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

Shirokov, Nikita M.

Research paper

The Bondi Dipole in Full Numerical Relativity

A Self-Accelerating Positive–Negative Mass Binary

Author: Nikita M. Shirokov
Published: 25 Aug 2026
arXiv: 2608.24577
Subjects: gr-qc · astro-ph.HE

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In brief

In 1957 Hermann Bondi showed that a negative mass and a positive mass, released side by side, set off together on their own: the positive mass pulls the negative one toward it, the negative mass pushes the positive one away, and the pair accelerates as one — with nothing expelled and no conservation law broken.

We put this “Bondi dipole” through full 3+1 numerical relativity, with a phantom scalar star as the negative mass. Across a campaign of thirty-four evolutions — runaway pairs, controls and parameter scans — the mixed pair accelerates as a unit, and the force behind it measures as gravity: inverse-square in separation, linear in mass.

Simulation time

t = 000

Pair speed

0.000c

AnimationThe phantom star (hill) chases the canonical star (well) that it repels, after Fig. 3. The motion follows the paper’s numbers — released at rest, constant acceleration, separation held at 10; the surface itself is schematic.

Why it matters

First 3D simulation of the Bondi dipole
Bondi’s 1957 runaway pair, evolved in full 3+1 numerical relativity across thirty-four runs.
First negative-mass star in numerical relativity
The phantom scalar star has negative ADM mass and survives on its own to t = 1000; we know of no earlier example.
Self-acceleration, confirmed in full general relativity
The pair reaches 0.056c from rest with nothing expelled, and its total momentum stays at zero.
The force is gravity
Inverse-square in separation and linear in mass, and the runaway radiates no detectable gravitational waves.

Speed the pair reaches from rest by t = 400, with the acceleration steady to 2%.: 0.056c
Measured force law over separations d = 8 to 20 — gravity’s inverse square, ±0.01.: d^−2.03
Swap which star is the phantom and the acceleration inverts, to two parts in 10⁵.: 2×10⁻⁵
Fall-off of the ℓ = 2 amplitude — a near-zone field, not a flux. The runaway radiates nothing detectable.: r^−4.8

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 ± 0.01 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 ≲ 1%. The force is gravity on both of its axes: a ∝ d^{−2.03 ± 0.01} over d = 8 to 20, a ∝ M^{0.97 ± 0.06} over a factor 2.5 in mass, and a d²/M̄ = 1 within 2.4% on the equal-mass ladder. Swapping the sectors inverts the acceleration to two parts in 10⁵; gauge, solver-depth and mesh variations move the drift by ≲ 0.01% and box doubling by 4%; same-sign control pairs hold their centroids to ≲ 8 × 10⁻⁴ even while merging. The runaway carries no detectable gravitational radiation: the signed dipole cannot radiate, the quadrupole’s Q̈ is constant, and the measured ℓ = 2 amplitude falls as r^−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.

13 pages, 10 figures, 4 ancillary movies · gr-qc, astro-ph.HE

Figures from the paper

Full paper

The Bondi dipole, Fig. 1 — Bondi’s sign rules. — Fig. 1Bondi’s sign rules. Inertia is positive for every body; only the active mass carries a sign. Like pairs fall together or push apart with the barycentre fixed — the mixed pair runs away.

The Bondi dipole, Fig. 3 — The runaway in the headline cell at t = 0, 60, 120, 200. — Fig. 3The runaway in the headline cell at t = 0, 60, 120, 200. Bottom row: χ − 1 — the phantom star is a hill, the canonical star a well. Both move toward +x while the gap holds to 1%.

The Bondi dipole, Fig. 4 — Core worldlines; a single constant-acceleration fit to the midpoint drift; the signal against every null run; and the Bondi signature — sector momenta grow while their signed sum holds at zero. — Fig. 4Core worldlines; a single constant-acceleration fit to the midpoint drift; the signal against every null run; and the Bondi signature — sector momenta grow while their signed sum holds at zero.

The Bondi dipole, Fig. 5 — Fig. 5Both axes of a = M̄/d², measured: acceleration ∝ d^{−2.03 ± 0.01} at fixed mass, and ∝ M̄^{0.97 ± 0.06} at fixed separation.

Simulation movies

Playlist

Video 1

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If this work or its code is useful to you, please cite the arXiv preprint.

Code & campaign data

BibTeX

@misc{shirokov2026bondi,
  title         = {The Bondi Dipole in Full Numerical Relativity: a Self-Accelerating Positive-Negative Mass Binary},
  author        = {Shirokov, Nikita M.},
  year          = {2026},
  eprint        = {2608.24577},
  archivePrefix = {arXiv},
  primaryClass  = {gr-qc},
  url           = {https://arxiv.org/abs/2608.24577}
}

The Bondi Dipole in Full Numerical Relativity

In brief

Abstract

Figures from the paper

Simulation movies

A positive and a negative mass star accelerate themselves

The runaway does not stop: 400 time units

Control: two positive-mass stars

Control: two negative-mass stars

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