Web Tutorials on Relativity

These are some good places to begin your study of relativity. (Some readers may prefer the Popular Science sites listed at the bottom of this page.)

• The sci.physics.relativity FAQ is well written and authoritative, with excellent contributions by Michael Weiss, John Baez, and other usenet luminaries. Graphicsfree, but highly recommended as a first stop.
• Relativity Tutorial, by Ned Wright (Astronomy, UCLA). A brief overview featuring some fine graphics.
• Spacetime 101, by Patricia Schwarz (Physics, Cal Tech) offers a gentle illustrated introduction to the basic concepts of special and general relativity.
• The Light Cone: an illuminating introduction to relativity, by Rob Salgado (Physics, Syracuse University). An impressive site with nice gif pictures and also several Java applets. Topics include Minkowski space and light cones, the Equivalence Principle, and the Schwarzschild solution.
• A Brief Introduction to Einstein's Gravity, by Anonymous (Astronomy, Penn State), offers a nice visual overview of the basic concepts of gtr.
• Black Holes FAQ, by Ted Bunn (UC Berkeley) offers a nontechnical (graphics free) overview of some of the most important properties of black holes.
• The Cambridge Relativity HomePage has links to some brief tutorials on black holes and cosmic strings.
• How do we see Black Holes?, by John Blondin (Theoretical Astrophysics and Physics, North Carolina State University). A classy, gentle introduction to black hole physics.
• General Relativity, a masterful overview by John Baez (Mathematics, University of California at Riverside). More demanding than the previous items, but highly recommended.
• The Thermodynamics of Black Holes: Limited Events on the Horizon, by Gary N. Felder and Charles E. Stebbins (Physics, Oberlin College). An html document.

Visualizing Relativity

There are a large number of sites which offer computer simulations of what you would experience during relativistic space travel, falling into a black hole, etc. I think these are some of the best.

• Relativistic Starflight, by Steve VanDevender is a C program which in effect draws a movie on your computer screen, showing the view out the front window of a relativistic starship. I was able to download, compile, and run this program without trouble. Try running it with the command



xrel -accel 100 -bounce -viewangle 150 -limit 0.75 -max

This will display constant acceleration at 10g with a wide angle view, at one real time second = one day of spaceship time; at v = 0.75, deceleration at 10g is initiated, and so forth. You should be able to see very clearly how boosting effects a hyperbolic linear fractional transformation on the celestial sphere. Such an LFT has two fixed points (opposite points on the celestial sphere) and as you boost with positive acceleration toward the North star, stars flow from South Pole along longitude lines toward the North star. Since LFT's are conformal transformations, even though the night sky becomes rather "distorted" overall, small shapes (e.g. constellations) are preserved.

If the starship were rotating along the axis of motion as well, you would see stars spiraling toward one fixed point and away from the other.

• Relativistic Flight Simulator, by Wade Lutgen, a recent graduate of the University of Wisconsin. This is a delightful C program which "puts you in the pilot seat of a near light speed capable space vehicle" moving through a nicely simulated starfield. "This basically means that you have an infinite amount of fuel to accelerate your craft up to (but not including, of course) the speed of light. The relativistic effects of Doppler shift, stellar abberation, and mass increase have been taken into account. The rest of the physics should also be correct." There are versions for DOS and X windows. I was able to download, compile and run the second version in our X windows environment without any trouble.
• Going Potty!, (i.e., "crazy", a pun on the teapot used in the illustrations) by Ronan Bonan (Computer Science, Trinity College, Dublin). Illustrates the Penrose-Terrel "rotation", an optical effect predicted in objects moving at relativistic speeds.
• Gravitational Lensing, by Pete Newbury (Mathematica, University of British Columbia). Features mpeg movies illustrating lensing, Einstein rings, etc., and some good background information. Don't miss your opportunity to run your own computer simulation of a gravitational lens! I recommend that you star with the far star somewhat above the close star, and trying running the program several times, lowering the distant star a bit each time. Note the marked displacement of the image of the distant star away from its actual position (relative to the closer star). As the distant star gets closer to the near star, notice that its image gets smeared out in a crescent. As you line up the two stars more and more accurately, you should see a new "secondary image", also crescent shaped, forming on the other side of the near star, and perhaps even an "Einstein cross". Once the stars are almost perfectly lined up, you see a beautiful ring shaped image of the far star. Last but not least, compare with the HST photographs (see links below) of real gravitional lenses. Highly recommended!
• Falling Into a Black Hole, by Andrew Hamilton (Astrophysical and Planetary Sciences, University of Colorado). If you've ever wondered what you'd see near a black hole, this is the site for you! A very well done "tour" illustrating would you would see approaching, orbiting and then falling into a black hole, featuring attractive animated gifs illustrating such effects as the "Schwarzschild bubble". (Don't miss the comparison of Schwarzschild, Eddington, and Kruskal coordinates). Highly recommended!
• Black Holes with Java, by Peter Musgrave (Physics, Queensland University). Start with the related Central Force with Java page and then try the black hole orbit applet, which allows the user to explore the effect of changing various orbital parameters.
• Solving Einstein's Equation in Three Dimensions, by Patricia Schwarz (Physics, Caltech), is a Mathematica notebook exploring Einstein's equation, curvature computations, null geodesics, etc.
• Orbits in Strongly Curved Spacetime, a Java applet by John Walker (Fermilab), shows the effective potential and embedding profile as well as the trajectory of a test particle orbiting a nonrotating hole. (I have one quibble: for some reason Walker wrote the applet so that the ball hugs the "effective potential" curve (first introduced by Misner) whereas of course energy is conserved and the ball should move back and forth on a horizontal line--- see for instance the textbook by Misner, Thorne, & Wheeler or the applet by Musgrave.)
• Geometry Around Black Holes, by Michael Cramer Andersen (Astronomy, University of Copenhagen). Features some good (advanced undergraduate level) background and some VRML images and mpeg/Quicktime movies. An excellent discussion of frame dragging and light paths near a rotating (Kerr) black hole.
• Relativistic Simulations from the Physics Department at the University of Tuebingen. Features a half dozen magnificent mpeg movies of pulsars, orbiting a neutron star, orbiting a black hole, a relativistic flight to Polaris (compare with Lutgens program) , embeddings of "space" near a black hole, etc.
• Numerical Relativity Exhibitions, from the NCSA Relativity Group. Features stills from production quality movies of black holes, gravitational waves, colliding black holes, etc. For mpeg versions of some black hole simulations, see this page. What more need be said! Go and look, you won't be disappointed!
• Particle Trajectories Near Black Holes, by Peter Diener. Features a form enabling you to compute effective potentials and three dimensional trajectories for the motion of test particles near a rotating black hole. (Unfortunately, you can only see the trajectories if you have VRML or Inventor software.) Includes good background material at the undergraduate level.
• Black Hole Simulations by Sam Hart (Physics, University of Arizona). Features fabulous gif images of fluid flow in an accretion disk, etc.
• Magnetic Field Lines in a Black Hole Plasma Disc, by Boris Gudikson and Bjorn Ostman (Physics, University of Copenhagen). Features a brief introduction to black hole electrodynamics and accretion disks, with very nice pictures.
• Relativistic Ray-Tracing Simulating the appearance of rapidly moving objects. This work arose as a result of a discussion with Dr Sandy Dance , in which the question arose: "If we happened to be flying past an object at nearly the speed of light, what would that object look like"?
• Falling Into a Black Hole
• Simulation of a Black Hole by Raytracing

Lecture Notes and Articles

• Relativity and Cosmology, undergraduate course notes by Jose Wudka (Physics, UC Riverside). Topics covered include both special relativity (e.g., spacetime, Lorentz invariance, various "paradoxes") and general relativity (e.g., equivalence principle, black holes, gravitational waves, experimental tests of gtr). Apparently a survey course for non-majors, with little math but some very nice graphics.
• Colliding Black Holes, five lectures by Jorge Pullin (Physics, Penn State). Topics covered include curvature, the Einstein equation, black holes, and gravitational waves.
• Tensors and Relativity, by Peter Dunsby (Mathematics, University of Cape Town). This is a fabulous site featuring a complete course available for free over the web as html documents. (Registered students can also download postscript and dvi versions.) At the level of Schutz, A First Course in General Relavitity, i.e. more challenging than the previous site. Topics covered include vectors and tensors in flat spacetime, the conceptual basis of general relativity, curved spacetime, the field equation, and the Schwarzschild solution. Highly recommended!
• A First Look at Relativity and Gravitation, by Clifford Johnson (Physics, University of Kentucky). At the level of D'Inverno, Introducing Einstein's Relativity. About half of two dozen lecture notes are currently available, in both html and postscript, including exercises! Topics include tidal forces, geodesic deviation, the matter tensor (aka stress-energy tensor) for dust, fluid, and EM fields, as well as Schwarzschild's solution, weak field theory, and methods for solving Einstein's equation. When complete, this will basically be a full course of lecture notes on gtr, including cosmology. At a level similar to or a bit above the preceding site.
• Black Holes: The inside story, by Serge Droz (Physics, University of Guelph), Werner Israel (Physics, University of Alberta), and Sharon M Morsink (Physics, University of Wisconsin-Milwaukee). Originally appeared in Physics World. A beautifully illustrated html article focusing on the interior geometry of black holes.
• Simulating Relativistic Orbits About a Black Hole, by Steven Bell (Lockheed Martin). Discusses a computer program which computes the paths of test particles in orbit about a rotating (Kerr-Newman) black hole. First published in Computers in Physics and recently updated by the author.
• Differential Forms in Electromagnetic Theory, by Richard H. Selfridge, David V. Arnold and Karl F. Warnick (Electrical and Computer Engineering, Brigham Young). A very well done site, with extensive teaching materials available as dvi documents. A good introduction to an essential tool in relativity.

Graduate Level Lecture Notes and Articles

To really appreciate the beauty and subtleties of general relativity, you must grapple with the mathematics, which lies, unfortunately, just beyond the undergraduate level. Here are some fine graduate level courses, and also some expository articles on topics of particular interest.

• A Short Course on GR, by William L. Burke, (Physics, UC Santa Cruz). Topics include weak field theory, gravitational waves, radiation damping, cosmology, the Friedmann and Lemaitre dusts, singularities, black holes, the Schwarzschild metric and Kruskal's extension of it. There is an appendix on mathematical notation. This is a single postscript document (about 75 pages).
• General Relativity, by Petr Hadrava, (Astronomical Institute, Academy of Sciences of the Czech Republic). Lecture notes (in English) on str and gtr. Topics include the Equivalence Principle, the field equations, weak-field theory, the Schwarzschild exterior (vacuum) and interior (stellar "fluid") solutions, the Friedmann cosmological solutions. Two mathematical appendices sketch the mathematics of tensor algebra, exterior algebra, connection, Lie derivatives, Killing vectors, and variational principles. This is a single postscript document (50 pages).
• Lecture Notes on General Relativity, by Sean M. Carroll (Institute for Theoretical Physics, University of California Santa Barbara). From a course taught at MIT. Topics covered include str, manifolds, covariant derivatives, connections, curvature, Lie derivatives, pullbacks, Killing vectors, the Equivalence Principle, the matter tensor, the field equation of gtr (Einstein's equation), the initial value and variational principle formulations of the field equation, weak field theory, gravitational waves, a complete discussion of the Schwarzschild solution, cosmology and the Friedmann solutions. Carroll's careful discussion of the geometry of the Kerr solution is particularly noteworthy. The lectures are available as either html or postscript documents (about 200 pages total).
• Black Holes, by Paul Townsend (Applied Mechanics and Theoretical Physics, Cambridge). A very thorough introduction, studies the Schwarzschild, Reissner-Nordstrom, and Kerr solutions using a variety of coordinate systems. Additional topics include gravitational collapse, horizons, singularities, Carter-Penrose diagrams (aka conformal compactification), Hawking radiation and black hole thermodynamics. This is a 145 page postscript document.
• A Description of the Initial Value Formulation of Vacuum General Relativity for the Non-Specialist, by Mark Miller (Syracuse University). Available either as html pages or as a 15 page postscript document.
• Differential Geometry, by Sergei Yakovenko (Weizmann Institute). A complete set of lecture notes. Topics include manifolds, diffeomorphisms, partitions of unity, the Whitney embedding theorem, tangent bundle, algebra of vector fields, Lie derivatives, commutators, points as maximal ideals, derivations, local rings, differentiable forms, etc.
• Riemannian Geometry and General Relativity, the problem sets (with solutions) from a course taught by Michael Shubin (Mathematics, Northeastern).

Research Frontiers

• Living Reviews in Relativity is a fabulous resource for students. "Living Reviews in Relativity is a solely WWW-based, peer reviewed journal for physics. The journal publishes invited reviews of research in all areas of relativity. The journal is offered as a free service to the scientific community. Articles are invited pieces from specialists in their fields and are directed toward physicists at the graduate student level and beyond. Articles appearing in Living Reviews provide peer-refereed, carefully edited, current and insightful overviews of what is happening in the fields they cover." Presently, this site features very high quality review articles on Loop Quantum Gravity, Local and Global Existence Theorems, Stationary Black Holes, Rotating Stars, and more. The list of forthcoming articles is even more mouth watering!
• The Los Alamos Preprint Server is the place to look for the very latest current research in all areas of physics, including general relativity and quantum gravity. Note that this site features a search tool.

Symbolic Tensor Manipulation Software

Anyone who studies some of the courses in the previous section will probably appreciate the desirability of software capable of computing connections and curvatures.

• GRTensorII, a freeware symbolic tensor manipulation package available for both Maple V and Mathematica, developed by the Physics Department, Queen's University at Kingston, Ontario, and the Faculty of Mathematical Studies at the University of Southampton. Note in particular the demonstration page devoted to general relativity!



Which version should you use? For what it is worth, I have found that the Mathematica version is much better at trig manipulations, but is thoroughly outclassed by the Maple version whenever constraint equations appear. The Maple version also has somewhat better documentation. I use both versions.

• Ricci, a freeware Mathematica package for doing symbolic tensor computations that arise in differential geometry. Developed by Jack Lee (Mathematics, University of Washington). Complete with a 90 page manual.
• Information on a number of commericial tensor manipulation packages is available at CAIN (Computer Algebra Information Network), a consortium of eight European organizations.

Experimental and Observational Evidence

The evidence in favor of gtr is overwhelming; nonetheless some important predictions still await confirmation (in particular, gravitational waves have yet to be directly detected, although there is already very strong indirect evidence for their existence).

• Seeing is Believing! In recent years the Hubble Space Telescope (HST) has made spectacular observations of many phenomena predicted by gtr, including:
• The actual accretion disks of supermassive black holes at the center of the galaxies NGC 6251 (300 million ly distant, in the constellation Ursa minor) and NGC 4261 (in the constellation Virgo) have been photographed by the HST. The first of these is, remarkably, warped like the brim of a Stetson hat: it is about 1000 ly wide, and a 3 million ly long double jet of hot gas is being flung out orthogonally to the disk. Supermassive black holes turn out to be very common; HST has found evidence of one at the center of most galaxies so far examined. In particular, one has just been found at the center of the galaxy M84 and twin jets have been tracked in real time near a suspected supermassive black hole at the center of the galaxy NGC 4151. See also Observational Evidence for Black Holes (by the Cambridge Relativity Group).
• A dozen or so gravitational lenses have been photographed by the HST. You can see several crescent shaped images of distant galaxies lying behind a cluster of much closer galaxies called Abell 2218; this crescent shape is the unmistakable sign of a gravitational lens. As it happens, the most distant galaxy known is also seen by HST through a gravitational lens. HST has also photographed a spectacular example of an Einstein cross, consisting of four images of a single distant galaxy lying behind the closer galaxy 2237+0305
• The fate of the universe is eternal expansion, according to recent observations of distant supernovae by the HST. Two teams of astronomers have independently concluded that their observations of distant supernovas show that our universe is modeled locally be a Friedmann model with spacelike slices having constant negative curvature. Stories in the popular press entirely overlooked the possibility (widely appreciated by specialists) that these spacelike slices need not be a hyperbolic plane but might be a quotient space with finite volume. Confusingly, press reports of this remarkable convergence of results were followed within days by apparently contradictory reports that in fact the very same observations show that the Hubble expansion is actually increasing with time, rather than decreasing as all the Friedmann models predict. Such behavior is consistent with cosmological solutions to the field equation with the so-called "cosmological constant" included, which led to a spate of news stories announcing the discovery of "antigravity". See Ned Wright's Cosmology Tutorial for more information.
• Early protogalaxies, some 11 billion ly distant (and ll billion years old) much smaller than the modern kind, have been photographed by the HST. In fact, the HST has shown a whole sequence whereby closer galaxies look more and more like the modern variety.
• An isolated neutron star (400 million ly distant, in the southern Constellation Coronae) has been photographed by the HST. We know what it is because it is so small (16 miles in diameter), so hot (one million degrees) and so dim that it cannot be any other kind of star.
• A bit off topic, but while you're at it, don't miss these gorgeous pictures of exploding stars.
• You can read about various astronomical observations of black holes in my compilation of miscellaneous press releases. See also a recent study of gas plunging violently onto the surface of a neutron star.
• Recently, a team of astronomers led by Dr. Ignazio Ciufolini (coauthor with Wheeler of the recent book, Graviation and Inertia) were able to confirm the prediction of frame dragging by a careful examination of two low earth orbiting satellites.
• General Relativity in the Global Positioning System, by Neil Ashby (Physics, University of Colorado) describes how the 24 GPS satellites were designed to take account of the gravitational red shift predicted by gtr; this system simply wouldn't work if gtr were not extremely accurate, since the red shift turns out to be quite significant.
• The cosmic background radiation has been mapped in great detail by the COBE (Cosmic Background Explorer) satellite.
• Experiments in General Relativity, (STEP, Stanford University) Describes current attempts to directly confirm frame dragging and other effects using an earth orbiting satellite--- this will be one of the most sensitive physical experiments ever performed! Features considerable background information.
• Tests of General Relativity at Jodrell Bank radio observatory use astronomical observations of binary pulsars.
• LIGO Home Page. "The Laser Interferometer Gravitational-Wave Observatory (LIGO) project is a pioneering effort to design and construct a novel scientific facility - a gravitational-wave observatory - that will open a new observational window on the universe." Links to other gravitational wave sites, explanatory pages,etc.

Popular Science Sites

These are good places to go to if you just want to get some idea of what relativity is all about.

• (19 September 1998) Foundations, by Greg Egan, offers a gentle introduction to some of the basic
• ideas of str and gtr.
• Spacetime Wrinkles, a beautifully realized web museum exhibit from the University of Illinois. Features excellent gif images and mpeg movies.
• Our Hierarchical Universe, another classy web museum exhibit from the University of Illinois, featuring discussion of the Hubble expansion, cosmic background radiation, cosmological horizons, etc.
• Scientific American has some relevant feature articles at their website, including:
• Twist and Shout: discusses recent observations of an apparent anisotropy in the universe.
• Constructing the Cosmos: discusses large scale structure of the universe.
• All in the Timing: discusses satellite observations of neutron stars.
• Black Holes. Information leaflets from the Royal Greenich Observatory, collected by Mike Guidry (Physics and Astronomy, The University of Tennessee, and Oak Ridge National Laboratory).
• Cosmology for Beginners, by John Gribbin. Includes extensive discussion of Alan Guth's inflationary universe scenario.
• Brief Answers to Special and General Relativity Questions, by Dr. Sten Odenwald This is part of the site Ask the Astronomer. See also his black holes page.
• Special Relativity, a brief history by John J. O'Connor and Edmund F. Robertson (School of Mathematical and Computational Sciences, University of St Andrews, Scotland.) Part of the MacTutor History of Mathematics archive. See also the brief history of General Relativity and Non-Euclidean geometry.

For More Information

• RELATIVITY bookmarks from Syracuse University. A comprehensive list of links to sites on the history of str and gtr, popular science type relativity sites, visualization sites, lecture notes, research journals, academic departments, relevant software sites, etc. Scroll down, the best stuff is at the bottom!

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