Given the metric: 1, x 1 , x 2
We have:
R (Ricci) =
g 11
( R 11 ) + g 22 ( R 22 ) + g 33 ( R 33 )
Where: R 11 = g 22 R
2112
R 22 = g 11 R 1221 + g 33
R 3223
R 33
= g 22 R
2332
Then: g 11 = 1, g 22 =
1/ x 1 , g 33
=
1/ x 2
R 1221
= -
1/ x 1
R 2332
= - 1/ x 2
And: R 3223 =
- 1/ x 2
Further: R 2112 = - 1/ x 1
And: R 2332 =
- 1/ x 2
So:
R
11 = g 22 R
2112 = 1/ x 1 (-
1/ x 1 )
R 11 = - 1 / (x 1 ) 2
R 22 = g 11 R 1221 + g 33
R 3223
= ( -
1/ x 1 ) +
( 1/ x 2
) (-
1/ x 2 )
R 22
= -
1/ x 1 - 1 /
(x 2 ) 2
R 33
= g 22 R
2332 = (
1/ x 1 ) (-
1/ x 2)
R 33 =
- 1 / x 1 x 2
Then: g 11
( R 11 ) + g 22 ( R 22 ) + g 33 ( R 33 ) =
( 1) [- 1 /
(x 1 ) 2 ]
+ 1/ x 1 [-
1/ x 1 - 1 /
(x 2 ) 2 ]
+
(1/ x 2
) [-
1/ x 1 x 2
] =
-2 / (x 1 ) 2 - 1 / x 1 (x 2 ) 2 - 1/ [x 2 ( x 1 x 2 )]
-2 / (x 1 ) 2 - 1 / x 1 (x 2 ) 2 - 1/ [x 2 ( x 1 x 2 )]
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