Question

Is the Matter energy-momentum tensor zero identically for the following specific case?

Answer 1

The Schwarzschild metric describes that portion space-time which contains no matter, and in which the stress-energy tensor is therefore identically zero. There is no 'mass at the origin with energy Mc^2'; the metric cannot be continued to r=0 as the curvature diverges as r tends to 0, so there is no point in this space-time with r=0. 

More realistically, the Schwarzschild metric for r sufficiently large describes the space-time geometry around a spherically symmetric mass distribution (but not inside the mass, where there is a  non-zero stress energy tensor).

The term 2GM/c^2 arises mathematically as a constant of integration which is identified with the mass of a spherically symmetric object by requiring that the predictions of general relativity reduce to those of Newtonian gravity in the limit of slow motion in a weak field.

I don't know which derivations you've been looking at, but all this is explained clearly and explicitly in every derivation of the Schwarzschild metric that I've seen: the first three I pulled off my shelf were in the textbooks of Mould (Basic Relativity); Adler Bazin and Schiffer (Introduction to General Relativity); and Rindler (Relativity).

Answer 2

Explanation:

In order to construct the gravitational field equations (24.36) we now need the energy-momentum tensor T n k of the cosmic matter.