General Relativity Notes

My notes on general relativity.

My aim is for them to be in a story-telling tone, and to be more on the intuitive-understanding side of things.

The content is largely based on Profesor Leonard Susskind's special relativity and general relativity lectures in his amazing series The Theoretical Minimum.

The content is also based on Sean Caroll's classic text on relativity Spacetime and Geometry.

The notes directory contains all the notes. The solutions directory contains the solutions to the exercises that appear in the notes.

Below is the list of topics and their structure.

0. Prelude

• 0.0 Prelude
• 0.5 The chain rule

1. Motivation

• 1.1 Motivation : losing track of time
• 1.2 Motivation : frame-invariant equations, and the Equivalence Principle
• 1.99 Motivation summary

2. Geometry 1: straight lines (and a little bit of curvature)

• 2.1 What is a vector; contravariant and covariant vectors
• 2.2 What is a tensor; the metric tensor
  • 2.2.1 An interlude on tensor products
• 2.3 Everything is locally flat
• 2.4 The covariant derivative

The following is still in construction. The outline can change based on how it will actually go.

• 2.5 The geodesic equation
  • 2.5.1 A slight interlude regarding parametrization: affine parametrization
  • 2.5.2 The geodesic equation only works for an affine parametrization
• 2.6 The Riemann curvature tensor
• 2.99 Flashcard on important geometry equations

3. Special relativity 1: particle kinematics (so four-vectors/velocities, Lorentz transforms, Minkowski metric)

• 3.1 The Lorentz Transform; The Minkowski Metric
  • 3.1.1 Quick results from the Lorentz Transform: time dilation, length contraction, velocity addition, the Doppler Effect
  • 3.1.2 Quick results regarding 4-velocities
• 3.2 Using the geodesic equation in Minkowski spacetime
• 3.3 The Equivalence Principle revisited: using the geodesic equation to derive the fictitious gravity in an accelerating frame

4. Geometry 2: straight lines again

• 4.1 Of course, a straight line is the shortest path between two points: the Euler-Lagrange equation
• 4.2 The free particle Lagrangian in Minkowski Spacetime; why it has a minus sign: the "Twin Paradox"
• 4.3 Applying the Euler-Lagrange equation to the Minkowski free particle Lagrangian; Lagrangians don't require an affine parametrization!
• 4.9 The four principles of Laws Of Physics
• 4.99 Lagrangian of a particle in a scalar field in Minkowski spacetime, and its Newtonian limit ("KE-PE")

5. Lagrangian tricks

Almost all courses in advanced classical mechanics start with telling you "there's this thing called the Euler-Lagrange equation, and the Lagrangian for a system is KE-PE". From there on we can apply this method to most classical systems. I will explicitly do a couple of examples just for illustrating the power of Lagrangian mechanics, but they are not necessary for general relativity. Skip it if you wish.

• 5.0 Examples of classical Lagrangian: Newton's second law, harmonic oscillator, inclined wedge sliding on a plane, the double pendulum

From here on it's necessary.

• 5.1 Symmetry and conserved quantities
• 5.2 Conservation of energy: the Hamiltonian

6. Special relativity 2: particle dynamics (so four-momentum, energy)

7. Lagrangian for fields

• 7.1 The Euler-Lagrange equation for fields
• 7.2 The free field Lagrangian in Minkowski spacetime; the wave equation
• 7.3 Particle-field interactions: the Poisson equation

8. Special relativity 3: electromagnetism

• 8.1 The magnetic vector potential and gauge invariance
• 8.2 The Lorentz Force Law with a classical Lagrangian
• 8.3 The Lorentz Force Law with a relativistic Lagrangian
• 8.4 What is A0? The electromagnetic field strength tensor
• 8.5 The electromagnetic source 4-vector
• 8.6 Maxwell's equations

9. Lagrangian for fields: field momentum and field energy

10. The Schwarzschild metric (so black holes)

• 10.0 A reminder about the equivalence principle
• 10.1 Postulating the Schwarzschild metric
• 10.2 Watching your friend falling into a black hole
• 10.3 The photon sphere
• 10.4 Kruskal coordinates
• 10.5 Penrose diagrams

11. The Einstein field equations

• 11.0 A reminder about the Riemann curvature tensor
• 11.1 The Einstein field equation
• 11.2 The Einstein field equation from the Einstein-Hilbert action
• 11.99 Solving the Einstein field equations for the Schwarzschild metric

Computational Files

A list of computational files are available to compute stuff, or to explicitly illustrate some of the phenomena we encounter.

• connection.py: a python script calculating the connection coefficients from a given metric
• stuck_on_horizon.py: a python script showing that it takes an infinite amount of coordinate time (or lab time) for an object to reach the horizon