# Length contraction problem

suppose you had a planet with a train going all the way around the circumference. The train is a complete loop so you just have an endless train. now speed the train up to near c.
Because of length contration would the train begin to squeeze the planet?

Originally posted by revesz
Because of length contration would the train begin to squeeze the planet?

No. The train doesn't contract like a noose, it just rips apart: you can't have a perfectly rigid body in relativity. This is related to the question of the rigid rotating disc:

http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

If an object were to go exactly c would it lose its 3rd dimention?

what i mean is, would it have any length whatsoever?

A massive body cannot travel at c, but its length can contract to arbitrariliy small values as it comes arbitrarily close to c.

A massive body cannot travel at c, but its length can contract to arbitrariliy small values as it comes arbitrarily close to c
Why arbitrary?

jcsd
Gold Member
It's another way of saying as v -> c, l -> 0. The word 'arbitarily' is used as whatever theoretically possible value of l we choose there will always be another value of theoretically possible value of l' closer to 0 and whatever theoretically possible value for v we choose there will always be another theoretically possible value of v closer to c (but l can never equal 0 and v can never equal c).

It's another way of saying as v -> c, l -> 0. The word 'arbitarily' is used as whatever theoretically possible value of l we choose there will always be another value of theoretically possible value of l' closer to 0 and whatever theoretically possible value for v we choose there will always be another theoretically possible value of v closer to c (but l can never equal 0 and v can never equal c)
Ah.

This is a very valid point. Due to the small curvature of the earth, each section of the train is, at an instant, contracted in its direcetion of motion.

This means that the total length of the train must be foreshortened.

We need not think of the train as travelling at a speed very close to c. Say it travels at 1080km/h = 0.30km/s = 300m/s

Let it 'rest' length = L'

Its contracted length is now

L=L'*sqrt(1-v^2/c^2)

= L'*sqrt(1 - (300/3*10^8)^2)
= L'*sqrt(1- (10^-6)^2)
= L'*sqrt(1 - 10^-12)

L = L'(0.999999999999499999999999875)

A negligable amount?

The circumference of the earth is approximatly 40,000km or 40,000,000m

Thus the length of the train would be

L' = 40,000,000m

Thus L = 40,000,000 * 0.999999999999499999999999875

L = 39,999,999.999979999999999995m

This is still less than a millimeter of a difference. And the train would have to travel at 1080km/h to obtain even this contraction. But this speed is approximatly the speed of rotation of the earth about the equator. Well the earths rotational speed is actually 1600km/h.

So the earths circumference is contracted by less than a millimeter.

It would be interesting to see what happens for other planets.

But seriously, relativity says nothing about the streaths of materials. So the train might squeeze the earth

jcsd
Gold Member
Obsessive, in Newtonian mechanics it's perfectly possible to have a perfectly rigid body that doesn't expand or contract, this is impossible in relativity as a wave through this material would be instanteous.

a perfectly riged train and perfectly rigid planet would pose problems for relativity. the train would contract with infinite strength, while the planet would repel the squeazing with infinite strength.

Eepl
While you're on technicalities.

Since the train is a 3 dimensional object only the top of the train would be able to travel at the speed of light. So the train would have to squeeze itself before it could start to squeeze the planet.

Since the train is a 3 dimensional object only the top of the train would be able to travel at the speed of light. So the train would have to squeeze itself before it could start to squeeze the planet.
Not true. While the top of the train may be travelling slightly faster, the bottom of the train would still contract at almost the same rate.

If a rocket were traveling at c-5mph, and a passenger on board were to run to the front at 10mph, would he be traveling at c+5mph?

Eepl
Even though this is purly hypothetical, it is impossible. As soon as the man running hit 5mph he would be traveling at c. Once c is obtained time stops, then the man would no longer be able to run. Here's the next one for you.
-Because the man has to move independantly to obtain c, when he reachs c he cannot move anymore. So would he slow down, or always be stuck going c until some outside force slowed him down?

Originally posted by Eepl
Even though this is purly hypothetical, it is impossible. As soon as the man running hit 5mph he would be traveling at c.

No, he can't ever reach c. He can run at 5 mph, 10 mph, or 1000000 mph relative to the rocket, but he'll still be running at less than c relative to the observer who sees the rocket moving at c - 5 mph. This follows from the SR velocity addition formula:

http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html

marcus
Gold Member
Dearly Missed
Originally posted by revesz
suppose you had a planet with a train going all the way around the circumference. The train is a complete loop so you just have an endless train. now speed the train up to near c.
Because of length contration would the train begin to squeeze the planet?

even before considering relativistic effects I want to imagine the tension resulting from "centrifugal force"
this would tend to stretch each car of the train and create strain in the couplings connecting one car to the other

if the train did not break apart by this tension, then it would be stretching out like taffy-------and then the contrary effect of SR length-contraction would be factored in. So, depending on the young's modulus or strength of the traincars, the two effects could even cancel????

At 1/2 the speed of light the centripetal acceleration

(1/4) 9E16/6.4E6

3.5 E9 meters per second per second.
So if each traincar masses E4 kilogram, each car must be experiencing a downwards force of 35 trillion newtons. This translates into tension throughout the train----which therefore needs supernatural tensile strength not to snap and even so it would stretch. This is before we introduce the SR length contraction.

The SR length contraction is only down to 0.87 of original length at that speed. I am thinking the train would be lucky not to have already stretched out more than enough to compensate for that!

I understand that the train would have in real life ripped appart, but then again in real life the train wouldnt be going at relative speed. also the length contraction would not neutralize the centrifical force at all, the tension on the train would still remain the same even if the train did contract a considerable amount, and so the train would ripp appart either way.
But now I have another question to do with light, and the effect of length contraction. When an objects undergoes length contraction, its mass is not effected and so it has the same number of atoms. So all of the atoms that it had prior to contraction exist within less space. If light comes from exited electrons jumping from one orbit to another and back, then because of the length contraction the length of the electrons orbit jump is also contracted in the direction of the movement. The frequency of light is related directly to how far the electron jumps, so any light shone in the direction that the rocket is traveling will have an altered frequency. Is that true, or is there something that I overlooked?
Also if this were true, would it be a red, or blue shift?

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On the wrong track?

Well, of course, if the train around the equator was earthbound and travelling west-wards it would experience a special relativity length expansion., though presumably not enough to take it off the rails. ;-)

Dennis Revell

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