## Monday, May 23, 2022

### Revisiting The Ricci Tensor

The Riemann curvature tensor was defined in an earlier blog post, e.g.

A Look At General Relativity and Tensors (Part 3)

as:   n b m n

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:

b m =   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    bn  R  nm

The number of independent components of this tensor  in a space of N- dimensions is:  ½ N (N + 1 ).  Where the  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.

Consider first the simplest case for N = 2 and let the metric of interest be *:

11    =   1,     22 =  ,

For this (N=2) case:

R =  11  ( R’ 11 ) +  g  22    ( R’ 21   )

Where:  11      =   1,     g  22 =  1/  ,

(Note:  Value of  22   is not the same as  g  22  !)

11  =  g  22   ( R’ 21   ) =    (1/  1  )( -1/ 1 )

22 = 11  ( R’ 12   )  =  (1)( -1/ 1 )

R (Ricci) =  11  ( R  11 )  g  22    ( R 22   )

R (Ricci) =   11  (1/  1  )( -1/ 1 )  g  22    ( - 1/ 1 )

R (Ricci) =

= (1) [- 1/  () 2 ]  +  (1/   ) [- 1/ 1  ]

= - 1/  () 2     - 1/  () 2      =   - 2 /  () 2

More complex, if N= 3, one has six components and the final equation is written:

R (Ricci) =  11  ( R 11 )  +  g  22  ( R 22 ) +  g   33   ( R 33 )

Where:  R 11 =    g  22     R 2112

22 =   11   1221 +    g   33    3223

33 =    g  22     2332

----------------------------
*  Assume a metric given by the 'g' values:

11    =   1,     22 =  ,    g 33 =  2

Then the nonzero Christoffel symbols have values:

G 1 22    =  -1 ,    G 2 12    =  G 2 21    = 11

G 3 23    =  G 3 32     =   - 12

Example (1):   Find:
11   G 2 21

11   G 2 21      =  (1) (11) =  11

Example (2):  Find:     22 G 3 32

22 G 3 32      (12 ) =  -  / x 2

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 :

11    =   1,     22 =  ,   33 =  2.