The river model of gravity, turned around
Space isn’t falling in. Matter is falling behind.
The river model pictures space pouring into every planet and star like water into a drain. Density Field Dynamics reads the same picture the other way round: the universe is stretching, bound matter is not, and the river is what that lag looks like from outside.
One scene, two viewpoints
From the river to the lag.
What the river model says
Space pours into matter like a drain.
Picture space as a river flowing into every planet and star. Things “fall” because the flowing space carries them in. Near a black hole the river flows faster than light, so nothing can swim out. It’s a great picture, and the mathematics behind it is real general relativity: the Gullstrand–Painlevé form of the Schwarzschild solution.
What it leaves unanswered: why does space flow in, and where does the river go?
Density Field Dynamics
Nothing flows in. Matter falls behind.
Start with the fact everyone agrees on: the universe is stretching. Every bit of empty space is slowly growing. Now put a lump of matter in it, like a planet or a galaxy. The lump holds itself together, so it does not stretch. The universe keeps stretching all around it, and the lump just doesn’t keep up.
A drain in the fabric
Everywhere is stretching except the lump. Compared with where stretching would have put things, the space near the lump keeps ending up closer to it than it should. The lump looks like a drain that space is pouring into. That’s the river.
Holding still
Nothing is being swallowed. The lump is sitting still and refusing to stretch, while the rest of the universe streams outward past it and moves on without it.
Same scene, two viewpoints. The river model draws matter’s lag as a current flowing in. DFD draws it as matter expanding less than the universe. Neither picture has anything actually being sucked. Step 4 of the tour does the conversion: take the mass’s view, subtract the uniform stretching, and the river appears.
So what is gravity?
It’s the mismatch.
In DFD the expansion of space and gravity are carried by the same field, ψ. The extra density that sources the field is exactly the matter’s accumulated failure to stretch with the universe:
$$1+\delta=\exp\!\int\big(3H-\theta\big)\,dt$$
Here $3H$ is how fast space stretches and $\theta$ is how fast the matter stretches. For ordinary matter this is the continuity equation; DFD’s step is reading the expansion and gravity as one field. Things near the lump drift toward it because the stretching that would carry them away is held back there. That drift is what we feel as falling.
The bonus
The lump keeps its own clock.
Because bound matter isn’t stretching with the universe, its clocks and rulers stay slightly out of step with the universe’s. You can’t notice this from inside, since every clock and ruler in your lab is out of step together. You only see it when you compare a ruler made out here, among bound matter, with one made in the early universe, before anything had clumped.
In DFD that comparison is where the Hubble tension lives: about 72 km/s/Mpc from nearby yardsticks against about 68 from early-universe yardsticks, a gap of roughly 7%.
Space isn’t flowing into matter. Matter is falling behind the universe’s expansion, and the river is just what that lag looks like from outside.