Quick question to clear up some confusion on Riemann tensor and contraction

In summary: This is done by multiplying by g^{\beta\gamma} = g_{\beta\gamma} and contracting the indices on the left...In summary, the conversation discusses the calculation of the Ricci tensor in terms of the contractions of the Riemann tensor and the use of metrics to lower indices. The Riemann tensor is defined using the derivative operator and its trace yields the Ricci tensor. To lower the upper index of Riemann or to get the scalar-curvature from Ricci, a metric is used by multiplying and contracting the indices on the left.
  • #1
Deadstar
104
0
Let's say I want to calculate the Ricci tensor, [tex]R_{bd}[/tex], in terms of the contractions of the Riemann tensor, [tex]{R^a}_{bcd}[/tex]. There are two definitions of the Riemann tensor I have, one where the [tex]a[/tex] is lowered and one where it is not, as above.

To change between the two all that I have ever seen written is 'we lower the indices' but I don't think I fully understand this. Does this mean...

[tex]R_{abcd} = g_{aa} {R^a}_{bcd}[/tex]

So the answer to my original question of finding the Ricci tensor is...

[tex]R_{bd} = g^{ac} g_{aa} {R^a}_{bcd}[/tex]

Following this, I also have written before me that...

[tex]{R^b}_{bcd} = 0[/tex] since [tex]R_{abcd}[/tex] is symmetric on a and b. Shouldn't this be antisymmetric on a and b?

Sorry if these are basic questions but I'm finding the vagueness of 'lowering the indices' a bit confusing...

Cheers.
 
Physics news on Phys.org
  • #2
Deadstar said:
[tex]R_{abcd} = g_{aa} {R^a}_{bcd}[/tex]
You can't have the same index appearing 3 times in an expression, what you mean is

[tex]R_{abcd} = g_{ae} {R^e}_{bcd}[/tex]​
 
  • #3
The Riemann tensor

[tex]R^{\alpha}_{\;\beta\gamma\delta}=g^{\alpha\lambda}R_{\lambda\beta\gamma\delta}[/tex].

The raising and lowing of indices is done using the fundamental metric tensor [itex]g_{\alpha\beta}[/itex] and the inverse metric [itex]g^{\alpha\beta}[/itex].

The Ricci tensor

[tex]R^{\alpha}_{\;\beta\gamma\delta}\rightarrow R^{\alpha}_{\;\beta\alpha\delta}=R_{\beta\delta}[/tex]

is a contraction of the first and third indices.
 
  • #4
I would argue that [tex]
R^{\alpha}_{\;\beta\gamma\delta}
[/tex]
is the more fundamental expression for the Riemann tensor
since this is defined from the derivative operator [without reference to a metric].
Its trace yields the Ricci tensor [as jfy4 wrote]... and this too doesn't make use of a metric.

To lower the upper index of Riemann or to get the scalar-curvature from Ricci now requires a metric.
 

1. What is the Riemann tensor?

The Riemann tensor is a mathematical object that describes the curvature of a space. In the context of general relativity, it is used to describe the curvature of spacetime.

2. What is the significance of the Riemann tensor in physics?

The Riemann tensor is a crucial component in Einstein's theory of general relativity. It helps us understand how mass and energy affect the curvature of spacetime, which in turn determines the motion of objects in the universe.

3. How is the Riemann tensor calculated?

The Riemann tensor is calculated using the Christoffel symbols, which are derived from the metric tensor. It involves taking multiple derivatives of the metric tensor and performing various contractions and permutations.

4. What is the importance of contractions in the Riemann tensor?

Contractions in the Riemann tensor allow us to simplify the mathematical expression and make it easier to interpret physically. They also reveal important information about the curvature of spacetime, such as the Ricci tensor and scalar curvature.

5. How does the Riemann tensor relate to the curvature of spacetime?

The components of the Riemann tensor represent the rate of change of the metric tensor, which describes the curvature of spacetime. This means that the Riemann tensor is directly related to the curvature of spacetime and is essential in understanding the relationship between mass and gravity.

Similar threads

  • Special and General Relativity
Replies
10
Views
664
  • Special and General Relativity
Replies
1
Views
874
Replies
13
Views
522
Replies
1
Views
949
  • Special and General Relativity
Replies
2
Views
2K
  • Special and General Relativity
Replies
28
Views
6K
  • Special and General Relativity
Replies
22
Views
2K
  • Special and General Relativity
Replies
10
Views
2K
  • Special and General Relativity
Replies
1
Views
512
  • Special and General Relativity
Replies
6
Views
3K
Back
Top