Undergrad An issue with length contraction

[member 728827]

The terminology is what it is. Contraction may not be the best word, but it is the word used to describe the measurements I described above. Those measurements are not just perception. They describe the measured geometry of physical objects.

If we could easily change bad terminology then we wouldn’t still have relativistic mass.

I would like to point that a strain gauge inside the moving object would not record anything special, except for the acceleration and deacceleration phases perhaps (and some tiny geometric relativistic tension if is constantly accelerating), and that a beautiful object, designed by Leonardo da Vinci golden ratio proportions, would be apparently deformed in the direction of movement, so the "measures" of physical objects seem just perceptions of a geometry that changes its rules by the relative speed of observers. Is like looking through a glass. Nevertheless, time is other kind of stuff.

 

[Ibix]

a beautiful object, designed by Leonardo da Vinci with aural proportions,

I suspect a translation failure here. By "aural proportions" I guess you mean $(1\pm\sqrt 5)/2$? That's called the Golden Ratio in English; aural means "related to hearing".

the "measures" of physical objects seem pretty must just perceptions of a geometry that changes its rules by the relative speed of observers.

Not really. You are just measuring different parts of a 4d entity when you use different frames. It's quite closely analogous to a unit square, which is 1 wide. Rotate yourself 45° and the square is now a diamond which is $\sqrt 2$ wide. Neither the rules of geometry nor the square have changed, we just changed our mind about what we were calling "width". The same happens when we change frame in relativity - we change our mind about what space is, and hence about the intersection of the object's worldtube and space.

General comment, not just to Lluis Olle: note that there's potential for confusion over the word "object". Is it the 4d worldtube, or the 3d slice of the worldtube that intersects our "now"? Make sure you know which one you mean and make sure you consider the possibility that someone you are talking to means the other one.

 

[member 728827]

I suspect a translation failure here. By "aural proportions" I guess you mean $(1\pm\sqrt 5)/2$? That's called the Golden Ratio in English; aural means "related to hearing".

Sure, I mean the golden ratio, which is "la proporción aurea" for me. Is a false friend word :) Here you have a relativistic effect in action.

Rotate yourself 45° and the square is now a diamond which is 2 wide. Neither the rules of geometry nor the square have changed

It continues to be a square, all its four sides are equal (I don't say the same length, I mean that can't be distinguished and are interchangeable).

 

[Ibix]

It continues to be a square, all its four sides are equal (I don't say the same length, I mean that can't be distinguished and are interchangeable).

Exactly. But the width is different because it's not purely a property of the square: width also depends on the orientation of the square/diamond with respect to you. Similarly, length contraction arises because you measure a different cross section of the worldtube and still call it length.

 

[member 728827]

Exactly. But the width is different because

Sorry, what is "the width"? ... a square is "a plane figure with four equal straight sides and four right angles".

And by cross section you mean this? Because being this a Physics forum, some words like "fragile", and others have an agreed meaning in the community.

 

[Ibix]

Sorry, what is "the width"? ... a square is "a plane figure with four equal straight sides and four right angles".

It depends which way you're looking at it, which is my point. If we've drawn the figure on this screen and you're holding your screen vertically, I'd say it's the horizontal extent of it in the plane of your screen. Rotate your screen in its own plane to vary the width.

And by cross section you mean this?

I mean that "the object, now" is the intersection between a worldtube and a flat spacelike plane, in particular the one you are calling "now" in your frame. That's a cross-section in the same sense that a slice through a sausage gives you a circular or elliptical cross section of the cylindrical sausage.

 

[jbriggs444]

Sorry, what is "the width"? ... a square is "a plane figure with four equal straight sides and four right angles".

I think that a "square" it not as helpful an analogy as could have been achieved. A "two-edged measuring tape" would be better so that we do not have any distracting ends. Just the two sides.

How do we measure the width of a tape? By placing a ruler on it crosswise.
But what if the ruler is placed at an angle? We obtain a different measurement.

So we are careful. We place the ruler on the tape so that it is at the "1 centimeter" marker on both edges of the tape.

One would be tempted to use a try-square to measure this Euclidean width but relativity does not provide us with the equivalent of a "right angle" between space and time, so we are forced to this alternate mechanism in order to keep the analogy apt.

Then we worry about the careful details of the alignment in markings on the two edges of the tape. This is analogous to the relativity of simultaneity.

 

[student34]

student34 should carefully ponder this physical example of length contraction (and time dilation):

Cosmic rays are known to strike the upper atmosphere at $\sim15000\mathrm{m}$ altitude, producing downward-directed muons traveling at nearly the speed-of-light $c$. Since muons at rest have a mean lifetime of $\overline{\tau}=2.2\mathrm{\mu s}$, their maximum travel distance in that time is only $c\overline{\tau}=660\mathrm{m}$. This is much less than the distance to the ground, yet, due to relativity, these muons are readily detected at the Earth's surface.

In the muon's rest frame, the Earth is seen to approach at nearly $c$ with the depth of its atmosphere foreshortened (length-contracted) by a factor of $1/\gamma$, where $\gamma\approx21$. This is thin enough that the muon easily reaches the ground before decaying, in agreement with experiment. (See the left illustration below, where the depth of the atmosphere is symbolized by the height of the mountain.)

Alternatively, at rest on Earth we observe the "internal clock" of the muon to be slowed-down by time-dilation, increasing its apparent lifetime $\overline{\tau}$ by the same $\gamma$-factor of $\sim21$. This gives the muon ample time to reach the surface, again just as experimentally observed. (See the right illustration below.)

I leave it to the OP and the philosophers to debate whether this contraction of atmospheric-depth or dilation of half-life (depending on the reference frame) is "actual" or "apparent", but operationally both effects certainly seem "real" to me.

View attachment 315451
(Martin Bauer on Twitter)

It is still sort of an illusion. The muon just takes a shorter route through spacetime to get to the ground.

If we were to make a time over distance graph and display the muon trip horizontally, we see that the event where the muon takes off from and the event where the muon impacts the Earth are along two parallel lines extending only through time (as seen by an observer at rest). Then the world line of the muon just takes a shorter spacetime interval to get to the Earth.

The muon is the green worldline. And some object that takes off where the muon takes off from, but at a slower speed, is the blue line.

 

[Ibix]

It is still sort of an illusion. The muon just takes a shorter route through spacetime to get to the ground.

Why is the green line more illusory than the blue one?

 

[PeterDonis]

The muon just takes a shorter route through spacetime to get to the ground.

Shorter compared to what?

ome object that takes off where the muon takes off from, but at a slower speed

There is no such object in the scenario. What's the point of this?

 

[PeterDonis]

I just wanted to know if I was misinterpreting something about length contraction. Now I don't think I am.

Your post #68 strengthens the impression I expressed in post #57.

 

[Dale]

It is still sort of an illusion. The muon just takes a shorter route through spacetime to get to the ground.

Why would you call a shorter route an illusion? Is it also an illusion that it is shorter to drive from Miami to New York through Washington DC than through Denver? In what way is the word "illusion" appropriate here?

 

[student34]

Why is the green line more illusory than the blue one?

I meant that the length contraction for the muon's journey is an illusion, at least in the strict sense of the term contraction.

 

[student34]

Shorter compared to what?There is no such object in the scenario. What's the point of this?

Shorter than compared to the slower object in blue.

 

[Ibix]

I meant that the length contraction for the muon's journey is an illusion, at least in the strict sense of the term contraction.

No, it's just a different measurement. Again, you are focusing on the exact meanings of words in their general context, words that we repeatedly tell you are not and cannot be precise. The sooner you accept that the words are jargon and do not have their customary meanings, the sooner you can get on with learning relativity.

 

[student34]

Why would you call a shorter route an illusion? Is it also an illusion that it is shorter to drive from Miami to New York through Washington DC than through Denver? In what way is the word "illusion" appropriate here?

If I am riding the muon from its resting point through its journey to the Earth, it is going to seem like the distance contracts for me once I start accelerating. But I am actually taking a geomterically different route than the route which I saw when I was at rest. It is an illusion for humans because we don't see the temporal dimension.

 

[PeterDonis]

I am actually taking a geomterically different route than the route which I saw when I was at rest.

No, you are not. The muon's route through spacetime is an invariant.

I meant that the length contraction for the muon's journey is an illusion, at least in the strict sense of the term contraction.

In other words, we are right back to where we started with this thread. None of the discussion after 75 posts has made any impression on you at all. So what's the point of discussing?

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

After Mentor discussion, the thread will remain closed.