I propose this equation, which is new, but dimensionally-consistent as being;

P^aP^bη_ab=η^2_0 v^2_f(P_μ,η)-η^2_i v^2_i(P_μ,η)

This is mathematically-consistent as being a dispersion formula.

The left-hand side is in fact the physical equivalance of

P^aP^bη_ab=v^2 (\hbar c/G)

so that

E^2/c^2 - ∏_(n=1)^3 |p^2| = v^2 (\hbar c/G)

Then the equation can become:

E^2-|p^2c^2| = (v^2)^2 \hbar c/G

The relation can be found:

G(\hbar c/G)/ \hbar c = Fvt/Mc^2

and

P^aP^bη_ab = E^2/c^2 - p^2