This tensor , as noted in that post, is of key importance in general relativity. The Ricci tensor of the first kind is defined simply as a contraction of the Riemann tensor of the 2nd kind, i.e. the above curvature tensor. Thus:
R b m = R n b m n
Raising an index yields the Ricci tensor of the second kind, e.g. R b m
It is completely determined by knowing the quantity R b m for all vectors V i of unit length. The tensor is obtained by defining a Ricci tensor of the 2nd kind thus:
R b m = g bn R nm
The
number of independent components of this tensor
in a space of N- dimensions is:
½
N (N + 1 ). Where the g bn denote the g- tensor components with indices raised.
Hence,
there will be three components if N =
2, six components if N = 3 and ten components
if N = 4. In the latter we have the case
for relativistic 4 – dimensional space-time.
We consider
here the simplified case for N = 2 and let the metric of interest be *:
g 11 = 1,
g 22 = x 1 ,
For
this (N=2) case:
R
= g 11 ( R’ 11 ) + g 22 ( R’ 21 )
Where: g 11
= 1,
g 22
= 1/ x 1
,
(Note: Value of g 22 is not the same as g 22 !)
(Note: Value of g 22 is not the same as g 22 !)
R 11 = g 22
( R’ 21 ) = (1/ x 1 )( -1/ x 1 )
R 22 = g 11 ( R’ 12 )
= (1)( -1/ x 1 )
R
(Ricci) = g 11 ( R 11 ) + g 22 ( R 22 )
R
(Ricci) = g 11 (1/ x 1 )( -1/ x 1 ) + g 22 ( - 1/ x 1 )
R
(Ricci) =
=
(1) [- 1/ (x 1 ) 2 ]
+ (1/ x 1
) [- 1/ x 1 ]
= - 1/
(x 1 ) 2 -
1/ (x 1 ) 2 =
- 2 / (x 1 ) 2
The
reader should bear in mind this is the simplest form of the calculation and
while in this instance the components are proportional to the components of the
metric tensor, this is not true for spaces of higher dimension. For example, if N= 3, one has six components and
the final equation is written:
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
----------------------------
* Assume a metric given by the 'g' values:
g 11 = 1,
g 22 = x 1 , g 33 = x 2
Then the nonzero
Christoffel symbols have values:
G 1 22 = -1 ,
G 2 12 = G 2 21 = 1/ x 1
G 3 23 = G 3 32 = - 1/ x 2
E. g. Find: g 11 G 2 21
g 11 G 2 21 = (1) (1/ x 1)
= 1/ x 1
Find: g 22 G 3 32
g 22
G 3 32 = x 1
(- 1/ x 2 ) = - x 1
/ x 2
Recall the relations of Riemann tensors to Christoffel values, e.g.
Recall the relations of Riemann tensors to Christoffel values, e.g.
R 1 212 = - 1 - G 1 22 G 1 11 - G 2 21 G 1 22 = 0
Suggested Problem:
Obtain the Ricci tensor for the metric :
g 11 = 1,
g 22 = x 1 , g 33
= x 2.
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