Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: jeffreyH on 17/09/2014 01:52:35
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If we have a disk where the outer edges is rotating at relativistic speeds could we assume greater length contraction at the edge than at the centre? If so how can we square this with physical reality where you would expect compression to originate from the cetre of an object?
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That's the general idea. Einstein referred to this in The Foundation of the General Theory of Relativity (http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity):
"This can be seen easily when we regard the whole measurement-process from the system K and remember that the rod placed on the periphery suffers a Lorentz-contraction, not however when the rod is placed along the radius..."
But length contraction is a bit odd. It isn't actually what people think it is. Let's just say that we're both 2m long, and we pass each other by at a relativistic speed. You look like you're 1m long to me, and I look like I'm 1m long to you. But we're each carrying big butterfly nets 1.1m wide. We can't scoop each other up, because what you see isn't always how it is. When your measurement of something changes, sometimes it's because the thing changed. But sometimes, it's because you changed. For example, you're two metres long, and Chris decides he's going to zoom past you at some significant fraction of the speed of light. I watch as it happens. Does your length change from 2m to 1m? No. Your length doesn't change a bit.
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If we have a disk where the outer edges is rotating at relativistic speeds could we assume greater length contraction at the edge than at the centre? If so how can we square this with physical reality where you would expect compression to originate from the cetre of an object?
What do you mean by compression to originate from the cetre of an object?
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It's a interesting question. some say it should fracture, I believe it should, assuming it to spin close to lights speed, all hypothetically this is. what we might be able to do is to look at spinning black holes to find out what happens, for example if there is a gas cloud getting captured by it. Well, maybe? :)
and yes, what was you thinking of writing "compression to originate from the center of an object" ? Something spinning, defined in its own frame of reference, have no compression that I can see?
http://www.space.com/4843-black-holes-spin-speed-light.html
Or maybe it has? Were you thinking of a centrifugal force acting on the particles making up that disk? wondering at what spin that would break it up, defining it such as the highest 'speed' would be at the rim of the disk?
Then again, if so, that's not a length contraction.
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disk where the outer edges is rotating at relativistic speeds
Such a disk would shatter from centripetal/centrifugal force long before you had to worry about relativistic effects due to the outer edge approaching the speed of light and rotating more slowly than the center of the disk.
spinning black holes...the highest 'speed' would be at the rim of the disk?
If the disk is the accretion disk of a black hole, the highest speed would be at the inner edge of the disk, not the outer edge. This is part of the effect which tears apart infalling objects and turns them into a plasma.
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Such a disk would shatter from centripetal/centrifugal force long before you had to worry about relativistic effects due to the outer edge approaching the speed of light and rotating more slowly than the center of the disk.
If the radius is made large enough then the centripetal acceleration can be made as small as you'd like, small enough to prevent too much stress in the material to shatter it.
Jeff - What you're looking for is known as the Ehrenfest paradox. There's a lot of material online about this such as
http://en.wikipedia.org/wiki/Ehrenfest_paradox
http://simple.wikipedia.org/wiki/Rigidly_rotating_disk_paradox
http://ehrenfestparadox.wordpress.com/
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True Evan, was thinking of a plate when I wrote 'rim'.
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Such a disk would shatter from centripetal/centrifugal force long before you had to worry about relativistic effects due to the outer edge approaching the speed of light and rotating more slowly than the center of the disk.
If the radius is made large enough then the centripetal acceleration can be made as small as you'd like, small enough to prevent too much stress in the material to shatter it.
Jeff - What you're looking for is known as the Ehrenfest paradox. There's a lot of material online about this such as
http://en.wikipedia.org/wiki/Ehrenfest_paradox
http://simple.wikipedia.org/wiki/Rigidly_rotating_disk_paradox
http://ehrenfestparadox.wordpress.com/
Thanks Pete. I'll have a look at it over the weekend.
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Pete the point that is most interesting to me is "The paradox has been deepened further by Albert Einstein, who showed that since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the circumference, which would thus measure greater than 2πR." I think this is what most people miss about the properties of relativistic geometry.
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The train traveling around a set radius can be shown to exhibit no time dilation relative to other points on its path. If no time dilation is present then that brings into question the validity of length contraction along this path. So does angular momentum, because of its curvature cancel out length contraction? For Born rigidity this is an important question to answer.
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People think quantum mechanics is weird. This problem is just right up there above it.
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If we consider all the directions of the tangents to the circumference then can we show a length contraction out along the radius? This would then have to be a consideration when examining galactic rotation.
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We should keep in mind that the Ehrenfest paradox is about a circle and not necessarily something that is made of material. All we need do is to set up points along the circumference of the ring and measure the distances between the points and let the number of points become very large compared to the circumference.
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From Wikipedia (http://en.wikipedia.org/wiki/Ehrenfest_paradox):
Any rigid object made from real materials, which is rotating with the transverse velocity close to the speed of sound in this material, must exceed the point of rupture due to centrifugal force because centrifugal pressure can not exceed shear modulus of material
An interesting linkage of two subjects which I thought would have no connection: The outer rim of a spinning disk cannot exceed the speed of sound in that material!
The speed of sound (http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html) in various materials:
- Iron: 5km/sec
- Diamond: 12km/sec
- Beryllium: 13 km/sec
So to carry out this experiment, you need a substance where the speed of sound in this material approaches the speed of light in a vacuum.
Apparently a disk made of neutronium (http://en.wikipedia.org/wiki/Neutronium) might just do the trick: The speed of sound in the stuff of neutron stars is thought to be a fraction of the speed of light.
Anyway, it's an interesting thought experiment, even if it's not practical with materials we have at hand!
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We should keep in mind that the Ehrenfest paradox is about a circle and not necessarily something that is made of material. All we need do is to set up points along the circumference of the ring and measure the distances between the points and let the number of points become very large compared to the circumference.
Isn't that just fitting more into the available space? A parabolic surface would be a better metaphor in that case. It would appear initially that fracture of the material is the solution but it exists in a spacetime that is itself curving due to the momentum of the mass. The curve of the curve so to speak.
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If we have a disk where the outer edges is rotating at relativistic speeds could we assume greater length contraction at the edge than at the centre? If so how can we square this with physical reality where you would expect compression to originate from the cetre of an object?
What do you mean by compression to originate from the cetre of an object?
Sorry Pete missed this one. I was thinking about large spheres when writing this involving internal pressures. I didn't make that clear.
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From Wikipedia (http://en.wikipedia.org/wiki/Ehrenfest_paradox):
Any rigid object made from real materials, which is rotating with the transverse velocity close to the speed of sound in this material, must exceed the point of rupture due to centrifugal force because centrifugal pressure can not exceed shear modulus of material
An interesting linkage of two subjects which I thought would have no connection: The outer rim of a spinning disk cannot exceed the speed of sound in that material!
The speed of sound (http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html) in various materials:
- Iron: 5km/sec
- Diamond: 12km/sec
- Beryllium: 13 km/sec
So to carry out this experiment, you need a substance where the speed of sound in this material approaches the speed of light in a vacuum.
Apparently a disk made of neutronium (http://en.wikipedia.org/wiki/Neutronium) might just do the trick: The speed of sound in the stuff of neutron stars is thought to be a fraction of the speed of light.
Anyway, it's an interesting thought experiment, even if it's not practical with materials we have at hand!
I tell you what I'll generate the relativistic speeds if you can supply the neutronium disk.
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When a star collapses its angular momentum increases as the radius contracts. I would assume the relativistically spinning disk would have to undergo this same contraction of radius. This would accommodate the length contraction and there would no longer be a paradox. To an external observer the rotation would appear to slow due to the contraction and exhibit time dilation.
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I can see no way in which this would be distinguishable from a neutron star. Or should I say neutronium disk.