Gravity meets the same fork as charge: under Gauss's law in 4D no orbit is stable. Six routes, what each costs, the lens assumed for symmetry with charge, and orbits under each law.
Chapter 2 pinned the force between charges at 1/r and left gravity open. Gravity hardly matters to atoms, but it makes the stars and holds the planets. Its problem is the same as charge's. A field obeying Gauss's law in four dimensions falls as 1/r², and for 1/r we need a propagator of 1/k³. So the same forks appear (charge's, for comparison), with gravity's own twists. Gravity pulls on everything alike, it's nonlinear in our world already, and it has waves and black holes of its own.
The four-dimensional version of Einstein's gravity, whose slow, weak limit is 1/r². It keeps everything: locality, relativity, the equivalence principle, waves at c.
costs: no stable orbit (below), and a star of gas is exactly marginal (chapter 16). Stiff bodies like rock could hold themselves together, but nothing would orbit anything. No solar systems.
As for charge, a fractional power of the wave operator gives 1/r statics and keeps relativity.
costs: negative probabilities, by the same unitarity bound as charge's route 2.
Give gravity's vacuum a response that grows with wavenumber, as charge's route 3 does.
costs: gravitational waves disperse with no speed limit. Long waves outrun short ones without bound.
Make gravity's Gauss law nonlinear, ∇·(|g|^(1/2) g) ∝ ρ in 4D, the kind of law Bekenstein and Milgrom wrote for our galaxies. A point mass then pulls as 1/r. Gravity is already nonlinear in our world, though not in its weak, everyday limit. This law keeps locality, and in our world such laws have been given relativistic completions.
costs: masses don't add. A body's pull grows as its mass to the 2/3 power, so two stars together pull less than twice one, and a planet's motion feels the galaxy it sits in. In exchange the divergence theorem still holds: the shell theorem survives. A round star acts as a point, its outer layers don't press inward, and orbits round it are exactly Kepler's.
Let gravity see three dimensions while matter sees four.
costs: coherence. Gravity pulls on all energy, including motion along w, so it can't ignore a dimension that matter moves in. This route breaks the premise.
Gravity's static pull seen through the same kind of lens, a 1/k³ kernel, so 1/r between masses. Everything still falls alike, and masses add. Gravity's waves can stay as 4D Einstein's.
we'd pay: relativity for gravity's pull, an absolute "now" for gravity as for charge, and no black holes as Einstein's are defined. No shell theorem either: a star's outer layers press inward, and orbits round a star slowly precess.
Routes 4 and 6 are the real choice, and they predict different skies. Under the lens, orbits round a star precess and masses add. Under the nonlinear law, orbits close exactly but a cluster of stars pulls less than its parts. Both give bound stars, because both make a star's gravitational energy scale as 1/R. The book assumes the lens from here on, for its symmetry with charge, and marks where route 4 would differ. The choice is open.
What the choice does to orbits decides whether there are planets at all. Under the pinned 1/r, two bodies orbit as Kepler said. A central pull keeps an orbit in one plane, even in 4D, and a 1/r pull closes it into an ellipse in any number of dimensions (chapter 4). Under Gauss's 1/r² no orbit is stable. Give a circular orbit 1% more speed and it runs away:
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