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/