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:

- The vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns.
- The gravitational constant G = 6.67430(15)*10**(-11) (MKS units) is the same as for Newton's law of universal gravitation.
- The cosmological constant Λ
which Einstein instroduced in
1917 for
cosmology and used
to obtain the
Einstein universe (1917).
It is still only needed for
cosmology, but including
in cosmology the
large-scale structure of the universe
when the highest level of accuracy is needed for the
large-scale structure of the universe.
It was a new hypothetical
fundamental constant.
Einstein did
**NOT**initially call Λ the cosmological constant, but the name got attached to it later somewhere. Einstein called Λ the cosmological member in 1945 (Cormac O'Raifeartaigh, Historical and Philosophical Aspects of the Einstein World, 2019, p. 18). - The Einstein tensor G_ij = R_ij-(1/2)Rg_ij which is a tensor describing the geometry of spacetime. The Einstein tensor is constructed using the Ricci curvature tensor R_ij and the Ricci scalar curvature R.
- The metric tensor of general relativity g_ij.
- The energy-momentum tensor T_ij.

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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2016
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