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General relativity (GR) is a description of gravity in terms of spacetime geometry, characterizing acceleration due to gravitational force as constant motion within a curved spacetime. In other words, what I feel as the Earth's gravitational force is an artifact of the space I'm occupying moving in that direction relative to the Earth (and since I'm standing still on Earth, I'm moving "upward" relative to that space, and in fact, accelerating within an inertial reference frame).
General relativity consists of the mathematical equations that provide such a characterization that is virtually consistent with the observed effects of Newton's laws on our more familiar (Euclidean) space-time concept, and with special relativity. GR relegates gravitational force to be an artifact, like centrifugal force or Coriolis force. Einstein completed its development 1916 after years of effort, his earlier special relativity having been developed in 1905. General relativity remains the current favored model of gravity, based on its unique ability to explain the common, intuitive behavior of gravity as well as observed extreme cases that defy that "common" behavior. It is encapsulated in Einstein's field equation aka Einstein's equation:
1 8πG Rμν - ——— gμνR + gμνΛ = ———— Tμν 2 c4
μ and ν each indicate one of the four dimensions of spacetime, i.e., Rμν, gμν and Tμν each indicate a relationship between the four individual scalar values (a tensor, e.g., a 4×4 matrix).
Of interest is Λ, the cosmological constant, which Einstein included to compensate for the equation's implication that the universe would otherwise be expanding or contracting. Upon Edwin Hubble's later discovery of the visible expansion of the universe, Einstein no longer saw reason to include it. More recent observations showing expansion not following general relativity's predictions has induced physicists to revive the term to make the equation to match observed reality, hypothesizing dark energy to explain it.
GR is proven and accepted (current everyday technology requires GR theory to be taken into account, including your phone's GPS), and despite occasional news headlines announcing "Einstein was right again", physicists have long known that the workings of the real world are not explained without GR. However, modifications to GR are explored in hopes of meshing GR with quantum theory and in explaining various unexplained phenomena. Such modified GR theories are currently of interest because advances, such as those in observing black holes can offer tests that could confirm some modified version of GR (and "Einstein right again" announcements are made when proposed modifications fail to improve on existing GR in such a test). The term Einstein gravity is sometimes used to distinguish his version of GR from proposed modifications.