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if it were P^{μ}P_{μ} instead of P^aP^b, and without η_{ab} (what is it? you haven't defined it), then it would have been an obvious relation. If you don't define the terms and what you exactly mean, it's not so easy to understand your post...

Then it is obvious: you have written E^{2} = (mc^{2})^{2} + (cp)^{2}, which is a well known equation.

I still don't understand what you mean.Don't know, maybe it's me...