What's the basic idea?
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Both space and spacetime can either be curved or flat.
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We've described the Euclidean (or Euclidean-Mesopotamian :-) metric in two space dimensions:
dL2 = dX2 + dY2
and we've discussed at some length the complications that arise with the addition of time to space to give the Minkowski metric (shown here in just one space and one time dimension):
dS2 = c2 dT2 - dX2
What else can we do to our spacetime distance function to make life more interesting (and hopefully solve the problem with Newtonian gravity discussed in the last section)?
What if we play around with the form of the Minkowski metric? It turns out that if the spacetime metric is arranged in the right manner, we can get something called spacetime curvature. And that is what the General Theory of Relativity is all about.
For example, suppose we add some extra space and time dependence to the Minkowski metric to make a new spacetime distance function
dS2 = gTT(T,X) c2 dT2 - gXX(T,X) dX2
Using differential geometry, taking the right combination of first and second derivatives of gTT(T,X) and gXX(T,X), we could calculate the what is called the curvature tensor Ruv for this choice of spacetime distance function. The subscripts on Ruv are called tensor indices and refer back to the coordinates used in the above metric. The Minkowski metric corresponds to the choice gTT = gXX = 1 and it has Ruv = 0 for all values of the tensor indices. This is why the Minkowski metric is known also as flat spacetime - because the spacetime curvature calculated from this distance function is zero.
In Einstein's time they were already learning about differential geometry, but Einstein motivated this field of mathematics even more when he came up with an equation relating the curvature tensor of the spacetime distance function to the distribution of matter and energy in spacetime, encoded in a tensor Tuv called the stress-energy tensor.
This equation is now called the Einstein equation:
Ruv - (1/2) guv R = (8 Pi G/c4) Tuv
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This equation (or actually, set of equations, for there is an equation for every combination of tensor indices u and v) models a lot of phenomena in the Universe that was impossible to describe with mathematics just using Newton's law of gravity. For example, observations of the bending of light by gravity, gravitational radiation emitted by pulsars, new observations of black holes and the observed expansion of the visible Universe can all be modelled rather successfully using this elegant formalism uncovered by Einstein.
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