Riemann tensor: indipendent components

[AdeBlackRune]

Hi, thanks for the attention and excuse for my bad english.
I'm studying general relativity and I have a doubt about the number of indipendent component of the riemann curvature tensor.
We have two kind of riemann tensor:

type (3,1) Rⁱ_kml

type (4,0) R_rkml

There are also some symmetry rule:

first skew-symmetry

R_rkml = -R_krml​

secondo skew-symmetry

R_rkml = -R_rklm​

(Rⁱ_kml = -Rⁱ_klm)​

block symmetry

R_rkml = R_mlrk​

Bianchi's first identity

R_rkml+R_rlkm+R_rmlk=0​

(Rⁱ_kml+Rⁱ_lkm+Rⁱ_mlk=0)​

Now, the (4,0) R-type has
n²(n²-1)/12
indipendent components. What about the (3,1) R-type? The indipendent component are also n²(n²-1)/12 or they are n²(n²-1)/3?

(In the case n=4, (4,1) R-type has 20 indipendent component; (3,1) R-type are 20 or 80?)

 

[haushofer]

Why would acting with the metric on the Riemann tensor change the number of independent components?

 

[Phrak]

Why would acting with the metric on the Riemann tensor change the number of independent components?

I give up. Why would it not, though it does seems reasonable that raising or lowering indices would have no effect?


Come to think of it, we could take our N independent variables, and transform to Riemann normal coordinates, where the metric is Minkowskian, without loss of generality. With the metric now having constant entries, indices are raised and lowered without changing the number of independent variables.

 

[AdeBlackRune]

I suppose that raising or lowering indices leaves unchanged the number of indipendent components, but i was not able to find a confirm of that. I thought that, given the metric tensor, the Rⁱ_kml are (linear?) functions of the R_rkml component. So, can 20 indipendent quantities generate more than 20 indipendent quantities?

 

[paweld]

Of course rising and lowering indicies does not change the number of independent
components of any tensor. However the number of independent components
should depend of the choice of base. For example in coordinate (holonomic)
base the number of independent components of metric tensor is 10 but
in ortonormal base it's of course 0. The number of independent components
of Riemann tensor in 4 dimension in coordinate base is 20 but what about
ortonormal base. Does the extra condidtion imposed on base give in this example any
extra condition which have to be fullfiled by Riemann tensor?

 

[Phrak]

We cast around this word 'indendent' so casually at times, it's sometimes hard to tell at times what is independent of what. In the case of a vector, we like to cast it as an object that is independent of coordinate system or metric. The same applies to any tensor object.

In the case of the components of a tensor, I believe we are looking for the linear independence of tensor element, and perhaps eliminate additional degrees of freedom for other reasons.