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Indefinite Metric

Metrics with mixed signature — bilinear forms allowing both positive and negative lengths.

Mathematics • Linear Algebra / Differential Geometry🔥 Intermediate

An indefinite metric is a symmetric bilinear form that is not positive-definite, assigning lengths that can be positive, zero, or negative. Learners will grasp signatures, canonical forms, and key applications such as Minkowski spacetime and pseudo-Riemannian geometry.

⏱️
30 minTypical time
📋
3Prerequisites
🎯
4Learning outcomes

📋Prerequisites

  • Linear algebra: vector spaces, matrices, change of basis
  • Bilinear forms and inner product basics
  • Multivariable calculus or an introduction to manifolds (helpful but not required)

🎯What You'll Learn

  • ✓Define an indefinite metric and compute its signature and index
  • ✓Bring a quadratic form to canonical form using change of basis and apply Sylvester's law of inertia
  • ✓Differentiate between positive-definite, negative-definite, and indefinite forms
  • ✓Recognize and apply indefinite metrics in contexts like Minkowski space and pseudo-Riemannian manifolds

🔗Related Concepts

builds-onPseudo-Riemannian Manifoldapplication-ofMinkowski SpaceprerequisiteQuadratic FormprerequisiteSylvester's Law of Inertiasimilar-toInner Product Space

🔗Learn More

📖Indefinite quadratic form — Wikipedia📖Pseudo-Riemannian manifold — Wikipedia🔗Minkowski space — Encyclopedia Britannica📖Sylvester's law of inertia — Wikipedia🔗Linear Algebra (MIT OpenCourseWare) — Lecture notes🔗Lecture notes: Indefinite metrics and signature (PDF)
📐

Introduction to Indefinite Metric

High-level overview, motivation, and historical context for indefinite metrics in geometry and physics.

⏱️12 min

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🎥Short Introductory Lecture
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Mathematical Definition

Formal definitions, notation, and mathematical consequences of an indefinite metric.

⏱️20 min

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Mathematical Formula

Variables:

Type: mathematical

Signature and Indexing

🧮

Examples & Computations

Worked examples, computations, and code snippets illustrating indefinite metrics in practice.

⏱️25 min

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🌌

Physical Applications

How indefinite metrics appear in physics (e.g., special and general relativity) and applied settings.

⏱️18 min

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Common Application Examples

🎥Lecture: Metrics in Physics
✅

Exercises & Assessment

Practice problems, quizzes, flashcards, and channels for expert feedback.

⏱️22 min
Multiple-Choice Quiz
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💬AI Expert Chat
COURSE

Further Reading: Advanced Course

Full course available