 |
|
| Science Forum Index » Math - Symbolic Forum » Kahn-Penrose metric and Maxima... |
|
Page 1 of 1 |
|
| Author |
Message |
| pbillet... |
 Posted: Fri Oct 23, 2009 9:51 pm |
|
|
|
Guest
|
Hello,
from the book "Relativity demystified", one considers Kahn-Penrose metric :
ds^2=2dudv-(1-u)^2dx^2-(1+u^2)dy^2
One wants to calculate the Christoffel symbol of second kind : Gamma(2,3,3)
In the book, the solution is : 1-u
With Maxima :
[ 0 1 0 0 ]
[ ]
[ 1 0 0 0 ]
[ ]
[ 2 ]
[ 0 0 - (1 - u) 0 ]
[ ]
[ 2 ]
[ 0 0 0 - (u + 1) ]
(%i10) ct_coords;
(%o10) [u, v, x, y]
(%i15) mcs[2,3,3];
(%o15) 0
Where is truth ? |
|
|
| Back to top |
|
|
|
| ... |
| Posted: Sat Oct 24, 2009 8:54 pm |
|
|
|
Guest
|
pbillet schrieb:
[quote]
Hello,
from the book "Relativity demystified", one considers Kahn-Penrose metric :
ds^2=2dudv-(1-u)^2dx^2-(1+u^2)dy^2
One wants to calculate the Christoffel symbol of second kind : Gamma(2,3,3)
In the book, the solution is : 1-u
With Maxima :
[ 0 1 0 0 ]
[ ]
[ 1 0 0 0 ]
[ ]
[ 2 ]
[ 0 0 - (1 - u) 0 ]
[ ]
[ 2 ]
[ 0 0 0 - (u + 1) ]
(%i10) ct_coords;
(%o10) [u, v, x, y]
(%i15) mcs[2,3,3];
(%o15) 0
Where is truth ?
[/quote]
Wow - Questions about pseudo-differential operators, and now
differential geometry!
I can't find any independent source for the Kahn-Penrose metric, so I
have to assume simply that your matrix is correct. Also I hope there are
no disagreements as to variable order and index offset (0 or 1) between
your Maxima input and the demystifying book.
Using Hans Dudlers's tensor library for Derive I obtain
g_ij_:=[[0,1,0,0],[1,0,0,0],[0,0,-(1-u)^2,0],[0,0,0,-(u+1)^2]]
g__ij_:=g_ij_^(-1)
g__ij_:=[[0,1,0,0],[1,0,0,0],[0,0,-1/(u-1)^2,0],[0,0,0,-1/(u+1)^~
2]]
x_:=[u,v,x,y]
C_ij_k_:=CHRIS2(g_ij_,g__ij_,x_)
C_ij_k_:=[[[0,0,0,0],[0,0,0,0],[0,0,1/(u-1),0],[0,0,0,1/(u+1)]],~
[[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]],[[0,0,1/(u-1),0],[0,0,~
0,0],[0,u-1,0,0],[0,0,0,0]],[[0,0,0,1/(u+1)],[0,0,0,0],[0,0,0,0]~
,[0,u+1,0,0]]]
NZEL_T(C_ij_k_)
[[[1,3,3],1/(u-1)],[[1,4,4],1/(u+1)],[[3,1,3],1/(u-1)],[[3,3,2],~
u-1],[[4,1,4],1/(u+1)],[[4,4,2],u+1]]
for the non-zero elements of the Christoffel symbol of the second kind.
Here, the index offset is 1, i.e. allowed indices are 1,2,3,4.
This result confirms that of Maxima, and there is no 1-u among the
elements at all. I read that the definition of some common tensors (and
related objects: the Christoffel symbol is no tensor) in differential
geometry varies from authority to authority with respect to sign, but I
don't think this affects the Christoffel symbol. Perhaps the author of
the demystifying book meant to write Gamma(3,3,2) = u-1 ?
Martin. |
|
|
| Back to top |
|
|
|
|
|
All times are GMT - 5 Hours
The time now is Tue Dec 08, 2009 10:43 am
|
|