Length Contraction vs. Distance Contraction

1. Dec 13, 2012

arindamsinha

Consider observer A at rest, seeing observer B traveling in a rocket at high speed towards a star C. Star C is at rest w.r.t. A.

A in his frame sees B's rocket to be length contracted. B considers his distance to C to be shorter than (contracted) compared to what A sees.

I seem to be finding the term 'length contraction' being used for describing both phenomena. I feel these are two separate phenomena (though perhaps 2 sides of the same coin) as they are observed in different observers' frames.

Are there separate terms to describe these two contractions (length and distance), or are they considered the same phenomenon?

2. Dec 13, 2012

Staff: Mentor

They are both the same thing, part of the Lorentz transform.

3. Dec 13, 2012

arindamsinha

I am wondering how they can be the same thing. These two are happening in two different frames. In the rest frame, the moving body is affected. In the moving frame, something external (a distance) is affected.

I am not able to understand how they can be the same.

(Understand that they are both Lorentz transformation related, but so are many other aspects like time dilation, which is not the same manifestation of LT as the above phenomena).

4. Dec 13, 2012

PAllen

Every frame is a rest frame. Picture two lights on the end of a ruler. If it is moving towards you and one light reaches you have have:

- ruler shorter
- light at end of it closer

Same thing, as Dalespam said.

5. Dec 13, 2012

Staff: Mentor

Suppose that A and C are separated by two light years, as measured in the frame in which both are at rest. B is midway between them, one light year from C. Now let us imagine that there is a rod, a full light year long, sticking out from C towards A. Because B is at the halfway point, he has just reached the end of that rod.

In the frame in which A and C are at rest (you called it "the rest frame", and that's a bad habit) only B is moving and contracted.

In the frame in which B is at rest, A, C, and the rod are all moving, so are contracted. And B is at one end of the rod and C is at the other, so the length of the rod is the distance between B and C. So if the rod is contracted in the frame in which B is at rest, so is the distance to C.

6. Dec 13, 2012

Staff: Mentor

Length contraction requires two material points at rest wrt each other. Then, according to the Lorentz transform, the distance between them in their mutual rest frame is greater than the distance in any other frame.

7. Dec 13, 2012

arindamsinha

OK. If I understand correctly, the ruler is shorter and light is closer 'in the direction of motion'. In the reverse direction (opposite the motion) ruler is longer and light is farther. Is that a correct interpretation?

8. Dec 13, 2012

arindamsinha

Isn't the AB distance now larger than the BC distance according to B's frame?

9. Dec 13, 2012

arindamsinha

I think this works for the length of B's rocket but not for the BC distance... there is still length contraction of BC in spite of them not being at rest to one another...

10. Dec 13, 2012

PAllen

No. Doppler is different for approach and recession. Length and distance contraction are the same.

11. Dec 13, 2012

ghwellsjr

But the length/distance contraction only occurs along the direction of motion, not perpendicular to it.

12. Dec 13, 2012

Staff: Mentor

No. Set up a second rod one light year long extending from A towards C, so that the ends of the two rods are touching in the middle where B is. The AB and the CB rods are both moving relative to B so are both contracted; same γ factor.

13. Dec 13, 2012

Staff: Mentor

No, the math of the Lorentz transform is symmetric this way.

14. Dec 13, 2012

Staff: Mentor

Correct, the front and back of B's rocket are at rest in B's frame. A and C are at rest in A's frame. So A measures the rocket length shorter than B does, and B measures the AC distance shorter than A does.

Well, the Lorentz transform still applies, but as you mentioned earlier, the Lorentz transform includes other effects (time dilation and relativity of simultaneity) as well. In the case of the material points at rest then all of the other effects cancel out and you are left with the simple length contraction formula, but in general they do not. Specifically, since B and C are moving relative to each other, the relativity of simultaneity effects do not cancel so the formula is more complicated than just the length contraction formula.

Last edited: Dec 13, 2012
15. Dec 14, 2012

Mike Holland

From A's point of view, everything at rest in B's frame is Lorentz-contracted in the direction of relative motion.
From B's point of view, everything at rest in A's frame is Lorentz-contracted in the direction of relative motion.

So B sees the A-C distance contracted. A sees B's rocket contracted.
B thinks he gets there sooner because the distance is less. A says he didn't get there sooner, but he thinks he did because his clock was time-dilated.

Edit: Just a note to say that both views are equally valid as long as B keeps travelling at uniform speed. The situation changes when B slows down and comes to a stop at C. But that's another story - see the Twin Paradox.

Last edited: Dec 14, 2012