The Surgical Criterion: A Geometric Invariant for Detecting Causal Singularities (Intuitive Follow up on Surgery on Lorentzian Manifolds Mathematical Monograph)

The surgical criterion detects when/where to surgically remove spacetime regions before singularities form. The covariance is what makes it a genuine geometric trigger for surgery, valid for all observers, in all reference frames.No horizon finding needed, the cut surface is the emerging horizon. 1. Modified Curvature ∥Rμν∥g Measures departure from smooth soliton geometry Includes matter + geometric flow terms Blows up at singularities 2. Causal Distance dg(p,∂J−(p)) Maximum proper time from point pp to its past light cone boundary Shrinks as region becomes causally isolated Zero at singularities (light cone pinches off) 3. Threshold Θ Dimensionless constant Calibrated so surgery happens just before classical singularity ~1 in Planck units Curvature × Distance² is: Dimensionless → universal threshold Scale-invariant → detects shape, not size Monotonic during collapse → guaranteed trigger Finite at singularities → triggers at finite value ∞ (curvature) × 0 (distance^2) → finite Physical During gravitational collapse: Curvature increases (star compresses) Causal distance decreases (light cones narrow) Product grows When ≥Θ→ cut along ∂J−(p) → replace with smooth soliton The boundary ∂J−(p) becomes the surgical surface, automatically the apparent horizon. One invariant does: Singularity detector (flags bad regions) Horizon locator (finds cut surface) Surgical trigger (tells when to cut) Eliminates: Horizon-finding elliptic solves, singularity-handling hacks, angular momentum loss during excision.