General Relativity Solutions & Resources
Solutions to exercises from "Spacetime and Geometry" by Sean Carroll and comprehensive video tutorials
About These Resources
General Relativity requires mastering differential geometry before tackling Einstein's field equations. These resources provide step-by-step solutions and video explanations to help you build intuition and mathematical fluency.
Carroll Textbook
The standard graduate GR text
Eigenchris Videos
Best tensor tutorials online
Worked Solutions
Exercise-by-exercise help
Eigenchris Video Series
The most highly recommended YouTube series for learning tensor calculus and general relativity. Eigenchris provides crystal-clear explanations that don't skip steps.
Tensors for Beginners
Start here if you're new to tensors. Covers vectors, covectors, tensor transformation laws, and index notation with excellent visual intuition.
Tensor Calculus
Christoffel symbols, covariant derivatives, parallel transport, geodesics, and curvature tensors. Essential mathematics for GR.
Relativity
From special relativity through Einstein's field equations. Covers the equivalence principle, stress-energy tensor, and Schwarzschild solution.
Carroll Textbook Solutions
Worked solutions to exercises from Sean Carroll's "Spacetime and Geometry: An Introduction to General Relativity".
General-Relativity.net
Comprehensive solutions organized by chapter, covering topics from tensor algebra through gravitational waves.
- • Chapter 1: Special Relativity
- • Chapter 2: Manifolds
- • Chapter 3: Curvature
- • Chapter 4: Gravitation
- • And more...
Carroll's Lecture Notes
Sean Carroll's original lecture notes from MIT's 8.962 course - the foundation for his textbook. Free PDF.
- • 238 pages of comprehensive notes
- • Worked examples throughout
- • "No-Nonsense Intro" (24 pages) also available
Problem Book in Relativity
Classic collection by Lightman, Press, Price, and Teukolsky with fully worked solutions.
- • 500+ problems with full solutions
- • Princeton University Press
- • Covers all areas of GR
UC Santa Cruz Problem Sets
Complete homework sets with solutions from Physics 226, using Carroll's textbook.
- • Problem sets 1-9
- • Full solution sets provided
- • Graduate level
Key Problem Topics
Differential Geometry
- • Metric tensor calculations
- • Christoffel symbol computation
- • Riemann tensor components
- • Geodesic equations
Einstein's Equations
- • Stress-energy tensor
- • Weak field approximation
- • Newtonian limit recovery
- • Linearized gravity
Exact Solutions
- • Schwarzschild metric
- • Kerr black holes
- • FLRW cosmology
- • Gravitational waves
Essential Formulas
Einstein Field Equations
$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$
Relates spacetime curvature to matter/energy content
Christoffel Symbols
$\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu})$
Connection coefficients from the metric
Riemann 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 intrinsic curvature of spacetime
Geodesic Equation
$\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$
Equation of motion for free particles