Every pair of bodies attracts under Newtonian gravity, F = G m1 m2 / r², directed along the line joining them. The acceleration of body i from all others is:
a_i = Σ_j G · m_j · (r_j − r_i) / |r_j − r_i|³
Note mi itself cancels out of ai — only the other body's mass sets the acceleration. That is the whole trick of negative mass:
- A positive body next to a negative body is pushed by a mass with the "wrong" sign, so it accelerates away — it is repelled.
- The negative body feels the positive body's ordinary attractive pull, but F = m·a with m < 0 flips its acceleration, so it accelerates toward the very body that is fleeing it.
The negative-mass body chases, the positive-mass body flees at the same rate — both keep accelerating in the same direction forever, "bootstrapping" the pair to ever higher speed. Total momentum Σ m·v is still exactly conserved — Newton's third law (F₁₂ = −F₂₁) holds regardless of the sign of either mass, so the readout stays pinned at its starting value (zero for a pair spawned at rest). What is not conserved is kinetic energy: ½m·v² keeps growing for the positive body while the negative body's ½m·v² keeps shrinking (going more negative), so the pair's total energy budget balances even as both bodies' speeds climb without bound — a real and correctly-derived (if deeply strange) consequence of allowing negative mass into F = ma. This constant-acceleration runaway is the textbook "negative-mass propulsion" thought experiment; small-scale analogues of effective negative mass have been observed in Bose–Einstein condensates and exciton–polariton fluids.
The "Add Random Field" button drops several bodies of mixed sign into the scene at once — watch pairs spontaneously lock into chases while same-sign bodies simply orbit or scatter under ordinary attraction/repulsion.