From exchanges to spacetime · Epistemic Physics

A light pulse reaches a small courier. The courier changes its motion, carries its parcel farther, and adds another moment to its journey.

Let us build that journey from the same two-column arithmetic we used for belief. The sum will give energy, the difference momentum, and their balance the slope of a path through spacetime. Even the clock on board will join the picture.

Give the two columns a physical role

Our evidence notebook began with raw support and opposition weights $Y$ and $N$ . Its Beta parameters are $α = Y + 1 / 2$ and $β = N + 1 / 2$ when we include the Jeffreys prior.

Now try a physical model. Read $Y$ and $N$ as positive rightward and leftward energy weights carried by the courier. Choose one unit of energy per unit of weight, and units where light speed is $c = 1$ . The sum gives its energy $E$ ; the difference gives its signed momentum $p$ .

These columns summarize what the courier carries into its next encounter. They can take fractional values. We start the game at $Y = N = 1$ : energy 2, momentum 0, a balanced body at rest.

The belief notebook places its prior underneath the raw trials. Each new rightward or leftward contribution then makes the same change in both pictures.

\begin{aligned} E & = Y + N, \\ p & = Y - N . \end{aligned}

One update, seen through two readings
Arrival	Raw weights	Beta parameters
Right / support	$Y \to Y + 1$	$α \to α + 1$
Left / opposition	$N \to N + 1$	$β \to β + 1$

A pulse changes the balance

Send the courier a rightward light pulse with energy 1 and momentum $+ 1$ . Absorbing it adds one to $Y$ . The two columns become $(Y, N) = (2, 1)$ , so the courier now carries energy 3 and momentum 1.

A leftward pulse instead adds one to $N$ . Its energy is also positive, while its momentum is negative. Adding the columns therefore conserves exactly what the received pulse contributes.

Coasting adds nothing to either column. The courier keeps carrying its existing energy and momentum. In the game, each push delivers a pulse before the next one-second movement begins; Coast lets the current motion continue.

\begin{aligned} Δ E & = Δ Y + Δ N, \\ Δ p & = Δ Y - Δ N . \end{aligned}

Let the balance draw a journey

The balance ratio now becomes a velocity $v$ . Relativistic particle motion gives $v = p / E$ , so our courier moves according to the difference of its two columns divided by their sum.

After that first rightward pulse, its velocity is $(2 - 1) / (2 + 1) = 1 / 3$ . Light arrives at speed 1, then the body receiving it moves at speed $1 / 3$ . During the next lab second, it travels one third of a light-second. A light-second is the distance light travels in one second.

Record the courier's position $x$ and lab time $t$ after each move. Each pair $(t, x)$ marks an event. Put later times above earlier ones and connect the dots: you have drawn a worldline, the courier's path through spacetime.

A balanced pair makes a vertical trail. An imbalance tilts the trail. Coasting extends its current slope; a new pulse can bend it. Try a rightward push followed by Coast and watch the two-column balance become a route.

\begin{aligned} v & = \frac{Y - N}{Y + N}, \\ \frac{d x}{d t} & = v . \end{aligned}

The same pair sets mass and clock rate

Take the difference of squares: $(Y + N)^{2} - (Y - N)^{2} = 4 Y N$ . The familiar two-column product now gives the courier's rest mass $m$ through $m^{2} = E^{2} - p^{2}$ .

There is another connection hiding here. Proper time $s$ is the elapsed time measured by an ideal clock traveling with the courier. Relativity gives $d s / d t = \sqrt{1 - v^{2}} = m / E$ . Substitute the two columns and the same geometric mean, $\sqrt{Y N}$ , sets the clock ratio.

When $Y = N$ , the courier is at rest and both clocks advance together. As one column dominates, the speed approaches the lightlike boundary and the courier's clock advances less per lab second. At speed 0.6, it records 0.8 seconds per lab second.

Absorbed energy stays aboard, so later pulses can change the courier's rest mass as well as its velocity. Its clock adds the proper time of each segment.

\begin{aligned} m & = 2 \sqrt{Y N}, \\ \frac{d s}{d t} & = \frac{2 \sqrt{Y N}}{Y + N} . \end{aligned}

Your turn

Reach a gold beacon using light pulses and coasting. Then restart and compare right-then-left with left-then-right: both finish with the same Y and N, while their routes and stopping positions differ.

A little journey through spacetime

Pilot the light courier

Carry a parcel to the gold beacons. Push the courier with a light pulse, or let it coast. Every move takes one second on the lab clock. Watch what its own clock records.

Lab clock0.00 s

Courier clock0.00 s

A push changes your motion. Coast keeps it. Buttons work with Enter or Space.

N · left1

Y · right1

Reach either gold beacon. You can change direction whenever you like.

Each dot below marks a second of your route. Upward means later; sideways means a new position. New pushes change the next segment. Earlier events stay on the page. The view expands to keep your whole journey visible.

How the pulses and clocks work

Each push delivers one light packet with energy 1 and momentum in the chosen direction, absorbed before the next move. Its energy stays aboard. The courier becomes a new composite, so its rest mass can grow. Coast adds no packet.

A rightward pulse adds one to Y; a leftward pulse adds one to N. Their sum gives energy, their difference gives momentum, and their balance sets the next movement.

\begin{aligned} E & = Y + N \\ p & = Y - N \\ m & = 2 \sqrt{Y N} \end{aligned}

Energy E: 2.00
Momentum p: 0.00
Rest mass m: 2.00

A light-second is the distance light travels in one second. These units make light speed $c = 1$ . The courier starts at rest with energy and mass 2. Its position is $x$ and the lab time is $t$ . Its own elapsed time, called proper time, is $s$ .

\begin{aligned} Δ t & = 1 \\ Δ x & = p / E \\ Δ s & = m / E \end{aligned}

On each segment, $Δ s = \sqrt{(Δ t)^{2} - (Δ x)^{2}}$ . Add those clock readings along the route. Going faster leaves less time on the courier's clock for the same lab second.

An incoming packet changes the courier's energy and momentum at an event. Between pulses, those quantities describe motion along its next segment. Position and time accumulate along the route:

\begin{aligned} E & = m d t / d s \\ p & = m d x / d s \end{aligned}

The route remembers the order

Restart. Push right once, then left once. Restart again and reverse the order.

Both runs finish with $(Y, N) = (2, 2)$ : energy 4, momentum 0, and rest mass 4. Both use two lab seconds and give equal courier-clock readings. But the courier stops on opposite sides of its starting point. The same final columns can grow through different journeys.

The notebook has the same opportunity: retain the sequence of trials as well as their totals. For the courier, the starting event, ordered pulses, and time between encounters let us reconstruct the whole worldline. Each retained update becomes part of a history.

Could exchanges build a world?

We have a small working synthesis. The two columns receive an exchange, their balance sets the next direction of motion, and their product connects mass with elapsed proper time. Add the successive movements and a journey takes shape.

The same pattern appears in the belief notebook, with $τ_{E} = α + β$ and $x_{E} = α - β$ . Sum, difference, and product organize both readings. Including the prior shifts the notebook's starting point; the incoming trial arrows still follow the shared two-column rule.

Now imagine extending the game. Perhaps a particle could be a lasting pattern of exchanges, and a clock could count recurring events within that pattern. A network of such patterns might supply its own clocks and rulers. To build it, we would choose interactions that sustain those patterns and make both participants agree on their meeting events.

Our courier gives us a first construction to play with: a two-column history shaping a body, a clock reading, and a path through spacetime. What kinds of worlds could a whole network of those histories make?

Follow the connection

Worldlines, proper time, and relativistic momentum