The physics of length contraction
Question 1
- Is length contraction the same if you push or pull an object (rod or space ship)
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Question 2
- What is the time if you move one end of a rod slightly that the other end will also move ? (assume that the length of the rod is 300000 km)
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Question 3
- What is the definition of a rigid body ?
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Background question 1
The reason of question 1 is how easy (or difficult) it actual is to perform and demonstrate length contraction. In actual fact all questions are related to that. In order to answer this question we will use an Experiment
Background question 2
The answer on question 2 is very closely related to question 1. The longer this time the larger the difference between pushing or pulling a rod.
Background question 3
For a definition of a rigid body see:
http://math.ucr.edu/home/baez/physics/rigid_disk.html
It is important to note that in that article they make a distinction between a rigid body and rigid motion. They write: If we accelerate a rod rigidly in the longitudinal direction, then the rod suffers the usual Lorentz contraction.. The question now becomes: How do you really do that. It seems that there two ways to accelerate a rod: one with and one without Lorentz contraction. IMO all of this is not very clear.
Description of an Experiment
In order to answer the first question consider the following experiment which shows what happens if you push an object towards the right. The length of the rod is 300000km and it is assumed that the answer on question 2 is 3 seconds i.e. the speed with which the disturbance moves is 100000 km/sec.
In the following sketch the rod is subdivided in three equal parts: A,B and C
0 <--------><--------><--------> a=0 v=0 v=0
1 <--------><--------><--------> a=1 v=0 v=0
2 <-------><--------><--------> a=1 v=1 v=0
3 <------><-------><--------> a=1 v=2 v=0
4 <-----><------><-------> a=0 v=3 v=0
5 <-----><-----><------> a=0 v=3 v=1
6 <-----><-----><-----> a=0 v=3 v=2
7 <-----><-----><-----> a=0 v=3 v=3
8 <-----><-----><-----> a=0 v=3 v=3
9 <-----><-----><-----> a=0 v=3 v=3
10 <-----><-----><-----> a=-1 v=3 v=3
11 <------><-----><-----> a=-1 v=2 v=3
12 <-------><------><-----> a=-1 v=1 v=3
13 <--------><-------><------> a=0 v=0 v=2
14 <--------><--------><-------> a=0 v=0 v=1
15 <--------><--------><--------> a=0 v=0 v=0
16 <--------><--------><--------> a=0 v=0 v=0
t A B C vA vC
The above sketch shows the behavior of the rod under three conditions:
- Acceleration with a = 1, when you push from the back to the right. The speed increases. The sketch shows that the front is delayed.
- Acceleration is zero and speed is constant. Total length is contracted.
- Acceleration with a = -1, when you pull at the back to the left. The speed decreases.
The sketch shows that the front is delayed.
The next sketch shows what happens if you pull a rod towards the left.
0 <--------><--------><--------> a=0 v=0 v=0
1 <--------><--------><--------> a=1 v=0 v=0
2 <---------><--------><--------> a=1 v=1 v=0
3 <----------><---------><--------> a=1 v=2 v=0
4 <-----------><----------><---------> a=0 v=3 v=0
5 <-----------><-----------><----------> a=0 v=3 v=1
6 <-----------><-----------><-----------> a=0 v=3 v=2
7 <-----------><-----------><-----------> a=0 v=3 v=3
8 <-----------><-----------><-----------> a=0 v=3 v=3
9 <-----------><-----------><-----------> a=0 v=3 v=3
10 <-----------><-----------><-----------> a=-1 v=3 v=3
11 <----------><-----------><-----------> a=-1 v=2 v=3
12<---------><----------><-----------> a=-1 v=1 v=3
13<--------><---------><----------> a=0 v=0 v=2
14<--------><--------><---------> a=0 v=0 v=1
15<--------><--------><--------> a=0 v=0 v=0
16<--------><--------><--------> a=0 v=0 v=0
t A B C vA vC
The above sketch shows (like previous) the behavior of the rod under three conditions:
- Acceleration with a = 1, when you pull at the back to the left. The speed increases. The sketch shows that the front end is delayed.
- Acceleration is zero and speed is constant. Total length is expanded
- Acceleration with a = -1, when you push from the back to the right. The speed decreases.
The sketch shows that the front end is delayed.
Answer question 1
What the above experiment shows, assuming you agree with it, that the effects of pushing and pulling are different. It demonstrates that the effects depent what you do first.
- if you push first and then pull length contraction is observed.
- if you pull first and then push the opposite effect is observed i.e. length becomes longer.
The cause of both effects is that mechanical movement "slowly" propagates through the rod.
You could argue that the above demonstration shows what happens when you use a non rigid body. In case of a rigid body this is not the case and than propagation acts instantaneous.
The problem is than that there is then no length contraction
Reflection
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Created: 4 February 2001
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