## 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">&nbsp;&nbsp;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

$$x0 =3D r[/itex] cosh \xi $x1 =3D r$ sinh $\xi sin \theta cos \phix2 =3D r$ sinh $\xi sin \theta sin \phix3 =3D r$ sinh $\xi cos \theta$$ But there is trouble as the image of cosh is [1,oo) and not [itex][-1,1]$ as
needed. Is there any other way or do I have to wick rotate to euclidean
signature first?

$$Ren=E9$$

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[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 > > $x0 =3D r$ cosh \xi > $x1 =3D r$ sinh $\xi sin \theta cos \phi$ > $x2 =3D r$ sinh $\xi sin \theta sin \phi$ > $x3 =3D r$ sinh $\xi cos \theta$ > > But there is trouble as the image of cosh is [1,oo) and not $[-1,1]$ 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 $-oo,$ 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 $$x0 = r[/itex] sinh \xi $x1 = r$ cosh $\xi sin \theta cos \phix2 = r$ cosh $\xi sin \theta sin \phix3 = r$ cosh $\xi cos \theta$$ Now you're done. In short, use the first set of equations when [itex]x0^2-x1^2-x2^2-x3^2 >$ and the second set of equations for $x0^2-x1^2-x2^2-x3^2 <$ . Hope this helps. Igor

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