Searching for Ricci Scalar in Schwarzschild Metric

[touqra]

I searched the net for the Ricci scalar for the Schwarzschild metric but in vain. Can anyone tell me what's the Ricci scalar?
Are there any standard list or tables that records down the properties of any metric for GR?

 

[pervect]

Because it's a vacuum solution, the Ricci Scalar for the Schwarzschild metric is zero.

It's immediately obvious that the Einstein tensor is zero from $G_{\mu\nu} = 8 \pi T_{\mu\nu}$, and it can be shown that the Ricci scalar must also be zero.

The simplest way to illustrate is to look at

http://math.ucr.edu/home/baez/gr/outline2.html

part 13 in the section that says

But what does it mean? To see this, let's do some "index gymnastics". Stand with your feet slightly apart and hands loosely at your sides. Now, assume the Einstein equation!

to see the derivation of $R = -T^{\mu}{}_{\nu}$, and then it's immediately obvious that when $$T_{\mu\nu}=0$$ (a vacuum solution), R is also zero.

GRTensorJ-Books at http://grtensor.org/teaching/ has a list of various metrics and the various tensors and scalars from textbooks, but it actually calculates them and to calculate them it needs a non-free program, Maple.

 

[mijoon]

I searched the net for the Ricci scalar for the Schwarzschild metric but in vain. Can anyone tell me what's the Ricci scalar?
Are there any standard list or tables that records down the properties of any metric for GR?

The Ricci scalar is the contraction of the Ricci tensor which is a contraction of the Riemann tensor. It appears in the Einstein field equations, one of the solutions of which is the Schwartzschild Solution.