Why does
matter fall?

A whirlpool is a recognizable thing, even though water continually flows through it. What lasts is an organized pattern of activity.

RS takes this kind of persistence seriously at the level of matter. In its light-pattern account, a material object persists through recurring physical changes and relationships. The object’s continued existence is something the dynamics must accomplish, moment after moment.

Here, recognition means a physical distinction and comparison: a later state can carry information about an earlier one. It does not require a mind watching. And “light pattern” is a proposed microscopic account, not a claim that an atom is a tiny whirlpool of ordinary visible light.

A pendulum swings. An atom oscillates. A clock uses a repeatable change to count duration. To compare two clocks, ask: how many cycles does one complete while the other completes a fixed number?

Distance is also a physical comparison. For example, send a light pulse to a mirror and back. Half the round-trip time, multiplied by the local speed of light, gives the distance in that observer’s measurement.

RS seeks to build these clock and distance relationships from its physical events and their connections. The resulting network of times, lengths and angles is what physicists call geometry. It is more than a drawing of dots on a grid: which dots are connected does not, by itself, say how many metres lie between them.

Suppose two neighbouring states have a positive ratio x. Perfect agreement is x = 1. RS’s canonical comparison rule assigns a number to their mismatch. It gives the same number whether you compare A with B or B with A.

Now connect many comparisons. Changing one location changes the comparisons on its links. Those links connect to other locations. A locally imposed source can therefore require a field extending beyond the source. A field is simply a value at every location.

Here is a small, exact version of that idea. The two ends are held fixed. A source at the centre favours a lower central value; the comparison rule resists abrupt differences. The balanced shape fits these requirements together.

Why does the balance bind things together?

A field has a positive energy cost. That does not prevent binding, because the matter energy changes too. In the weak-field balance, the matter energy falls by twice the field’s positive cost. The combined change is therefore negative.

Try making too little or too much field around a fixed source. At first, the reduction in matter energy wins. Eventually the extra field cost wins. The minimum is the balanced field.

Now place two positive sources near each other. Each responds to the other’s field through the same clock coupling. After subtracting their separate self-energies, the three-dimensional weak-field interaction is negative and proportional to 1/separation. Halve the separation and this interaction energy becomes twice as negative. Its derivative gives attraction.

“Lower energy” specifies the static balance and force; it is not a claim that nature looks ahead or instantly settles down. Actual field formation requires time-dependent equations and boundary conditions. The next chapter shows the motion of a test object once the field is present.

The important step is from one mismatch to a consistent pattern of comparisons. The empty locations do not need their own matter source to take part in the field.

For gravity, the next physical identification is to connect this field to clock and distance measurements. In the weak-field construction, it becomes the gravitational potential: a quantity whose spatial differences determine free-fall acceleration. The comparison rule plus a specified source gives a response law; choosing what physically supplies that source and how the field is measured is also part of the model.

Read the same field as clocks and motion.

Let’s make that physical identification explicit. In this weak-field example, we identify the balanced field, after a small change of scale, with gravitational potential. Lower potential means a slower clock rate relative to the clocks at the ends. This clock reading is an input connecting the comparison model to physical measurements.

Look at the same three central locations. The centre has the lowest value and counts the fewest cycles. On the left, the potential falls as you move right; on the right, it falls as you move left. Free-fall acceleration points toward lower potential, so it points toward the source from both sides. The slope, or change per unit distance, sets the acceleration. A constant clock-rate offset everywhere would produce no slope and no such acceleration.

Now zoom in on either side of the source. Over a small region, the potential has an almost constant slope. We have shown the inward acceleration arrows; next we explain them through the motion of a matter wave. A held clock is a way to measure the rate field, not a separate device pulling on an object.

Phase is position within a wave’s repeating cycle. Think of an angle on a clock face. A matter wave’s phase changes at a rate set by its energy. A slower physical clock rate therefore changes how much phase it accumulates compared with the same matter at a higher rate.

Near Earth, the clock rate increases upward. Over the same reference time, different heights acquire different extra phases. This makes a spatial phase gradient: a tilt in phase across the wave.

That is the missing step between timing and motion. In quantum mechanics, a spatial phase gradient carries momentum. With the clock-energy coupling used here, the extra phase gradient gives momentum downward. As time passes, that downward momentum grows: the object accelerates toward lower clock rate.

Clocks are not pulling on the object. The same physical rate enters its matter-wave equation, and that equation changes its motion. A constant rate offset everywhere would shift the phase uniformly, producing no such force.

Why do different masses fall together?

A mass three times larger acquires three times the phase tilt and three times the momentum change. But it also takes three times the momentum change to give it the same change in velocity. The same mass appears in both places and cancels. That is why the two acceleration readings agree.

This cancellation uses a substantive physical condition: the energy coupled to the rate field and the mass governing inertia belong to the same matter dynamics. It would not follow merely from calling an arbitrary source coefficient “mass.”

What changes when you fall with it?

Picture a small room falling with the object. In that room’s frame, the local first-order timing difference disappears. This is another description of the same acceleration. The RS coherent-continuation result identifies this cancellation for a supplied potential.

A floor prevents that freely falling motion by pushing upward on you. That supporting push is what you feel as weight. Try finding the falling frame below, then add a tidal difference that cannot be removed across the whole room.

At free fall, all three marks line up if the field is uniform over this small region. Add a tidal difference: the centre can still be in free fall, but one common acceleration cannot erase the remaining variation across the whole object.

This is the difference between free fall here and curvature across a region. Two neighbouring free objects can change their separation even while neither feels a supporting push.

We have connected energy balance, clock rates and motion in a weak static field. Full gravity also includes distances, directions, pressure and propagating disturbances. These must belong to the same coupled description.

RS’s comparison approach asks both questions using one mathematical rule. You have already used a simple example: the chain’s total cost. Test a small change in its shape, and the change in cost tells you whether it is balanced.

For moving systems, the rule assigns a number to a whole possible history. That number is called the action. Its equations come from testing tiny changes to the history and finding where the score has zero initial slope for each allowed change. This is called stationarity.

Now ask two questions of that same action. Change the matter while keeping geometry fixed: this gives the matter equation. Change the geometry while keeping matter fixed: this gives the matter’s gravitational source.

Why change geometry? Because the same physical pattern is compared differently if you change lengths, angles or clock rates. The change in its action tells you how strongly it responds. This includes directional flow and pressure, not only “how much stuff is here.”

Stretch the setting. Keep the pattern difference.

Imagine a field whose value changes from one side of a small region to the other. Keep that difference fixed. Stretch the region horizontally: the same change now happens over a longer distance, so the horizontal gradient becomes smaller. The region also becomes larger. Both effects enter the matter action.

For this particular action, doubling the horizontal length halves the score when the field varies horizontally. When the field varies vertically, the horizontal stretch leaves that gradient unchanged and doubles the score through the larger volume. The response remembers direction. That is why a gravitational source carries more information than a single amount of mass.

Clocks, distances and angles belong to one geometry. The same field that affects timing must also give consistent spatial measurements. Matter and geometry are then solved together; changing the matter changes the problem the geometry must satisfy.

Once a gravitational field is established, Newton’s law describes its slow-motion, weak-field effects. For an isolated spherical source in that limit, the field spreads through three-dimensional space. Double the distance from its centre and the same total flux is spread over four times the area: the acceleration is one quarter as large.

An apple falls almost straight down because it has little sideways speed. The Moon is continually falling too, but its sideways motion carries it around Earth. A space station and its occupants fall together, so the occupants do not press on a supporting floor.

Einstein’s geometric description makes the larger connection: freely moving matter follows the locally straight paths of spacetime, while relative free fall reveals curvature. RS asks how the patterns and comparisons beneath that description can produce it. The sourced-field construction and the coupled example above are concrete parts of that work.