A Summary of General Relativity
By Bruce R. Linnell, PhD


The following files represent a summary of general relativity, broken into bite-size pieces of roughly 10 pages each (except GR2f, which is more like 20). In all these files, items in blue text are the most important things, items in green text are "gotchas" for the beginning student of general relativity (except for in GR1f), and items in gold are things I am 99% (but not 100%) sure of.

DISCLAIMER #1 : these were originally created as my own private notes, so sometimes they explain things in detail, sometimes they just present the results. This is not a tutorial or a textbook! Some of the labels in the figures don't always exactly match the text, but I tried to indicate that where possible.

The first set of files contain the minimum amount of information you need in order to understand what Einstein's "field equation" means. They assume the reader is already familiar with college-level math including vectors, derivatives, partial derivatives, and matrix math, as well as introductory physics. If you want to skip the math, the end of GR1c contains the explanations and GR1e contains many cool visuals. There are references in these files to the GR2 files, but you do not need to read the second set in order to understand these.

GR1a – a brief review of vectors and coordinate systems; an introduction to tensors, index notation, and Einstein

summation; definitions of manifolds, tangent vectors and tangent planes

             GR1b – a brief refresher of special relativity; basic terms and concepts of general relativity; the metric tensor; an

introduction to covariant and contravariant tensors; introduction to parallel transport

             GR1c – the Riemann, Ricci, and Weyl tensors; the Ricci scalar; an introduction to the stress-energy tensor or

energy-momentum tensor (it goes by both names); Einstein’s general relativity equation and some

explanations/interpretations of it

             GR1d – tensor notation is incredibly “dense”, so this shows Einstein’s equation (in terms of the metric) expanded

into its full glory

             GR1e – some visualizations of common spacetimes including Schwarzschild (non-rotating mass), Kerr (rotating

mass) and a moving mass (the visualizations of the Kerr spacetime are my own, and try to show in

several different ways how spacetime is stretched around a spinning mass); other spacetimes such as

wormholes and warp-drives; a discussion of the many tests general relativity correctly predicts

             GR1f – general relativity as applied to cosmology, and why we think the universe is filled with “dark matter” and “dark

energy” (and all the assumptions that go into that)

 

These next files go into much more mathematical depth and detail, explaining things well enough so that you could (hopefully) understand scholarly papers and articles about general relativity.

             GR2a – miscellaneous background info : more on vectors and tensors (wedge product, dyadic product, Levi-Civita symbol,

etc.); more general relativity terms and definitions; intrinsic/internal vs. extrinsic/external curvature; various kinds of “flat”

spacetimes

             GR2b – much more on special relativity : spacetime diagrams; simultaneity and cause/effect; light cones; Minkowski

diagrams; Lorentz transformations; doing physics in special relativity (velocity, acceleration, momentum, force, etc.)

             GR2c – all about derivatives : path tangent vectors; scalar field derivatives; directional scalar derivatives; vector field

derivatives; affine connections; Christoffel symbols; covariant derivatives; directional covariant derivatives; Lie

derivatives; the explanations of the Christoffel symbols and covariant derivatives contain

as-clear-as-I-can-make-them explanations with good visuals

             GR2d – all about covariant and contravariant tensors : how they are defined and how they transform; the Jacobian;

one-forms; exterior derivatives; and a recap of contravariant/covariant formulas

             GR2e – electromagnetism : classical formulas; special relativity formulas; general relativity formulas

             GR2f – general relativity, take two : differences from special relativity; affine parameters; proper acceleration; a recap of

parallel transport and geodesics; an in-depth as-clear-as-I-can-make-it look at the stress-energy-momentum tensor

(including my own standard-physics explanation of why “pressure” draws matter and energy towards it); energy

conditions; more on the Weyl tensor; geodesic deviation; a discussion of the pitfalls when converting line elements

to metrics; the meaning of off-diagonal terms in the metric (which is rarely discussed anywhere!)

 

I have in the back of my mind an idea for a third set of documents which would include enough information so that you could actually work problems in general relativity, but that probably won’t happen for another year or two.  Some of the GR2 files reference these non-existent GR3 files.

 

DISCLAIMER #2 : while I have copied text and figures from other people’s web sites to create these notes, much of it has been heavily edited, rearranged and merged, and some of it is entirely my own. If you find any of your own material here that you would like me to reference, please let me know, I would be glad to do so (since these were originally intended as my own notes, I didn’t keep track of what came from where). If you would like me to remove your material, I will, but please realize that if I used it that means that I thought it was exceptionally clear and helpful.

As I continued to look into general relativity, I discovered an amazing thing : Einstein's field equation is a simplification! The equation, as complicated as it is, ignores something called "torsion" (which is what happens when you turn a screwdriver or wrench). Including torsion adds a second, equally-as-complicated, equation to the mix!