Not every 3-metric is conformally flat



Hello,

On Monday 26 May 2003 00:00, Camm Maguire wrote:
CM>forwarded 191671 maxima@www.ma.utexas.edu
CM>thanks
CM>
CM>
CM>Greetings!  This bug has been filed by a Debian maxima user and looks
CM>to be an issue with maxima proper.  I'm away from my email until 6/1,
CM>and cannot receive sourceforge mail as my ISP blocks it as spam.  I
CM>therefore cannot usefully register the bug on the website.  Please
CM>excuse therefore the submission to this list instead.
CM>
CM>
CM>Package: maxima
CM>Version: 5.9.0-10
CM>
CM>The output of computing the Weyl tensor on an arbitrary 3 metric says
CM>that the spacetime is conformally flat.
If in short, it seems to me this is not a bug. The maxima's reply could be 
made more accurate, however.

CM>(C2) g=tsetup();
Sorry, I don't understand this. Function tsetup() make so many things at once 
so we can not use the equation assignment
Just as advice, we must compute the components of the Christoffel symbols and 
components of Ricci tensor before to compute the Weyl tensor.

CM>(C3) weyl(g);
The rigth syntax here weyl(dis) .if  dis:true then non zero component of the 
Weyl tensor will be shown 
CM>
CM>this spacetime is conformally flat
This is a standard reply for the case  and if the components of the Ricci 
tensor are zero.
The message is likely not correct in the general case, however. Rigth below 
the "rigth" sequence to compute the Weyl tensor. Another example can be found 
in the "swartz.dem"
1) tsetup(), Note the coordinates names (case sensitivity!!) have to be in 
correspondence with the metric defined
2) christof(all);
3) lriccicom(all);weyl(true);
CM>On a related note, xmaxima does not respond to the tsetup(); command, it
CM>never responds with the line
This behaviour with tsetup() is typical for xmaxima on gcl & cmucl as well. 
Everything is ok for clisp. Actually. for the tensor packages in maxima the 
clisp is the best choice. The issue with tsetup() could be overcame however.
What's tsetup() designed for? It is designed to define the metric.
So instead to use the function tsetup(). You could 1)define coordinates 
through omega:[x,y,z], 2)define the metric via lg :   
matrix([x,0,0],[0,y,0],[0,0,z]), 3)compute inverse metric ug:invert(lg). 
These steps are usually done with tsetup().
rgds,
v