The Inspiral: Matter Spiraling In

The BPT law

BPT Part 3.10 (After the Final Well: the Pulsing Decay): "The final well closes the census; it does not end the universe. What follows is its decay." "Stars and remnants are flung out of their galaxies or spiral into the central wells as their orbits shed energy," and "the heaviest wells go last." Before that, in the same Part: "every bit returns through a singularity on its loop home, the final well is the last of those returns, and a universe ends when all its bits have returned" (quoted at matter.js:146–148). BPT Part 3.14 adds that "Ascendant Wells grow in the decay, when the wells take in all that is left." What the orbits shed goes to the space layer: BPT Part 3.13's Gravitational-Wave Power, P_GW = (32/5) · G⁴ · m₁² · m₂² · (m₁ + m₂) / (c⁵ · a⁵), "the power two orbiting masses give the space layer."

The established physics

A circular binary shrinks by gravitational radiation as d(a⁴)/dt = −(256/5)·G³·m·M·(m + M)/c⁵ (Peters 1964). The energy given up at the last orbit follows the Barausse, Morozova & Rezzolla 2012 fit (Part 3.10).

The game's math

Matter.inspiral(wells, dtau) matter.js:152, called each decay tick in the matter stage (stepDecay() app.js:2666), with dtau the tick's growing span of flips. For each body:

1. Its well is the non-site well that pulls it hardest, M/d², with d at least 10⁻⁶.
2. a⁴ ← a⁴ − (256/5)·G³·m·M·(m + M)/𝒞→⁵ · dtau.
3. If a⁴ ≤ r_s⁴, the body is in. The well gains m − E, with E = gwFraction(m, M)·(m + M), and E is booked to that well's gwBy and then to radiate('gwIn').
4. Otherwise the body is placed at a_new = (a⁴)^(1/4) on the same line, on a circular orbit of speed min(𝒞→, √(GM/a_new)), keeping its sense of turn.

A ripple is drawn for the largest shed if it passes 10⁻⁴ of the matter (app.js:2667).

In the game

The decay begins when the player presses "⌛ Continue the universe" on the dive prompt (app.js:2884). On the torus the stars draw in round their wells and vanish into them, galaxy by galaxy; at each decay tick the well that took in the most sends out the cyan wave train of Part 3.10 from where it sits. When no bodies are left the galaxies are cleared from the scene ("the matter is in: no galaxy is left to draw", app.js:2675), the census is read again (Part 3.14), and the wells themselves begin to evaporate, each ending in a flash (app.js:2686–2690). The life lines of the wells show the growth: their mass "a → b" and their peak, with their share of the universe.

The scale model

G, 𝒞→ and masses are in the game's units, and dtau is in flips. The law is applied exactly at the model's scale. The decay's clock (DECAY_GROW) belongs to the decay chapter.

Where the game and the theory differ

• Peters 1964 is used in its circular-orbit form, and after each tick the body is set back on a circular orbit, so any eccentricity a body had is dropped. BPT Part 3.10 also has bodies "flung out of their galaxies"; the inspiral step only draws bodies in.

Fixed numbers