Length Contraction in Special Relativity

[ubergewehr273]

Could somebody explain the logical reasoning behind lorentz contraction when an object travels faster and faster ?

 

[russ_watters]

It's the other side of the coin from time dilation: d=s*t. So if "t" varies and "s" (speed of light) stays the same, "d" must vary.

Or, from an observational point of view: if one person sees another's clock running slow, the other must see the first person's distance contracted. This is what causes the muon decay observation:
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html

Muons decay. Time dilation makes them live longer, which makes them travel further (in our frame) or, rather, the distance they travel becomes shorter (from their frame).

 

[Orodruin]

the distance they travel becomes shorter (from their frame).

The distance they travel in their own rest frame is zero. It is the Earth that travels, but as it is length contracted it needs to travel a shorter distance.

 

[Chestermiller]

Could somebody explain the logical reasoning behind lorentz contraction when an object travels faster and faster ?

It's a geometric effect, related to the geometry of 4D space-time. It's sort of like what happens when you look at an object that has been rotated, such that it looks shorter. But, there's more a little more to it than this. You're seeing one end of the object at a later time (as reckoned from the object's frame of reference) and the other end of the object at an earlier time (as reckoned from the object's frame of reference). As a result of both these features, the object looks to be in the correct location in your frame of reference, but it looks shorter.

Chet

 

[Khashishi]

It all comes out of the fact that light travels at the same speed in all frames of reference.

 

[phinds]

In addition to what kashishi said, keep in mind that length contraction is a result of remote observation, not something that actually happens to an object. You, right now as you read this, are massively length contracted from the frame of reference of an accelerated particle at CERN. Do you feel any shorter / thinner ?

 

[pervect]

Could somebody explain the logical reasoning behind lorentz contraction when an object travels faster and faster ?

The logical reasoning is that we assume there is a linear and invertible transformation between the coordinates (t,x) of a stationary object and the coordinates (t'. x') of a moving object, (here we are only considering two dimensions of space-time, one time and one space), and that this transformation must have the property that the speed of light is equal to "c" for all observers.

Note that the familiar transformation, called the Galilean transformation, that we use in pre-relativity physics between coordinates (t,x) and (t', x') is linear , but does not preserve the speed of light. In fact if light moves at "c" in the (t,x) frame, we expect it to move at (c-v) in the (t', x') frame. So we need a different transformation, and we look first for one that is linear.

These two requirements allow us to deduce the required coefficients of the linear transform. Mathematically, the constancy of the speed of light results in an eigenvector problem, the requirement that we map a vector to a scaled version of itself is the eignenvector problem from linear algebra. Letting c represent the speed of light, specificaly we say that a point on the light cone (t=a, x=c*a) which we can represent in vector notation as a*(0,c) must map to another point on the light cone k1*a*(0,c). Similarly a point on the light cone a*(0,-c) must map to a point k2*a*(0,c)

In matrix notation ##\Lambda \vec{x} = k \vec{x}## when ##\vec{x}## is lightlike, i.e. (1,c) or (1, -c).

Solving the eigenvector problem for the linear transformation matrix ##\Lambda## yields the Lorentz transform, and the lorentz transform has the property that moving objects lorentz contract.

Sorry if this is too technical, but any short answer to the question would have to be technical, I think.

 

[DaveC426913]

In addition to what kashishi said, keep in mind that length contraction is a result of remote observation, not something that actually happens to an object.

Isn't this contradicted by Bell's Spaceship paradox?

 

[PeterDonis]

Isn't this contradicted by Bell's Spaceship paradox?

It depends on what you are referring to as "length contraction". The normal usage of that term refers to the effect of transforming between frames; you aren't doing anything physically to the object itself, you're just changing the frame you use to describe it. That is (I believe) what phinds was referring to.

In the Bell Spaceship Paradox, the breaking of the string between the ships is sometimes attributed to "length contraction" of the string (note that we have a forum https://www.physicsforums.com/threads/what-is-the-bell-spaceship-paradox-and-how-is-it-resolved.742729/ on this that discusses why that terminology might not be entirely apt), but the term here refers to something different: it's an actual physical process to which the object is subjected, which can be described using a single frame and has nothing to do with how the description of the object changes if you change frames. This physical process is indeed real, but it's important to keep it conceptually distinct from the effect of changing frames.

 

[phinds]

RIght. Thanks for that clarification Peter. I was, as you realized, referring to the frame-dependent observations, but I do tend to forget about the confusion you pointed out.

 

[harrylin]

Could somebody explain the logical reasoning behind lorentz contraction when an object travels faster and faster ?

When an object (either a particle or an instrument) travels faster and faster, its speed as determined with an inertial reference system changes. According to relativity theory, its length as determined with that reference system decreases with speed. Here's a non-technical, historical explanation.

Based on strong experimental evidence it was accepted that light propagates like a wave. Consequently, the experimentally determined speed of light is independent of the motion of the source or anything else that it encounters on its path; that's also where the label "c" comes from, similar to the "c" of sound.

However, it is only expected that the speed c will be measured relative to a single reference system - the "rest frame". It was therefore thought that we can determine our velocity relative to that "rest frame" by means of optical experiments. This turned out to be wrong: the results of optical measurements are exactly the same, independent of the speed of the instrument (the most famous experiment is called the "Michelson Morley experiment", you likely heard of it). Poincare concluded that the relativity principle of mechanics must also be valid for optics, and that a new theory was needed to account for this fact.

Lorentz as well as Einstein took up the challenge and combined these two assumptions (raised by Einstein to postulates) in order to come up with a new theory; this thing called "length contraction" logically followed from the starting assumptions. Einstein phrased it as follows:
"A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a state of motion -viewed from the stationary system- the form of an ellipsoid"
- http://fourmilab.ch/etexts/einstein/specrel/www/

If you merely want to understand the necessity of length contraction, then you only need to consider the Michelson Morley experiment: the interferometer changes velocity in the course of time. Else it may be handier to follow Einstein's "simple derivation" of the Lorentz transformations (click for Fig.2 on the link to section 11):

https://en.wikisource.org/wiki/Rela...mple_Derivation_of_the_Lorentz_Transformation

Length contraction and time dilation directly follow form these transformations, as well as the fact that the speed of light is the limit speed for particles in acceleration experiments.

 

[bhobba]

Could somebody explain the logical reasoning behind lorentz contraction when an object travels faster and faster ?

You have some nice explanations here.

However I am with Chestermiller on this.

It's basically caused by space-time geometry similar to rotating a rod to fit through a door. The length of the rod doesn't change but its geometrical relation to the door does so its 'effective' length is smaller. In fact when you work through the math the length contraction is a hyperbolic rotation.

Why is space-time like that? Good question. Its really got nothing to do with light etc. Its the symmetries implied by the principle of relativity:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

This is a very abstract way of viewing it but for me its the most natural and convincing - its very hard to doubt its assumptions.

Thanks
Bill