Learning to Simplify the Curvature Tensor

[TimeRip496]

I just watched susskind video on EFE but he didnt show us how to convert curvature tensor(the one with 4 indices) to that of Ricci tensor.
Can anyone help me with this? Try to simplify it as I just started this.

 

[PeterDonis]

The Ricci tensor is the contraction of the Riemann tensor on its first and third indexes:

$$
R_{\mu \nu} = R^{\rho}{}_{\mu \rho \nu}
$$

This means each component of the Ricci tensor is the sum of multiple components of the Riemann tensor; for example, ##R_{11} = R^0{}_{101} + R^1{}_{111} + R^2{}_{121} + R^3{}_{131}## (assuming we are in 4-dimensional spacetime).

 

[TimeRip496]

The Ricci tensor is the contraction of the Riemann tensor on its first and third indexes:

$$
R_{\mu \nu} = R^{\rho}{}_{\mu \rho \nu}
$$

This means each component of the Ricci tensor is the sum of multiple components of the Riemann tensor; for example, ##R_{11} = R^0{}_{101} + R^1{}_{111} + R^2{}_{121} + R^3{}_{131}## (assuming we are in 4-dimensional spacetime).

I know. But how do you contract it? If I m not wrong one has to double derivative the curvature tensor. Besides what is the geometrical meaning of the ricci tensor?

 

[PeterDonis]

how do you contract it?

Just the way I described; you sum components of the Riemann tensor to get components of the Ricci tensor. No derivatives are involved. (The Riemann tensor already includes second derivatives of the metric tensor; that's where the derivatives are involved.)

what is the geometrical meaning of the ricci tensor?

Basically, the Ricci tensor is the piece of the Riemann tensor that is directly linked to the presence of matter and energy, via the Einstein Field Equation. John Baez gives a good description of it in this overview of GR:

http://math.ucr.edu/home/baez/einstein/

 

[sardar]

Sure, I can help with this! The curvature tensor is a mathematical tool used to describe the curvature of spacetime in Einstein's theory of general relativity. It has four indices, which can be thought of as representing directions in spacetime.

To convert the curvature tensor to the Ricci tensor, we first need to understand what the Ricci tensor represents. The Ricci tensor is a contraction of the curvature tensor, meaning that it is formed by summing over certain combinations of the indices of the curvature tensor. This contraction is what simplifies the curvature tensor into the Ricci tensor.

In order to perform this contraction, we need to use a mathematical operation called a contraction operator, which is represented by the symbol "g". This operator essentially takes two indices of the curvature tensor and contracts them together, resulting in a simplified object with two indices, the Ricci tensor.

To visualize this, think of the curvature tensor as a matrix with four rows and four columns. The Ricci tensor is then formed by summing over the diagonal elements of this matrix, resulting in a new matrix with only two rows and two columns. This new matrix is the Ricci tensor.

I hope this helps simplify the concept of converting the curvature tensor to the Ricci tensor. It may seem daunting at first, but with practice and further study, you will become more familiar with the mathematical operations involved. Good luck with your studies!