Physics Relativistic Mass From Jcost

Relativistic mass from cost

Relativistic mass is the speed-dependent mass of an object in motion, a concept from Einstein's special relativity. As an object's speed v approaches the speed of light c, its relativistic mass increases without bound. The factor that governs this growth is the Lorentz factor, γ = 1/√(1 - v²/c²). At everyday speeds this factor is nearly 1, but at v = c/φ, where φ is the golden ratio, γ equals about 1.272, which is the square root of φ. This particular speed is notable because it is where the golden ratio appears naturally in the relativistic mass formula.

The Lorentz factor was introduced by Hendrik Lorentz in 1904 to explain the null result of the Michelson-Morley experiment, and it became a key part of Einstein's 1905 special relativity. The factor also appears in the transformation of time and length, not just mass. A key property is that γ is always at least 1, reaching 1 only when v = 0. The function is smooth and continuous, but it grows rapidly as v approaches c, creating a barrier that prevents massive objects from reaching the speed of light.

In Recognition Science, the framework models the relativistic mass ratio through a cost function, a measure of the effort required for a recognition event. The framework's core cost function is J(x) = (x + 1/x)/2 - 1, which is forced by five plain conditions. The module defines domainCost as J(m/e), where m is the moving mass and e is the rest mass. This definition directly connects the mass ratio to the universal cost function, suggesting that the cost of recognition grows with the mass ratio in a specific, forced way.

The module proves three general facts about this cost function. First, the cost is zero when the mass equals the rest mass, meaning no cost is incurred for a stationary object. Second, the cost is never negative for positive masses, so it cannot decrease below zero. Third, the threshold φ - 3/2 is positive, which is a technical condition about the golden ratio. These results are general properties of the cost function J, not specific to relativistic mass, because the module does not define m and e in physical terms.

The framework's library, a machine-checked collection of formal theorems, proves these three facts without any framework-specific axioms. The module also defines a structure called RelMassCert that packages these proofs, and it shows this structure is inhabited. However, the module does not prove that the cost function actually equals the relativistic mass ratio; that would require a physical definition of m and e. The natural relativistic speed v = c/φ corresponds to a recognition rung in the framework, where γ equals the square root of φ, a connection that remains a research note rather than a theorem.

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

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

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]

What this page does not claim

The module does not prove that the cost function equals the relativistic mass ratio. The module does not derive the Lorentz factor from the cost function. The natural speed v = c/φ is a research note, not a proved result.

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

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