How to solve EFE without using the vacuum solution like how you did in schwarzschild metric?
Do you mean, how to solve the EFE for a general situation? If that is what you mean, probably using numerical methods and a computer.
If you mean something else, please say what.
I am not sure why can we just use the vacuum solution to solve EFE? Why cant we use a nom zero stress momentum tensor to solve?
The space outside the surface of an object floating in space (like a star or a planet) is vacuum so the stress-energy tensor there is zero. Thus, if we want a solution to the field equations that applies there we set ##T## equal to zero.
In other problems, for example a ball of dust collapsing under its own gravitational attraction to form a star or planet (google for "Oppenheimer-Snyder"), we don't have a vacuum and the stress energy tensor is not zero. Then we have to solve the field equation for a non-zero stress-energy tensor.
Another example is the solutions describing the universe as a whole.
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