A note on the index notation used in posts on relativity. In many of my posts, I was lazy and used Roman rather than Greek letters for tensor indices, as they are easier to type in Latex. The convention in most textbooks on relativity is to use Greek indices for the four components of spacetime, and Roman indices when referring only to spatial coordinates. Apologies for this, but it would be far too much work to go through all the posts and convert everything to Greek letters. In most cases, there shouldn't be any confusion, as in almost all posts, we're talking about the four dimensions of spacetime.

The conventions for distinguishing between 4-vectors (with four spacetime components) and 3-vectors (in regular 3-d space) vary from one book to another, and I usually use the convention that was used in the book that was my primary source for a post. Moore represents 4-vectors by boldface font, which is usually reserved for 3-vectors in other texts. Schutz uses ordinary font for 4-vectors and a little arrow over the letter for 3-vectors. Always check the conventions for whatever book you're using.

- A flawed theory of gravity
- Apparent size of a black hole
- Are black holes really black?
- Area and volume in spacetime
- Area of unit spheres in higher dimensions
- Bianchi identity for the Riemann tensor
- Black hole entropy
- Black hole entropy - it's pretty large
- Black hole in equilibrium with a thermal reservoir
- Black hole evaporation - how long will a black hole live?
- Black hole evaporation - remnants of the big bang
- Black holes and the Large Hadron Collider
- Black hole heat engine
- Black hole radiation - energy of emitted particles
- Black hole radiation - energy of a particle from a solar mass black hole
- Black hole radiation - energy at infinity of radiated particle
- Black hole radiation - mass as a function of time
- Black hole temperatures at different distances
- Black hole with static charge; Reissner-Nordström solution
- Boyer-Lindquist coordinates and curvature of space
- Christoffel symbols - symmetry
- Christoffel symbols defined for a sphere
- Christoffel symbols for Schwarzschild metric
- Christoffel symbols and the covariant derivative
- Christoffel symbols in noncoordinate bases
- Christoffel symbols in sinusoidal coordinates
- Christoffel symbols in terms of the metric tensor
- Circular orbit around a supermassive black hole
- Circular orbit - appearance to a falling observer
- Circular orbits - 3 measurements of the period
- Circular orbits - Kepler's law
- Circular orbits - relation between radius and angular momentum
- Circular orbits - Schwarzschild vs Newton
- Components of one-forms and vectors
- Composition of velocities in relativity
- Conformally flat spaces
- Conservation of four-momentum implies the geodesic equation
- Contravariant tensors
- Coordinate transformations - the Jacobian determinant
- Cosmic strings
- Covariant and mixed tensors
- Covariant derivative is a tensor
- Covariant derivative - commutativity
- 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 derivative of a general tensor
- Covariant derivative in semi-log coordinates
- Curvature of 2-dimensional space
- Curvature of a sphere
- Deflection of light by a mass
- Deflection of light by the sun
- Delay of light passing a mass - Shapiro delay
- Derivatives of a one-form field in polar coordinates
- Derivatives of a vector field in polar coordinates
- Diagonalizing the metric
- Distance from external radius up to the event horizon
- Divergence in curved space
- Divergence of vector field in polar coordinates
- Dominant energy condition
- Doppler effect
- Doppler effect and four-momentum
- Einstein equation - trying the Ricci tensor as a solution
- Einstein tensor and Einstein equation
- Einstein equation - alternative form
- Einstein equation in the Newtonian limit
- Einstein equation solution for the interior of a spherically symmetric star
- Einstein tensor of zero implies a zero Ricci tensor
- Einstein equation for a perfect fluid
- Einstein equation on the surface of a sphere
- Einstein equation for an exponential metric
- 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 - invariance of inner product
- Electromagnetic field tensor - invariance under Lorentz transformations
- Electromagnetic field tensor - justification
- Electromagnetic field tensor - Lorentz transformations
- Electromagnetic stress-energy tensor
- Electron-positron collision must produce at least 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
- Equality of gravitational and inertial masses
- Escape velocity near an event horizon
- Eskimo mites and their metric
- 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 terms of the stress-energy tensor
- Four-acceleration
- Four-momentum conservation
- Four-momentum conservation - a trip to alpha centauri
- Four-momentum conservation in the electron-electron collision
- Four-momentum conservation in the electron-photon collision
- Four-momentum conservation in electron-positron annihilation
- Four-momentum of photons
- Four-vectors - basics
- Four-vectors - summation convention
- Four-velocity
- Four-velocity - an example
- Four-velocity - another example
- Geodesic deviation in a locally inertial frame
- Geodesic equation and four-velocity
- Geodesic equation - 2-d space-time
- Geodesic equation in 2-d with exponential metric
- Geodesic equation - polar coordinates
- Geodesic equation - geodesics on a sphere
- Geodesic equation on a paraboloid
- Geodesics - paths of longest proper time
- Globally parallel vector field
- Gradient as covector - example in 2-d
- Gradient as a one-form
- Gravitational lensing and the Einstein ring
- Gravitational lensing - image brightness
- Gravitational lensing - large angles
- Gravitational lensing - the twin quasar
- Gravitoelectric and gravitomagnetic densities
- Gravitoelectric and gravitomagnetic densities for the vacuum
- Gravitomagnetic acceleration is perpendicular to velocity
- Gravitomagnetic acceleration near a rotating star
- Gravitoelectric and gravitomagnetic acceleration for a moving wire
- Gravitoelectric and gravitomagnetic acceleration for parallel plates
- Gravity can't exist in 2 spacetime dimensions
- Gravity can't exist in 3 spacetime dimensions either
- Higher order derivatives are not tensors
- Hyperbolic coordinates in flat space
- Hyperbolic motion and acceleration
- Inertia tensor
- Invariance of spacetime intervals
- Invariant hyperbolas
- Kruskal-Szekeres coordinates and the event horizon
- Kruskal-Szekeles metric - what can you see as you fall into a black hole?
- Kruskal-Szekeres diagrams - saving a space shuttle
- Kruskal-Szekeres diagrams - another space ship disaster
- Kruskal-Szekeres metric - more fun with photons
- Length contraction
- Light cones near the event horizon
- Linearity of Lorentz transformations
- Local flat coordinate systems - four-momentum of photons
- Local flatness theorem
- Local flat frame for a circular orbit
- Lorentz invariance of electric charge
- Lorentz transformations
- MACHOs and seeing distant objects with a gravitational lens
- Manifolds, curves and surfaces
- Maxwell's equations in cylindrical coordinates
- Mercator map metric
- Metric tensor and basis vectors
- Metric tensor for surface of a sphere
- Metric tensor in parabolic coordinates
- Metric tensor in semi-log coordinates
- Metric tensor in sinusoidal coordinates
- Metric tensor in spherical coordinates
- Metric tensor - inverse and raising and lowering indices
- Metric tensor under Lorentz transformation
- Metric tensor as a stress-energy tensor
- Momentum and energy
- Newtonian tidal effect
- No contraction in directions perpendicular to the motion
- Noncoordinate bases
- Normal vectors and unit normal one-forms
- One-form basis
- One-forms in polar coordinates
- Orbit of a comet around a black hole
- Painlevé-Gullstrand (global rain) coordinates
- Painlevé-Gullstrand metric - photon paths inside the event horizon
- Painlevé-Gullstrand coordinates - derivation using a local flat frame
- Parallel transport and the geodesic equation
- Parallel transport and the scalar product
- Parallel transport around a spherical triangle
- Particle orbits - conserved quantities
- Particle falling towards a mass - two types of velocity
- Particles falling towards a mass
- Penrose diagrams in flat spacetime
- Perihelion shift in planetary orbits
- Perihelion shift - contribution from the radial coordinate
- Perihelion shift - contribution of the time coordinate
- Perihelion shift - a couple of examples
- Perihelion shift - numerical solution
- Photon equations of motion
- Photon orbits - speed measured at two places
- Photon path in flat space
- Plane symmetric spacetime
- Pole in a barn paradox
- Postulates of special relativity
- Projection operator in spacetime
- Proper time to fall through the event horizon
- Red-shifts and blue-shifts
- Relativistic units
- Ricci tensor and curvature scalar
- Ricci tensor and curvature scalar for a sphere
- Ricci tensor for a spherically symmetric metric - the worksheet
- Riemann tensor - derivation
- Riemann tensor from parallel transport
- Riemann tensor - symmetries
- Riemann tensor - counting independent components
- Riemann tensor - counting components in general
- Riemann tensor for an infinite plane of mass
- Riemann tensor in 2-d polar coordinates
- Riemann tensor in 2-d curved space
- Riemann tensor in 2-d flat space
- Riemann tensor in an exponential 2-d curved space
- Riemann tensor for surface of a sphere
- Riemann tensor for 3-d spherical coordinates
- Riemann tensor in the Schwarzschild metric
- Riemann tensor in the Schwarzschild metric - observer's view
- Riemann and Ricci tensors in the weak field limit
- Schwarzschild metric - four-momentum of a photon
- Schwarzschild metric - acceleration
- Schwarzschild metric - gravitational red shift
- 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 metric - finding the metric; Birkhoff's theorem
- Schwarzschild metric - the Newtonian limit & Christoffel symbol worksheet
- Schwarzschild metric with negative mass
- Schwarzschild metric with non-zero cosmological constant
- Schwarzschild radius
- Shapiro delay - the twin quasar
- Spacetime diagrams
- Spacetime diagrams - particle detector
- Spacetime diagrams - two observers
- Spacetime diagrams - events and world lines
- Spherical metric - distance in 2-d curved space
- Spherically symmetric solution to the Einstein equation
- Squashed sphere
- Stereographic projection of the sphere
- Stress-energy tensor - conservation equations
- Stress-energy tensor for dust
- Stress-energy tensor for a perfect fluid at rest
- Stress-energy tensor for perfect fluid - general coordinates
- Stress-energy tensor in a local orthonormal frame
- Stress-energy tensor - negative pressure revisited
- Stress-energy tensor - symmetry
- Stress-energy tensor at the centre of the sun
- Stress-energy tensor of a slowly rotating star
- Stress-energy tensor - relativistic perfect fluid
- Stress-energy tensor for a photon gas
- Stress-energy tensor in the weak field limit
- The sun as a gravitational lens
- Symmetric and anti-symmetric tensors
- Tensor equations are valid in all coordinates
- Tensor index notation
- Tensor product - numerical example
- Tensor trace
- Tidal effect for objects in freefall near the Earth's surface
- Time and space interchange as we cross the event horizon
- Time dilation and proper time
- Torus metric
- Trace of metric tensor
- Transforming derivatives of four-vectors and scalars
- Twin paradox
- Twin paradox with a black hole
- Unit spheres in higher dimensions
- Vacuum stress-energy and the cosmological constant
- Vectors and the metric tensor
- Vectors in polar coordinates
- Vertical particle motion
- Volume element in terms of metric determinant
- Wave solution of the weak-field Einstein equation