Why does length contraction occur and how does it relate to time dilation?

[syfry]

Was learning about length contraction of muons that rain down from cosmic particles colliding in our upper atmosphere, and how the distance for the muon is shorter than we measure.

But looking up any further info, found a number of sources describing how the muon's own length is contracting, and from what I've learned about relativistic travel, the distance becomes shorter. I didn't find anything on a search of distance contraction.

Now curious about some of the particulars of length contraction after contemplating a few scenarios.

My questions are:

If a long pole at relativistic speed would strike an object, would we calculate from the front of that object as if the rest of its length had shrunk toward the front, or, would we calculate as if the front had shrunk back toward the middle of pole?

Would its front shrinking toward the middle counteract some of the pole's gain in arrival time?

Do we separately calculate the contracted distance and combine that with calculations of the object's own contracted length?

And is time dilation the same as the effect of length contraction, but in another reference frame?

For example, the muon had lived a longer time in our frame of reference (by time dilation), but instead the muon's trip was shorter in its own frame (by length contraction).

 

[Ibix]

Your questions are rather confused.

Nobody cares about the length contraction of muons because they are point particles anyway. In the case of a rod colliding with something I'm not sure what you are trying to get at. Are you asking how much later the back of the rod hits the target than the front? That would be its length contracted length divided by its speed.

the muon had lived a longer time in our frame of reference (by time dilation), but instead the muon's trip was shorter in its own frame (by length contraction).

Bith of these explanations are valid, yes.

 

[syfry]

Are you asking how much later the back of the rod hits the target than the front?

Nope. Thanks for the tidbit about the back of rod though!

What I'm wondering is if the front would be delayed or would hit later because it's now slightly farther away from the target after having shrunk toward the middle. (by contraction... if that's how it works)

And if that would counteract any of the distance contraction. (for example if the target was 500 kilometers away before the pole's launch, but now in its relativistic reference frame after launch the target is instead 100 kilometers away... does the delay from the object's length contraction counteract the gain in timing from the distance's length contraction)

Yeah my question was worded a bit convoluted. Hopefully it's more clear.

(Also thanks for confirming about length contraction equal to time dilation from different frames)

 

[hutchphd]

And if that would counteract any of the distance contraction. (for example if the target was 500 kilometers away before the pole's launch, but now in its relativistic reference frame after launch the target is instead 100 kilometers away... does the delay from the object's length contraction counteract the gain in timing from the distance's length contraction)

You cannot skip willy-nilly between reference frames. In its own frame, the rod does not change in length. In the frame of the target it travels 500 km. The Lorentz transformations quantify the mathematics of the transformation between frames.

 

[syfry]

You cannot skip willy-nilly between reference frames. In its own frame, the rod does not change in length. In the frame of the target it travels 500 km. The Lorentz transformations quantify the mathematics of the transformation between frames.

Ok, that's where I erred. Had thought that the rod's length and the distance between the target were contracted in the same reference frame of the rod.

So let's see if I got this right:

The distance to target is contracted only in the rod's frame. The time to arrive there is dilated / reduced only in an observer's reference frame. The rod's own length is contracted only in an observer's frame.

Is that correct?

 

[phinds]

The distance to target is contracted only in the rod's frame.

yes

The time to arrive there is dilated / reduced only in an observer's reference frame.

no --- in the rod's frame

The rod's own length is contracted only in an observer's frame.

yes

 

[syfry]

no --- in the rod's frame

Ibix had confirmed the quoted text below as valid. Is there an aspect I'm missing? Or is your reply contradicting theirs? (maybe you're both right, I've got no idea):

the muon had lived a longer time in our frame of reference (by time dilation), but instead the muon's trip was shorter in its own frame (by length contraction)

 

[syfry]

Also, I was trying to find the difference between contraction of length vs distance. I found only references to length contraction. Is there a specific wording for distance contraction?

For example, the link below addresses only the length contraction of the object itself.

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/tdil.html

It's about calculating length contraction, but I'm trying to find a source that similar discusses distance contraction. (unless it's supposed to use the same calculations as time dilation does)

Today is my first time learning that time dilation and distance contraction were related. Maybe people mostly stick to time dilation and there isn't a word for distance dilation?

 

[jbriggs444]

Ibix had confirmed the quoted text below as valid. Is there an aspect I'm missing? Or is your reply contradicting theirs? (maybe you're both right, I've got no idea):

Time dilation is the difference between two time measurements, one in the Earth frame and one in the muon frame. The elapsed time measured by the muon will be shorter ("dilated") by comparison with a measurement in the Earth frame.

It is important to realize that the Earth measurement is [conceptually] the result of comparing two Earth clocks. One at rest in the upper atmosphere and one lower down where the muon decays. The two clocks are synchronized using an Earth-based synchronization standard. The Earth measurement is the difference in the two clock readings at their respective events. This is a measurement of "coordinate time" -- a time computed by comparing two time coordinates.

By contrast, there is only one clock on the muon. The muon clock reading is the elapsed time on the muon clock. This is a measurement of "proper time" -- a time measured directly by a single clock.

Proper time measurements will be reduced compared to coordinate time measurements. We call this "dilated" because moving clocks take longer to tick as compared to an array of synchronized clocks at rest in a chosen coordinate system.

 

[jbriggs444]

Also, I was trying to find the difference between contraction of length vs distance. I found only references to

There is no important difference between length and distance. Any distance can be interpreted as a length by putting a measuring tape across the distance. The distance is the length of that tape.

The length of a moving object is defined as the distance between its end points at the same time. If a ruler, meter stick or measuring tape is moving, the "at the same time" part becomes important.

 

[Vanadium 50]

willy-nilly

There is a lot of willinilliness going on.

OP, you would do well to think of things in terms of events. The Lorenyz transformation lets you transform the coordinates of events. To get the length of a rod, it's the difference between the coordinates at one end and the other at the same time in that frame.

That avoids the willinilliness.

It's really no more complicated than what you are doing. Nothing to be afraid of. No need to get, ahem, the willies.

 

[syfry]

There is no important difference between length and distance. Any distance can be interpreted as a length by putting a measuring tape across the distance. The distance is the length of that tape.

I'm probably making an error from not knowing what I don't know.

What I've gathered is there are two types of contraction.

In one type, the object experiences the contraction (a 5 meter object becomes 4 meters, as experienced in an observer's frame).

In the other type, the distance of travel is contracted (a distance of 10,000 km is now 5.000 km, in the traveler's frame).

I'm trying to find somewhere that discusses the differences between those two types, if any exist.

The length of a moving object is defined as the distance between its end points at the same time. If a ruler, meter stick or measuring tape is moving, the "at the same time" part becomes important.

If I'm understanding correctly, you're saying we must include both the front and back of the object in our measurement of it. But that sounds obvious so maybe I'm erring.

What I'm trying to figure out is, does an object's contraction affect its reach to the finish line in a race.

Say a long bus were traveling at relativistic speed. If its length contracts from the front and the back at the same time, toward its middle, now that'll delay its reaching the finish line (its nose breaking the tape), as compared to if instead the front of bus had stayed put so its rear contracts forward.

Logically, both ends would probably contract inward instead of the rear shrinking toward the front, but I'd like to verify.

 

[syfry]

There is a lot of willinilliness going on.

OP, you would do well to think of things in terms of events. The Lorenyz transformation lets you transform the coordinates of events. To get the length of a rod, it's the difference between the coordinates at one end and the other at the same time in that frame.

That avoids the willinilliness.

It's really no more complicated than what you are doing. Nothing to be afraid of. No need to get, ahem, the willies.

I tried to learn about coordinate transforming from a minute physics video below, but they went way too fast even with pausing constantly. Didn't really make sense.



Maybe I'll fare better at brilliant .org since it'll probably have interactive ways of learning that.

 

[Dale]

Would its front shrinking toward the middle counteract some of the pole's gain in arrival time?

Length contraction doesn’t have any sense of “shrinking” and it doesn’t have any point towards which it shrinks.

If you have an inertially moving rod and two inertially moving observers then they will disagree on the length of the rod. But for each observer it has always been that long, so it isn’t a matter of “shrinking”.

In one type, the object experiences the contraction (a 5 meter object becomes 4 meters, as experienced in an observer's frame).

In the other type, the distance of travel is contracted (a distance of 10,000 km is now 5.000 km, in the traveler's frame).

I'm trying to find somewhere that discusses the differences between those two types, if any exist

They are the same. You can always consider a rod laid to match the distance of travel. So you can convert one to the other.

 

[Ibix]

Ibix had confirmed the quoted text below as valid. Is there an aspect I'm missing? Or is your reply contradicting theirs? (maybe you're both right, I've got no idea):

I confirmed what was written, which may or may not be what you understood.

Measured from a frame in which the Earth is at rest, clocks travelling with the muons are time dilated and tick slowly. Hence enough muons survive long enough that we receive them at ground level. We would also expect that rulers travelling with the muons would be length contracted compared to ground-based rulers, but this result is irrelevant to the experiment.

Measured from a frame in which the muons are at rest the atmosphere is length contracted so it is only a short distance to the Earth's surface. Enough muons therefore survive until the ground reaches them that they can be detected there. Clocks at rest on the Earth would also tick slowly in this frame, but that is not relevant to this experiment.

 

[Mister T]

If a long pole at relativistic speed would strike an object, would we calculate from the front of that object as if the rest of its length had shrunk toward the front, or, would we calculate as if the front had shrunk back toward the middle of pole?

If you replace the long pole with a shorter pole would you have the same set of questions?

 

[jbriggs444]

I'm probably making an error from not knowing what I don't know.

What I've gathered is there are two types of contraction.

In one type, the object experiences the contraction (a 5 meter object becomes 4 meters, as experienced in an observer's frame).

In the other type, the distance of travel is contracted (a distance of 10,000 km is now 5.000 km, in the traveler's frame).

There is only the one sort of contraction. A distance measured in one frame will differ from a distance measured in another.

There is little or nothing physical going on here. This is very much analogous to measuring the distance across a ribbon with a ruler. It depends on what angle you hold the ruler.

The analogy is that you have to pick two points on opposite sides of the ribbon to do your measurement. Once you pick those two points, the angle that you hold the ruler is determined and the measurement is simple.

When you are measuring the distance between two ends of a moving object, it is very similar. The object traces out a "world tube" in four dimensional space time. You want to measure the distance between the two sides of the tube. But that means picking out one event on one side and a matching event on the other side. We use "at the same time" to pick out the matching event. But "at the same time" means different things to different frames of reference.

Two frames of reference will measure between different events on the boundary of the world tube and get different results. It's just 4 dimensional geometry.

 

[Mister T]

What I've gathered is there are two types of contraction.

In one type, the object experiences the contraction (a 5 meter object becomes 4 meters, as experienced in an observer's frame).

In the other type, the distance of travel is contracted (a distance of 10,000 km is now 5.000 km, in the traveler's frame).

Imagine a 10 000 km rod used to measure the distance of travel. That rod would contract in the same way the 5 meter object contracts to 4 meters.

In other words, these two types of contraction are the same phenomenon.

 

[syfry]

If you replace the long pole with a shorter pole would you have the same set of questions?

Yeah, had added on 'long' for a better chance that my question wasn't misinterpreted. But then after using the word 'pole' for the traveler and 'object' for the target, I still messed up anyway and referred to traveler as an object.

 

[syfry]

The reason I said long pole is because of the muon experience. If the muon itself were contracted, then logically that seems like it'd have to travel a greater distance. But if the distance of travel is contracted, then it'll arrive sooner at its destination.

I'm still confused by that aspect. Do we contract everything? (the pole and its route)

Or only one? (pole alone, or route alone, but not both)

Since the muon is too small to be contracted, I then emphasized long pole for the most clarity from an answer.

 

[Vanadium 50]

I tried to learn about coordinate transforming from a minute physics video below, but they went way too fast

I took a speed reading class and then was able to read War and Peace in twenty minutes.

It's about Russia.

 

[syfry]

I took a speed reading class and then was able to read War and Peace in twenty minutes.

It's about Russia.

Fair enough. "Learn about (something)" isn't the same as saying "learn the (something)". Had meant in a general way learn the gist of its possibilities, what things mean. But even that was too fast. I'd have to write down what they're saying word for word to slowly read at my own pace, that's all.

 

[jbriggs444]

The reason I said long pole is because of the muon experience. If the muon itself were contracted, then logically that seems like it'd have to travel a greater distance. But if the distance of travel is contracted, then it'll arrive sooner at its destination.

I'm still confused by that aspect. Do we contract everything? (the pole and its route)

None of the above. You are starting from an incorrect mental picture. We need to tear down your intuitions before we can build them back up.

Do you know what a frame of reference is?

 

[syfry]

None of the above. You are starting from an incorrect mental picture. We need to tear down your intuitions before we can build them back up.

Do you know what a frame of reference is?

Glad to tear down my misconceptions! To my understanding, a frame of reference is a place or object from which we perceive an apparent motion. In special relativity that would readjust what observers measure space and time to be.

 

[jbriggs444]

Glad to tear down my misconceptions! To my understanding, a frame of reference is a place or object from which we perceive an apparent motion. In special relativity that would readjust what observers measure space and time to be.

No. That is not correct.

A "frame of reference" is an agreed upon standard of rest. It reaches everywhere. If one picks an origin, a set of directions for x, y and z, and units for time and distance, it amounts to a coordinate system.

Most of the time, I think of "coordinate system" and "frame of reference" interchangably.Many times you find textbooks or presentations talking about things like an "observer" or the "rocket's rest frame". This is a shorthand way of using the observer or the rocket as an anchor to define a state of rest. One can then extend a coordinate system and the associated standard of rest to the remainder of the universe by laying out an imaginary rigid and non-rotating grid of clocks and rulers using the "observer" or other object as a starting point.

It is worth keeping the image of an imaginary rigid non-rotating grid of synchronized clocks and rulers in mind when thinking about a coordinate system in special relativity.

[In General relativity, things get more difficult. A single object is not enough to specify a frame of reference and there is no such thing as an inertial frame of reference in curved space time]

Given a coordinate system, we can use coordinates [such as (x, y, z, t)] to tell when and where an event takes place. The same event will have different coordinates when we write the coordinates down in different coordinate systems.

"Time dilation" and "length contraction" reflect what happens when you rotate your coordinate system. The physical stuff does not change. The numbers that appear in the coordinates do.

 

[Vanadium 50]

I'd have to write down what they're saying

There is a new invention when this is all done for you. It's called "Books".

I recommend Spacetime Physics by Taylor and Wheeler, 1st Edition if you can find it.

 

[syfry]

A "frame of reference" is an agreed upon standard of rest. It reaches everywhere. If one picks an origin, a set of directions for x, y and z, and units for time and distance, it amounts to a coordinate system.

Most of the time, I think of "coordinate system" and "frame of reference" interchangably.

Ok great starting point! So when people say "from which frame of reference?" should I interpret that as "from which agreed upon standard of rest"? (alternatively: "from which agreed upon coordinate system")

Does the agreement have to all come from only the same reference frame?

It is worth keeping the image of an imaginary rigid non-rotating grid of synchronized clocks and rulers in mind when thinking about a coordinate system in special relativity.

So even though the Earth is rotating, you'd treat it as not rotating for the purpose of that grid?

Given a coordinate system, we can use coordinates [such as (x, y, z, t)] to tell when and where an event takes place. The same event will have different coordinates when we write the coordinates down in different coordinate systems.

So each coordinate system will look at a particular coordinate system through its own lens, so to speak.

"Time dilation" and "length contraction" reflect what happens when you rotate your coordinate system. The physical stuff does not change. The numbers that appear in the coordinates do.

In that sense, rotate has nothing to do with planetary type of rotation, right? It's more like rotate the grid or part of it, sounds like.

Can you elaborate on "the physical stuff does not change"? Does that mean a length contraction isn't about resizing an object, but only about changing the x, y, z, t values?

Thanks a bunch!

 

[jbriggs444]

Ok great starting point! So when people say "from which frame of reference?" should I interpret that as "from which agreed upon standard of rest"? (alternatively: "from which agreed upon coordinate system")

Yes. So when you report a velocity, you need to tell us what frame of reference that velocity is relative to.

What is less obvious is that when you report an elapsed time, you need to be clear on what frame of reference that is relative to.

Does the agreement have to all come from only the same reference frame?

You can supply pieces of data taken relative to different frames. You just have to be clear which frame goes with which data.

When asking for results, you need to be clear about which frame of reference you want the answer relative to.

So even though the Earth is rotating, you'd treat it as not rotating for the purpose of that grid?

Most of the time the Earth is rotating slowly enough and light moves fast enough that we can pretend that the Earth is not rotating.

There are instruments (e.g. laster ring gyroscopes) that can detect the rotation. See the Sagnac effect. So sometimes one does need to distinguish between a rotating frame anchored to the Earth and an inertial frame anchored to the Earth's center.

Technically, the space time near the earth is curved. That is how General Relativity models gravity. So technically, the non-rotating frame anchored to the Earth's center is not truly inertial.

So each coordinate system will look at a particular coordinate system through its own lens, so to speak.

Maybe. Tread with care when trying to personify physics. It does not think. It does not perceive. It does not desire. It just is.

In that sense, rotate has nothing to do with planetary type of rotation, right? It's more like rotate the grid or part of it, sounds like.

If we were in empty space, we could tell whether our coordinate system were rotating. We could put some small test objects out there and see whether their coordinates changed over time. If we observe patterns of motion consistent with centrifugal force and Coriolis acceleration then we can deduce that our coordinate system is rotating.

[Spatial] rotation for a coordinate system is just like a planet or a moon rotating. It is nothing special.

One does have to be aware of the idea of "rotating" a coordinate system in the fourth dimension (time). That is sufficiently different from an ordinary spatial rotation that we give it a different name. We call it a "boost" instead. This corresponds to changing one's standard of rest -- giving the new coordinate system a non-zero velocity relative to the original coordinate system.

It is still a rotation, but it is a rotatin in hyperbolic geometry, not spherical geometry. Not something you need to understand completely at the moment.

I suppose that I should agree that there is a distinction between a inertial coordinate system being rotated through a discrete angle (using the Lorentz transforms or a coordinate rotation matrix) to obtain a new coordinate system on the one hand and a continuously rotating non-inertial coordinate system on the other. We do use the same word, "rotation" to refer to both situations, even though they are rather different. [Albeit related -- the inertial forces in the one situation can be derived as the result of a series of infinitesimal rotations in the other situation]

Can you elaborate on "the physical stuff does not change"? Does that mean a length contraction isn't about resizing an object, but only about changing the x, y, z, t values?

Right. This figures into the barn-pole paradox or the bug-rivet paradox. We have a physical pole. We view it against one frame of reference (the barn) and measure it to be short. We view it against another frame of reference (the pole itself) and measure it to be long.

Both measurements are correct. "Length" is not an intrinsic attribute of a pole. It is an attribute of a pole in relation to a frame of reference. Various frames of reference can disagree about the measured value.

That said, "proper length", the length of the pole as measured in its own rest frame is a fixed quantity. Every frame will agree on that quantity.

If you accelerate the pole to a new velocity (and do not hold it rigid -- see Bell's paradox), its structure will work to maintain its original proper length in its new rest frame.

I hope that at least some of this is aimed at the right level.

 

[syfry]

is relative to.

That phrase often throws my mind for a loop. To confirm, should I interpret it as meaning "x speed according to an agreed upon standard of rest"?

"Length" is not an intrinsic attribute of a pole. It is an attribute of a pole in relation to a frame of reference. Various frames of reference can disagree about the measured value

That said, "proper length", the length of the pole as measured in its own rest frame is a fixed quantity. Every frame will agree on that quantity.

That makes sense. Since physics works as it's normally expected to in every reference frame, without a proper length or agreement, then maybe length contraction would produce strange or bizarre results in some things such as neutrinos, which might contract many objects to thinner than their Swartzchild radius and therefore could get stuck in an event horizon that exists only in their own frame of reference.

Or, volatile reactive ingredients that aren't near enough in their own frame could be next to each other in the neutrinos frame.

I hope that at least some of this is aimed at the right level.

Layperson level, and yes thanks! That cleared up some stumbling blocks.

 

[jbriggs444]

That phrase often throws my mind for a loop. To confirm, should I interpret it as meaning "x speed according to an agreed upon standard of rest"?

Yes.

That makes sense. Since physics works as it's normally expected to in every reference frame, without a proper length or agreement, then maybe length contraction would produce strange or bizarre results in some things such as neutrinos, which might contract many objects to thinner than their Swartzchild radius and therefore could get stuck in an event horizon that exists only in their own frame of reference.

Nothing quite that mysterious or mystical. Let's not talk Schwarzschild radii yet. We're still doing special relativity, not general relativity.

Or, volatile reactive ingredients that aren't near enough in their own frame could be next to each other in the neutrinos frame.

This does not sound descriptive of anything real.

 

[syfry]

Ok I'll make note of your points and will check out the Spacetime Physics book.

Thanks all who chimed in and contributed to helping me understand this better!

 

[PeroK]

Fair enough. "Learn about (something)" isn't the same as saying "learn the (something)". Had meant in a general way learn the gist of its possibilities, what things mean. But even that was too fast. I'd have to write down what they're saying word for word to slowly read at my own pace, that's all.

Try this:

https://scholar.harvard.edu/david-morin/special-relativity

The first chapter is free.

It takes more than a minute to learn SR!

 

[syfry]

I recommend Spacetime Physics by Taylor and Wheeler, 1st Edition if you can find it.

Over $200 for 1st edition

(found a beat up one for $12 -ish)

 

[Vanadium 50]

The 2nd edition is free to download, but I think the 1st edition is better. But not $200 better.

 

[PeroK]

Over $200 for 1st edition

(found a beat up one for $12 -ish)

Are you a student or a collector?