# Calculate the elements of the Riemann tensor

1. Nov 10, 2012

### zardiac

1. The problem statement, all variables and given/known data
Compute 21 elements of the Riemann curvature tensor in for dimensions. (All other elements should be able to produce through symmetries)

2. Relevant equations
$R_{abcd}=R_{cdab}$
$R_{abcd}=-R_{abdc}$
$R_{abcd}=-R_{bacd}$

3. The attempt at a solution
I don't see how 21 elements can be enough,
for example
abcd
0000
1111
2222
3333
1110
1100
1000
2220
2200
2000
3330
3300
3000
0012
0120
0023
0230
1112
1122
1212
2223
2233
2323

I am already up to 24 elements and I am not even sure if this is enough to derive all the other ones. Am I missunderstanding something?
According to http://mathworld.wolfram.com/RiemannTensor.html it should be possible, but I really don't see it right now.
Any hints would be appreciated.

2. Nov 11, 2012

### clamtrox

There are 20 components of the tensor which can be non-zero. For example R0000 is certainly 0 (why?)

3. Nov 11, 2012

### andrien

see 8th heading in the following,it will guide you how there are only 20 independent components and not 256 as it might be think(4×4×4×4)
http://www.mathpages.com/rr/appendix/appendix.htm

4. Nov 11, 2012

### zardiac

Ah of course, there are some elements that are zero. I don't know why I didn't checked that first. Thanks for quick replies, I think I got it right now :)