Gravity Gravitational Wave Phase3 From Jcost

Gravitational wave phase

Gravitational waves are ripples in spacetime that travel at the speed of light, emitted when massive objects accelerate. The most famous source is a pair of black holes spiraling into each other. As they orbit, they lose energy by radiating these waves, and the signal's frequency rises in a pattern called a chirp. The phase of this chirp, the precise timing of the wave crests, is one of the most accurately predicted quantities in physics, and its measurement lets astronomers extract the masses of the merging objects.

The standard description of this phase comes from post-Newtonian (PN) theory, a systematic approximation to general relativity for slow-moving, weakly gravitating systems. The leading-order term, the one that dominates the signal, scales as the minus five-thirds power of the frequency. The coefficient of this term involves the chirp mass, a combination of the two black hole masses. This leading coefficient is what a detector like LIGO measures most cleanly, and it sets the overall amplitude of the phase evolution.

In Recognition Science, the framework models the universe as maintaining a discrete record of recognition events, with a forced cost for each recognition. The central result is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. From this cost function, the framework derives constants and scaling laws. For gravitational waves, the framework identifies the chirp mass function with phi^(D-1), where phi is the golden ratio and D is the number of spatial dimensions. For three spatial dimensions, this gives phi^2 = 2.618, which the framework proposes as the leading-order PN phase coefficient in its own units.

The module GravitationalWavePhase3FromJCost in the machine-checked library of formal theorems establishes three general facts about the cost function applied to a mass-to-energy ratio. It proves that the cost vanishes when the mass equals the energy, that the cost is nonnegative for positive inputs, and that phi minus 3/2 is positive. These are structural properties of the cost function itself, not specific to gravitational waves. The module defines a domain cost as J(m/e), but it does not define what m and e mean in this context. The connection to the gravitational wave phase, the identification of the chirp mass with phi^2, is a research note recorded in the docstring, not a proved theorem.

What this means in plain language is that the framework has a general tool, the cost function, and it has shown that this tool has some basic, sensible properties. It has not yet shown that this tool actually describes gravitational waves. The specific claim about the phase coefficient is a target, not a result. The gap is the definition of m and e in terms of the physical system. Until that definition exists, the module proves nothing about gravity.

The consequence for the reader is a clear picture of where the framework stands on this topic. The mathematical machinery is in place, and the general properties are proved. The physical application, the derivation of the chirp mass from the cost function, remains open. This is an honest and precise statement of what the module does and does not establish.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module does not prove any specific fact about gravitational waves. The identification of the chirp mass with phi^2 is not a proved theorem. The framework does not derive the full post-Newtonian expansion in this module.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/GravitationalWavePhase3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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