Riemann-Tensor have in n- dimensional space?

  • Thread starter Thaakisfox
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How many independant components does the Riemann-Tensor have in n- dimensional space?
 
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How many independant components does the Riemann-Tensor have in n- dimensional space?
this is related to the number of components related to riemanian tensor and the space that contains it
as a simple example if u have a mixed tensor of 5 components 3 contarvariant and 2 covariant in 4 dimensional space so u have 4^5 compnents i.e 1024
u see how many equations are contracted to single one that's why einstein begin his general relativity by studying tensors with his friend Grassmann

my name is mina, i study QFT
 
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How many independant components does the Riemann-Tensor have in n- dimensional space?
for more information see Shaum vector analysis chap8
 

Chris Hillman

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Independent components?

How many independant components does the Riemann-Tensor have in n- dimensional space?
Just thought I'd stress that you are probably asking about the number of algebraically independent components. To understand the physical significance of a "geometric" field equation such as the Einstein field equation, you also need to appreciate a crucial differential relation, the (differential) Bianchi identity, which is crucial to understanding, for example, how it can happen in gtr that fluid motion inside some fluid filled region can give rise to gravitational radiation which propagatges as a wave across a vacuum region.
 

Chris Hillman

Science Advisor
2,337
8
this is related to the number of components related to riemanian tensor and the space that contains it
as a simple example if u have a mixed tensor of 5 components 3 contarvariant and 2 covariant in 4 dimensional space so u have 4^5 compnents i.e 1024
u see how many equations are contracted to single one that's why einstein begin his general relativity by studying tensors with his friend Grassmann

my name is mina, i study QFT
Hi, Mina, I think you are confusing Marcel Grossmann with Hermann Grassmann here! The latter was a fellow graduate student with Einstein at ETH (but Grossmann studied math not physics); the former was the completely different mathematician who introduced what is now called Grassmann or exterior algebra, later adopted by Cartan to give exterior calculus, aka the study of differential forms.

Also, the example you gave overlooks the possibility of algebraic symmetries which will in general reduce the number of algebraically independent components. For example the Riemann tensor (more or less by definition) satisfies [itex]R_{abcd} = -R_{bacd}[/itex].
 
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