General Relativity
A comprehensive graduate-level course on Einstein's theory of gravity—from tensor calculus and differential geometry through the Einstein field equations to black holes, cosmology, and gravitational waves.
Course Overview
General Relativity, formulated by Albert Einstein in 1915, is the modern theory of gravitation. It describes gravity not as a force, but as a manifestation of the curvature of spacetime caused by mass and energy. This course provides a rigorous mathematical treatment of GR, starting from differential geometry and culminating in applications to black holes, cosmology, and gravitational wave astronomy.
What You'll Learn
- • Tensor analysis and differential geometry
- • Riemann curvature tensor and Einstein equations
- • Schwarzschild and Kerr solutions
- • Black hole physics and thermodynamics
- • Cosmological models (FLRW universe)
- • Gravitational waves and LIGO detections
- • Singularity theorems and quantum gravity
- • Numerical relativity and modern applications
Prerequisites
- • Special Relativity
- • Advanced Mathematics (calculus, linear algebra)
- • Vector calculus and differential equations
- • Basic tensor notation
- • Classical mechanics (Lagrangian formulation)
- • Some exposure to Quantum Mechanics helpful
Course Structure
5 Parts covering 29 chapters • From differential geometry to quantum gravity • Includes detailed derivations, worked examples, and observational evidence • Suitable for advanced undergraduates and graduate students
Course Parts
Part I: Differential Geometry
Mathematical foundations: manifolds, tensors, metric, connections, and covariant derivatives. Master the geometric language needed for General Relativity.
Part II: Curvature of Spacetime
The Riemann tensor, Ricci curvature, Bianchi identities, and geodesic deviation. Learn how curvature encodes gravity.
Part III: Einstein Field Equations
Derive Einstein's field equations from the Einstein-Hilbert action. Study the stress-energy tensor, weak field limit, and Newtonian approximation.
Part IV: Classic Solutions
Exact solutions: Schwarzschild (spherical black holes), Kerr (rotating black holes), Reissner-Nordström (charged), FLRW cosmology, and gravitational waves.
Part V: Advanced Topics
Singularity theorems, black hole thermodynamics, ADM formalism, numerical relativity, gravitational lensing, and the path to quantum gravity.
Key Equations
Einstein Field Equations
The fundamental equations of General Relativity: spacetime curvature (left side) is determined by matter-energy content (right side).
Schwarzschild Metric
Describes spacetime outside a spherically symmetric mass. Event horizon at r_s = 2GM/c².
Geodesic Equation
Free particles follow geodesics—the straightest possible paths in curved spacetime.
Related Courses
Special Relativity
Foundation for GR: Lorentz transformations, spacetime, and relativistic mechanics
Black Holes
In-depth study of Schwarzschild, Kerr, thermodynamics, and observations
Cosmology
Apply GR to the universe: expansion, dark matter, dark energy, CMB
Quantum Gravity
Beyond classical GR: string theory, loop quantum gravity, black hole entropy
"Matter tells spacetime how to curve, and spacetime tells matter how to move."— John Archibald Wheeler
Learning Path & Prerequisites
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