Cosmic adventure 5.9 Length Contraction at Rest in Visonics

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THE PERCEPTION OF A LONG IMAGE AT REST

Cosmic Adventure 5.9


Looking at a Ruler in Olden Days

We are looking at a ruler at a distance.

In olden days when light speed is not considered, we could see the ruler in its entirety at any time. But now a finite light speed has to be considered, the situation is slightly different.


A B

Continuous chain of emitted images from B

Continuous Emission or Reflection from Ruler

In taking the speed of light into consideration, we must first recognize the fact that continuous emission from the body of the ruler happens all the time.

Continuous chain of emitted images from A


A B

Images from B

Only simultaneous images become Visual

But in actual observations, only those images which reach the eye simultaneously will be visually perceivable.

Images from A


So the sighting of a long object relies only on those images which reach the eye at the same time (simultaneously) will be investigated. This kind of simultaneity is denied by the Theory of Relativity.

A B

Concurrent images that form the whole image of the ruler

Simultaneous Sighting

B

A


Ruler at Rest

In the case of studying a length which is not moving, we start with a surveyor looking at a stationary ruler of length 𝑙0 at a distance 𝑠 through a telescope.

A B

𝑙0𝑠


Vison in classical optics

The visual appearance of an object is determined by the light from all points on the objects that arrives simultaneously at where the eye is.


In all scales

It takes place in all scales of material bodies or environment, from a microbe to a galaxy.


Images of Different Times

To have a visual image of the entire ruler, both images of ends A and B have to arrive at the observer simultaneously.

A B

BA



This image of B will covered the distance 𝐿0/𝑐 in time 𝑡0.

B

Earlier image of B

B

A

𝑡0 =𝐿0𝑐



Then this image of B will travel together with an image of A together to the Observer.

B

Earlier image of B

Image of A

B

A

𝑡0 =𝐿0𝑐


End B at 𝑡1 + 𝑡2 ago

A B𝑙0

𝑡2 =𝑙0𝑐𝑠 = 𝑐𝑡1

End A at 𝑡1 ago

This image of B arrives at the same time with the image of A because it was emitted earlier

I can see the whole image of the ruler

Historic


That is how we are looking at a long object with ends of different history.

End B at𝑡1 + 𝑡2 ago

End A at 𝑡1 ago

𝑡1 =𝑠

𝑐𝑡1 + 𝑡2 =

𝑠

𝑐+𝑙0𝑐


Ruler at Rest

For a rigid ruler at rest, the observer does not see any difference in length because the images are flowing in a continuous stream. The old images and new ones are in the same position. There is only a difference in time between the images of each end.



If the image of end A takes time 𝑡1 to reach the surveyor, the image of B has to take a bit longer. This longer time is 𝑡2 which takes light to go from B to A, that is 𝑡2 = 𝑙0/𝑐.

A B𝑙0

𝑡2 =𝑙0𝑐𝑐𝑡1

Earlier image of B

Image of A


𝑡1 =𝑠

𝑐𝑡1 + 𝑡2 =

𝑠

𝑐+𝑙0𝑐

The difference in time is:

𝑡1 + 𝑡2 − 𝑡1 =𝑠

𝑐+𝑙0𝑐−𝑠

𝑐=𝒍𝟎𝒄

For a ruler at rest, the actual length experiences no change under visual observation. There is only time difference if clocks are present.


A Ruler in Motion

However for a ruler that is on the move along the line of sight, the situation is different, because the position of the ends are changing all the time.


MEASURING LENGTH IN MOTION

To be continued on Cosmic Adventure 5.9

Cosmic adventure 5.9 Length Contraction at Rest in Visonics

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