Rapidly rotating wire

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BlackSails
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Rapidly rotating wire

Postby BlackSails » Fri Jan 23, 2009 6:49 pm UTC

Lets say you have a long wire out in space, and you start spinning it very quickly around one end, so it looks sort of like a rope with a weight on the end being spun by a person.

Now, the wire is metal. The electrons in it are constantly diffusing all around. However, when you spin it, the outermost segments of wire are moving faster than the innermost segments. Because of time dilation, this means the electrons there will diffuse more slowly, which ought to induce a charge seperation in the wire (because in each second from an external observer, an electron is more likely to move outward than inward)


Where does the energy come from? Would the wire simply slow down over time?

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Re: Rapidly rotating wire

Postby Charlie! » Fri Jan 23, 2009 7:54 pm UTC

Huh, that's interesting. I'd assume that the energy comes from the only place it CAN come from - the energy it takes to accelerate the wire to its final angular speed. Once you're at that speed an equilibrium should be set up, with no more energy needed.
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Re: Rapidly rotating wire

Postby frezik » Fri Jan 23, 2009 10:09 pm UTC

Meteorswarm wrote:Might it come from an accumulation of mass/energy at the tips of the wire via electron motion, lowering the rotational energy of the system, just like rotating sliding blocks on a rope lowers the energy by sliding out?


You haven't actually lost energy in that situation, though. You've lost speed and gained mass at the outside of the rotating body. Overall energy is conserved.
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Re: Rapidly rotating wire

Postby idobox » Fri Jan 23, 2009 11:04 pm UTC

I'm not sure I understand what you mean.
On each point of the wire, the electrons diffuse at the same speed in every direction, so the net current in every point is null, and no charge builds up.
On the other hand, in order to have a significant time dilatation, you ought to spin the wire really fast, and even though the mass of electrons is small, the centrifugal force might be significant.
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Re: Rapidly rotating wire

Postby BlackSails » Fri Jan 23, 2009 11:09 pm UTC

Lets say the wire just has two segments.

The outer segment is faster than the inner. Therefore, electrons move (and diffuse) more slowly than the inner segment. At the border between the two segments, assuming no time dilation, you have equal amounts moving across the border in each direction. However, since the outer segment is slower, the electrons cross the boundary less often. This leads to a net flux of electrons towards the outer segment.

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Re: Rapidly rotating wire

Postby ThinkerEmeritus » Fri Jan 23, 2009 11:10 pm UTC

You don't need relativistic effects. Electrons in a metal are pretty much free to move relative to the metal ions. In the spinning wire, the electrons are going around in circular motion (as seen from the stationary frame). There needs to be a force to hold them in circular motion, and binding to the metal ions doesn't provide one. The electrons move outward along the wire, charging the outer end negatively and the inner end positively. The resulting electric force provides the force needed for the electrons' centripetal acceleration. Ask where the energy comes from this effect, and you are asking something that can be addressed successfully both theoretically and experimentally.

Now if you want a really good puzzle: what is holding the positive ions in circular motion? The overall electric field is the wrong sign. Note you can't get out of the puzzle by reversing the electric field, because then I will ask you what is keeping the electrons in circular motion.
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Re: Rapidly rotating wire

Postby idobox » Sat Jan 24, 2009 2:12 am UTC

Lets say the wire just has two segments.

The outer segment is faster than the inner. Therefore, electrons move (and diffuse) more slowly than the inner segment. At the border between the two segments, assuming no time dilation, you have equal amounts moving across the border in each direction. However, since the outer segment is slower, the electrons cross the boundary less often. This leads to a net flux of electrons towards the outer segment.


If that worked like that, you'd have a current between the hot and cold parts of a metal wire. If you take a infinitesimally short segment of the wire, you can consider there is no time distortion along it. Now if you count the electrons diffusing at one side, and the electrons diffusing at the other side, the numbers are the same, so overall current is null.

Now if you want a really good puzzle: what is holding the positive ions in circular motion? The overall electric field is the wrong sign. Note you can't get out of the puzzle by reversing the electric field, because then I will ask you what is keeping the electrons in circular motion.


Since the wire is metallic, I'd say all the free electrons move to wherever they want, but the covalent ones stay where they are and provide the cohesion between the atomic nuclei.
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Re: Rapidly rotating wire

Postby Carnildo » Sat Jan 24, 2009 3:37 am UTC

idobox wrote:If that worked like that, you'd have a current between the hot and cold parts of a metal wire.


See "thermoelectric effect"

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Re: Rapidly rotating wire

Postby BlackSails » Sat Jan 24, 2009 7:27 am UTC

idobox wrote:If that worked like that, you'd have a current between the hot and cold parts of a metal wire. If you take a infinitesimally short segment of the wire, you can consider there is no time distortion along it. Now if you count the electrons diffusing at one side, and the electrons diffusing at the other side, the numbers are the same, so overall current is null.


Zeno's paradox. You dont have no time distortion. Across an infitesimal segment you have an infitesimal amount of time dilation.

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Re: Rapidly rotating wire

Postby idobox » Sat Jan 24, 2009 9:56 am UTC

See "thermoelectric effect"

The thermoelectric effect happens at a junction between two materials.

An other way to see the problem: let's take a plane that cuts the wire perpendicularly. The metal on both sides diffuses at the same speed. If a part of the metal diffuses slower because of time dilatation, it also means the electrons diffusing into it move slower. If you think of it as a gas of free electrons, with the observed speed of electrons being different depending on the place, you find that for a point the electrons go in all directions with the same speed, resulting in a net displacement of zero. You might think the electrons that move faster are "hotter" and build up pressure, but this isn't the case. The electrons all move at the same speed in their relative frame of reference, and so have the same kinetic energy. Okay, not the exact same speed, but they are described by the same Bolztman equation.
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Re: Rapidly rotating wire

Postby Charlie! » Sat Jan 24, 2009 7:08 pm UTC

idobox wrote:An other way to see the problem: let's take a plane that cuts the wire perpendicularly. The metal on both sides diffuses at the same speed. If a part of the metal diffuses slower because of time dilatation, it also means the electrons diffusing into it move slower. If you think of it as a gas of free electrons, with the observed speed of electrons being different depending on the place, you find that for a point the electrons go in all directions with the same speed, resulting in a net displacement of zero. You might think the electrons that move faster are "hotter" and build up pressure, but this isn't the case. The electrons all move at the same speed in their relative frame of reference, and so have the same kinetic energy. Okay, not the exact same speed, but they are described by the same Bolztman equation.

Here's an example of why your argument has hole: Let's take a vertical line that cuts the function y=x2. The function on both sides of the vertical line has the same value. Therefore, at any point there's no change in y=x2. Wait, what?

To fix this, you need calculus. Not just a plane of width zero, but two planes an infinitesimal distance apart. Then describe diffusion as a function of r and dr and integrate.
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Re: Rapidly rotating wire

Postby idobox » Sun Jan 25, 2009 10:05 am UTC

Okay, I'll try another way.

You've got two parts of the wire, some distance apart, with the same temperature, and in both, an electron diffuses in direction of the other. Since the wire has the same temperature everywhere, we know that the average velocity of electrons is the same everywhere in the local referential.
The first electron moves from the 'fast time' zone, to the ' slow time' zone without loosing or gaining kinetic energy. It's speed in the local referential is unchanged, but its speed seen by an external observer decreases.
The second electron goes from slow to fast, and its apparent speed increase. Finally, the two electrons take the same amount of time to travel the distance one way or the other.
The two parts of the wire have the same number of free electrons, with the same energy distribution, so the exact number of electrons moving to the other zone, at with the exact same kinetic energy, and as we just saw, same apparent speed. Net current is null.
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Re: Rapidly rotating wire

Postby BlackSails » Sun Jan 25, 2009 2:37 pm UTC

idobox wrote:Okay, I'll try another way.

You've got two parts of the wire, some distance apart, with the same temperature, and in both, an electron diffuses in direction of the other. Since the wire has the same temperature everywhere, we know that the average velocity of electrons is the same everywhere in the local referential.
The first electron moves from the 'fast time' zone, to the ' slow time' zone without loosing or gaining kinetic energy. It's speed in the local referential is unchanged, but its speed seen by an external observer decreases.
The second electron goes from slow to fast, and its apparent speed increase. Finally, the two electrons take the same amount of time to travel the distance one way or the other.
The two parts of the wire have the same number of free electrons, with the same energy distribution, so the exact number of electrons moving to the other zone, at with the exact same kinetic energy, and as we just saw, same apparent speed. Net current is null.


That is incorrect. A slowdown in time in one section is equivalent to an increase in velocity in the other.

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Re: Rapidly rotating wire

Postby idobox » Sun Jan 25, 2009 6:39 pm UTC

uh?

I'll try to be more clear. What I call the "fast zone" is the zone near the center, with low speed and low time distortion, and the "slow zone", the zone near the edge, which has stronger time dilatation, and in which events seem to happen slower to an external observer.

Anyway, that doesn't change anything to the idea.
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Re: Rapidly rotating wire

Postby Seraph » Sun Jan 25, 2009 11:17 pm UTC

idobox wrote:
See "thermoelectric effect"

The thermoelectric effect happens at a junction between two materials.

While you are technically correct I would still call this statement wrong, or misleading at best. The thermoelectric effect is actaully the result of several different effects.
We can measure one of those, the Thompson effect, without any junctions. So while the thermoelectric effect does happen at junctions, it also happens at non-junctions.

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Re: Rapidly rotating wire

Postby Charlie! » Mon Jan 26, 2009 9:01 pm UTC

idobox wrote:Okay, I'll try another way.

You've got two parts of the wire, some distance apart, with the same temperature, and in both, an electron diffuses in direction of the other. Since the wire has the same temperature everywhere, we know that the average velocity of electrons is the same everywhere in the local referential.
The first electron moves from the 'fast time' zone, to the ' slow time' zone without loosing or gaining kinetic energy. It's speed in the local referential is unchanged, but its speed seen by an external observer decreases.
The second electron goes from slow to fast, and its apparent speed increase. Finally, the two electrons take the same amount of time to travel the distance one way or the other.
The two parts of the wire have the same number of free electrons, with the same energy distribution, so the exact number of electrons moving to the other zone, at with the exact same kinetic energy, and as we just saw, same apparent speed. Net current is null.

I think the bold is the problematic statement. You say it, but you neglect the effects of switching how fast your clock is going on the speed of diffusion of the electrons next to you. Instead, you interpret it to mean "everything is the same at all points on the wire, from any point of view." However, this is inconsistent, since from a non-accelerating point of view you can clearly see that some electrons are diffusing faster than others.

Would it help if I struggled through the calculus?
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Re: Rapidly rotating wire

Postby doogly » Mon Jan 26, 2009 9:29 pm UTC

I have not solved it out exactly yet but let me give you my thought: the energy stored in the field is going to matter. You've got accelerating particles so some energy radiating is gonna happen. It is an interesting problem though so maybe I will want to do the whole thing.
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Re: Rapidly rotating wire

Postby Goemon » Tue Jan 27, 2009 6:42 am UTC

Hmmm...

Suppose, for the sake of argument, that the average speed of electrons along the length of the wire when it's not spinning is s. Assume there's no net diffusion when the wire's at rest. Start the wire spinning. We've now added a component of velocity perpendicular to the wire.

So what?

This has no effect on the component of velocity along the wire. MY clocks are still ticking at the same rate. The electrons are still drifting along the wire at the same speed, according to MY clocks and measuring rods. There's no change in the rate of diffusion, according to MY measurements.

If the electrons happen to be carrying tiny little clocks along with them as they drift hither and yon, I might note that those moving clocks at the spinning ends of the wire are ticking more slowly now that they have a new velocity = (their original, unchanged velocity up and down the wire) + (velocity perpendicular to the wire due to spinning).

The point is that the diffusion rate I measure depends on the speed the electrons are moving up and down the wire (which is unchanged by spinning the wire), not how fast their clocks are ticking relative to mine.
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Re: Rapidly rotating wire

Postby Rentsy » Tue Jan 27, 2009 6:46 am UTC

Relativistic effects on electrons are called magnetism.

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Re: Rapidly rotating wire

Postby doogly » Tue Jan 27, 2009 7:18 am UTC

Spinning the wire will cause the electrons to drift towards the outside. The centrifugal force on an electron at r will have to balance with the electric contributions from the rest of the wire. So you need
[math]m\omega^2 r - a\int_0^r \frac{\lambda(r')}{r'^2} dr' +q \int_r^R \frac{\lambda(r')}{r'^2} dr' = 0[/math]
I have a sneaking suspicion that Volterra?... Laplace?... might be able to solve this, but I am having some trouble and need to do other work. Maxwell's Laws are already Lorentz invariant, so I'd say just solve everything where the electrons aren't moving and then boost to any other frame of interest. Assuming you are better at the above integral equation, of course. I even tried to solve it with a series solution for you guys, and it doesn't have one. Do you see how much I love you, forum? I do not do series at 2:30 unless there is love. And yet I may not do series well at 2:30 either, so I encourage everyone else to check.
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Re: Rapidly rotating wire

Postby Charlie! » Tue Jan 27, 2009 7:20 am UTC

Rentsy wrote:Relativistic effects on electrons are called magnetism.

How would you go about explaining this effect with magnetism? It seems hard, although I dunno, it might be possible.

Goemon wrote:The point is that the diffusion rate I measure depends on the speed the electrons are moving up and down the wire (which is unchanged by spinning the wire), not how fast their clocks are ticking relative to mine.
Speed IS how fast their clocks are ticking. Speed = d/t. Change how something measures time and you must, must, must change how fast it's moving or else it could simply measure some universal, "more real" time by measuring velocity. This is exactly what einstein says doesn't exist.
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Re: Rapidly rotating wire

Postby idobox » Tue Jan 27, 2009 11:45 am UTC

I think the bold is the problematic statement

What bold part?

We can measure one of those, the Thompson effect, without any junctions

The Thompson effect doesn't cause current or potential difference.

Relativistic effects on electrons are called magnetism.

I'd love to hear the explanation of that. Magnetism also happens with non relativistic electrons.

Speed IS how fast their clocks are ticking. Speed = d/t. Change how something measures time and you must, must, must change how fast it's moving or else it could simply measure some universal, "more real" time by measuring velocity. This is exactly what einstein says doesn't exist.

That's the whole point, you don't change how you measure time. You just measure time at different places, that's the whole point of relativity. An observer anywhere on the wire will measure the same electron speed.
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Re: Rapidly rotating wire

Postby BlackSails » Tue Jan 27, 2009 1:28 pm UTC

idobox wrote:That's the whole point, you don't change how you measure time. You just measure time at different places, that's the whole point of relativity. An observer anywhere on the wire will measure the same electron speed.


What? The speed of light is constant. Other particles move at different speeds in different frames.

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Re: Rapidly rotating wire

Postby idobox » Tue Jan 27, 2009 5:51 pm UTC

What? The speed of light is constant. Other particles move at different speeds in different frames.

Get two observers with identical clocks and rulers, and place them on different parts of the wire. When they observe electrons near them, they measure the same average speed. Because the clock ticks slower, but the ruler also shrinks, so the ratio stays the same.

And as Goemon stated, relativistic effects will only occur in the direction of motion, which is perpendicular to the wire, and will not affect diffusion in the radial direction.
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Re: Rapidly rotating wire

Postby Charlie! » Tue Jan 27, 2009 11:40 pm UTC

idobox wrote:
Charlie wrote: I think the bold is the problematic statement
What bold part?
The part of the quote that looked like this.


idobox wrote:
What? The speed of light is constant. Other particles move at different speeds in different frames.

Get two observers with identical clocks and rulers, and place them on different parts of the wire. When they observe electrons near them, they measure the same average speed. Because the clock ticks slower, but the ruler also shrinks, so the ratio stays the same.

And as Goemon stated, relativistic effects will only occur in the direction of motion, which is perpendicular to the wire, and will not affect diffusion in the radial direction.

Read this: http://en.wikipedia.org/wiki/Gravitatio ... e_dilation
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Re: Rapidly rotating wire

Postby Goemon » Wed Jan 28, 2009 12:47 am UTC

Charlie! wrote:
Goemon wrote:The point is that the diffusion rate I measure depends on the speed the electrons are moving up and down the wire (which is unchanged by spinning the wire), not how fast their clocks are ticking relative to mine.
Speed IS how fast their clocks are ticking. Speed = d/t. Change how something measures time and you must, must, must change how fast it's moving or else it could simply measure some universal, "more real" time by measuring velocity. This is exactly what einstein says doesn't exist.


Electron A is moving east at 1m/s
Electron B has velocity with eastward component = 1m/s and northward component = 298,289,729 m/s (gamma = 10)
After one second has passed, which electron is farther east?

Answer: they've both moved eastward one meter.

B's clock may be ticking ten times slower than A's, but the eastward component of both velocities is 1m/s, so their eastward progress is exactly the same. I don't see that spinning the wire has any affect on the rate of diffusion...
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Re: Rapidly rotating wire

Postby Charlie! » Wed Jan 28, 2009 1:30 am UTC

Goemon wrote:
Charlie! wrote:
Goemon wrote:The point is that the diffusion rate I measure depends on the speed the electrons are moving up and down the wire (which is unchanged by spinning the wire), not how fast their clocks are ticking relative to mine.
Speed IS how fast their clocks are ticking. Speed = d/t. Change how something measures time and you must, must, must change how fast it's moving or else it could simply measure some universal, "more real" time by measuring velocity. This is exactly what einstein says doesn't exist.


Electron A is moving east at 1m/s
Electron B has velocity with eastward component = 1m/s and northward component = 298,289,729 m/s (gamma = 10)
After one second has passed, which electron is farther east?

Answer: they've both moved eastward one meter.

B's clock may be ticking ten times slower than A's, but the eastward component of both velocities is 1m/s, so their eastward progress is exactly the same. I don't see that spinning the wire has any affect on the rate of diffusion...

You're right in the case you're thinking of, but the case you're thinking of isn't the same as the case relevant to this thread. Imagine a chunk of matter at the bottom of a gravitational potential sooo big that time, for this object, is going only half as quickly as it is for some observer in space. This observer in space will observe ALL POSSIBLE MEASURES OF TIME to be slower for the chunk of matter at the bottom, because THAT'S THE BLOODY POINT. This includes speed of particles, as shown by the many experiments using particle speed-based clocks (i.e. clocks) to confirm the predictions of general relativity.

http://en.wikipedia.org/wiki/Gravitatio ... e_dilation
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Re: Rapidly rotating wire

Postby Goemon » Wed Jan 28, 2009 4:38 am UTC

Charlie! wrote:You're right in the case you're thinking of, but the case you're thinking of isn't the same as the case relevant to this thread. Imagine a chunk of matter at the bottom of a gravitational potential sooo big that time, for this object, is going only half as quickly as it is for some observer in space. This observer in space will observe ALL POSSIBLE MEASURES OF TIME to be slower for the chunk of matter at the bottom, because THAT'S THE BLOODY POINT. This includes speed of particles, as shown by the many experiments using particle speed-based clocks (i.e. clocks) to confirm the predictions of general relativity.

http://en.wikipedia.org/wiki/Gravitatio ... e_dilation


Actually, there's no need to call up general relativity or gravity wells for this problem - there's no spacetime curvature involved. We can analyze it using a global SR inertial frame, in which clocks everywhere tick at the same rate. And analyzing in this frame (per previous posts) sez there's no impact to the diffusion rate caused by time dilation.

Alternatively, the problem can be analyzed in a rotating frame of reference centered on the wire's midpoint, in which the wire remains stationary but feels a centripetal acceleration that increases with radius. (Note: this analysis also does not require GR; it's still SR). In this reference frame, clocks farther from the center of rotation tick more slowly. An observer located at the center would measure (according to his local clock and meter stick) that the drift velocity of electrons at the wire ends is slower than those near the center. But the ratio of velocities is exactly cancelled by the fact that clocks at the wire ends are also ticking more slowly. The observer at the center (or anywhere else) calculates that the drift velocity of electrons is the same at every point along the wire, when measured by HIS clocks and rulers located AT THE POINT in question.

The analyses in the two reference frames must agree, and do: Time dilation doesn't affect the diffusion rate.

Of course, the acceleration from spinning the wire should cause some electrons to be pulled a bit further toward the wire ends just via centrifugal force. But the accumulation of excess charge would repel any more electrons from joining them, so it would quickly reach equilibrium at any fixed rate of rotation.
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Re: Rapidly rotating wire

Postby Rentsy » Wed Jan 28, 2009 4:50 am UTC

I mean to say that magnetism is, itself, a relativistic effect.

Maybe this will explain it better?

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Re: Rapidly rotating wire

Postby idobox » Wed Jan 28, 2009 12:52 pm UTC

The part of the quote that looked like this.

Okay, the difference between bold and normal characters isn't very clear on my computer. Sorry for the trouble.

I mean to say that magnetism is, itself, a relativistic effect.

I never thought of it that way. Does it mean moving masses create a magnetic-like gravitational field ?
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Re: Rapidly rotating wire

Postby Charlie! » Wed Jan 28, 2009 5:43 pm UTC

Goemon wrote:
Charlie! wrote:You're right in the case you're thinking of, but the case you're thinking of isn't the same as the case relevant to this thread. Imagine a chunk of matter at the bottom of a gravitational potential sooo big that time, for this object, is going only half as quickly as it is for some observer in space. This observer in space will observe ALL POSSIBLE MEASURES OF TIME to be slower for the chunk of matter at the bottom, because THAT'S THE BLOODY POINT. This includes speed of particles, as shown by the many experiments using particle speed-based clocks (i.e. clocks) to confirm the predictions of general relativity.

http://en.wikipedia.org/wiki/Gravitatio ... e_dilation


Actually, there's no need to call up general relativity or gravity wells for this problem - there's no spacetime curvature involved. We can analyze it using a global SR inertial frame, in which clocks everywhere tick at the same rate. And analyzing in this frame (per previous posts) sez there's no impact to the diffusion rate caused by time dilation.

Alternatively, the problem can be analyzed in a rotating frame of reference centered on the wire's midpoint, in which the wire remains stationary but feels a centripetal acceleration that increases with radius. (Note: this analysis also does not require GR; it's still SR). In this reference frame, clocks farther from the center of rotation tick more slowly. An observer located at the center would measure (according to his local clock and meter stick) that the drift velocity of electrons at the wire ends is slower than those near the center. But the ratio of velocities is exactly cancelled by the fact that clocks at the wire ends are also ticking more slowly. The observer at the center (or anywhere else) calculates that the drift velocity of electrons is the same at every point along the wire, when measured by HIS clocks and rulers located AT THE POINT in question.

The analyses in the two reference frames must agree, and do: Time dilation doesn't affect the diffusion rate.

Of course, the acceleration from spinning the wire should cause some electrons to be pulled a bit further toward the wire ends just via centrifugal force. But the accumulation of excess charge would repel any more electrons from joining them, so it would quickly reach equilibrium at any fixed rate of rotation.


I'll try and take this one thing at a time. Acceleration is equivalent to gravity. http://en.wikipedia.org/wiki/Equivalence_principle . So yes, there is a reason to call up general relativity for this problem.
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Re: Rapidly rotating wire

Postby Goemon » Thu Jan 29, 2009 6:33 am UTC

Charlie! wrote:I'll try and take this one thing at a time. Acceleration is equivalent to gravity. http://en.wikipedia.org/wiki/Equivalence_principle . So yes, there is a reason to call up general relativity for this problem.


[I think this is getting a bit off topic, so I'll hide it for those who aren't interested in following the side conversation...]


Spoiler:
Special Relativity can be used by any observer who is in uniform motion (feels no acceleration), and who is able to construct a perfectly regular frame of reference that extends throughout all space and time. That means he/she can lay out measuring sticks in the x, y, and z directions which form a perfect grid, and all the clocks AT REST IN THIS REFERENCE FRAME - ie, attached to the grid points - remain synchronized.

That's what the "Special" in "Special Relativity" means; the laws of SR only apply to observers who meet these requirements.

An observer sitting "at rest" nearby the spinning wire meets all these requirements. He can observe the motion and behavior of all the particles in the wire, (using measurements based on his universal grid and the stationary clocks attached to it) and apply the ordinary laws of SR to all of his measurements. This observer will find that time dilation of the various electrons doesn't impact the diffusion rate. (Ah ha!)

An observer sitting on the end of the spinning wire is not in uniform motion. She is not permitted to apply the ordinary laws of SR to her observations. BUT - our first observer, who IS in uniform motion in a global Lorentz frame, is able to take complete measurements of everything in the experiment, including the dizzy observer. From this, we can easily reconstruct exactly what the dizzy observer will see and measure at every instant. And furthermore, we can deduce what equations should be used by a rotating observer in flat spacetime. In other words, we can deduce what an accelerating observer will see and measure, using nothing but the equations of SR and a little brainpower.

Now consider an entirely different scenario: an observer located near the Sun will find that there's no such thing as a global inertial reference frame. No matter how you try to construct it, test particles in one region or another insist on flying off hither and yon due to "gravity". (This is actually another way of saying it's impossible to lay out measuring sticks in the x, y, z directions throughout the entire region in any way that will enable them to mesh perfectly. Just like trying to cover a sphere with a grid of uniform squares, it can't be done). Solving problems in a region of spacetime near a gravitational source requires an entirely new set of cogs and wheels. It can't be done with SR, because there's no such thing as a global inertial observer anywhere.

The oft-misquoted Principle of Equivalence provides the link between SR and GR. It states that LOCALLY (read carefully through the description on the wiki page) gravitation is equivalent to acceleration. That means that we can begin to understand how to build the machinery of GR by starting with the assumption that tiny little slices of spacetime near a gravitational source behave exactly like empty space - IF we apply the rules about acceleration that we learned from watching the rotating wire from a distance to each separate slice individually. (They're only LOCALLY equivalent.)

The trick is figuring out how to join the equations describing each little slice together into a description of the entire region as a whole. That's what General Relativity does. GR is the study of curved spacetime near gravitating sources.

No planet, no GR.

And that's far too long a post for anyone to have read the entire thing...
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Re: Rapidly rotating wire

Postby doogly » Thu Jan 29, 2009 6:56 am UTC

Common mistake! I think the misconception is due to people only using a metric formulation of relativity in the general case, but different choices of flat metric are all special. Special relativity works fine in accelerating frames. Uniform cceleration is not the same as gravity when you consider a whole spacetime - imagine the ever popular elevator is rather large, and you can weigh yourself at the top and at the bottom. If they disagree, you know you are falling due to gravity and not uniform acceleration.

So it is fine to use special relativity with a flat, accelerating metric (as in Rindler space). You can use a rotating frame without any trouble. (You can also use this frame to get the first approximation to the Lens Thirring effect, and see how much of the actual effect is due to the mass.)
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Re: Rapidly rotating wire

Postby mfb » Thu Jan 29, 2009 2:42 pm UTC

Lets reduce the problem to two wires, one in rest (relative to the observer), one moving in vertical direction:

---___ t=0

------- t=1

---¯¯¯ t=2

For an observer in the right wire, the average velocity in the right wire is v. For us, the average velocity in the left wire is v, too, and the average velocity in the right wire is v/gamma due to time dilation. It looks like electrons have to move to the right wire (for us), even if the problem is symmetric. But the heigth of the wire is contracted with gamma, too, so the electron density is higher - without any moving electrons! The product of density and speed is constant, so there is no reason for a flux.

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Re: Rapidly rotating wire

Postby PM 2Ring » Thu Jan 29, 2009 5:58 pm UTC

Goemon wrote:Alternatively, the problem can be analyzed in a rotating frame of reference centered on the wire's midpoint

Can we? According to the OP:

BlackSails wrote:Lets say you have a long wire out in space, and you start spinning it very quickly around one end, so it looks sort of like a rope with a weight on the end being spun by a person.

So the wire is being spun around one end, not the centre.

Still, this problem reminds me of that old chestnut, the relativistic spinning rigid disk.

(Sorry I don't have anything more constructive to add to this thread).

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Re: Rapidly rotating wire

Postby Charlie! » Thu Jan 29, 2009 8:38 pm UTC

Goemon wrote:
Charlie! wrote:I'll try and take this one thing at a time. Acceleration is equivalent to gravity. http://en.wikipedia.org/wiki/Equivalence_principle . So yes, there is a reason to call up general relativity for this problem.


[I think this is getting a bit off topic, so I'll hide it for those who aren't interested in following the side conversation...]


Spoiler:
Special Relativity can be used by any observer who is in uniform motion (feels no acceleration), and who is able to construct a perfectly regular frame of reference that extends throughout all space and time. That means he/she can lay out measuring sticks in the x, y, and z directions which form a perfect grid, and all the clocks AT REST IN THIS REFERENCE FRAME - ie, attached to the grid points - remain synchronized.

That's what the "Special" in "Special Relativity" means; the laws of SR only apply to observers who meet these requirements.

An observer sitting "at rest" nearby the spinning wire meets all these requirements. He can observe the motion and behavior of all the particles in the wire, (using measurements based on his universal grid and the stationary clocks attached to it) and apply the ordinary laws of SR to all of his measurements. This observer will find that time dilation of the various electrons doesn't impact the diffusion rate. (Ah ha!)

An observer sitting on the end of the spinning wire is not in uniform motion. She is not permitted to apply the ordinary laws of SR to her observations. BUT - our first observer, who IS in uniform motion in a global Lorentz frame, is able to take complete measurements of everything in the experiment, including the dizzy observer. From this, we can easily reconstruct exactly what the dizzy observer will see and measure at every instant. And furthermore, we can deduce what equations should be used by a rotating observer in flat spacetime. In other words, we can deduce what an accelerating observer will see and measure, using nothing but the equations of SR and a little brainpower.

Now consider an entirely different scenario: an observer located near the Sun will find that there's no such thing as a global inertial reference frame. No matter how you try to construct it, test particles in one region or another insist on flying off hither and yon due to "gravity". (This is actually another way of saying it's impossible to lay out measuring sticks in the x, y, z directions throughout the entire region in any way that will enable them to mesh perfectly. Just like trying to cover a sphere with a grid of uniform squares, it can't be done). Solving problems in a region of spacetime near a gravitational source requires an entirely new set of cogs and wheels. It can't be done with SR, because there's no such thing as a global inertial observer anywhere.

The oft-misquoted Principle of Equivalence provides the link between SR and GR. It states that LOCALLY (read carefully through the description on the wiki page) gravitation is equivalent to acceleration. That means that we can begin to understand how to build the machinery of GR by starting with the assumption that tiny little slices of spacetime near a gravitational source behave exactly like empty space - IF we apply the rules about acceleration that we learned from watching the rotating wire from a distance to each separate slice individually. (They're only LOCALLY equivalent.)

The trick is figuring out how to join the equations describing each little slice together into a description of the entire region as a whole. That's what General Relativity does. GR is the study of curved spacetime near gravitating sources.

No planet, no GR.

And that's far too long a post for anyone to have read the entire thing...

Wait, but obeying that rule, GR wouldn't resolve the twin paradox at all, if you just put it in some hypothetical gravity-free zone. A stationary observer should definitely see accelerating objects slow their clocks, even if the acceleration is not from gravity.
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Re: Rapidly rotating wire

Postby Rentsy » Fri Jan 30, 2009 4:00 am UTC

Damnit, there's no such thing as a paradox. Not in science. The "Twin Paradox" is only a "paradox" when taught badly. Bad teaching causes bad thinking.

You've got to optimize thinking because there is just so much of it to do.

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Re: Rapidly rotating wire

Postby Charlie! » Fri Jan 30, 2009 4:45 am UTC

Rentsy wrote:Damnit, there's no such thing as a paradox. Not in science. The "Twin Paradox" is only a "paradox" when taught badly. Bad teaching causes bad thinking.

You've got to optimize thinking because there is just so much of it to do.

I had a post, but I realized it could be summarized in two words.

And then I realized one of those words wasn't very nice.

But the other one was "semantics."
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Re: Rapidly rotating wire

Postby Goemon » Sat Jan 31, 2009 3:37 am UTC

Charlie! wrote:Wait, but obeying that rule, GR wouldn't resolve the twin paradox at all, if you just put it in some hypothetical gravity-free zone. A stationary observer should definitely see accelerating objects slow their clocks, even if the acceleration is not from gravity.


Not sure what you're saying, but:

You don't need GR to resolve the twins paradox;

If you're at rest in empty space, and see Carl Sagan accelerating across the solar system, then at any given instant the rate his clock ticks depends ONLY on his velocity. If he's moving at 99.5% of the speed of light, his clock ticks ten times slower than yours - whether he's accelerating at 0.1g or 200g or not at all.

If YOU are accelerating in empty space, clocks will tick faster than yours in front of you and slower than yours behind you. And the rate can be deduced directly from the laws of Special Relativity; so long as there are no planets nearby it's not necessary to pull out the big gun.

PM 2Ring wrote:So the wire is being spun around one end, not the centre.


Good point, missed that :)

Rentsy wrote:Damnit, there's no such thing as a paradox.


Try looking up the definition of the word paradox sometime. I don't think it means what you think it means :)
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Re: Rapidly rotating wire

Postby Charlie! » Sat Jan 31, 2009 7:34 am UTC

Goemon wrote:
Charlie! wrote:Wait, but obeying that rule, GR wouldn't resolve the twin paradox at all, if you just put it in some hypothetical gravity-free zone. A stationary observer should definitely see accelerating objects slow their clocks, even if the acceleration is not from gravity.


Not sure what you're saying, but:

You don't need GR to resolve the twins paradox;

If you're at rest in empty space, and see Carl Sagan accelerating across the solar system, then at any given instant the rate his clock ticks depends ONLY on his velocity. If he's moving at 99.5% of the speed of light, his clock ticks ten times slower than yours - whether he's accelerating at 0.1g or 200g or not at all.

If YOU are accelerating in empty space, clocks will tick faster than yours in front of you and slower than yours behind you. And the rate can be deduced directly from the laws of Special Relativity; so long as there are no planets nearby it's not necessary to pull out the big gun.

Consistency with GR means that the accelerating frame does exactly what gravity does, no matter the observer. Yes, you can work out the twin paradox with doppler stuff, but you must also be able to apply GR to it.

I swear, this stuff works even then the observer is completely neutral.
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