- The spacetime of experience [at the macroscopic level] is well-characterized
as a smooth four-dimensional manifold with a metric and a connection compatible
with that metric.
- The metric is locally Minkowskian, so that for nearby points in spacetime
we recover the effects of Special Relativity.
- The effects of a permanent gravitational field are encoded in the Curvature
of spacetime and are discernable by the relative acceleration of observers
moving along geodesic paths in the existing spacetime.
- The motion of test particles [one's with negligible mass-energy] is given
by the geodesic paths in the spacetime
- This geometrical structure is not static; the local curvature is affected
by the distribution of matter at that point. The matter-geometry connection
constitutes one of the central tenants of the theory and it relates some
aspects of curvature to some aspects of material phenomena:
Contracted Curvature <=> Energy-Momentum per unit Volume
The Einstein program: Given the descriptors for the energy and momentum
content of the physical system one wishes to characterize, then
- Solve the 10 nonlinear partial differential equations for the components
of the metric that satisfy the Einstein field equations
- Use the solution's metric components to calculate the full curvature and,
as a result, characterize the gravitational field throughout the spacetime
- Use the solution's metric compatible connection to determine the motion of
test particles, including the motion of light rays.