The Surgical Criterion: A Geometric Invariant for Detecting Causal Singularities (Intuitive Follow up on Surgery on Lorentzian Manifolds Mathematical Monograph)
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2026
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| _version_ | 1866901967527215104 |
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| author | Rayan D. Peru |
| author_facet | Rayan D. Peru |
| contents | <h1> </h1> <p>The surgical criterion detects <strong>when/where to surgically remove</strong> spacetime regions <strong>before</strong> singularities form. The covariance is what makes it a <strong>genuine geometric trigger</strong> for surgery, valid for all observers, in all reference frames.<br>No horizon finding needed, the cut surface <strong>is</strong> the emerging horizon.</p> <h3><strong>1. </strong>Modified Curvature ∥Rμν∥g</h3> <ul> <li> <p>Measures <strong>departure from smooth soliton geometry</strong></p> </li> <li> <p>Includes matter + geometric flow terms</p> </li> <li> <p>Blows up at singularities</p> </li> </ul> <h3><strong>2. </strong>Causal Distance dg(p,∂J−(p))</h3> <ul> <li> <p>Maximum proper time from point pp to its <strong>past light cone boundary</strong></p> </li> <li> <p>Shrinks as region becomes causally isolated</p> </li> <li> <p>Zero at singularities (light cone pinches off)</p> </li> </ul> <h3><strong>3. </strong>Threshold Θ</h3> <ul> <li> <p>Dimensionless constant</p> </li> <li> <p>Calibrated so surgery happens <strong>just before</strong> classical singularity</p> </li> <li> <p>~1 in Planck units</p> </li> </ul> <p><strong>Curvature</strong> × <strong>Distance²</strong> is:</p> <ul> <li> <p><strong>Dimensionless</strong> → universal threshold</p> </li> <li> <p><strong>Scale-invariant</strong> → detects shape, not size</p> </li> <li> <p><strong>Monotonic during collapse</strong> → guaranteed trigger</p> </li> <li> <p><strong>Finite at singularities</strong> → triggers at finite value</p> </li> </ul> <p>∞ (curvature) × 0 (distance^2) → finite</p> <h2>Physical</h2> <p>During gravitational collapse:</p> <ol> <li> <p><strong>Curvature increases</strong> (star compresses)</p> </li> <li> <p><strong>Causal distance decreases</strong> (light cones narrow)</p> </li> <li> <p><strong>Product grows</strong></p> </li> <li> <p><strong>When</strong> ≥Θ→ cut along ∂J−(p) → replace with smooth soliton</p> </li> </ol> <p>The boundary ∂J−(p) becomes the <strong>surgical surface,</strong> automatically the <strong>apparent horizon</strong>.</p> <p><strong>One invariant does:</strong></p> <ol> <li> <p><strong>Singularity detector</strong> (flags bad regions)</p> </li> <li> <p><strong>Horizon locator</strong> (finds cut surface)</p> </li> <li> <p><strong>Surgical trigger</strong> (tells when to cut)</p> </li> </ol> <p><strong>Eliminates:</strong> Horizon-finding elliptic solves, singularity-handling hacks, angular momentum loss during excision.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18301245 |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The Surgical Criterion: A Geometric Invariant for Detecting Causal Singularities (Intuitive Follow up on Surgery on Lorentzian Manifolds Mathematical Monograph) Rayan D. Peru <h1> </h1> <p>The surgical criterion detects <strong>when/where to surgically remove</strong> spacetime regions <strong>before</strong> singularities form. The covariance is what makes it a <strong>genuine geometric trigger</strong> for surgery, valid for all observers, in all reference frames.<br>No horizon finding needed, the cut surface <strong>is</strong> the emerging horizon.</p> <h3><strong>1. </strong>Modified Curvature ∥Rμν∥g</h3> <ul> <li> <p>Measures <strong>departure from smooth soliton geometry</strong></p> </li> <li> <p>Includes matter + geometric flow terms</p> </li> <li> <p>Blows up at singularities</p> </li> </ul> <h3><strong>2. </strong>Causal Distance dg(p,∂J−(p))</h3> <ul> <li> <p>Maximum proper time from point pp to its <strong>past light cone boundary</strong></p> </li> <li> <p>Shrinks as region becomes causally isolated</p> </li> <li> <p>Zero at singularities (light cone pinches off)</p> </li> </ul> <h3><strong>3. </strong>Threshold Θ</h3> <ul> <li> <p>Dimensionless constant</p> </li> <li> <p>Calibrated so surgery happens <strong>just before</strong> classical singularity</p> </li> <li> <p>~1 in Planck units</p> </li> </ul> <p><strong>Curvature</strong> × <strong>Distance²</strong> is:</p> <ul> <li> <p><strong>Dimensionless</strong> → universal threshold</p> </li> <li> <p><strong>Scale-invariant</strong> → detects shape, not size</p> </li> <li> <p><strong>Monotonic during collapse</strong> → guaranteed trigger</p> </li> <li> <p><strong>Finite at singularities</strong> → triggers at finite value</p> </li> </ul> <p>∞ (curvature) × 0 (distance^2) → finite</p> <h2>Physical</h2> <p>During gravitational collapse:</p> <ol> <li> <p><strong>Curvature increases</strong> (star compresses)</p> </li> <li> <p><strong>Causal distance decreases</strong> (light cones narrow)</p> </li> <li> <p><strong>Product grows</strong></p> </li> <li> <p><strong>When</strong> ≥Θ→ cut along ∂J−(p) → replace with smooth soliton</p> </li> </ol> <p>The boundary ∂J−(p) becomes the <strong>surgical surface,</strong> automatically the <strong>apparent horizon</strong>.</p> <p><strong>One invariant does:</strong></p> <ol> <li> <p><strong>Singularity detector</strong> (flags bad regions)</p> </li> <li> <p><strong>Horizon locator</strong> (finds cut surface)</p> </li> <li> <p><strong>Surgical trigger</strong> (tells when to cut)</p> </li> </ol> <p><strong>Eliminates:</strong> Horizon-finding elliptic solves, singularity-handling hacks, angular momentum loss during excision.</p> |
| title | The Surgical Criterion: A Geometric Invariant for Detecting Causal Singularities (Intuitive Follow up on Surgery on Lorentzian Manifolds Mathematical Monograph) |
| url | https://doi.org/10.5281/zenodo.18301245 |