General relativity is simple
I like the brief summary of the basic mathematics in general relativity given by John Baez: http://math.ucr.edu/home/baez/gr/outline1.html although his more detailed explanation here: http://math.ucr.edu/home/baez/gr/outline2.html is less encouraging; it seems to omit both physics and the process by which general relativity was obtained. What can anyone learn from the conventional teaching of general relativity? The answer is mathematical manipulation.
On my page http://nigelcook0.tripod.com/ I give a discussion from a different perspective, considering the physical processes and historical development of general relativity. I also give simplified differential equations for the various tensors and their units very briefly. I do not give the full expansion of the Riemann tensor, for example. Students have to go elsewhere for that.
What seems more important for quantum gravity is the fact that the spacetime fabric can be treated as a perfect fluid.
‘… the source of the gravitational field can be taken to be a perfect fluid…. A fluid is a continuum that ‘flows’... A perfect fluid is defined as one in which all antislipping forces are zero, and the only force between neighboring fluid elements is pressure.’ – Bernard Schutz, ‘General Relativity’, Cambridge University Press, 1986, pp. 89-90.
‘The Michelson-Morley experiment has thus failed to detect our motion through the aether, because the effect looked for – the delay of one of the light waves – is exactly compensated by an automatic contraction of the matter forming the apparatus…. The great stumbing-block for a philosophy which denies absolute space is the experimental detection of absolute rotation.’ – A.S. Eddington, Space Time and Gravitation, Cambridge, 1921, pp. 20, 152.
So the contraction of the Michelson-Morley instrument made it fail to detect absolute motion. This is why special relativity needs replacement with a causal general relativity:
‘According to the general theory of relativity space without ether is unthinkable.’ – Albert Einstein, Leyden university lecture ‘Ether and Relativity’, 1920. (A. Einstein, Sidelights on Relativity, Dover, 1952, p. 23.)
An overdose of the tensor properties of general relativity can give the reader the false impression that nature is mathematical not physical. This seems to be the case with the current interest in string theory. I don't doubt that extra dimensions, particularly the 4th dimension and possibly the M-theory compactified dimensions, have a mathematical equivalence to reality. But the answer might not be found by getting ever more mathematical. Electromagnetism is weird enough, with forces being propagated along electric and magnetic field lines, to believe that there could be some extra dimensional significance behind it. But you need a physical theory tied to experimental facts even more if there are extra dimensions, because you will get in a real tangle without sticking to physical reality. The vast number of ideas about branes and ways to deal mathematically with extra dimensions show this.
‘Children lose interest … because a natural interest in the world around them has been replaced by an unnatural acceptance of the soundness of certain views, the correctness of particular opinions and the validity of specific claims.’ – David Lewis, You can teach your child intelligence, Book Club Associates, London, 1982, p. 258.