Non-accelerated observers follow geodesics in spacetime.
In Special Relativity, spacetime is flat, so these geodesics are
simply straight worldlines. In General Relativity, the effect
of matter on space (gravity) is taken into account. Gravity is
measured as a curvature of spacetime, which is represented in
Geodesics follow a rule called the Geodesic Equation. The Geodesic
This equation gives the shape of the geodesic in an arbitrary
coordinate system in terms of a parameter (such as the proper
In General Relativity, gravity is seen as a distortion of spacetime,
which is referred to as a curvature (because of its similarities
to a curved 2-dimensional surface in 3-dimensional space). However,
in the case of gravity, a 4-dimensional spacetime is being curved,
and, as far as is known, there is no "super-space" in
which it resides.
One representation of curvature is the Metric Tensor,
The metric tensor explains how distances relate to changes in
Another, more direct representation, is the Riemann Curvature
Tensor. The curvature tensor is given by
and is the four-dimensional equivalent of the curvature of a two-dimensional
surface. The Riemann tensor measures how quickly geodesics separate
from one another.
Some part of the curvature of spacetime is generated by the local
presence of matter and energy. That part is measured by the Einstein
The Einstein Tensor, therefore, is only affected by local considerations, and is unaffected by gravitational waves from distant sources (which result in a change to the Riemann Tensor).