Derivation of length contraction

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Discussion Overview

The discussion revolves around the derivation of length contraction in the context of special relativity. Participants explore the correct methodology for measuring the length of an object from different reference frames, addressing the nuances of simultaneity and the transformation of events and worldlines.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant outlines a method for measuring length in both rest and moving frames, questioning the validity of their approach due to simultaneous measurements of endpoints.
  • Another participant suggests that the ends of the rod should be considered as worldlines rather than events, indicating a need to transform worldlines instead of events.
  • A different participant emphasizes the necessity of using three events for accurate length measurement, where two events are simultaneous in one frame and the third event is simultaneous in another frame.
  • Concerns are raised about the assumption that two events can suffice for length measurement, with a participant insisting that three events are required to account for simultaneity in different frames.
  • One participant identifies an error in assuming that the length measured in the moving frame can be equated directly when the times are not simultaneous, highlighting the importance of correct simultaneity in measurements.
  • Another participant references the book's treatment of the problem, questioning how the concept of three events can be applied in their context.
  • A participant discusses the implications of equating spacetime intervals in different frames, providing a mathematical perspective on the relationship between lengths measured in different reference frames.

Areas of Agreement / Disagreement

Participants express disagreement regarding the number of events required for accurate length measurement, with some advocating for two events while others argue for three. The discussion remains unresolved as different viewpoints are presented without consensus.

Contextual Notes

Participants note the importance of simultaneity in different frames and the implications of measuring lengths based on the choice of events. There is an acknowledgment of the complexities involved in transforming worldlines and events, which may not be fully addressed in the book's treatment of the topic.

Pushoam
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Problem : To measure length of a scale
Rest frame : The frame w.r.t. which scale is at rest
Moving frame : The frame w.r.t. which the scale is moving with speed v along +ve x-axis

In rest frame ,the positions of the two ends of the scale are (measured simultaneously ) x1 and x2. So length L = x2 -x1
and t1 =t2=t

In moving frame,
the positions of the two ends of the scale are (measured simultaneously ) x'1 and x'2. So length L' = x'2 -x'1
and t'1 =t'2=t

Now,
x'2 -x'1 = γ ( x2 -x1 - v(t2 - t1) )
⇒ L' = γ L as t2 = t1

and

x2 -x1 = γ ( x'2 -x'1 - v(t'2 - t'1) )
⇒ L= γ L' as t'2 = t'1

What is wrong here?
I am asking this as I often do this mistake.

I think,

What I am doing here is I am measuring (x2 ,x1),and (x'2 ,x'1) simultaneously.
So, I am having four events.

What I should do:
I am supposed to have only two events for measuring the length and these two events should be such that these are simultaneous in the moving frame while non - simultaneous in the rest frame. Let's denote the two events as A and B.
Now, the coordinates of A and B are in rest frame [(t2,x2 ),(t1,x1)] respectively and the coordinates of A and B are in moving frame [(t'2,x'2),(t'1,x'1)] respectively and t'1 =t'2
Now,
x'2 -x'1 = γ ( x2 -x1 - v(t2 - t1) )
Now, expressing t2 and t1in prime coordinates using Lorentz transformation and doing further calculation gives L= γ L'
and
x2 -x1 = γ ( x'2 -x'1 - v(t'2 - t'1) )
⇒ L= γ L' as t'2 = t'1

Is this correct?
 
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Pushoam said:
What is wrong here?
I think that the main problem is that you are thinking of the ends of the rod as two events, but they are two worldlines. So you should be thinking of transforming the worldlines rather than transforming events.
 
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Pushoam said:
I am supposed to have only two events for measuring the length and these two events should be such that these are simultaneous in the moving frame while non - simultaneous in the rest frame. Let's denote the two events as A and B.
No, you are supposed to have two pairs of events. As Dale has already pointed out, you have two worldlines - one for each end of the rod.

To do your length measurements, you need three events: A, which is somewhere on the worldine of one end of the rod, and B and C which are events on the worldline of the other end. B and C are chosen so that A and C are simultaneous in one frame and A and B are simultaneous in the other. Now A and B give us the position of both ends of the rod at the same time in one frame, and A and C give us the position of both ends at the same time in the other frame.
 
lc1.png
lc2.png
lc2.png


The book is considering only two events.
How to apply the concept of three events?
 
Pushoam said:
Now,
x'2 -x'1 = γ ( x2 -x1 - v(t2 - t1) )
⇒ L' = γ L as t2 = t1
This is the error x'2-x'1 is not L' when t2 = t1. Because then t'2 <> t'1 so the measuring of the end positions in S' is not at the same time and that's not a correct length measurement.
 
Pushoam said:
The book is considering only two events.

What is your description of those two events?
 
Pushoam said:
The book is considering only two events.
How to apply the concept of three events?
Look at what they were doing when they said "Now as measured in ##S ## these same two point events have ##x## coordinates equal to ##x_1## and ##x_2##, respectively, independent of the time as measured in ##S##." They are taking advantage of the fact that in the rest frame, and only the rest frame, you can skip the "at the same time" constraint.
 
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Consider equating two spacetime intervals for S and S' , A and B
A at rest at x1
B moving to right at v from x0 to x1 l = x1 - x0
From A the time between events is l/v and l = 0
From B the time between events is l0/v and distance = l0
So l2/v2 = lo 2 /v2 - lo2
l = lo √1-v2
 

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