Anti-gravity necessitates the involvement of negative mass. Scientists speculating what the effects of negative mass would be came to some interesting conclusions. The funniest one is the runaway motion. It's kinda complex, but I'll give you the long story short of it through analogy and metaphor.
Look at the drawings below:
What the image depicts does not happen because you can't violate Thermo's Second. However, if instead of magnets you used a portion of positive mass (regular, everyday mass we all deal with all the time) and a portion of negative mass, then you would get exactly the troll physics effects depicted above!
Don't trust me just because I am saying it. Check the Wikipedia entry on negative mass. The further you read, the more the troll nature of Nature itself becomes apparent (all emphasis are mine):
Although no particles are known to have negative mass, physicists (primarily Hermann Bondi in 1957, William B. Bonnor in 1989, then Robert L. Forward) have been able to describe some of the anticipated properties such particles may have. Assuming that all three concepts of mass are equivalent the gravitational interactions between masses of arbitrary sign can be explored, based on the Einstein field equations and the equivalence principle:
For two positive masses, nothing changes and there is a gravitational pull on each other causing an attraction. Two negative masses would repel because of their negative inertial masses. For different signs however, there is a push that repels the positive mass from the negative mass, and a pull that attracts the negative mass towards the positive one at the same time.
Hence Bondi pointed out that two objects of equal and opposite mass would produce a constant acceleration of the system towards the positive-mass object, an effect called "runaway motion" by Bonnor who disregarded its physical existence, stating:
“I regard the runaway (or self-accelerating) motion […] so preposterous that I prefer to rule it out by supposing that inertial mass is all positive or all negative.”
— William B. Bonnor, in Negative mass in general relativity.
Such a couple of objects would accelerate without limit (except relativistic one); however, the total mass, momentum and energy of the system would remain 0.
This behavior is completely inconsistent with a common-sense approach and the expected behaviour of 'normal' matter; but is completely mathematically consistent and introduces no violation of conservation of momentum or energy. If the masses are equal in magnitude but opposite in sign, then the momentum of the system remains zero if they both travel together and accelerate together, no matter what their speed.
Forward used the properties of negative-mass matter to create the concept of diametric drive, a design for spacecraft propulsion using negative mass that requires no energy input and no reaction mass to achieve arbitrarily high acceleration.
The existence of anti-gravity therefore causes a violation of the second law of thermodynamics, which will probably win you a visit from the auditors of reality!