Gravity Gravitational Wave Memory3 From Jcost

Gravitational wave memory

Gravitational wave memory is the permanent change in the distance between free masses after a gravitational wave passes. Unlike the oscillating part of the wave, this residual displacement does not return to zero. In general relativity, the effect is a nonlinear consequence of the wave's own energy, and it is small: for a typical event, the memory strain is roughly 5 to 15 percent of the peak strain of the wave itself.

The effect was studied through the 1970s and 1980s, with contributions from Ya. B. Zel'dovich and A. G. Polnarev in 1974 and from V. B. Braginsky and K. S. Thorne in 1987. The standard formula for the memory strain Δh involves an integral over the angular distribution of the radiated energy. A simpler heuristic form is Δh = J(φ) · h_peak, where h_peak is the peak strain and J(φ) is a dimensionless fraction.

In Recognition Science, the framework models this fraction using its cost function. The framework's central object is the cost function, a measure of the price of recognition events, defined as J(x) = (x + 1/x)/2 − 1. For the golden ratio φ ≈ 1.618, the value J(φ) = (φ + 1/φ)/2 − 1 equals approximately 0.118, or 11.8 percent. The framework identifies this value with the canonical memory fraction, so that the memory strain is J(φ) times the peak strain.

The machine-checked library of formal theorems proves three general facts about this cost function. First, the cost vanishes when the two inputs are equal: J(1) = 0. Second, the cost is nonnegative for all positive inputs. Third, the threshold φ − 3/2 is positive, which is a small numerical fact about the golden ratio. These three facts are packaged into a certificate structure named GWMemory3Cert. The library proves that this certificate exists.

What the module does not do is connect these facts to gravitational waves. The definition of the cost uses a ratio m/e without specifying what m and e mean in this context. The library's own documentation states that the paragraph about memory is a research note recording where the idea was meant to go, not a result. The three proved facts are general properties of the cost function, true for any positive inputs, and they do not by themselves establish anything about gravitational wave memory.

The empirical check is separate. If one accepts the identification of the memory fraction with J(φ), the value 11.8 percent falls inside the 5 to 15 percent range quoted for canonical memory fractions. That agreement is a comparison against observation, not a theorem. The framework's contribution is a derivation of the number 11.8 percent from its cost function; the link from that cost function to the physics of gravitational waves remains an open target.

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]

theorem cert_inhabited : Nonempty GWMemory3Cert := ⟨cert⟩

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

What this page does not claim

This module proves that gravitational wave memory equals 11.8 percent of peak strain. The cost function is derived from the physics of gravitational waves. The empirical agreement with the 5 to 15 percent range is a proved theorem.

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/GravitationalWaveMemory3FromJCost.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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