What Does Ricci Equal to Zero Mean for Black Holes?

[Philosophaie]

The Schwarzenchild and the Kerr Metrics for Black Holes give a Ricci = {}.

What does a Ricci Flat mean?

It obliviously has no momentum density or flow because the Stress-Momentum Tensor is also = {}.

It could mean freefall geodesic into a Black Hole?

or it also could be in some sort of stable orbit: circular, ellipse or hyperbolic.

Any suggestions?

 

[WannabeNewton]

Ricci flat just means $R_{ab} = 0$ identically. In the context of GR it represents vacuum solutions to the field equations. Are you looking for something specific?

 

[Philosophaie]

Looking for a real world metric that will produce a non-zero Ricci, Ricci Scalar and therefore a Energy-Momentum Tensor.

 

[WannabeNewton]

Ah. Well there are a handful of standard examples of those. For example the FLRW metric: http://en.wikipedia.org/wiki/FLRW_metric#General_metric

 

[Philosophaie]

Thanx

 

[Dale]

Looking for a real world metric that will produce a non-zero Ricci, Ricci Scalar and therefore a Energy-Momentum Tensor.

FLRW as WannabeNewton mentioned. Also, any interior Schwarzschild solution.

 

[andrien]

Just as a side note,in 4-D vanishing of Ricci curvature tensor does not imply flat spacetime.It is the Riemann tensor which describes it.However in 3-D,if ricci is zero then riemann curvature also vanishes and so in 2-D vanishing of scalar curvature implies flatness.No further need of ricci or riemann curvature tensor.