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1. Guess the form of the tensor in special relativity in co-moving frame

(ρ+p)u

^{μ}u

^{v}+pη

^{μv}

Note that the pη

^{00}term cancels the p in u

^{0}u

^{0}, to simplify the understanding of the tensor.

2. Transform the tensor to arbitrary frame in special relativity

3. The η is replaced by g in general relativity

Problem -

1. In step 1, the pη could be 'plugged into' the u

^{μ}u

^{v}, why do we need to extract it out for the sake of fitting the form of GR tensor? It seems that it knows the answer before head and construct it. Is there any more vigorous proof?

2. You could replace g by η as a special case of g. But, replacing all η by g means "special relativity implies general relativity" which is not valid in a logical sense.

Could anyone please suggest a better derivation, or help me understand the proof better?

Thank you very much. =]