General Relativity

A rigorous graduate-level course on Einstein's theory of gravity—from tensor calculus and differential geometry through the Einstein field equations to black holes, gravitational waves, and numerical relativity.

Featured Documentary

Inside Einstein’s Mind · NOVA documentary on the path to general relativity

Course Overview

General Relativity, formulated by Albert Einstein in 1915, describes gravity not as a force but as the curvature of spacetime caused by mass and energy. This course develops the complete mathematical framework—from tensors and metrics to the Einstein field equations—and applies it to black holes, gravitational lensing, cosmology, and gravitational wave astronomy.

What You'll Learn

• Tensor calculus and the metric tensor
• Covariant derivatives and geodesic motion
• Riemann curvature and Ricci tensors
• Einstein field equations and energy-momentum
• Schwarzschild solution and black hole physics
• Kerr metric and rotating black holes
• Gravitational waves and cosmological solutions
• Numerical relativity and modern applications

Prerequisites

• Physical Mathematics (foundation course)• Special relativity • Differential geometry (helpful)• Multivariable calculus and linear algebra • Differential equations • Classical mechanics (Lagrangian formulation)• Electrodynamics (helpful)

References

• S. Carroll, Spacetime and Geometry
• R. Wald, General Relativity
• Misner, Thorne & Wheeler, Gravitation
• B. Schutz, A First Course in General Relativity

Course Structure

📐

Part I: Mathematical Foundations

Tensor calculus, metric tensor, covariant derivative, and geodesics.

🌌

Part II: Einstein's Theory

Riemann curvature, Ricci tensor, Einstein equations, and energy-momentum tensor.

⚫

Part III: Schwarzschild

Schwarzschild solution, orbital mechanics, gravitational lensing, and black hole thermodynamics.

🔭

Part IV: Advanced Topics

Kerr metric, cosmological solutions, gravitational waves, and numerical relativity.

Key Equations

Metric Tensor

$$ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu$$

The line element defining distances and causal structure in spacetime

Christoffel Symbols

$$\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu} \right)$$

Connection coefficients for the Levi-Civita connection

Riemann Curvature Tensor

$$R^\rho{}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$$

Measures the intrinsic curvature of spacetime

Einstein Field Equations

$$G_{\mu\nu} = 8\pi G \, T_{\mu\nu}$$

Mass-energy tells spacetime how to curve; spacetime tells matter how to move

Schwarzschild Metric

$$ds^2 = -\left(1 - \frac{2GM}{rc^2}\right)c^2 dt^2 + \left(1 - \frac{2GM}{rc^2}\right)^{-1} dr^2 + r^2 d\Omega^2$$

Exact vacuum solution for a spherically symmetric, non-rotating mass

Kerr Metric

$$ds^2 = -\left(1 - \frac{r_s r}{\Sigma}\right)c^2 dt^2 - \frac{2r_s r a \sin^2\theta}{\Sigma} \, c \, dt \, d\phi + \frac{\Sigma}{\Delta} dr^2 + \Sigma \, d\theta^2 + \frac{A \sin^2\theta}{\Sigma} d\phi^2$$

Exact vacuum solution for an axially symmetric, rotating black hole

General Relativity in the Prize Record

Einstein never received a Nobel Prize for relativity itself, but the field has accumulated honours through every generation since: Crafoord recognitions for cosmology, Dirac and Max Planck Medals for theoretical foundations, the Nobel for gravitational-wave detection (2017) and black-hole imaging (2020).

Nobel Physics →

The benchmark global physics prize since 1901 — 35 laureate lectures (2013–2025).

Crafoord Astronomy →

Royal Swedish Academy honours in astronomy and astrophysics.

Dirac Medal of ICTP →

ICTP Trieste annual award for theoretical physics, since 1985 — laureate lectures + ICTP Distinguished Conversations.

Max Planck Medal →

DPG’s highest honour for theoretical physics, since 1929 — laureate lectures plus the Lise Meitner Lecture series.

Fields Medal & Abel Prize →

The two top mathematics honours, with the complete Abel Prize laureate-interview archive (2003–2025).

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