The Landscape Sets the Reading

What the picture gives of the geodesic principle. What it names. What it does not derive.

Claim. The crossing record between two sites is the route light takes (AP52 §3). Put a pile between A and B and light no longer crosses on the flat route. It is redirected by the pile. That is the author’s ruling. A redirected route is a longer crossing record. So the same two sites are further apart than they were before the pile landed. Distance, in the corpus’s sense, changed. Nothing moved. Mass landed. For a grain it is the same: the landscape sets where it can move. The pile changes the slope. The change of slope across places is what a distant reader calls curvature. Gravitational time dilation is named here and not derived.

Intuition. On a flat plane the shortest route from A to B is a straight line. That is the route light takes. Now make a pile of sand between A and B. Light is directed round the pile. For a grain it is the same. If the landscape is flat, the roll is short. If there is a big dent between A and B, the grain must follow the landscape. It rolls on it. The landscape determines where it can move.

Givens. AP52 §3 (distance is the crossing record; light is the ruler). AP52 §4 (a site’s clock is its own count). AP52 §5 (motion is settling relative to the landscape; accelerated settling refused there and taken up here). AP50 §1 (where a grain lands depends on the landscape). The author’s describing.

Ruling (the author). “It influences their relative distance from one another. On a flat plane the shortest route between point A and point B is a straight flat line — the shortest route light can travel to transfer information from A to B. Now if you make a pile of sand between A and B, the shortest route from A to B is no longer a straight line — the light is now directed — the distance — by the concentration of mass piled between them. Mass determines the trajectory possibilities of the light. The inverse is the same. For a grain of sand to travel from A to B it must roll across the landscape. If it is nice and flat, the destination is in sight and the roll is easy and the shortest possible roll. Now if there is a big dent in the landscape between A and B, the grain of sand has to follow the curvature of the landscape. It must roll on the landscape. The landscape determines where it can move.” Is the pile’s slope what a distant reader calls curvature? “Yes.”

Theorem — a changed route is a changed distance. AP52 §3: a site’s only measure of another’s distance is the crossing record, the ticks a grain took to arrive. That light is redirected by the pile is the author’s ruling above. Given the ruling, the crossing between the same two sites is longer than the same two sites’ crossing before the pile landed. That is a comparison inside the record history, and it is readable. So distance changed and nothing moved. Both sites read the crossing longer. That there is a delay, and that it is symmetric, follows from the ruling. It is the kind of delay a reader far away calls Shapiro’s. Its amount is the metric’s, AP08’s, not this paper’s.

Slope, and the change of slope. The pile changes the slope of the landscape around it. The slope at a place is the pull there. A slope alone is not curvature: an incline has slope and no curvature, and a reader settling freely on one incline reads nothing. What a reader far away calls curvature is the change of slope across places. Two grains dropped a little apart converge where the slope changes. The author’s “yes” is read as that: the pile changes the slope, and the change of slope is curvature. That is a ruling of the picture. The metric that makes it exact is not here.

Gravitational dilation, named. A site in a dent is read from outside as counting slowly. It reads the outside as counting fast. That is gravitational dilation. A longer crossing record does not give it. A longer route delays every arrival by the same amount and leaves the spacing of arrivals unchanged. Dilation is the spacing changing, and it is asymmetric. The picture names it: the deep site keeps its grains against a steeper pull, and its count is its own ticks (AP52 §4). The picture does not yet derive it. The amount and the direction are AP08’s, by §7.

Why this is not a force, and what it is not yet. A force acts on a thing and moves it. The landscape does not move the grain. It is where the grain can be. “The landscape determines where it can move.” That is the geodesic principle said in sand, on the author’s ruling. It is AP52 §5’s settling with the landscape given a shape. What it is not yet is a motion law: which route, of all the routes on the landscape, the grain takes, and why light takes the route it takes. That needs a metric. The metric is the fourth sentence, §7.

Grade. That light is redirected by the pile, and that the landscape sets where a grain can move: Ruled. That a changed route is a changed distance, and the symmetric delay: Derived from AP52 §3. The change of slope as curvature: Ruled. Gravitational dilation: Named; AP08’s. The motion law: not here.

Kill switches

KS-GRAV.2 — The landscape sets the crossing. Fires if the crossing record between two sites at rest to each other is shown unchanged by a pile landing between them. The observables are the standard ones: the deflection of light by a mass, and the round-trip delay of a signal passing near a mass, each against its own null. LIVE — STRUCTURAL.

The pile changes the slope. The change of slope is what you call curvature from far away. Up close it is where the grain can go.