Kefka G
Jun25-04, 03:58 PM
<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>Here\'s the problem I was thinking about recently, which perhaps someone can\noffer some advice on. Suppose spacetime was, instead of 3+1 dimensional, m+n+1\ndimensional, where the n extra space dimensions are curled up on a torus of the\nappropriate dimensionality, and the m space dimensions are infinite (R^m x T^n\nx R spacetime). What would the scalar potential of a point charge look like?\nIn other words, what is a solution of Laplace\'s equation satisfying the\nappropriate boundary conditions?\n\nHere\'s what I know: In 3+1+1 dimensions, the answer is\n\nU(r,y) = (A / r) sinh(r) / (cos(y)-cosh(r))\n\nwhere r is the 3-dimensional distance, A is a constant, and y is the coordinate\nin the circular dimension. This is easily verified by checking that it\nsatisfies the differential equations (being careful about how the Laplacian\nlooks as a function of r...it is a function of dimensionality, too, if we\nseparate it like that) and boundary conditions (look at the r->0, y->0 and\nr->infinity limits), or by using the method of images and explicitly solving\nthe infinite sum that results.\n\nAs it turns out, if we take a derivative of this with respect to r and divide\nthe result by r we get the 5+1+1 result - in general, take 1/r times the r\nderivative of the n+1+1 solution to obtain the (n+2)+1+1 solution, as can be\nproven fairly easily from the differential equations (where r is reinterpreted\nin the formula as the n+2 dimensional distance instead of the n dimensional\ndistance), or using the image charges (the r-component of the electric field in\none dimensionality contains an identical infinite sum to the potential in a\nhigher dimensionality, so by expressing the electric field as a derivative of\nthe potential we can prove this result).\n\nFurthermore, we can drop down by one open dimension by integrating a solution\nover one cartesian variable, for example over x1 where r = sqrt(x1^2 + x2^2 +\nx3^2) (again, this follows fairly easily from the PDE\'s). Thus at least in\ntheory, combined with the above result we can handle n+1+1 spacetimes for any\nn, although I haven\'t yet checked if we can perform the integrals explicitly.\n\nHowever, try as I might, I can\'t figure out how to come up with an explicit\nformula for the potential when the compact dimensions are any more complicated\nthan just a circle. Ideally it would be nice to have a recursive way to at\nleast find the potential if some of the dimensions are curled on an n-torus,\ngiven some lower dimensional solution, but I can\'t find any way to build up to\nextra compact dimensions. Anybody have any ideas, keeping in mind that what\nI\'m looking for is a finite formula, not an infinite series (which is\nstraightforward to obtain as either a Laurent or Fourier series)? What I\'m\neventually interested in is the delayed piece of the self-force in higher\ndimensional "Maxwell" electrodynamics where some dimensions are periodic.\nThere is a failure of Huygen\'s principle in this case, regardless of\ndimensionality (it fails in even space dimensionalities no matter what, which\nfollows from the integration trick discussed above when we apply it to\nd\'Alembert\'s equation instead of Laplace\'s), and having explicit Coulomb\nsolutions is a reasonable first step to help analyze this.\n\nSorry this was so long,\nEric\n\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>Here's the problem I was thinking about recently, which perhaps someone can
offer some advice on. Suppose spacetime was, instead of 3+1 dimensional, m+n+1
dimensional, where the n extra space dimensions are curled up on a torus of the
appropriate dimensionality, and the m space dimensions are infinite (R^m x T^n
x R spacetime). What would the scalar potential of a point charge look like?
In other words, what is a solution of Laplace's equation satisfying the
appropriate boundary conditions?
Here's what I know: In 3+1+1 dimensions, the answer is
U(r,y) = (A / r) sinh(r) / (cos(y)-cosh(r))
where r is the 3-dimensional distance, A is a constant, and y is the coordinate
in the circular dimension. This is easily verified by checking that it
satisfies the differential equations (being careful about how the Laplacian
looks as a function of r...it is a function of dimensionality, too, if we
separate it like that) and boundary conditions (look at the r->0, y->0 and
r->infinity limits), or by using the method of images and explicitly solving
the infinite sum that results.
As it turns out, if we take a derivative of this with respect to r and divide
the result by r we get the 5+1+1 result - in general, take 1/r times the r
derivative of the n+1+1 solution to obtain the (n+2)+1+1 solution, as can be
proven fairly easily from the differential equations (where r is reinterpreted
in the formula as the n+2 dimensional distance instead of the n dimensional
distance), or using the image charges (the r-component of the electric field in
one dimensionality contains an identical infinite sum to the potential in a
higher dimensionality, so by expressing the electric field as a derivative of
the potential we can prove this result).
Furthermore, we can drop down by one open dimension by integrating a solution
over one cartesian variable, for example over x1 where r = \sqrt(x1^2 + x2^2 +x3^2) (again, this follows fairly easily from the PDE's). Thus at least in
theory, combined with the above result we can handle n+1+1 spacetimes for any
n, although I haven't yet checked if we can perform the integrals explicitly.
However, try as I might, I can't figure out how to come up with an explicit
formula for the potential when the compact dimensions are any more complicated
than just a circle. Ideally it would be nice to have a recursive way to at
least find the potential if some of the dimensions are curled on an n-torus,
given some lower dimensional solution, but I can't find any way to build up to
extra compact dimensions. Anybody have any ideas, keeping in mind that what
I'm looking for is a finite formula, not an infinite series (which is
straightforward to obtain as either a Laurent or Fourier series)? What I'm
eventually interested in is the delayed piece of the self-force in higher
dimensional "Maxwell" electrodynamics where some dimensions are periodic.
There is a failure of Huygen's principle in this case, regardless of
dimensionality (it fails in even space dimensionalities no matter what, which
follows from the integration trick discussed above when we apply it to
d'Alembert's equation instead of Laplace's), and having explicit Coulomb
solutions is a reasonable first step to help analyze this.
Sorry this was so long,
Eric
offer some advice on. Suppose spacetime was, instead of 3+1 dimensional, m+n+1
dimensional, where the n extra space dimensions are curled up on a torus of the
appropriate dimensionality, and the m space dimensions are infinite (R^m x T^n
x R spacetime). What would the scalar potential of a point charge look like?
In other words, what is a solution of Laplace's equation satisfying the
appropriate boundary conditions?
Here's what I know: In 3+1+1 dimensions, the answer is
U(r,y) = (A / r) sinh(r) / (cos(y)-cosh(r))
where r is the 3-dimensional distance, A is a constant, and y is the coordinate
in the circular dimension. This is easily verified by checking that it
satisfies the differential equations (being careful about how the Laplacian
looks as a function of r...it is a function of dimensionality, too, if we
separate it like that) and boundary conditions (look at the r->0, y->0 and
r->infinity limits), or by using the method of images and explicitly solving
the infinite sum that results.
As it turns out, if we take a derivative of this with respect to r and divide
the result by r we get the 5+1+1 result - in general, take 1/r times the r
derivative of the n+1+1 solution to obtain the (n+2)+1+1 solution, as can be
proven fairly easily from the differential equations (where r is reinterpreted
in the formula as the n+2 dimensional distance instead of the n dimensional
distance), or using the image charges (the r-component of the electric field in
one dimensionality contains an identical infinite sum to the potential in a
higher dimensionality, so by expressing the electric field as a derivative of
the potential we can prove this result).
Furthermore, we can drop down by one open dimension by integrating a solution
over one cartesian variable, for example over x1 where r = \sqrt(x1^2 + x2^2 +x3^2) (again, this follows fairly easily from the PDE's). Thus at least in
theory, combined with the above result we can handle n+1+1 spacetimes for any
n, although I haven't yet checked if we can perform the integrals explicitly.
However, try as I might, I can't figure out how to come up with an explicit
formula for the potential when the compact dimensions are any more complicated
than just a circle. Ideally it would be nice to have a recursive way to at
least find the potential if some of the dimensions are curled on an n-torus,
given some lower dimensional solution, but I can't find any way to build up to
extra compact dimensions. Anybody have any ideas, keeping in mind that what
I'm looking for is a finite formula, not an infinite series (which is
straightforward to obtain as either a Laurent or Fourier series)? What I'm
eventually interested in is the delayed piece of the self-force in higher
dimensional "Maxwell" electrodynamics where some dimensions are periodic.
There is a failure of Huygen's principle in this case, regardless of
dimensionality (it fails in even space dimensionalities no matter what, which
follows from the integration trick discussed above when we apply it to
d'Alembert's equation instead of Laplace's), and having explicit Coulomb
solutions is a reasonable first step to help analyze this.
Sorry this was so long,
Eric