This is a table of contents for my notes on relativity. The posts are arranged in an order that might be found in a textbook.

A note on the index notation used in posts on relativity. In many of my posts, I was lazy and used Roman rather than Greek letters for tensor indices, as they are easier to type in Latex. The convention in most textbooks on relativity is to use Greek indices for the four components of spacetime, and Roman indices when referring only to spatial coordinates. Apologies for this, but it would be far too much work to go through all the posts and convert everything to Greek letters. In most cases, there shouldn't be any confusion, as in almost all posts, we're talking about the four dimensions of spacetime.

The conventions for distinguishing between 4-vectors (with four spacetime components) and 3-vectors (in regular 3-d space) vary from one book to another, and I usually use the convention that was used in the book that was my primary source for a post. Moore represents 4-vectors by boldface font, which is usually reserved for 3-vectors in other texts. Schutz uses ordinary font for 4-vectors and a little arrow over the letter for 3-vectors. Always check the conventions for whatever book you're using.

- Measuring the speed of light using Jupiter's moons
- Postulates of special relativity
- Relativistic units
- Invariance of spacetime intervals
- Spacetime diagrams
- Spacetime diagrams - two observers
- Spacetime diagrams - events and world lines
- Spacetime diagrams - particle detector
- Penrose diagrams in flat spacetime
- Invariant hyperbolas
- Time dilation and proper time
- The light clock and time dilation
- Length contraction
- No contraction in directions perpendicular to the motion
- Twin paradox
- Length contraction and time dilation - a few examples
- Pole in a barn paradox
- Doppler effect
- Doppler effect and four-momentum
- Lorentz transformations
- Linearity of Lorentz transformations
- Lorentz transformations - derivation from symmetry
- Lorentz transformations and causality
- Lorentz group
- Generators of the Lorentz group for 4-vectors
- Four-velocity
- Four-velocity - an example
- Four-velocity - another example
- Four-acceleration
- Hyperbolic motion and acceleration
- Force in relativity
- Relativistic acceleration in terms of force
- Outrunning a light ray
- Acceleration under a constant force
- Relativistic energy revisited
- Composition of velocities in relativity
- Momentum and energy
- Relation between energy and velocity of observer
- Four-momentum conservation
- Compton scattering
- A free electron cannot absorb a photon
- Four-momentum conservation - a trip to alpha centauri
- Another trip to Alpha Centauri - more Lorentz transformation examples
- Four-momentum of photons
- Four-momentum conservation in electron-positron annihilation
- Electron-positron collision must produce at least 2 photons
- Four-momentum conservation in the electron-electron collision
- Four-momentum conservation in the electron-photon collision
- A flawed theory of gravity
- Lorentz invariance of electric charge

- Four-vectors - basics
- Four-vectors - summation convention
- Manifolds, curves and surfaces
- Coordinate transformations - the Jacobian determinant
- Tensors and one-forms
- One-form basis
- Tensor index notation
- Components of one-forms and vectors
- Vectors in polar coordinates
- One-forms in polar coordinates
- Contravariant tensors
- Covariant and mixed tensors
- Gradient as a one-form
- Tensor equations are valid in all coordinates
- Gradient as covector - example in 2-d
- Normal vectors and unit normal one-forms
- Noncoordinate bases
- Projection operator in spacetime
- Tensor trace
- Tensor product - numerical example
- Symmetric and anti-symmetric tensors
- Inertia tensor

- Metric tensor and basis vectors
- Orthonormal and coordinate bases
- Area and volume in spacetime
- Vectors and the metric tensor
- Metric tensor for surface of a sphere
- Modified spherical coordinates
- Metric tensor under Lorentz transformation
- Metric tensor - inverse and raising and lowering indices
- Trace of metric tensor
- Metric tensor in semi-log coordinates
- Metric tensor in sinusoidal coordinates
- Metric tensor in spherical coordinates
- Metric tensor in parabolic coordinates
- Spherical metric - distance in 2-d curved space
- Local flatness theorem
- Riemann normal coordinates
- Diagonalizing the metric
- Covariant derivative of the metric tensor
- Covariant derivative of the metric tensor - application to a coordinate transformation
- Eskimo mites and their metric
- Mercator map metric
- Boyer-Lindquist coordinates and curvature of space
- Curvature of 2-dimensional space
- Curvature of a sphere
- Volume element in terms of metric determinant
- Unit spheres in higher dimensions
- Area of unit spheres in higher dimensions
- Squashed sphere
- Stereographic projection of the sphere
- Conformally flat spaces
- Divergence in curved space
- Torus metric
- Killing vectors
- Conserved quantities from Killing vectors

- Electromagnetic field tensor - justification
- Electromagnetic field tensor - conservation of mass
- Electromagnetic field tensor - change in kinetic energy
- Electromagnetic field tensor - contractions with metric tensor
- Electromagnetic field tensor - invariance under Lorentz transformations
- Electromagnetic field tensor - a couple of Maxwell's equations
- Electromagnetic field tensor - Lorentz transformations
- Electromagnetic field tensor - invariance of inner product
- Electromagnetic field tensor - four-potential
- Electromagnetic field tensor - cyclic derivative relation
- Electromagnetic stress-energy tensor
- Maxwell's equations in cylindrical coordinates

- Transforming derivatives of four-vectors and scalars
- Derivatives of a vector field in polar coordinates
- Divergence of vector field in polar coordinates
- Derivatives of a one-form field in polar coordinates
- Christoffel symbols and the covariant derivative
- Christoffel symbols defined for a sphere
- Christoffel symbols in terms of the metric tensor
- Christoffel symbols for wormhole metric
- Christoffel symbols in noncoordinate bases
- Covariant derivative of a general tensor
- Covariant derivative is a tensor
- Christoffel symbols - symmetry
- Higher order derivatives are not tensors
- Globally parallel vector field
- Covariant derivative in semi-log coordinates
- Christoffel symbols in sinusoidal coordinates
- Covariant derivative - commutativity

- Geodesics - paths of longest proper time
- Geodesic equation - geodesics on a sphere
- Parallel transport and the geodesic equation
- Parallel transport around a spherical triangle
- Parallel transport and the scalar product
- Geodesic equation - polar coordinates
- Geodesic equation and four-velocity
- Geodesic equation - 2-d space-time
- Geodesic equation in 2-d with exponential metric
- Geodesic equation on a paraboloid
- Hyperbolic plane

- Schwarzschild metric - radial coordinate is circumferential
- Thickness of spherical shell derived from areas
- Schwarzschild radius
- Schwarzschild metric - time coordinate
- Schwarzschild metric - gravitational red shift
- Gravitational redshift from the Killing vector
- Schwarzschild metric - redshift of Sirius B
- Schwarzschild metric - four-momentum of a photon
- Christoffel symbols for Schwarzschild metric
- Schwarzschild metric - acceleration
- Covariant derivative of a vector in the Schwarzschild metric
- Schwarzschild metric equivalent to weak field solution for spherical object

- Particle orbits - conserved quantities
- Circular orbits - Kepler's law
- Circular orbits - relation between radius and angular momentum
- Circular orbits - Schwarzschild vs Newton
- Particles falling towards a mass
- Particle falling towards a mass - two types of velocity
- Vertical particle motion
- Circular orbit around a supermassive black hole
- Circular orbits - 3 measurements of the period
- Photon orbits - speed measured at two places
- Orbit of a comet around a black hole
- Twin paradox with a black hole

- Perihelion shift in planetary orbits
- Perihelion shift - contribution from the radial coordinate
- Perihelion shift - a couple of examples
- Perihelion shift - contribution of the time coordinate
- Perihelion shift - numerical solution

- Embedding a two-dimensional surface in three-dimensional space
- Embedding 2-d curved space in 3-d - the sphere
- Embedding a 2-d curved surface in 3-d - the cosh
- Embedding a 2-d curved surface in 3-d - the cosine
- Embedding a 2-d curved surface into 3-d - inverse cosh
- Wormhole metric

- Photon equations of motion
- Local flat coordinate systems - four-momentum of photons
- Apparent size of a black hole
- Apparent size of a black hole to a moving observer
- Red-shifts and blue-shifts
- Photon path in flat space
- Circular orbit - appearance to a falling observer
- Local flat frame for a circular orbit

- Deflection of light by a mass
- Deflection of light by the sun
- Gravitational lensing and the Einstein ring
- Gravitational lensing - image brightness
- Gravitational lensing - the twin quasar
- Gravitational lensing - large angles
- MACHOs and seeing distant objects with a gravitational lens
- Delay of light passing a mass - Shapiro delay
- Shapiro delay - the twin quasar
- The sun as a gravitational lens

- Distance from external radius up to the event horizon
- Proper time to fall through the event horizon
- Time and space interchange as we cross the event horizon
- Falling into a black hole
- Falling into a black hole - tidal forces
- Falling object observed near the event horizon
- Light cones near the event horizon
- Escape velocity near an event horizon
- Hyperbolic coordinates in flat space
- Are black holes really black?

- Painlevé-Gullstrand (global rain) coordinates
- Painlevé-Gullstrand metric - photon paths inside the event horizon
- Painlevé-Gullstrand coordinates - derivation using a local flat frame
- Kruskal-Szekeres coordinates and the event horizon
- Kruskal-Szekeles metric - what can you see as you fall into a black hole?
- Kruskal-Szekeres metric - more fun with photons
- Kruskal-Szekeres diagrams - saving a space shuttle
- Kruskal-Szekeres diagrams - another space ship disaster

- Black hole radiation - energy of emitted particles
- Black hole radiation - energy at infinity of radiated particle
- Black hole evaporation - how long will a black hole live?
- Black hole radiation - energy of a particle from a solar mass black hole
- Black hole evaporation - remnants of the big bang
- Black hole radiation - mass as a function of time
- Black holes and the Large Hadron Collider
- Black hole heat engine
- Black hole temperatures at different distances
- Black hole entropy
- Black hole entropy - it's pretty large
- Black hole in equilibrium with a thermal reservoir

- Newtonian tidal effect
- Tidal effect for objects in freefall near the Earth's surface
- Flat space next to an infinite plane of mass
- Riemann tensor - derivation
- Riemann tensor from parallel transport
- Riemann tensor for an infinite plane of mass
- Riemann tensor - symmetries
- Riemann tensor - counting independent components
- Riemann tensor - counting components in general
- Riemann tensor in 2-d polar coordinates
- Riemann tensor in 2-d curved space
- Riemann tensor in 2-d flat space
- Riemann tensor in an exponential 2-d curved space
- Riemann tensor for 3-d spherical coordinates
- Riemann tensor for surface of a sphere
- Riemann tensor in the Schwarzschild metric
- Riemann tensor in the Schwarzschild metric - observer's view
- Geodesic deviation in a locally inertial frame
- Bianchi identity for the Riemann tensor
- Ricci tensor and curvature scalar
- Ricci tensor and curvature scalar for a sphere
- Riemann and Ricci tensors in the weak field limit

- Energy (not mass) is the source of gravity
- Stress-energy tensor for dust
- Stress-energy tensor for a perfect fluid at rest
- Stress-energy tensor for perfect fluid - general coordinates
- Stress-energy tensor - conservation equations
- Stress-energy tensor - symmetry
- Stress-energy tensor at the centre of the sun
- Stress-energy tensor of a slowly rotating star
- Stress-energy tensor - relativistic perfect fluid
- Stress-energy tensor for a photon gas
- Force in terms of the stress-energy tensor
- Metric tensor as a stress-energy tensor
- Stress-energy tensor - negative pressure revisited
- Stress-energy tensor in a local orthonormal frame
- Dominant energy condition
- Conservation of four-momentum implies the geodesic equation
- Stress-energy tensor in the weak field limit

- Einstein equation - trying the Ricci tensor as a solution
- Einstein tensor and Einstein equation
- Einstein equation - alternative form
- Einstein equation in the Newtonian limit
- Einstein tensor of zero implies a zero Ricci tensor
- Einstein equation for a perfect fluid
- Gravity can't exist in 2 spacetime dimensions
- Gravity can't exist in 3 spacetime dimensions either
- Einstein equation on the surface of a sphere
- Einstein equation for an exponential metric
- Vacuum stress-energy and the cosmological constant
- Gravitoelectric and gravitomagnetic densities
- Gravitoelectric and gravitomagnetic densities for the vacuum
- Gravitomagnetic acceleration is perpendicular to velocity
- Wave solution of the weak-field Einstein equation
- Gravitomagnetic acceleration near a rotating star
- Gravitoelectric and gravitomagnetic acceleration for a moving wire
- Gravitoelectric and gravitomagnetic acceleration for parallel plates

- Spherically symmetric solution to the Einstein equation
- Ricci tensor for a spherically symmetric metric - the worksheet
- Schwarzschild metric - finding the metric; Birkhoff's theorem
- Schwarzschild metric - the Newtonian limit & Christoffel symbol worksheet
- Plane symmetric spacetime
- Schwarzschild metric with negative mass
- Schwarzschild metric with non-zero cosmological constant
- Black hole with static charge; Reissner-Nordström solution
- Einstein equation solution for the interior of a spherically symmetric star
- Cosmic strings