Friedmann, "On the curvature of space" (1922) and "On the possibility of a world with constant negative curvature" (1924), chose a hyperbolic line element in which the spatial part was the hemisphere model equipped with a Riemann metric. There were two unknown parameters, R2 and M 2, that multiplied the space and time components of the metric, respectively, and were required to satisfy the Einstein field equations.
The Friedmann model of the universe forms the basis of the Standard Model. Yet, it is usually considered not as a pair of evolution equations, but, rather, a condition where the Hubble parameter and average density of the universe are determined empirically and the sum of all the densities, the mass density, the curvature term, and the vacuum energy density (cosmological term), add to unity.
The Euler-Lagrange equation in general relativity contains an additional term to the acceleration involving the Christoffel connection coefficients. In the weak field limit, Newton's law of gravity "pops" out of this term yielding Newton's second law for gravitational attraction. This supposedly supports the notion that non-Euclidean geometry with non-constant curvature is somehow related to gravitation.
It is quite remarkable that no one has questioned the physics of Einstein's law of gravitation,
where Rik is the Ricci tensor, as a condition of "emptiness". "Emptiness" is taken to mean the absence of matter and energy, but the presence of gravity which supposedly does not destroy the [sic] "emptyness", according to Dirac.
Einstein's law of gravitation assumes that all components of the Ricci tensor,
No matter how hard one tries, the marriage between electromagnetism just can't be made to happen. Electromagnetism is a linear theory, gravity is not. Statically, both forces satisfy an inverse square law, and although there are two types of charge, there is a single mass. But, if we try to linearize gravity shouldn't there be some common ground?
The indefinite metric of general relativity supposedly determines how a 'test' particle will evolve in spacetime. The Schwarzschild solution is usually employed in the discussion of the formation of black holes since it is the simplest one known: no electric charge, non-rotating, and that consists of a ststic, central, mass M. M enters through any arbitrary constant of integration, but plays a dominant role thereafter.
The Schwarzschild solution had been around for decades before anyone ever dreamed of the idea that it could harbor a black hole and emit gravitational waves. The question is why? Surely, the mathematical prowess of the likes of Edmund Whittaker would have surely discovered these phenomenal properties of the Schwarzschild solution to the static Einstein field equations had they existed. So what is the true story of the discovery---or lack thereof---of black holes and gravitational waves?