The Bondi Dipole in Full Numerical Relativity: a Self-Accelerating Positive-Negative Mass Binary

Abstract: Bondi showed in 1957 that bodies of opposite active gravitational mass self-accelerate: the negative chases the positive it repels, and the pair runs off together. We evolve this "Bondi dipole" in $3+1$ numerical relativity: two complex scalars share identical Klein-Gordon dynamics; only the phantom enters Einstein's equations with a minus sign -- inertial and passive masses positive, active mass negative. Across a matrix of thirty-four evolutions -- runaway pairs, controls, and parameter scans -- a mass-matched pair released at rest accelerates as a unit. The midpoint moves $3.00\pm0.01$ by $t=200$ with the separation held to 1%, and reaches a speed of $0.056c$ by $t=400$ with the acceleration steady to 2%; the total signed momentum holds at zero to $\lesssim 1$%. The force is gravity on both of its axes: $a\propto d<sup>{-2.03\pm0.01}$ over $d=8$ to $20$, $a\propto M<sup>{0.97\pm0.06}$ over a factor $2.5$ in mass, and $a\,d<sup>2/\bar{M}=1$ within 2.4% on the equal-mass ladder. Swapping the sectors inverts the acceleration to two parts in $10^5$; gauge, solver-depth and mesh variations move the drift by $\lesssim 0.01$% and box doubling by 4%; same-sign control pairs hold their centroids to $\lesssim 8\times10<sup>{-4}$ even while merging. The runaway carries no detectable gravitational radiation: the signed dipole cannot radiate, the quadrupole's $\ddot{Q}$ is constant, and the measured $\ell=2$ amplitude falls as $r<sup>{-4.8}$ -- near zone, not flux. The phantom star is, to our knowledge, the first asymptotically flat body of negative ADM mass evolved in numerical relativity; alone it survives to $t=1000$, slowly relaxing outward.