Index - Relativity

• A flawed theory of gravity
• 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
• 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 hole in equilibrium with a thermal reservoir
• 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: energy of emitted particles
• Black hole radiation: mass as a function of time
• Black hole temperatures at different distances
• Black hole with static charge; Reissner-Nordström solution
• Black holes and the Large Hadron Collider
• Black holes: are they really black?
• Centre of momentum frame
• Christoffel symbols
• Christoffel symbols for Schwarzschild metric
• Christoffel symbols in sinusoidal coordinates
• Christoffel symbols in terms of the metric tensor
• Christoffel symbols: symmetry
• Circular orbit around a supermassive black hole
• Circular orbit: appearance to a falling observer
• Circular orbit: Schwarzschild vs Newton
• Circular orbits: 3 measurements of the period
• Circular orbits: Kepler's law
• Circular orbits: relation between radius and angular momentum
• Collision of a pion and a proton
• Collision of two identical particles
• Composition of velocities in relativity
• Composition of velocities in relativity
• Compound Lorentz transformations
• Compton scattering
• Congruence of curves
• Conservation of four-momentum implies the geodesic equation
• Contravariant gradient operator
• Contravariant tensors
• Coordinate transformations - the Jacobian determinant
• Cosmic strings
• Covariant and mixed tensors
• Covariant derivative and connections
• Covariant derivative in semi-log coordinates
• Covariant derivative of a general tensor
• Covariant derivative of a vector in the Schwarzschild metric
• Covariant derivative of covariant vector
• Covariant derivative of higher rank tensors
• Covariant derivative of the metric tensor
• Covariant derivative of the metric tensor: application to a coordinate transformation
• Covariant derivative: commutativity
• Decay of a pion into a muon and a neutrino
• Decay of a pion into two photons
• 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
• Doppler effect and four-momentum
• Einstein equation for a perfect fluid
• Einstein equation for an exponential metric
• Einstein equation in the Newtonian limit
• Einstein equation on the surface of a sphere
• Einstein equation solution for the interior of a spherically symmetric star
• Einstein equation: alternative form
• Einstein equation: trying the Ricci tensor as a solution
• Einstein tensor and Einstein equation
• Einstein tensor of zero implies a zero Ricci tensor
• Elastic collision of two identical particles
• Electromagnetic field tensor: a couple of examples
• Electromagnetic field tensor: a couple of Maxwell's equations
• Electromagnetic field tensor: change in kinetic energy
• Electromagnetic field tensor: conservation of mass
• Electromagnetic field tensor: contractions with metric tensor
• Electromagnetic field tensor: cyclic derivative relation
• Electromagnetic field tensor: four-potential
• Electromagnetic field tensor: invariance of inner product
• Electromagnetic field tensor: invariance under Lorentz transformations
• Electromagnetic field tensor: justification
• Electromagnetic field tensor: Lorentz transformations
• Electromagnetic field tensor: Lorentz transformations
• Electromagnetic Minkowski force
• Electromagnetic stress-energy tensor
• Electron-positron collision must produce at least 2 photons
• 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
• 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
• Force in relativity
• Force in terms of the stress-energy tensor
• Four-acceleration
• Four-acceleration and Minkowski force
• Four-momentum conservation
• Four-momentum conservation in electron-positron annihilation
• Four-momentum conservation: a trip to Alpha Centauri
• Four-momentum conservation: electron-electron collision
• Four-momentum conservation: electron-photon collision
• Four-momentum of photons
• Four-momentum: example
• Four-vectors: basics
• Four-velocity again
• Four-velocity of a particle in hyperbolic motion
• Four-velocity versus ordinary velocity: an example
• Four-velocity, momentum and energy
• Four-velocity: an example
• Four-velocity: another example
• Four-velocity's square is a universal constant
• Gauss's law for a relativistic point charge
• Geodesic deviation in a locally inertial frame
• Geodesic equation and four-velocity
• Geodesic equation in 2-d: exponential metric
• Geodesic equation in terms of Christoffel symbols
• Geodesic equation: 2-d polar coordinates
• Geodesic equation: 2-d space-time
• Geodesic equation: geodesics on a sphere
• Geodesic equation: paraboloid
• Geodesics: paths of longest proper time
• 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 acceleration for a moving wire
• Gravitoelectric and gravitomagnetic acceleration for parallel plates
• Gravitoelectric and gravitomagnetic densities
• Gravitoelectric and gravitomagnetic densities for the vacuum
• 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
• Kronecker delta as a tensor
• Kruskal-Szekeles metric: what can you see as you fall into a black hole?
• Kruskal-Szekeres coordinates and the event horizon
• Kruskal-Szekeres diagrams: another space ship disaster
• Kruskal-Szekeres diagrams: saving a space shuttle
• Kruskal-Szekeres metric: more fun with photons
• Lie brackets (commutators)
• Lie derivative in terms of the covariant derivative
• Lie derivative: higher-rank tensors
• Lie derivative: product rule
• Lie derivatives
• 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 in a rotating disk: Ehrenfest's and Bell's spaceship paradoxes
• Lorentz contraction: no contraction in directions perpendicular to the motion
• Lorentz transformation in two dimensions
• Lorentz transformations
• Lorentz transformations and simultaneity
• Lorentz transformations as a linear map
• Lorentz transformations for energy and momentum
• Lorentz transformations: forward and backward
• 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 as a stress-energy tensor
• Metric tensor under Lorentz transformation
• Metric tensor: inverse and raising & lowering indices
• Metric tensor: parabolic coordinates
• Metric tensor: semi-log coordinates
• Metric tensor: sinusoidal coordinates
• Metric tensor: spherical coordinates
• Metric tensor: trace
• Metric tensor: trace
• 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 coordinates: derivation using a local flat frame
• Painlevé-Gullstrand metric: photon paths inside the event horizon
• Pair annihilation with an electron and a positron
• Parallel transport of tensors
• Particle falling towards a mass: two types of velocity
• Particle orbits - conserved quantities
• Particles falling towards a mass
• Perihelion shift - a couple of examples
• Perihelion shift - contribution from the radial coordinate
• Perihelion shift - contribution of the time coordinate
• Perihelion shift in planetary orbits
• 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
• Rapidity
• Red-shifts and blue-shifts
• Relative energy in a particle collider
• Relativistic electromagnetic force
• Relativistic electromagnetic potentials
• Relativistic invariants involving electromagnetic fields
• Relativistic kinetic energy
• Relativistic momentum and energy
• Relativistic transformation of electric and magnetic fields: an example
• Relativistic transformation of electromagnetic waves; the Doppler effect
• Relativistic transformation of force, electric and magnetic fields
• Relativistic transformation of force, electric and magnetic fields
• 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 and Ricci tensors in the weak field limit
• Riemann tensor - commutator of rank 2 tensor
• Riemann tensor and covariant contraction
• Riemann tensor for 3-d spherical coordinates
• Riemann tensor for an infinite plane of mass
• Riemann tensor for surface of a sphere
• Riemann tensor in 2-d curved space
• Riemann tensor in 2-d flat space
• Riemann tensor in 2-d polar coordinates
• Riemann tensor in a 2-d curved space
• Riemann tensor in the Schwarzschild metric
• Riemann tensor in the Schwarzschild metric: observer's view
• Riemann tensor: counting components in general
• Riemann tensor: counting independent components
• Riemann tensor: derivation
• Riemann tensor: symmetries
• Schwarzschild metric equivalent to weak field solution for spherical object
• Schwarzschild metric with negative mass
• Schwarzschild metric with non-zero cosmological constant
• Schwarzschild metric: acceleration
• Schwarzschild metric: finding the metric; Birkhoff's theorem
• Schwarzschild metric: four-momentum of a photon
• Schwarzschild metric: gravitational redshift
• Schwarzschild metric: radial coordinate is circumferential
• Schwarzschild metric: redshift of Sirius B
• Schwarzschild metric: the Newtonian limit & Christoffel symbol worksheet
• Schwarzschild metric: time coordinate
• Schwarzschild radius
• Self-force on a dipole in hyperbolic motion
• Shapiro delay: the twin quasar
• Simultaneity in special relativity
• Simultaneity; seeing versus observing
• Space-time diagrams
• Space-time diagrams: two observers
• Space-time intervals: invariance
• Spacelike intervals
• Spacetime diagrams: an example
• 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 at the centre of the sun
• Stress-energy tensor for a perfect fluid at rest
• Stress-energy tensor for a photon gas
• Stress-energy tensor for perfect fluid: general coordinates
• Stress-energy tensor in a local orthonormal frame
• Stress-energy tensor in the weak field, low velocity limit
• Stress-energy tensor of a slowly rotating star
• Stress-energy tensor: conservation equations
• Stress-energy tensor: negative pressure revisited
• Stress-energy tensor: relativistic perfect fluid
• Summation convention
• Symmetry of a rank 2 tensor is preserved under Lorentz transformation
• Tangent space
• Tangent space: partial derivatives as basis vectors
• Tensor arithmetic
• Tensor index notation
• Tensor indices: Newton's law
• Tensor product: numerical example
• Tensor trace
• Tensors: symmetric and anti-symmetric
• The Bianchi identity for the Riemann tensor
• The dual electromagnetic field tensor
• The invariant interval: some examples
• The sun as a gravitational lens
• The twin paradox analyzed using Lorentz transformations
• Tidal effect for objects in freefall near the Earth's surface
• Time dilation
• Time dilation: a rocket passing Earth
• Time dilation: resolving the paradox
• Transforming derivatives of four-vectors and scalars
• Transforming the electric field in relativity
• Twin paradox with a black hole
• Vacuum stress-energy and the cosmological constant
• Vectors and the metric tensor
• Velocity addition formulas for all 3 directions
• Velocity addition in special relativity
• Velocity addition: chasing space pirates
• Velocity addition: chasing space pirates viewed in four reference frames
• Vertical particle motion
• Wave solution of the weak-field Einstein equation