Before stating Einstein’s equation, we need a little preparation. We assume the
reader is somewhat familiar with special relativity — otherwise general relativity
will be too hard. But there are some big differences between special and general
relativity, which can cause immense confusion if neglected.
In special relativity, we cannot talk about absolute velocities, but only reative velocities.
For example, we cannot sensibly ask if a particle is at rest,
only whether it is at rest relative to another. The reason is that in this theory,
velocities are described as vectors in 4-dimensional spacetime. Switching to a
different inertial coordinate system can change which way these vectors point
relative to our coordinate axes, but not whether two of them point the same
way.
In general relativity, we cannot even talk about relative velocities, except for
two particles at the same point of spacetime — that is, at the same place at the
same instant. The reason is that in general relativity, we take very seriously the
notion that a vector is a little arrow sitting at a particular point in spacetime.
To compare vectors at different points of spacetime, we must carry one over to
the other. The process of carrying a vector along a path without turning or
stretching it is called ‘parallel transport’. When spacetime is curved, the result
of parallel transport from one point to another depends on the path taken! In
fact, this is the very definition of what it means for spacetime to be curved.
Thus it is ambiguous to ask whether two particles have the same velocity vector
unless they are at the same point of spacetime.
It is hard to imagine the curvature of 4-dimensional spacetime, but it is easy
to see it in a 2-dimensional surface, like a sphere.
