1 · What the machine is
Freeman Dyson's 1963 "gravitational machine" is a gravity assist taken to its extreme. A spacecraft that swings past a moving mass leaves with a velocity change of up to twice the mass's velocity in the system frame — not twice its own. Point that mass at a companion so the two orbit each other, and every close pass steals a bite of orbital energy. The ceiling on the ship's final speed is therefore set by the orbital speed of the assisting bodies:
For two black holes, that orbital velocity is in geometric units — it rises without limit as the pair spirals together. A wide binary orbits slowly and gives a feeble kick; only a compact binary near merger, with separation of a few Schwarzschild radii, orbits at the tenths-of-c needed to fling a ship relativistically. That single fact forbids a purely Newtonian treatment of the engine and is the reason this sits in the strong-field regime.
2 · How the ship is integrated
The ship is a relativistic test particle. Its Hamiltonian is the special-relativistic kinetic term plus the superposed potential of the two holes:
Because is separable, the kick–drift–kick leapfrog is symplectic: with the holes held fixed it conserves the ship's energy to one part in 10¹¹ over thousands of M of integration (verified). When the holes move, that same Hamiltonian is explicitly time-dependent — energy is no longer conserved, and the difference is exactly the work the moving holes do on the ship. That pumped energy is the slingshot. The velocity is recovered from momentum by , which caps the ship below c by construction no matter how hard the engine burns.
The binary itself inspirals by the Peters quadrupole law, , merging after a time — so with radiation reaction switched on, later passes are faster than earlier ones, and there is a finite window before the holes coalesce. That countdown is the drama of the second figure: launch before the engine destroys itself.
3 · Modes & presets
Both-hole accelerator is the mechanism of Fig 13.4: a bound circumbinary orbit that loops past both holes while the inspiralling pair pumps it outward and faster — hit Launch burn for Fig 13.5. Circumbinary loop freezes the inspiral so you can watch the ship orbit both holes indefinitely (stable, but no net gain — that's Jacobi's theorem, not a bug). Single-pass slingshot is one strong scattering flyby; Grazing capture is what a too-deep whirl actually does — it plunges. Kerr zoom-whirl switches to an exact single-hole geodesic. Use Full orbit (zoom-out) to frame the whole trajectory.
Binary slingshot controls
4 · Boundary of validity mandatory
This entry is the Tier-B rung of the honesty ladder for this scenario — research-grade in the browser, with the strong-field engine treated relativistically, but explicitly not numerical relativity. The four rungs:
Full numerical relativity: Einstein's equations on a grid. There is no closed-form metric for two black holes, so this is the only truly first-principles route — and it cannot run in a browser.
Relativistic ship (SR momentum, c cap) in a superposed Paczyński–Wiita field; binary inspirals by Peters. Correct regime, correct orders of magnitude, symplectic. Field superposition is the approximation.
Newtonian point masses at relativistic orbital speed under Kepler's laws, no c cap. Shows the mechanism but misrepresents the very regime that makes it work.
Detection-quality waveforms, exact horizon-crossing dynamics, tidal-disruption of the ship — beyond any honest browser picture.
What this calculation reproduces — and what it does not
The assist bound
The ship's per-pass speed change tracks the Newtonian ceiling of twice the holes' orbital velocity, and never reaches c. The engine's dependence on a relativistic binary is exact in spirit.
Strong-field orbit shapes
Paczyński–Wiita places the ISCO at exactly 6M and the potential divergence at the horizon 2M, so zoom-whirl passes, plunges, and captures emerge from the geometry — verified against the analytic ISCO.
Inspiral timescale & countdown
Peters 1964 gives the correct circular-orbit merge time T = 5a⁴/(256 m₁m₂M) and rising v_orb. The finite launch window is real.
Why the accelerator needs the inspiral
Around a frozen pair the ship's Jacobi constant is conserved, so a bound orbit cannot secularly gain energy — the "Circumbinary loop" preset loops forever at fixed speed, correctly. Net acceleration of a bound orbit only appears once the binary evolves (inspiral) or the ship burns its engine. This is honoured exactly, not faked.
Deep multi-whirl orbits are fine-tuned
The idealised many-deep-whirl orbit drawn in the figure is capture-adjacent: in a blind scan of deep-pass initial conditions, ~80% plunged the horizon within one or two passes. Robust bound accelerators here are the gentler circumbinary loops; the tight orange whirls are the schematic limit, not a generic trajectory.
Superposition is not a metric
Adding two Paczyński–Wiita potentials is not a solution of Einstein's equations. Near either horizon, and in the region between the holes, the true spacetime differs; frame-dragging from orbital motion and the holes' own spins are absent here.
No gravitational-wave recoil on the ship
The ship feels a potential, not the full radiative metric. Tail terms, the near-zone GW field, and the energy the ship radiates are neglected — fine for a test mass, wrong for a heavy payload.
Tidal survival is not modelled
Tidal acceleration across a ship of length ℓ scales as 2mℓ/d³. Around stellar-mass holes it shreds anything at these radii; a survivable machine wants supermassive holes, whose safe-radius orbital speed is lower. The ship here is an indestructible point.
Bottom line. A faithful picture of the energetics and orbital shapes of a Dyson slingshot off a relativistic black-hole binary, integrated by a genuinely symplectic scheme with a hard light-speed cap. It is not a metric-level solution and not a survivability study. For the exact single-hole whirl, switch to Kerr geodesic mode, where the trajectory is a Kerr geodesic to numerical precision.
References
- Dyson 1963. Gravitational Machines. In Interstellar Communication (A.G.W. Cameron, ed.), ch. 12. PDF — the binary gravitational slingshot; ceiling at twice orbital velocity.
- Peters 1964. Gravitational Radiation and the Motion of Two Point Masses. Phys. Rev. 136, B1224. APS — circular-orbit inspiral rate and merge time.
- Paczyński & Wiita 1980. Thick accretion disks and supercritical luminosities. A&A 88, 23. ADS — the pseudo-Newtonian potential reproducing the ISCO.
- Bardeen, Press & Teukolsky 1972. Rotating Black Holes. ApJ 178, 347. ADS — equatorial Kerr ISCO used in whirl mode.
- Levin & Perez-Giz 2008. A periodic table for black hole orbits. Phys. Rev. D 77, 103005. arXiv — the taxonomy of zoom-whirl orbits.
- Dyson 1968. Interstellar Transport. Physics Today 21, 41. ADS — companion piece on relativistic interstellar flight.
- Yoshida 1990. Construction of higher order symplectic integrators. Phys. Lett. A 150, 262. ADS — the symplectic leapfrog family used here.
- Misner, Thorne & Wheeler 1973. Gravitation. Freeman. — gravity assist, geodesics, and tidal forces near black holes (§25, §31).