Riemann-Tensor have in n- dimensional space?

  • Thread starter Thaakisfox
  • Start date
  • #1
Thaakisfox
263
0
How many independant components does the Riemann-Tensor have in n- dimensional space?
 

Answers and Replies

  • #3
mina26
7
0
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
 
  • #4
mina26
7
0
How many independant components does the Riemann-Tensor have in n- dimensional space?

for more information see Shaum vector analysis chap8
 
  • #5
Chris Hillman
Science Advisor
2,353
9
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.
 
  • #6
Chris Hillman
Science Advisor
2,353
9
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].
 
Last edited:

Suggested for: Riemann-Tensor have in n- dimensional space?

Replies
2
Views
587
Replies
4
Views
217
  • Last Post
Replies
16
Views
949
Replies
17
Views
2K
  • Last Post
Replies
9
Views
892
  • Last Post
Replies
1
Views
189
Replies
36
Views
1K
  • Last Post
Replies
9
Views
990
Replies
8
Views
662
Replies
16
Views
432
Top