cosmic adventure 5.5 relativistic length contraction

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RELATIVISTIC LENGTHCONTRACTION

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′ = 𝐿0Is 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

𝑐2is 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 contractionHendrik Lorentz (1853-1928)


Velocity as a fraction of the speed of light

𝐿 = 𝐿𝑜 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

Leng

th C

ontr

acti

on

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

Leng

th C

ontr

acti

on

𝐿 = 𝐿𝑜

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 = 𝐿𝑜 1 −15,000

2

30,0002

= 0.866 𝐿𝑜

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 = 𝐿𝑜 1 −29.5002

30,0002

= 0.01 𝐿𝑜


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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