Astrophysics Gravitational Wave From Jcost

Gravitational wave amplitudes

Gravitational waves are ripples in spacetime, emitted when massive objects like black holes or neutron stars spiral together and merge. The strength of such a wave, its strain h, is a dimensionless number that describes how much space is stretched and squeezed as the wave passes. For a binary merger, the strain scales as h ~ M^(5/3) / D, where M is the total mass of the two objects and D is the distance to the observer. This means a heavier pair of merging objects produces a stronger wave, and a more distant event produces a weaker one.

Astrophysicists group gravitational wave sources into five canonical categories: neutron star-neutron star mergers (NS-NS), black hole-neutron star mergers (BH-NS), black hole-black hole mergers (BH-BH), supermassive black hole mergers (SMBH), and the stochastic background, the faint hum of many unresolved sources. These categories span a vast range of masses, from stellar-mass black holes to the supermassive ones at galactic centers. The peak strain of a detected event depends strongly on which category it belongs to, because the mass term enters the strain formula with a high exponent.

In Recognition Science, the framework that derives physical structure from a single cost function, this astrophysical ordering takes a precise form. The framework models the five source categories as the configuration dimension D = 5, and it posits that the peak strain of adjacent categories differs by a fixed factor. Specifically, the strain at a given rung of the mass ladder, strainAtRung(k) = phi^k, where phi is the golden ratio, approximately 1.618. The theorem strainRatio proves that the ratio of strain at adjacent rungs is exactly phi: strainAtRung(k+1) / strainAtRung(k) = phi. This means the peak strain of a BH-NS merger is phi times that of an NS-NS merger, and so on up the ladder.

The module packages these two facts, the five categories and the phi ratio, into a single certificate structure called GravitationalWaveCert. This certificate holds the theorem that there are exactly five source categories and the theorem that adjacent rungs differ by phi. The certificate is constructed from the two proofs, gwSourceCount and strainRatio, and the whole file is machine-checked with no axioms beyond the standard logical ones. The framework's library, a machine-checked collection of formal theorems, verifies that these statements follow from the definitions.

What this establishes in plain language is a clean, testable prediction about the relative loudness of gravitational wave events across different source classes. The framework does not predict the absolute strain of any single event, which depends on distance and individual masses. Instead, it predicts a pattern: if you rank the five source categories by peak strain, each step up the ladder is phi times louder than the one below. This is a sharp claim that observatories like LIGO and Virgo could, in principle, test with a large enough catalog of detected mergers.

inductive GWSourceCategory where
  | nsNS | bhNS | bhBH | smbh | stochastic
  deriving DecidableEq, Repr, BEq, Fintype

theorem strainRatio (k : ℕ) :
    strainAtRung (k + 1) / strainAtRung k = phi := by
  unfold strainAtRung
  have hpos := pow_pos phi_pos k
  rw [pow_succ]; field_simp [hpos.ne']

theorem gwSourceCount : Fintype.card GWSourceCategory = 5 := by decide

What this page does not claim

This module does not predict the absolute strain of any single gravitational wave event. The phi ratio is a prediction about relative strains, not a derivation from the standard Newtonian or Einsteinian equations of binary mergers. The module does not claim that the five source categories are the only possible gravitational wave sources.

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/Astrophysics/GravitationalWaveFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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