- 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
- Lorentz transformations as rotations
- Lorentz transformations as 2×2 matrices
- Lorentz transformations and the special linear group SL(2,C)
- Lorentz transformation as product of a pure boost and pure rotation
- 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

Kian MalekiThanks for you website

I am studying General Relativity and in order to understand things deeper I usually come to your website. This websites explains better than the books which I am currently studying .At one point I decided to study everything in your website about General Relativity but when the chapters are in alphabetical order which means I cannot study all the chapters from begging to the end.

Would you please provide us a logical order of the chapters to study?

With Regards

Kian Maleki

gwrowePost authorYou can follow the posts (more or less) in order by looking at the problem solutions for a particular textbook. The general relativity book that I’ve done the most work on is the one by Moore, so you could look at the index of problems for his book, which are organized by chapter. Also, I’ve tried to provide enough cross-links within each post so that you can review background material relevant to that post. You can also use the Search box to look up posts on a particular topic.

With close to 1300 posts, it would take far too much time for me to try to produce an ordered list, and often one post relies on several previous posts (rather than just one), so the structure would be more like a graph than a linear list.

José VictorMy friend, first of all, I want to congratulate you on the extraordinary work you give to students and even teachers, divulging the solution to problems of relativity that, as you do, represents a valuable instrument in the learning of relativity and physics, generally. Because you approach all areas. Impressive. Since I found this site I have not stopped visiting it, reading and studying problem problem, which are masterfully solved by you. But I’d like to make a note to the students. The students only acquire maturity in relativity, only learn in a definitive way, if they themselves try, the solution of the problems for themselves. And only after these attempts, in the last case, resort to the rich source of elimination of doubts that you make available to them. I repeat: a solid learning depends, first of all, on personal effort, using ready-made solutions only in cases of many doubts. Your site is currently my home page.

Sorry for English. I’m Brazilian and I do not write much in that language.

Grateful.

MitulHi, I often use your site for various courses as well the solutions on the many problems has been very useful to double check. I was wondering if you could be able to post solutions to problems from Caroll’s book on General relativity, please?

Thanks,

gwrowePost authorSorry, no, it’s not a book I’m currently working on.