- Acceleration in special relativity
- Affine geodesic and affine parameter
- Affine parameter transformation
- 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
- Composition of velocities in relativity
- Compton scattering
- Compton scattering by cosmic rays
- Congruence of curves
- Contravariant tensors
- Contravariant and covalent components: dual basis vectors
- Coordinate transformations – the Jacobian determinant
- Covariant derivative and connections
- Covariant derivative of covariant vector
- Covariant derivative of higher rank tensors
- Covariant and mixed tensors
- Doppler effect and four-momentum
- Doppler shift
- 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
- A flawed theory of gravity
- Four-acceleration
- 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, momentum and energy
- Four-velocity: an example
- Four-velocity: another example
- Galilean relativity
- Geodesics: paths of longest proper time
- 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
- Higher order derivatives are not tensors
- Hyperbolic angles in relativity
- Inertia tensor
- Invariant hyperbolas
- Kronecker delta as a tensor
- 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
- Lorentz transformations
- Lorentz transformations as a linear map
- Lorentz transformations: geometric derivation
- Lorentz transformations in three dimensions
- Manifolds, curves and surfaces
- Metric tensor
- Metric tensor: inverse and raising & lowering indices
- Metric tensor under Lorentz transformation
- Metric tensor: parabolic coordinates
- Metric tensor: semi-log coordinates
- Metric tensor: sinusoidal coordinates
- Metric tensor: spherical coordinates
- Metric tensor: trace
- Michelson-Morley experiment
- Michelson-Morley experiment: length contraction?
- Orbit of a comet around a black hole
- 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
- Photon orbits: speed measured at two places
- Postulates of special relativity
- Relativistic units: (c=1)
- Riemann tensor – commutator of rank 2 tensor
- Riemann tensor and covariant contraction
- Schwarzschild metric: four-momentum of a photon
- Schwarzschild metric: gravitational redshift
- Schwarzschild metric: radial coordinate
- Schwarzschild metric: redshift of Sirius B
- Schwarzschild metric: time coordinate
- Schwarzschild radius
- Simultaneity in special relativity
- Space-time diagrams
- 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
- Summation convention
- 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
- Time dilation
- Transforming derivatives of four-vectors and scalars
- Twin paradox with a black hole
- Vectors and the metric tensor
- Velocity addition in relativity
- Vertical particle motion
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