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hyper-spherical coordinates in Minkowski space? |
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| Feb4-05, 08:35 AM | #1 |
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hyper-spherical coordinates in Minkowski space?
<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>\nHi,\n\nis there any way to define global hyper-spherical coordinates in\nMinkowski space? Something compatible with the metric (+---) is\n\nx0 =3D r cosh xi\nx1 =3D r sinh xi sin theta cos phi\nx2 =3D r sinh xi sin theta sin phi\nx3 =3D r sinh xi cos theta\n\nBut there is trouble as the image of cosh is [1,oo) and not [-1,1] as\nneeded. Is there any other way or do I have to wick rotate to euclidean\nsignature first?\n\nRen=E9\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>Hi,
is there any way to define global hyper-spherical coordinates in Minkowski space? Something compatible with the metric (+---) is [tex]x0 =3D r[/itex] cosh \xi [itex]x1 =3D r[/itex] sinh [itex]\xi sin \theta cos \phix2 =3D r[/itex] sinh [itex]\xi sin \theta sin \phix3 =3D r[/itex] sinh [itex]\xi cos \theta[/tex] But there is trouble as the image of cosh is [1,oo) and not [itex][-1,1][/itex] as needed. Is there any other way or do I have to wick rotate to euclidean signature first? [tex]Ren=E9[/tex] |
| Feb4-05, 11:41 AM | #2 |
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<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[Please post in ASCII and avoid MIME quoting such as "=3D".]\n\nmeyr2@web.de wrote:\n> Hi,\n>\n> is there any way to define global hyper-spherical coordinates in\n> Minkowski space? Something compatible with the metric (+---) is\n>\n> x0 =3D r cosh xi\n> x1 =3D r sinh xi sin theta cos phi\n> x2 =3D r sinh xi sin theta sin phi\n> x3 =3D r sinh xi cos theta\n>\n> But there is trouble as the image of cosh is [1,oo) and not [-1,1] as\n> needed. Is there any other way or do I have to wick rotate to\neuclidean\n> signature first?\n\nWhat you point out is not a problem. However, your equations are\nincomplete. If you fix r in your formulas, you\'ll notice that they\nparametrize the future hyperboloid distance r from the origin. If you\nvary r from 0 to oo, you fill in the interior of the future half of the\nlight cone. Varying r from 0 to -oo, you will fill in the interior of\nthe past half of the light cone. However, you still have not\nparametrized the region at space-like separation from the origin,\noutside the light cone. To do that you need a second set of equations\n\nx0 = r sinh xi\nx1 = r cosh xi sin theta cos phi\nx2 = r cosh xi sin theta sin phi\nx3 = r cosh xi cos theta\n\nNow you\'re done. In short, use the first set of equations when\nx0^2-x1^2-x2^2-x3^2 > 0 and the second set of equations for\nx0^2-x1^2-x2^2-x3^2 < 0.\n\nHope this helps.\n\nIgor\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>[Please post in ASCII and avoid MIME quoting such as "=3D".]
meyr2@web.de wrote: > Hi, > > is there any way to define global hyper-spherical coordinates in > Minkowski space? Something compatible with the metric (+---) is > > [itex]x0 =3D r[/itex] cosh \xi > [itex]x1 =3D r[/itex] sinh [itex]\xi sin \theta cos \phi[/itex] > [itex]x2 =3D r[/itex] sinh [itex]\xi sin \theta sin \phi[/itex] > [itex]x3 =3D r[/itex] sinh [itex]\xi cos \theta[/itex] > > But there is trouble as the image of cosh is [1,oo) and not [itex][-1,1][/itex] as > needed. Is there any other way or do I have to wick rotate to euclidean > signature first? What you point out is not a problem. However, your equations are incomplete. If you fix r in your formulas, you'll notice that they parametrize the future hyperboloid distance r from the origin. If you vary r from to oo, you fill in the interior of the future half of the light cone. Varying r from to [itex]-oo,[/itex] you will fill in the interior of the past half of the light cone. However, you still have not parametrized the region at space-like separation from the origin, outside the light cone. To do that you need a second set of equations [tex]x0 = r[/itex] sinh \xi [itex]x1 = r[/itex] cosh [itex]\xi sin \theta cos \phix2 = r[/itex] cosh [itex]\xi sin \theta sin \phix3 = r[/itex] cosh [itex]\xi cos \theta[/tex] Now you're done. In short, use the first set of equations when [itex]x0^2-x1^2-x2^2-x3^2 >[/itex] and the second set of equations for [itex]x0^2-x1^2-x2^2-x3^2 <[/itex] . Hope this helps. Igor |
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