One of the most important changes general
relativity forced in physical theory was the transition from modelling
space-time as a flat (vector) space to modelling it as a curved space. The
former is globally Euclidean (a
formal description of flat space): the latter is locally
Euclidean, but deviates from this behaviour in large
regions. The
mathematical formalism for describing curved spaces is the smooth
manifold.
A smooth manifold is a topological space with an open cover in which the
covering neighbourhoods are all smoothly isomorphic to one another. One
generally defines smoothly
for these purposes in terms of a smooth atlas. As time goes by, I'll doubtless fill
that definition in with hyper-links to explain the jargon it contains. For the
present, what matters is that we obtain a notion of smoothness for mappings to
and from the manifold; notably, scalar functions of position (mappings from the
manifold to {scalars}) and paths, a.k.a. trajectories (mappings from {scalars}
to our manifold). From this we can obtain a full tensor bundle for the smooth
manifold and a notion of smoothness for tensor fields on the manifold.
Given the tensor bundle, we are in a position to describe a Riemannian or pseudo-Riemannian geometry on the manifold. This suffices to allow us the illusion of a Euclidean or pseudo-Euclidean view of all sufficiently small regions of the smooth manifold. The deviations from true (pseudo-) Euclidean geometry in insufficiently small regions can then be understood in terms of curvature, for the meaning of which we must look at differential operators on the manifold.
We will in any case need differential operators on our manifold if we are to build anything like our usual physical theories: we must be able to differentiate arbitrary smooth tensor fields (and the relevant derivatives must possess at least some sensible properties). This, in fact, leaves a fair amount of freedom to chose differential operators: however, if we insist on a differential operator which considers our metric (a tensor encoding the geometry) to be constant, this suffices to specify our differential operator uniquely (and there is such a differential operator).
A differential operator is a special case of a Leibniz operator: it can be shown that any
Leibniz operator which annihilates all scalar fields acts as a linear map on
each rank of the tensor bundle: and that the antisymmetric self-product of any
Leibniz operator is a Leibniz operator (with twice the rank, of course). Given
that one of the sensible properties
referred to above is that our chosen
differential operator's square must act symmetrically on all scalar fields, we
now have a linear map describing its antisymmetric self-product's action on
gradients: this is known as the Riemann
tensor; it encodes the curvature of our manifold. A contraction applied to
the (fourth-rank) Riemann tensor delivers the (second-rank) Ricci tensor, which General Relativity uses
to characterise the curvature of space-time.
Written by Eddy.