Kefka G
Sep1-04, 04:43 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nLately I\'ve been warming up to start learning about self action in general\nrelativity. Since one of the most important pieces of those studies is the\nfailure of Huygens principle in any dimensionality, I figured a good way to\nwarm up would be to study very simple E/M situations where Huygens principle is\nviolated. The most basic of these would have to be the cases where we have N\nopen spacetime dimensions and 1 compact dimension curled up on a circle.\n\nHuygens principle can then fail for two reasons - one, because of the total\nspacetime dimensionality being odd, or two, because even in even dimensions,\nthe light cone of a particle re-intersects the particle\'s world line. It is\nthe second effect that I\'m more interested in, so I really want to look at\ncases where we have an even number of spacetime dimensions (probably 4+1+1\ndimensions first) before tackling 3+1+1.\n\nIn particular, imagine that you give a stationary electron a bump (in the\ndirection of one of the open dimensions) at t = 0. For a little bit, it will\nactually recieve an extra bit of electric (the magnetic forces cancel out)\nforce in the direction you bump it, because from its point of view (i.e. at the\nretarded times), its images viewed around the compact dimension are still\nstationary. But eventually, the radiation from the initial bump will reach the\nelectron and "pull it back," so to speak. From there things keep getting more\ncomplicated, as eventually it will see the next further images get a bump,\netc., and all the while it\'s still getting varying amounts of electrostatic\nrepulsion from the images that still appear stationary. And the image charges\neach have to be assumed to move exactly like the "original" particle, so it all\ngets very recursive. You can see that this is a highly complicated problem,\nand although the basic equations are fairly simple to put down, I\'ve not had\nany luck in solving them.\n\nBut I don\'t necessarily even need to solve them completely for the motion of\nthe particle - really all I\'m interested in is the radiation signature that\nsuch a process would produce, and the characteristic time scale before the\nparticle settles into a steady state and is just moving with some velocity.\nI\'m also interested in approximately how much of the initial energy given would\nbe lost to the radiation fields. Presumably all these things would depend on\nthe dimensionality, the size of the compact dimension, and the mass and charge\nof the particle.\n\nSo does anybody know a clever way to extract this kind of information without\nactually solving the equations of motion (which I suspect are unsolvable in\ngeneral )? I\'d appreciate very much any help anyone can throw my way, even\njust as far as getting order of magnitude estimates (frequency of radiation,\nespecially). Ultimately I\'m curious as to how large a curled dimension would\nhave to be (classically, at least) in order to make the radiation unambiguously\nmeasurable. And Google netted me just about nothing on any of this, so it\'d be\ngreat if someone had any ideas. I\'m getting close to just coding it up and\ndoing this numerically, but it\'s a major pain to reliably simulate point\ncharges on a computer - plus, resorting to a numerical study has always felt\nvery much like giving up to me, and I\'d love an analytical answer if possible.\n\nThanks,\nEric\n\nPS - You can also get partially self-sustaining motion with compact dimensions\nby letting the particle oscillate in the direction of the curled dimension,\nsince it then interacts with its own field and gets a restoring force. These\nmotions should die out due to radiation, but it is an interesting problem to\ntry and solve for all the possible states of vibration if we neglect radiation\nreaction. For large amplitudes, some of the states are quite non-sinusoidal,\nespecially relativistically, and should also have interesting radiation\nsignatures (although I suspect that for reasonable sizes of dimension/particle,\nthey would not be observable).\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Lately I've been warming up to start learning about self action in general
relativity. Since one of the most important pieces of those studies is the
failure of Huygens principle in any dimensionality, I figured a good way to
warm up would be to study very simple E/M situations where Huygens principle is
violated. The most basic of these would have to be the cases where we have N
open spacetime dimensions and 1 compact dimension curled up on a circle.
Huygens principle can then fail for two reasons - one, because of the total
spacetime dimensionality being odd, or two, because even in even dimensions,
the light cone of a particle re-intersects the particle's world line. It is
the second effect that I'm more interested in, so I really want to look at
cases where we have an even number of spacetime dimensions (probably 4+1+1
dimensions first) before tackling 3+1+1.
In particular, imagine that you give a stationary electron a bump (in the
direction of one of the open dimensions) at t = . For a little bit, it will
actually recieve an extra bit of electric (the magnetic forces cancel out)
force in the direction you bump it, because from its point of view (i.e. at the
retarded times), its images viewed around the compact dimension are still
stationary. But eventually, the radiation from the initial bump will reach the
electron and "pull it back," so to speak. From there things keep getting more
complicated, as eventually it will see the next further images get a bump,
etc., and all the while it's still getting varying amounts of electrostatic
repulsion from the images that still appear stationary. And the image charges
each have to be assumed to move exactly like the "original" particle, so it all
gets very recursive. You can see that this is a highly complicated problem,
and although the basic equations are fairly simple to put down, I've not had
any luck in solving them.
But I don't necessarily even need to solve them completely for the motion of
the particle - really all I'm interested in is the radiation signature that
such a process would produce, and the characteristic time scale before the
particle settles into a steady state and is just moving with some velocity.
I'm also interested in approximately how much of the initial energy given would
be lost to the radiation fields. Presumably all these things would depend on
the dimensionality, the size of the compact dimension, and the mass and charge
of the particle.
So does anybody know a clever way to extract this kind of information without
actually solving the equations of motion (which I suspect are unsolvable in
general )? I'd appreciate very much any help anyone can throw my way, even
just as far as getting order of magnitude estimates (frequency of radiation,
especially). Ultimately I'm curious as to how large a curled dimension would
have to be (classically, at least) in order to make the radiation unambiguously
measurable. And Google netted me just about nothing on any of this, so it'd be
great if someone had any ideas. I'm getting close to just coding it up and
doing this numerically, but it's a major pain to reliably simulate point
charges on a computer - plus, resorting to a numerical study has always felt
very much like giving up to me, and I'd love an analytical answer if possible.
Thanks,
Eric
PS - You can also get partially self-sustaining motion with compact dimensions
by letting the particle oscillate in the direction of the curled dimension,
since it then interacts with its own field and gets a restoring force. These
motions should die out due to radiation, but it is an interesting problem to
try and solve for all the possible states of vibration if we neglect radiation
reaction. For large amplitudes, some of the states are quite non-sinusoidal,
especially relativistically, and should also have interesting radiation
signatures (although I suspect that for reasonable sizes of dimension/particle,
they would not be observable).
relativity. Since one of the most important pieces of those studies is the
failure of Huygens principle in any dimensionality, I figured a good way to
warm up would be to study very simple E/M situations where Huygens principle is
violated. The most basic of these would have to be the cases where we have N
open spacetime dimensions and 1 compact dimension curled up on a circle.
Huygens principle can then fail for two reasons - one, because of the total
spacetime dimensionality being odd, or two, because even in even dimensions,
the light cone of a particle re-intersects the particle's world line. It is
the second effect that I'm more interested in, so I really want to look at
cases where we have an even number of spacetime dimensions (probably 4+1+1
dimensions first) before tackling 3+1+1.
In particular, imagine that you give a stationary electron a bump (in the
direction of one of the open dimensions) at t = . For a little bit, it will
actually recieve an extra bit of electric (the magnetic forces cancel out)
force in the direction you bump it, because from its point of view (i.e. at the
retarded times), its images viewed around the compact dimension are still
stationary. But eventually, the radiation from the initial bump will reach the
electron and "pull it back," so to speak. From there things keep getting more
complicated, as eventually it will see the next further images get a bump,
etc., and all the while it's still getting varying amounts of electrostatic
repulsion from the images that still appear stationary. And the image charges
each have to be assumed to move exactly like the "original" particle, so it all
gets very recursive. You can see that this is a highly complicated problem,
and although the basic equations are fairly simple to put down, I've not had
any luck in solving them.
But I don't necessarily even need to solve them completely for the motion of
the particle - really all I'm interested in is the radiation signature that
such a process would produce, and the characteristic time scale before the
particle settles into a steady state and is just moving with some velocity.
I'm also interested in approximately how much of the initial energy given would
be lost to the radiation fields. Presumably all these things would depend on
the dimensionality, the size of the compact dimension, and the mass and charge
of the particle.
So does anybody know a clever way to extract this kind of information without
actually solving the equations of motion (which I suspect are unsolvable in
general )? I'd appreciate very much any help anyone can throw my way, even
just as far as getting order of magnitude estimates (frequency of radiation,
especially). Ultimately I'm curious as to how large a curled dimension would
have to be (classically, at least) in order to make the radiation unambiguously
measurable. And Google netted me just about nothing on any of this, so it'd be
great if someone had any ideas. I'm getting close to just coding it up and
doing this numerically, but it's a major pain to reliably simulate point
charges on a computer - plus, resorting to a numerical study has always felt
very much like giving up to me, and I'd love an analytical answer if possible.
Thanks,
Eric
PS - You can also get partially self-sustaining motion with compact dimensions
by letting the particle oscillate in the direction of the curled dimension,
since it then interacts with its own field and gets a restoring force. These
motions should die out due to radiation, but it is an interesting problem to
try and solve for all the possible states of vibration if we neglect radiation
reaction. For large amplitudes, some of the states are quite non-sinusoidal,
especially relativistically, and should also have interesting radiation
signatures (although I suspect that for reasonable sizes of dimension/particle,
they would not be observable).