Alejandro
Aug24-04, 07:39 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>Let me sketch a couple of classical arguments that I think can be\nargued to ask space time dimension to compactify to D<5.\n\nConsider a bound gravitational orbit around a mass m. Kepler third law\nfor generic dimension is T^2=R^(D-1). For D=4 this is the usual law.\nAnd if instead of the radious of the orbit we consider the total area\nof the orbit, we can also write it as T^2=A^((D-1)/2). Here we see\nthat something special happens at D=5: the period in this space time\ndepends linearly of the area.\n\nIf we consider in detail the area sweept by a particle as time goes,\nwe appreciate that D<5 and D>5 are different issues: The area goes\n\n\\$\\$2 A(t) = G^1/2 m^1/2 R^((5-D)/2) t \\$\\$\n\nThus for D>5, for a given time interval the area decreases with radius\nof the orbit, while for D<5, areas increase with the radius of the\norbit.\nIt should be a surprise if this change in the trend of gravitational\nbound states were not noticed in the spectrum of any theory having\nclassical newtonian gravity as a limit. Thus D<5 should be a natural\npoint for compatitification.\n\nIn favour of LQG/discrete area theories, it is worth to notice what\nhappens if we ask A(t) to be a integer multiple of Planck area when t\nis Planck time. Except for D=5, this requisite fixes a quantification\nof the radius, but only for D=4 the gravitational constant G cancels\nout, leaving just h, c, and m. This is another signal of the\nprivileged status of our current space time, or at least of its\ndelicate relationship with the concepts of time and area.\n\nAlejandro Rivero\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>Let me sketch a couple of classical arguments that I think can be
argued to ask space time dimension to compactify to D<5.
Consider a bound gravitational orbit around a mass m. Kepler third law
for generic dimension is T^2=R^(D-1). For D=4 this is the usual law.
And if instead of the radious of the orbit we consider the total area
of the orbit, we can also write it as T^2=A^((D-1)/2). Here we see
that something special happens at D=5: the period in this space time
depends linearly of the area.
If we consider in detail the area sweept by a particle as time goes,
we appreciate that D<5 and D>5 are different issues: The area goes
$$2 A(t) = G^1/2 m^1/2 R^((5-D)/2) t $$
Thus for D>5, for a given time interval the area decreases with radius
of the orbit, while for D<5, areas increase with the radius of the
orbit.
It should be a surprise if this change in the trend of gravitational
bound states were not noticed in the spectrum of any theory having
classical newtonian gravity as a limit. Thus D<5 should be a natural
point for compatitification.
In favour of LQG/discrete area theories, it is worth to notice what
happens if we ask A(t) to be a integer multiple of Planck area when t
is Planck time. Except for D=5, this requisite fixes a quantification
of the radius, but only for D=4 the gravitational constant G cancels
out, leaving just h, c, and m. This is another signal of the
privileged status of our current space time, or at least of its
delicate relationship with the concepts of time and area.
Alejandro Rivero
argued to ask space time dimension to compactify to D<5.
Consider a bound gravitational orbit around a mass m. Kepler third law
for generic dimension is T^2=R^(D-1). For D=4 this is the usual law.
And if instead of the radious of the orbit we consider the total area
of the orbit, we can also write it as T^2=A^((D-1)/2). Here we see
that something special happens at D=5: the period in this space time
depends linearly of the area.
If we consider in detail the area sweept by a particle as time goes,
we appreciate that D<5 and D>5 are different issues: The area goes
$$2 A(t) = G^1/2 m^1/2 R^((5-D)/2) t $$
Thus for D>5, for a given time interval the area decreases with radius
of the orbit, while for D<5, areas increase with the radius of the
orbit.
It should be a surprise if this change in the trend of gravitational
bound states were not noticed in the spectrum of any theory having
classical newtonian gravity as a limit. Thus D<5 should be a natural
point for compatitification.
In favour of LQG/discrete area theories, it is worth to notice what
happens if we ask A(t) to be a integer multiple of Planck area when t
is Planck time. Except for D=5, this requisite fixes a quantification
of the radius, but only for D=4 the gravitational constant G cancels
out, leaving just h, c, and m. This is another signal of the
privileged status of our current space time, or at least of its
delicate relationship with the concepts of time and area.
Alejandro Rivero