Physics Schwarzchild Radius From Rs

Schwarzschild radius

The Schwarzschild radius is the distance from the center of a non-rotating, uncharged mass at which the escape velocity equals the speed of light. In classical general relativity, it is given by r_s = 2GM/c², where G is the gravitational constant, M is the mass, and c is the speed of light. For the Sun, this radius is about 2.95 kilometers; for Earth, about 8.87 millimeters. The concept originates from Karl Schwarzschild's 1916 solution to Einstein's field equations, the first exact solution describing the spacetime around a spherical mass.

The radius marks the event horizon: a boundary inside which nothing, not even light, can escape to infinity. This is not a physical surface but a coordinate singularity in the Schwarzschild solution. The radius scales linearly with mass, so any object compressed within its own Schwarzschild radius must collapse into a black hole. This threshold is central to astrophysics, defining the compactness required for black hole formation and the size of stellar-mass, supermassive, and primordial black holes.

In Recognition Science, the framework derives the gravitational constant as G = φ⁵/π in its native units, where φ is the golden ratio. Substituting this into the classical formula gives the Schwarzschild radius (the boundary distance for a black hole) as r_s = 2φ⁵M/π. For a unit mass in these units, the radius is 2φ⁵/π ≈ 3.38. The framework's machine-checked library of formal theorems proves this factor is strictly positive, ensuring the radius is always a real, positive length for any positive mass.

The module also defines the Planck length (the scale at which quantum effects dominate gravity) in these units as ℓ_P = √(Għ/c³) = √(1/π) ≈ 0.564. It introduces a classification of five canonical black hole classes: stellar, intermediate, supermassive, primordial, and Planck-scale. The library proves this classification has exactly five members, and packages these facts into a single certificate object, establishing a coherent foundation for black hole physics within the framework.

The practical consequence is that the framework's derived constants plug consistently into known physics, reproducing the Schwarzschild radius formula without free parameters. This does not derive new physics but confirms that the framework's internal constants are compatible with classical gravitational results. The positivity proof is a formal guarantee that the framework never produces a negative or zero radius for a positive mass, a sanity check that any physical theory must satisfy.

/-- Schwarzschild radius = 2φ^5/π M. -/
noncomputable def schwarzschildFactor : ℝ := 2 * phi ^ 5 / Real.pi

theorem schwarzschildFactor_pos : 0 < schwarzschildFactor := by
  unfold schwarzschildFactor
  apply div_pos
  · apply mul_pos (by norm_num) (pow_pos phi_pos 5)
  · exact Real.pi_pos

/-- Planck length = 1/sqrt(π) in RS units. -/
noncomputable def planckLength_RS : ℝ := (Real.pi)⁻¹ ^ (1/2 : ℝ)

theorem blackHoleClassCount : Fintype.card BlackHoleClass = 5 := by decide

What this page does not claim

This does not claim that the framework derives the Schwarzschild radius from first principles; it substitutes derived constants into the classical formula. This does not claim that the five black hole classes are exhaustive in astrophysics; they are a definitional choice in the framework. This does not claim that the positivity proof applies to rotating or charged black holes, which have different radius formulas.

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/SchwarzchildRadiusFromRS.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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