 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
 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 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
 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
 Composition of velocities in relativity
 Compound Lorentz transformations
 Compton scattering
 Compton scattering by cosmic rays
 Congruence of curves
 Conservation of fourmomentum implies the geodesic equation
 Contravariant tensors
 Contravariant and covariant components: dual basis vectors
 Coordinate transformations – the Jacobian determinant
 Covariant derivative and connections
 Covariant derivative: commutativity
 Covariant derivative of covariant vector
 Covariant derivative of higher rank tensors
 Covariant derivative in semilog 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
 Deflection of light by a mass
 Deflection of light by the sun
 Delay of light passing a mass: Shapiro delay
 Dominant energy condition
 Doppler effect and fourmomentum
 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 on the surface of a sphere
 Einstein tensor and Einstein equation
 Einstein tensor of zero implies a zero Ricci tensor
 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: fourpotential
 Electromagnetic field tensor: Lorentz transformations
 Electromagnetic field tensor: invariance under Lorentz transformations
 Electromagnetic field tensor: invariance of inner product
 Electromagnetic field tensor: justification
 Electromagnetic stressenergy tensor
 Electronpositron collision must produce 2 photons
 Embedding a 2d curved surface in 3d: the cosh
 Embedding a 2d curved surface into 3d: inverse cosh
 Embedding a 2d curved surface in 3d: the cosine
 Embedding 2d curved space in 3d: 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 terms of the stressenergy tensor
 Fouracceleration
 Fourmomentum conservation
 Fourmomentum conservation: electronelectron collision
 Fourmomentum conservation: electronphoton collision
 Fourmomentum conservation in electronpositron annihilation
 Fourmomentum conservation: a trip to Alpha Centauri
 Fourmomentum: example
 Fourmomentum of photons
 Fourvectors: basics
 Fourvectors: basis vectors
 Fourvelocity again
 Fourvelocity versus ordinary velocity: an example
 Fourvelocity, momentum and energy
 Fourvelocity: an example
 Fourvelocity: another example
 Fourvelocity of a particle in hyperbolic motion
 Fourvelocity’s square is a universal constant
 Galilean relativity
 Geodesic deviation in a locally inertial frame
 Geodesics: paths of longest proper time
 Geodesic equation in terms of Christoffel symbols
 Geodesic equation: geodesics on a sphere
 Geodesic equation: 2d polar coordinates
 Geodesic equation: 2d spacetime
 Geodesic equation in 2d: exponential metric
 Geodesic equation: paraboloid
 Geodesic equation and fourvelocity
 Gradient as covector: example in 2d
 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
 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
 Kronecker delta as a tensor
 KruskalSzekeres coordinates and the event horizon
 KruskalSzekeres diagrams: another space ship disaster
 KruskalSzekeres diagrams: saving a space shuttle
 KruskalSzekeles metric: what can you see as you fall into a black hole?
 KruskalSzekeres metric: more fun with photons
 Length contraction and the poleinabarn paradox
 Lie brackets (commutators)
 Lie derivatives
 Lie derivative: higherrank tensors
 Lie derivative: product rule
 Lie derivative in terms of the covariant derivative
 Light cones near the event horizon
 Local flat coordinate systems; fourmomentum 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 transformations in three dimensions
 MACHOs and brown dwarf stars
 MACHOs and seeing distant objects with a gravitational lens
 Manifolds, curves and surfaces
 Maxwell’s equations in cylindrical coordinates
 Metric tensor
 Metric tensor: inverse and raising & lowering indices
 Metric tensor under Lorentz transformation
 Metric tensor: parabolic coordinates
 Metric tensor: semilog coordinates
 Metric tensor: sinusoidal coordinates
 Metric tensor: spherical coordinates
 Metric tensor: trace
 Metric tensor as a stressenergy tensor
 MichelsonMorley experiment
 MichelsonMorley experiment: length contraction?
 Newtonian tidal effect
 Orbit of a comet around a black hole
 PainlevéGullstrand (global rain) coordinates
 PainlevéGullstrand metric: photon paths inside the event horizon
 PainlevéGullstrand coordinates: derivation using a local flat frame
 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
 Postulates of special relativity
 Pulsar planets
 Rapidity
 Redshifts and blueshifts
 Relativistic kinetic energy
 Relativistic momentum and energy
 Relativistic units: (c=1)
 Ricci tensor and curvature scalar
 Ricci tensor and curvature scalar for a sphere
 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 2d flat space
 Riemann tensor in 2d polar coordinates
 Riemann tensor in 2d curved space
 Riemann tensor in another 2d curved space
 Riemann tensor for 3d 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: fourmomentum of a photon
 Schwarzschild metric: gravitational redshift
 Schwarzschild metric: radial coordinate
 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
 Spacetime diagrams
 Spacetime diagrams: an example
 Spacetime diagrams: two observers
 Spacetime intervals
 Spacetime intervals: invariance
 Special relativity and the wave equation
 Special relativity with the kcalculus
 Spherical metric: distance in 2d curved space
 Stressenergy tensor
 Stressenergy tensor: conservation equations
 Stressenergy tensor in a local orthonormal frame
 Stressenergy tensor for a perfect fluid at rest
 Stressenergy tensor for perfect fluid: general coordinates
 Stressenergy tensor: relativistic perfect fluid
 Stressenergy tensor: negative pressure revisited
 Stressenergy tensor for a photon gas
 Stressenergy tensor at the centre of the sun
 Stressenergy tensor of a slowly rotating star
 Stressenergy 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 antisymmetric
 Tensor trace
 Tensors: oneforms
 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 fourvectors and scalars
 The twin paradox analyzed using Lorentz transformations
 Twin paradox with a black hole
 Vacuum stressenergy and the cosmological constant
 Vectors and the metric tensor
 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 weakfield Einstein equation

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