I wonder if there is any possibility to evaluate theoretical characteristics of matter with negative mass? This is not thing of anti-mass but classical matter just with negated weight. I thing that if you would put it on surface of the Earth it would fly to heaven, wouldn't it? Could this be space-freedom redefined problem solution or am I just curious too much? Can't we simply use working functions/formulas with negate mark?
Well, one of the basic facts from General Relativity (the commonly accepted theory of gravity) is the so-called Equivalence principle, which basically states that all kinematic behaviour of a particle in the gravitational field does not depend on the particle's internal properties (like mass). So it would not fly to heaven, no.
But the gravitational field generated by this particle could differ from the one generated by positive mass. Or it could not. Actually, physics (very much like history) does not allow us to ask "what would be if ...".
Let's take a quick look at QFT (the commonly accepted theory of matter and fundamental forces - everything except gravity). For bosons, the mass-defining parameter in the Lagrangian is $m^2$. One can either make $m^2$ less than zero (which would make your field tachyonic) or greater than zero, which does not fix the sign of $m$. So basically, there is no way of having negative-mass bosons in QFT. Same goes for fermions, where you can specify $m$ explicitly, but the observable mass is just the absolute value of this $m$.
We see that in QFT, there is no such thing as negative mass. So it is very much likely that it is in fact unphysical.