- Acceleration in special relativity
- Affine geodesic and affine parameter
- Affine parameter transformation
- Apparent size of a black hole
- Apparent size of a black hole to a moving observer
- Apparent speeds greater than the speed of light
- Average of a vector over all directions
- Bianchi identity for the Riemann tensor
- Black holes: are they really black?
- Black hole entropy
- Black hole entropy: it’s pretty large
- Black hole evaporation: how long will a black hole live?
- Black hole evaporation: remnants of the big bang
- Black hole heat engine
- Black holes and the Large Hadron Collider
- Black hole with static charge; Reissner-Nordström solution
- Black hole radiation: energy of emitted particles
- Black hole radiation: energy at infinity of radiated particle
- Black hole radiation: energy of a particle from a solar mass black hole
- Black hole radiation: mass as a function of time
- Black hole temperatures at different distances
- Black hole in equilibrium with a thermal reservoir
- Centre of momentum frame
- Christoffel symbols
- Christoffel symbols: symmetry
- Christoffel symbols in terms of the metric tensor
- Christoffel symbols for a general diagonal metric
- Christoffel symbols for Schwarzschild metric
- Christoffel symbols in sinusoidal coordinates
- Circular orbit: appearance to a falling observer
- Circular orbits: Kepler’s law
- Circular orbits: relation between radius and angular momentum
- Circular orbits: 3 measurements of the period
- Circular orbit around a supermassive black hole
- Circular orbit: Schwarzschild vs Newton
- Collision of two identical particles
- Collision of a pion and a proton
- Composition of velocities in relativity
- Compound Lorentz transformations
- Compton scattering
- Compton scattering by cosmic rays
- Congruence of curves
- Conservation of four-momentum implies the geodesic equation
- Conservation of momentum
- Contravariant tensors
- Contravariant gradient operator
- Contravariant and covariant components: dual basis vectors
- Coordinate transformations – the Jacobian determinant
- Cosmic strings
- Covariant derivative and connections
- Covariant derivative: commutativity
- Covariant derivative of covariant vector
- Covariant derivative of higher rank tensors
- Covariant derivative in semi-log coordinates
- Covariant derivative of a general tensor
- Covariant derivative of the metric tensor
- Covariant derivative of the metric tensor: application to a coordinate transformation
- Covariant derivative of a vector in the Schwarzschild metric
- Covariant and mixed tensors
- Decay of a pion into a muon and a neutrino
- Decay of a pion into two photons
- Decomposition of a rank 2 tensor
- Deflection of light by a mass
- Deflection of light by the sun
- Deflection of light in Newtonian gravity
- Delay of light passing a mass: Shapiro delay
- Dominant energy condition
- Doppler effect and four-momentum
- Doppler shift
- Einstein equation: trying the Ricci tensor as a solution
- Einstein equation: alternative form
- Einstein equation for an exponential metric
- Einstein equation for a perfect fluid
- Einstein equation in the Newtonian limit
- Einstein equation solution for the interior of a spherically symmetric star
- Einstein equation on the surface of a sphere
- Einstein tensor and Einstein equation
- Einstein tensor of zero implies a zero Ricci tensor
- Elastic collision of two identical particles
- Electromagnetic field tensor: change in kinetic energy
- Electromagnetic field tensor: conservation of mass
- Electromagnetic field tensor: contractions with metric tensor
- Electromagnetic field tensor: a couple of Maxwell’s equations
- Electromagnetic field tensor: cyclic derivative relation
- Electromagnetic field tensor: four-potential
- Electromagnetic field tensor: Lorentz transformations
- Electromagnetic field tensor: invariance under Lorentz transformations
- Electromagnetic field tensor: invariance of inner product
- Electromagnetic field tensor: justification
- Electromagnetic stress-energy tensor
- Electron-positron collision must produce 2 photons
- Embedding a 2-d curved surface in 3-d: the cosh
- Embedding a 2-d curved surface into 3-d: inverse cosh
- Embedding a 2-d curved surface in 3-d: the cosine
- Embedding 2-d curved space in 3-d: the sphere
- Energy (not mass) is the source of gravity
- Escape velocity near an event horizon
- Event horizon: distance from external radius up to r = 2GM
- Event horizon: proper time to fall into r = 0
- Event horizon: time and space swap round
- Falling into a black hole
- Falling into a black hole: tidal forces
- Falling object observed near the event horizon
- Flat space next to an infinite plane of mass
- A flawed theory of gravity
- Force in relativity
- Force in terms of the stress-energy tensor
- Four-acceleration
- Four-acceleration and Minkowski force
- Four-momentum conservation
- Four-momentum conservation: electron-electron collision
- Four-momentum conservation: electron-photon collision
- Four-momentum conservation in electron-positron annihilation
- Four-momentum conservation: a trip to Alpha Centauri
- Four-momentum: example
- Four-momentum of photons
- Four-vectors: basics
- Four-vectors: basis vectors
- Four-velocity again
- Four-velocity versus ordinary velocity: an example
- Four-velocity, momentum and energy
- Four-velocity: an example
- Four-velocity: another example
- Four-velocity of a particle in hyperbolic motion
- Four-velocity’s square is a universal constant
- Galilean relativity
- Geodesic deviation in a locally inertial frame
- Geodesics: paths of longest proper time
- Geodesics on a sphere: an ant searching for honey
- Geodesic equation in terms of Christoffel symbols
- Geodesic equation: geodesics on a sphere
- Geodesic equation: 2-d polar coordinates
- Geodesic equation: 2-d space-time
- Geodesic equation in 2-d: exponential metric
- Geodesic equation: paraboloid
- Geodesic equation and four-velocity
- Gradient as covector: example in 2-d
- Gravitational lensing
- Gravitational lensing: image brightness
- Gravitational lensing: large angles
- Gravitational lensing: the twin quasar
- Gravitoelectric and gravitomagnetic densities
- Gravitoelectric and gravitomagnetic densities for the vacuum
- Gravitoelectric and gravitomagnetic acceleration for a moving wire
- Gravitoelectric and gravitomagnetic acceleration for parallel plates
- Gravitomagnetic acceleration is perpendicular to velocity
- Gravitomagnetic acceleration near a rotating star
- Gravity can’t exist in 2 spacetime dimensions
- Gravity can’t exist in 3 spacetime dimensions either
- Gravity express train
- Higher order derivatives are not tensors
- Hyperbolic angles in relativity
- Hyperbolic coordinates in flat space
- Inertia tensor
- Inertial frames
- Invariance of scalar product under Lorentz transformations
- Invariant hyperbolas
- The invariant interval: some examples
- Kepler’s first law revisited
- Kronecker delta as a tensor
- Kruskal-Szekeres coordinates and the event horizon
- Kruskal-Szekeres diagrams: another space ship disaster
- Kruskal-Szekeres diagrams: saving a space shuttle
- Kruskal-Szekeles metric: what can you see as you fall into a black hole?
- Kruskal-Szekeres metric: more fun with photons
- Length contraction and the pole-in-a-barn paradox
- Lie brackets (commutators)
- Lie derivatives
- Lie derivative: higher-rank tensors
- Lie derivative: product rule
- Lie derivative in terms of the covariant derivative
- Lie’s method of generating rotations
- Lie rotations in higher dimensions
- Light cones near the event horizon
- Local flat coordinate systems; four-momentum of photons
- Local flat frame for a circular orbit
- Locally inertial frames (LIFs)
- Lorentz (length) contraction: a simple example
- Lorentz contraction: no contraction in directions perpendicular to the motion
- Lorentz contraction in a rotating disk: Ehrenfest’s and Bell’s spaceship paradoxes
- Lorentz transformations
- Lorentz transformations: forward and backward
- Lorentz transformations as a linear map
- Lorentz transformations: geometric derivation
- Lorentz transformations and simultaneity
- Lorentz transformation in two dimensions
- Lorentz transformations in three dimensions
- MACHOs and brown dwarf stars
- MACHOs and seeing distant objects with a gravitational lens
- Manifolds, curves and surfaces
- Manifolds, the Jacobian matrix and coordinate transformations
- Map projections: equal angles and equal areas
- Maxwell’s equations in cylindrical coordinates
- Metric tensor
- Metric tensor: conversion from spherical to cylindrical coordinates
- Metric tensor: inverse and raising & lowering indices
- Metric tensor under Lorentz transformation
- Metric for the Mercator projection
- Metric tensor for a non-orthogonal coordinate system
- Metric tensor: parabolic coordinates
- Metric tensor: semi-log coordinates
- Metric tensor: sinusoidal coordinates
- Metric tensor: spherical coordinates
- Metric tensor for the stereographic projection of a sphere onto a plane
- Metric tensor: trace
- Metric tensor as a stress-energy tensor
- Michelson-Morley experiment
- Michelson-Morley experiment: length contraction?
- Minkowski force
- Motion under a constant Minkowski force
- Newtonian tidal effect
- Orbit of a comet around a black hole
- Outrunning a light ray
- Painlevé-Gullstrand (global rain) coordinates
- Painlevé-Gullstrand metric: photon paths inside the event horizon
- Painlevé-Gullstrand coordinates: derivation using a local flat frame
- Pair annihilation with an electron and a positron
- Parallel transport of tensors
- Particle orbits – conserved quantities
- Particle falling towards a mass: two types of velocity
- Particle orbiting a neutron star
- Particles falling towards a mass
- Perihelion shift in planetary orbits
- Perihelion shift – a couple of examples
- Perihelion shift – contribution from the radial coordinate
- Perihelion shift – contribution of the time coordinate
- Perihelion shift: numerical solution
- Photon equations of motion
- Photon orbits: speed measured at two places
- Photon path in flat space
- Plane symmetric spacetime
- Postulates of special relativity
- Pulsar planets
- Rapidity
- Red-shifts and blue-shifts
- Relative energy in a particle collider
- Relativistic kinetic energy
- Relativistic momentum and energy
- Relativistic units: (c=1)
- Ricci tensor and curvature scalar
- Ricci tensor and curvature scalar for a sphere
- Ricci tensor for a spherically symmetric metric: the worksheet
- Riemann tensor: counting independent components
- Riemann tensor: counting components in general
- Riemann tensor: derivation
- Riemann tensor: symmetries
- Riemann tensor for an infinite plane of mass
- Riemann tensor in 2-d flat space
- Riemann tensor in 2-d polar coordinates
- Riemann tensor in 2-d curved space
- Riemann tensor in another 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
- Riemann tensor – commutator of rank 2 tensor
- Riemann tensor and covariant contraction
- Riemann and Ricci tensors in the weak field limit
- Schwarzschild metric: acceleration
- Schwarzschild metric: finding the metric; Birkhoff’s theorem
- Schwarzschild metric: four-momentum of a photon
- Schwarzschild metric: gravitational redshift
- Schwarzschild metric with negative mass
- Schwarzschild metric: the Newtonian limit & Christoffel symbol worksheet
- Schwarzschild metric with non-zero cosmological constant
- Schwarzschild metric: radial coordinate is circumferential
- Schwarzschild metric: redshift of Sirius B
- Schwarzschild metric: time coordinate
- Schwarzschild metric equivalent to weak field solution for spherical object
- Schwarzschild radius
- Shapiro delay: the twin quasar
- Simultaneity in special relativity
- Simultaneity; seeing versus observing
- Spacelike intervals
- Space-time diagrams
- Spacetime diagrams: an example
- Space-time diagrams: two observers
- Space-time intervals
- Space-time intervals: invariance
- Special relativity and the wave equation
- Special relativity with the k-calculus
- Spherical metric: distance in 2-d curved space
- Spherically symmetric solution to the Einstein equation
- Stress-energy tensor
- Stress-energy tensor: conservation equations
- Stress-energy tensor in a local orthonormal frame
- Stress-energy tensor for a perfect fluid at rest
- Stress-energy tensor for perfect fluid: general coordinates
- Stress-energy tensor: relativistic perfect fluid
- Stress-energy tensor: negative pressure revisited
- Stress-energy tensor for a photon gas
- Stress-energy tensor at the centre of the sun
- Stress-energy tensor of a slowly rotating star
- Stress-energy tensor in the weak field, low velocity limit
- Summation convention
- The sun as a gravitational lens
- Tangent space
- Tangent space: partial derivatives as basis vectors
- Tensor arithmetic
- Tensor index notation
- Tensor indices: Newton’s law
- Tensor product: numerical example
- Tensors: symmetric and anti-symmetric
- Tensor trace
- Tensors: one-forms
- Tensors transform like tensors
- Tidal effect for objects in freefall near the Earth’s surface
- Time dilation
- Time dilation: resolving the paradox
- Time dilation: a rocket passing Earth
- Transforming derivatives of four-vectors and scalars
- The twin paradox analyzed using Lorentz transformations
- Twin paradox with a black hole
- Vacuum stress-energy and the cosmological constant
- Vectors and the metric tensor
- Vectors and rotations
- Velocity addition in relativity
- Velocity addition in special relativity revisited
- Velocity addition: chasing space pirates
- Velocity addition: chasing space pirates viewed in four reference frames
- Velocity addition formulas for all 3 directions
- Vertical particle motion
- Wave solution of the weak-field Einstein equation