Why does the Ricci tensor vanishes in Schwarzschild metric?

Question

If the Schwarzschild metric is suppose to describe the behaviour of a spherical object in flat space, so the Schwarzschild is different from the flat metric because it describes curved space so why then does the Ricci tensor equal zero. Also, if the metric describes a spherical object of mass $M$ in space why should the Energy-Momentum Tensor vanish. If there is mass then there is energy so why must it vanish.

Answer 1

This is an answer to the question as qualified in a comment.

The stress energy tensor is a tensor field so it is a function of position in spacetime. In the Schwarzschild coordinates the geometry is time independent so the local value of the stress-energy tensor is just a function of the position in space. Everywhere outside the spherical object it is zero because there is no mass there. Within the object the Ricci tensor is not zero. For the Schwarzschild black hole all the mass is concentrated at the singularity so the Ricci tensor vanishes everywhere except at the singularity (where it is undefined!).

The stress-energy tensor is a well defined local quantity. My own preferred way to understand the stress-energy tensor is to start with the stress-energy tensor for a point particle, because this is simply:

$$ T_{\mu\nu} = \gamma m v_\mu v_\nu $$

at the position of the particle and zero everywhere else. You build up macroscopic objects by (conceptually) adding up the stress-energy tensors of the point particles that make up those objects. Actually, maybe this is more confusing than helpful - if so ignore the last two paragraphs.

As for the self-energy of the gravitational field, well that's the way the Einstein equation is defined i.e. we don't include the self-energy in the stress-energy tensor. In any case the field self-energy is an evasive quantity and couldn't be written in a local invariant form like the stress-energy tensor.