- #1

- 254

- 5

Can anyone help me with this? Try to simplify it as I just started this.

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- Thread starter TimeRip496
- Start date

- #1

- 254

- 5

Can anyone help me with this? Try to simplify it as I just started this.

- #2

- 33,551

- 11,962

$$

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).

- #3

- 254

- 5

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?

$$

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).

- #4

- 33,551

- 11,962

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.)

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:what is the geometrical meaning of the ricci tensor?

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

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