Length contraction (objects & space or objects only)

1. Apr 19, 2010

tabarnard

Does length contraction apply only to objects, or to space as well?

On the one hand, sources like this seem to imply that it applies only to objects: http://en.wikipedia.org/wiki/Bell's_spaceship_paradox

On the other hand, most people on this forum seem to believe it applies to space as well.

For instance, assume that star 1 and star 2 are 1 lightyear apart, at rest with respect to each other. If pass star 1 at .95 the speed of light, heading toward star 2, and measure the distance to star 2, will it be 1 lightyear (no length contraction) or 0.31 lightyears (length contraction)?

2. Apr 19, 2010

utesfan100

Length contraction applies to any length measured in one inertial frame and translated into another.

Whether that length represents space or distance along an object does not matter. What does matter is that the times between when the ruler was simultaneously at the ends of the distance being measured in one frame is not simultaneous in the other frame.

3. Apr 19, 2010

Janus

Staff Emeritus
The distances between the stars contract.
The bell spaceship paradox does not imply that only objects undergo length contraction. The trick is in understanding that the distance between the ships stays constant in the rest frame and that this an arbitrary condition placed on the scenario.

What this means is that the distance does not stay constant in the ship's frame in this case. As far as they are concerned, they drift apart while the string remains the same length.
If you restate the problem so that the ships maintain the same distance in the ship's frame, then the distance between the ship's contracts as seen from the rest frame along with the string.

In your situation with the two stars, both stars, in their own frame, maintain the same distance, so in your frame the distance contracts.

4. Apr 19, 2010

JustinLevy

Your question has already been answered, but since the confusion seems to have stemmed from Bell's spaceship paradox, let me make some comments on this, as it is a really interesting though experiment.

That condition isn't really all that arbitrary. The condition is actually that the ships have the same acceleration sequence ... the condition is merely that: the ships are identical, and start simultaneously in their mutual/intial inertial rest frame.

Yes, from this you can get that the length between the ships is constant according to the initial inertial frame, but it comes about so in a natural way.

What Bell's spaceship paradox shows, is several interesting things:
1) For a rigid object to accelerate, the acceleration must be different along the length of the object (unlike in Newtonian mechanics). This is fairly counter-intuitive and interesting about relativity. It also leads to the idea of a Rindler horizon.

2) Bell originally made the paradox to argue that length contraction of an object was physical in at least this sense: from an inertial frame S, measure an inertially moving object ... have it accelerate to a new inertial frame ... if in equilibrium in the object's inertial frames in the beginning and end, the physics according to frame S must have included a physical contraction (in his spaceship paradox, an elastic string between the ships will shorten the distance ... and only has tension while accelerating). It is an interesting problem, but wording can be subtle sometimes (it has lead to arguments on this forum before, because even though everyone agrees on the measurements and outcomes, the wording can unfortunately be easily misconstrued as misleading statements. So be careful to work out the physics, so you aren't mislead by unintential semantics of describing this scenario. Its usually the semantics of how to present this that leads to arguments unfortunately.)

Bell's spaceship paradox, and the reason so many people get it wrong, is that often people are too casual with the "length contraction" formula. Lorentz transformation actually just relate labels for events in one coordinate system to labels for events in another coordinate system. You can use the coordinate transformation formula to relate lengths measured in two inertial frames:
$$L = L' \sqrt{1 - v^2 / c^2}$$
For the length of an object under arbitrary motion to follow the equation:
$$L = L_0 \sqrt{1 - v^2 / c^2}$$
is something separate. Don't get me wrong, it can be easily derived from the Lorentz transformations given assumptions about the object and equilibrium, but the point is that the equation (while having the same form) is relating a different situation. Because instead of just involving transformations of labels, if v is not constant, it is telling us something about how the object changes as it accelerates. Man, now I see why people argue about the semantics. I know what I'm trying to describe, and knowledgeable people would probably understand as well, but I really don't like how that is worded. Well, if someone else here wants to try to do it better, feel free ... don't worry, I won't argue semantics with you.

Bell used his spaceship example to make this distinction extra clear. It still catches people off gaurd to this day.

Last edited: Apr 19, 2010
5. Apr 19, 2010

cos

According to the concept of length contraction - if you were moving at c then the entire universe, obviously including distances, would contract to zero.

On that basis - if you were moving at a velocity slightly slower than c the entire universe, including the distances between stars and galaxies (et al), could all contract to a dimension of 1cm.

You wouldn't have all matter compressed and 'space' still extending to infinity.

6. May 17, 2010

Passionflower

Why is this counter-intuitive, how else would you explain length contraction?

7. May 17, 2010

matheinste

If we are talking about Lorentz contraction (SR) then acceleration is not necessary to explain it. Also it is binary relation between two relatively moving frames, not a contraction of an object with respect to itself when viewed from another reference frame when the relative velocity between the frames changes.

Matheinste.

8. May 18, 2010

Passionflower

A rod only contracts if the trailing end of this rod underwent a higher proper acceleration than the leading end. That is a pretty obvious and intuitive thing to me.

What puzzles me is how anyone could call this counter intuitive.

9. May 18, 2010

matheinste

Yes, acceleration can cause deformation due to physical stress but this is not Lorentz contraction which is stress free and is experienced between non accelerating relatively moving frames.

Matheinste.

10. May 18, 2010

Passionflower

You either did not read what I wrote or you do not understand it.

If the second is the case I suggest you study Born rigid motion and Rindler coordinate charts.

And in case your next question will be "Will an observer 'on' the rod measure a shorter rod", the answers is: no, he will not measure a shorter rod. Why? A clock positioned at the trailing end of this rod will run slower with respect to a clock at the leading end of this rod, because the trailing end of the rod underwent a higher proper acceleration than the leading end.

It is all very simply if you think about it with a clear mind!

Last edited: May 18, 2010
11. May 18, 2010

matheinste

To clear up any misunderstanding I was stating that Lorentz contraction in SR is not caused by acceleration.

Matheinste.