## Changing Length - part 1

### Question 1

Is the length of train, when the speed drastically increases, getting any shorter?

### Question 2

How many lamps are ON: 0, 1 or 2?

### Question 3

Is the answer on question 2 dependent on the "whole" speed of the experiment setup?

### Question 4

When I perform this experiment on the northpole and when the train moves in the same direction as the Earth rotates, is the length of the train getting shorter ?

### Question 5

When I perform this experiment on the northpole and when the train moves in the opposite direction as the Earth rotates, is the length of the train getting shorter ? Is the answer on question 4 identical as on question 5 ?

### Purpose of the questions.

The purpose of the questions 1 and 2 is to challenge Length Contraction . This is done with a thought experiment. The idea of the thought experiment is to mimick a real experiment i.e. the reality as close as possible.
Length contraction is the effect that length of a rod decreases, as described by Lorentz Transformation: l = l0 * SQR(1 - v˛/c˛)
In order to make the thought experiment visible, a simulation in the form of a program in Quick Basic is supplied.
To order to get a copy select:TRAIN.BAS

The purpose of the questions 4 and 5 is to challenge what is called a rest frame.

### Thought experiment

Consider a very big circle. You are standing in the centre of this circle. The circle is actual a rail track. On the rails stands a train. The length of the train is half the circumference. The train used is an ideal train. That means when the front of the train moves also the back moves.

In order to detect if the length of the train changes, two lamps are used.
One lamp (L1) is place behind the track just before the beginning of the train.
From the observer point of view the lamp L1 is ON.
The second lamp (L2) is place behind the track just behind the end of the train.
From the observer point of view the lamp L2 is not visible i.e. is OFF.
That means from the observer point of view only one lamp is ON.

```            ---->                Initial Position
t                  O = The observer
t         t             t = train   T = start of train
t               t          x = track
t                 t         L1 = Lamp 1 = ON
T        L2 = Lamp 2 = Off
L2 t         O         x L1

x                 x
x               x
x         x 3
x
<----
```
The train moves in the direction of the arrow.

When the train starts to move, from the observer point of view the lamp L1 will be covered by the train i.e. will go off.
The lamp L2, which was covered by the train, will become visible i.e. will go ON.
That means from the observer point of view only one lamp is ON.

This situation will continue until the train has moved half the circumference.
At that moment L1 will go ON and L2 will go OFF etc.

Question one now becomes slightly different:
When the speed of the train increases, and approaches the speed of light, from the point of view of the observer, how many lamps are (sometimes) ON: 0, 1 or 2.

In order to do this test and to see both lights simultaneous, put two mirrors close before your nose. One mirror should point in the right direction. The other mirror should point in the left direction.

Maybe some will raise the objection that such a long train can not exist. To solve that objection the train can be half as long. In that case the number of lamps you need is four. In starting position three lamps are ON.

This process can be repeated as many times as one likes, but the question basically stays the same.

### Answer question 1 and 2

• Accordingly to Special Relativity (SR) the Answer is: Sometimes one or two lamps are on.
• IMO, always one, even when the train reaches the speed of light. The length of the train does not change.
This is in conflict with the Special Relativity theory (Lorentz transformation) which states that length should decrease.
• Consider that the train has the full length of the track, and that the front end touches the back end. Accordingly to SR when the train starts to move its length should decrease and the distance between front and back end should increase.
• Consider that the train has the full length of the track, and that the front and the back end are connected together i.e. the train forms a closed ring. What will happen now accordingly to SR?
• Whatever your answer, when the train moves, you will not see the train at the position where the train actual is. There is a delay.
When the speed of the train is v and the speed of light is c, then the angle in degrees is:
v * 360 / (c * 2 * pi)
The maximum angle, when v = c, is approximate 60 degrees.

### Answer on question 3

The answer on Question 3 is: Yes
Consider the following:
The train moves rather slowly with a constant speed.
When the whole experiment has no speed and the observer stands in the middle when the lamp in front goes OFF, the lamp at the back end (on the opposite site) goes ON.
When the whole experiment has a speed or the observer stands not in the middle, then this is not the case.
When the observer stands towards L1 (See figure), then light from lamps on the right side will reach the observer earlier as from light from lamps on the left side
When the observer stands in the center and the whole experiment setup moves to the right the same happens.
However this effect (the amount) is not constant and is a function of the position of the train along the track.
That means, that there are certain parts on the track where the lamp goes OFF before the opposite light goes ON and other parts where the lamp goes OFF after the opposite light goes ON. All related to the velocity of the observer.
You have to subtract this effect from the experiment.

### Answer question 4 and 5

Assume that the answer on question 4 is: Yes. Length contraction is as described by Lorentz contraction.

When the answer on question 4 is Yes, then the answer on question 5 is also Yes, however IMO the amount, with which the length decreases, will not be the same.
In order to compare both experiments the speed v of the two experiments (i.e. one in forward direction and one in backward direction) has to be the same.
The easiest way to do that is, remember this is a thought experiment, when you perform both experiments simultaneous in two parallel tracks around the equator. Both trains have the same speed v, when you start the experiment at a point P and when both trains meet point P, after one revolution, simultaneous. The important point is that both trains will have travelled the same distance in the same time delta t. When you perform the experiment under these conditions at different average speeds, starting from low to high, you will discover:

• That the length of the train, which moves in the direction of the rotation of the earth, always decreases (How faster how more)
• That the length of the train, which moves in opposite direction as to the rotation of the earth, first, at low average speed, will increase in length and secondly, at higher average speeds, will start to decrease.
This is not in accordance to Lorentz Transformation. Accordingly to Lorentz Transformation, if you start from a rest frame, a rod which moves at a speed v in any direction, should contract.

Your reply could be that the (rotating) surface of the Earth is not a rest frame. See also: Changing Length part 3

### Feedback

8 Nov 1996
You wrote: "The length of the train does not change". This is in conflict with the relativity theory ( Lorentz transformation ) which states that length should decrease."
However I do not think that it really is in conflict with the Theory of Relativity.
Theory of Relativity states that length contraction only occurs in that dimension of an object which is parallel to the direction of its motion. The other dimensions remain unchanged. If the train travels in circular path, as it did in your problem, no dimension of the train is parallel to the direction of its motion, because it is moving in a circle. That is why no length contraction occurs.
12 May 1998
(from feedback)
8 Nov 1996
You wrote: "The length of the train does not change".
Etc. See Above.
----------------------------
(Note: The writer's analysis is incorrect. In circular motion, there is always some dimension of an object aligned with the instantaneous direction of motion of the object. Unless the object is also rotating synchronously, the relationship changes continuously but it still exists.)

Actually, the theorized change in length is a function of speed relative to an observer. An example: a ball (sphere) is in front of me. Lets call the left-right direction (relative to me) the x axis, the up down direction the y axis and the away from me-toward me direction the z axis.

The ball moves away from me, along the z axis, at high speed. I'm supposed to see it getting shorter in it's z dimension, but it's diameter along the x and y axes should remain unchanged. I see it turn into a "disk", flat side toward me.

If the ball is also rotating, I'll still see a disk FLAT SIDE TOWARD ME that does not appear to be rotating. There is always some dimension of the rotating ball that is parallel to it's direction of motion relative to me, and that dimension will be shortened from my point of view. If the ball has some marks on it that allow it's rotation to be observed, I'll see those marks moving across the surface of the "non-rotating" disk. It would look quite unreal, like a computer animation.

An observer moving with the ball would see no dimensional changes at all, no matter how fast the ball goes, so dimensions parallel to motion change relative to some observers but not to others. That's why it's called the theory of RELATIVITY.

LB

Created: 12 May 1998
Last modified: 10 January 2000

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