Gravity as a scalar field

Newton's potential, Poisson's equation, and why three bodies break everything — an interactive lecture

1. One number per point in space

Newton originally described gravity as a force between pairs of masses. There is an equivalent, and in many ways deeper, formulation: describe gravity by a scalar field Φ — a single number attached to every point in space, called the gravitational potential, measured in joules per kilogram. It answers the question: how much energy per kilogram does it take to bring a mass to this point from infinitely far away? Near a massive body Φ is very negative — a deep "well" — and it approaches zero far away.

The force reappears as the slope of this landscape. The gravitational field g (an acceleration, m/s²) is the negative gradient of the potential:

g = −∇Φ masses roll "downhill" toward lower potential; steeper slope, stronger pull

And the potential itself is generated by matter through Poisson's equation, which is the field-theoretic statement of the entire Newtonian theory of gravity:

∇²Φ = 4πG ρ mass density ρ tells the potential how to curve; the potential tells masses how to move

For a set of point masses, Poisson's equation has an exact solution — each mass contributes a simple well, and the wells add up:

Φ(r) = − Σi G mi / |r − ri| superposition: the total potential is the sum of each body's well

Taking the gradient of this recovers the familiar inverse-square law. The "field" picture and the "force" picture are the same theory in two costumes.

2. Two bodies: solvable. Three bodies: chaos.

Knowing the field at one instant is easy. The hard part is the dynamics: the masses move, which reshapes Φ, which changes how they move — a feedback loop. For two bodies this loop closes into an exact, timeless answer: Kepler's ellipses, derived in closed form. For three bodies, no such formula exists. Henri Poincaré proved in the 1880s–90s that the general three-body problem has no closed-form solution in the classical sense, and in doing so discovered what we now call deterministic chaos: the equations are fully deterministic, yet arbitrarily small differences in starting conditions grow exponentially, making long-term prediction impossible in practice.

"Unsolvable" does not mean "unpredictable." Karl Sundman even found a convergent series solution in 1912 — but it converges so slowly it is useless for computation. What we lack is a formula like the two-body ellipse. What we have instead is numerical integration: step time forward in tiny increments, recomputing g = −∇Φ at every step. That is exactly what the simulation below does.

3. The simulation

Three bodies, full Newtonian gravity, integrated with the leapfrog (velocity-Verlet) scheme — a symplectic integrator that conserves energy well over long runs (watch the drift readout). The background heatmap is the scalar field itself: the potential Φ(r) computed live over the plane. Brighter regions are deeper potential wells. The bodies are literally rolling around in the landscape they collectively create.

Body 1 Body 2 Body 3

Figure-8 orbit. A celebrated special solution in which three equal masses chase each other along a single shared curve. Found numerically by Cris Moore in 1993 and proven to exist by Chenciner and Montgomery in 2000, it is one of the rare stable periodic solutions — a needle in the haystack of initial conditions.

Chaotic trio. The generic case. Each restart adds a small random perturbation to the starting positions and velocities — run it several times and watch how completely different the histories become. Close encounters act as slingshots, and typically one body is eventually ejected. This sensitivity to initial conditions is the fingerprint of chaos and the very reason no general formula can exist.

Star + 2 planets. The hierarchical case, like our solar system: one mass dominates, so each planet feels almost pure two-body physics with tiny mutual perturbations. Orderly for a long time — though over millions of simulated years even the solar system is technically chaotic.

4. Where the field picture earns its keep

For three bodies, computing pairwise forces directly is trivial — three pairs per step — and mathematically identical to solving Poisson's equation and taking the gradient. But pairwise cost grows as N². Simulating a galaxy of a million stars this way would require half a trillion force evaluations per step. So large simulations flip to the field view directly: deposit the mass onto a grid, solve ∇²Φ = 4πGρ numerically on that grid (fast, via the FFT), then read off g = −∇Φ everywhere at once. These particle-mesh methods are the workhorses of cosmological simulation. The scalar field stops being a mere reformulation and becomes the algorithm.

The field picture also points beyond Newton. In general relativity, Einstein replaced the single scalar Φ with the ten components of the spacetime metric, and Poisson's equation with his field equations — but in the weak-field, slow-motion limit, the metric component g₀₀ ≈ −(1 + 2Φ/c²) and Einstein's equations collapse back to ∇²Φ = 4πGρ. Newton's potential lives on inside relativity as the first whisper of curved spacetime.