Resolving the Incompatibilities Between Special Relativity and Newtonian Gravity

Inspiring the search for General Relativity
There are two manifest incompatibilities between Einstein’s special theory of relativity and Newton’s theory of Gravity.
1) In special relativity, nothing, not even information, can travel faster than c, the speed of light. If this was allowed, then it would be possible to violate causality, leading to effects preceding their causes. However, in Newtonian Gravity, the gravitational field propagates instantly. If we somehow wiggled the sun up and down, this effect would instantly appear in the gravitational field felt on Pluto. We could essentially send Morse Code messages to Pluto (or anywhere else in the universe) instantly by wiggling the sun up and down (actually, it wouldn’t need to be the sun – with sufficiently sensitive gravity detectors, a much smaller objects gravity would do just as well).
2) In Newtonian gravity, F_g=Gm₁m₂/|x₁-x₂|², which retains its form under a Galilean Transformation. However, in special relativity, the Galilean transform is not a true symmetry of space-time. Instead, we have the Lorentz Transform.
Space-time coordinates are now represented as a four dimensional vector:
x^ì=(x⁰,x¹,x²,x³)
x⁰=ct
x`=M^mu_nux^nu where M^mu_nu is the four by four Lorentz matrix that describes “boosts” (transforms into frames moving relative to the original frame, which mix the space and time coordinates) and rotations. Maxwell’s equations of Electromagnetism are invariant under the Lorentz Transform, but not under the Galilean Transform. This fact was part of the evidence that lead Einstein to Special Relativity.
But Newton’s law of gravity is simply incompatible with the Lorentz Transform.
Like every major breakthrough, general relativity came from the intellectual tension caused by the clash of incompatible theories, in this case Special Relativity and Newtonian Gravity. These events closely follow Kuhn’s theories of a paradigm shift.