Riemann-Tensor have in n- dimensional space?

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In summary: So if you have two copies of the Riemann tensor with the same coefficients except for the first and last indices which are swapped, then the two tensors are still algebraically independent but the number of components is now reduced to 3 (the number of indices).
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Thaakisfox
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How many independant components does the Riemann-Tensor have in n- dimensional space?
 
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  • #3
Thaakisfox said:
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
Thaakisfox said:
How many independant components does the Riemann-Tensor have in n- dimensional space?

for more information see Shaum vector analysis chap8
 
  • #5
Independent components?

Thaakisfox said:
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
mina26 said:
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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1. What is the Riemann Tensor in n-dimensional space?

The Riemann Tensor, also known as the Riemann curvature tensor, is a mathematical object used in the study of differential geometry and general relativity. It describes the curvature of a space at every point and is defined by a set of numbers that represent the curvature at each point in n-dimensional space.

2. How is the Riemann Tensor calculated?

The Riemann Tensor is calculated using the Christoffel symbols, which are derived from the metric tensor of the space. The Christoffel symbols represent the connection between the curvature of a space and the coordinates used to describe it. By combining the Christoffel symbols in a specific way, the Riemann Tensor can be calculated.

3. What is the significance of the Riemann Tensor?

The Riemann Tensor is important in the study of general relativity because it represents the intrinsic curvature of a space. In other words, it shows how space is curved even in the absence of matter or energy. This is crucial in understanding the behavior of objects in a gravitational field, as described by Einstein's theory of general relativity.

4. How does the Riemann Tensor change in higher dimensions?

In higher dimensions, the Riemann Tensor becomes more complex, as there are more possible ways for space to curve. In 3-dimensional space, the Riemann Tensor has 20 independent components, but in n-dimensional space, it has n(n-1)(n^2 - 1)/12 independent components. This means that the Riemann Tensor becomes increasingly important in higher dimensions, as it describes the curvature of space more thoroughly.

5. Can the Riemann Tensor be used to study spaces with non-Euclidean geometry?

Yes, the Riemann Tensor can be used to study any space, regardless of its geometry. In fact, it was originally developed to describe the curvature of non-Euclidean spaces, such as the 2-dimensional surface of a sphere. The Riemann Tensor allows for the study of curved spaces, which is essential in many areas of mathematics and physics.

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