Independent components of tensor

cscott · Jun 9, 2008

1. The problem statement, all variables and given/known data

For indices running from 0-3:
[tex]R_{iklm} = -R_{kilm} = -R_{ikml}[/tex]

With the above conditions how do I know the number of independent component is reduced from 4^4 = 256 to 36.

No idea how to figure this out.

HallsofIvy · Jun 9, 2008

cscott said: ↑

1. The problem statement, all variables and given/known data

For indices running from 0-3:
[tex]R_{iklm} = -R_{kilm} = -R_{ikml}[/tex]

With the above conditions how do I know the number of independent component is reduced from 4^4 = 256 to 36.

No idea how to figure this out.

When in doubt, look at a simpler case. If you know that [tex]R_{ij}= -R{ji}[/tex], then knowing [tex]R_{ij}[/tex] for i< k immediately gives you the value of [tex]R_{ji}[/tex]: Of the 16 possible values of [tex]R_{ij}[/tex], there are 4 with i= j so 16- 4= 12 where i is not equal to j and so 12/2= 6 where i< j. Knowing those 6 tells you the other 6. Further, if i= j then [tex]R_{ii}= -R_{ii}[/tex] so that must be 0: The 4 "diagonal" terms must be 0. You can choose 6 of the possible 16 terms of [tex]R_{ij[/tex].

Now, for this problem you know that R_{iklm} = -R_{kilm}[/itex]. Of the 4⁴= 256 terms, how many have i< k? Knowing that many automatically gives you an equal number. How many have i= k? You know those are 0.

You also know [tex]R_{iklm}= -R_{ikml}[/tex]. Do the same thing with m and l.}

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Independent components of tensor

cscott

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