The Einstein field equations displayed:
8πG G_ik = ---- T_ik Original form. c**4 8πG G_ik + Λ*g_ik = ---- T_ik Cosmological constant form. c**4The terms and factors are:
We will NOT explicate tensors here, but they are a compact way of writing a set of real numbers with a rather complex relationship to each other. But just for some insight, the tensors in the Einstein field equations can be represented by 4 X 4 matrices.
The Einstein field equations are, in fact, as set of differential equations---written very compactly---that are posited as true at every point in spacetime. Their solutions give the geometry of spacetime and this geometry is the manifestation of gravity in general relativity.
The general relativity geodesic equation tells mass-energy how to move in spacetime.
But, of course, mass-energy affects the geometry of spacetime via the Einstein field equations. So a real solution of geometry and motion for a system must be a simultaneous self-consistent solution of the Einstein field equations and the GR geodesic equation for all mass-energy in the system This is hard and this why general relativity is hard to apply. To current knowledge circa 2022, there are, in fact, only 6 top-level (i.e., general and important) exact solutions in general relativity (see Relativity file: general_relativity_exact_solutions.html).
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