortening there will be.
Relativists maintain that
this effect is physical and
real.

Cannot be Definitely
Explained
Length contraction cannot be
explained in a clean way by
common sense because the
theory is based on the
assumptions of the Michelson-
Morley experiment which to us is
still controversial. However,
because the theory of Special
Relativity had many apparent
achievement in other aspects,
the argument keeps on being hot
until these days.

TIME DILATION IN RELATIVITY
To be continued on Cosmic Adventure 5.6

Cosmic Adventure 5.5 Relativistic Length Contraction

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