General relativity, 2

Newton’s law of gravitostatics is incompatible with special relativity. To see this, suppose at time t in frame O m1 is at x1(t) and m2 is at x2(t). Newton’s gravitational force law says F1on 2(t) = Gm1m2 / [x2(t) − x1(t)]2 and relativistic dynamics says dp2 /dt = F1on 2. Transforming to another frame Oʹ moving relative to O leads to dp′2/dt′ = F′1on 2. But what is F′1on 2 ? If x1(t) and x2(t) are the simultaneous positions of m1 and m2 in O, the transformed positions x′1 and x′2 are not simultaneous in Oʹ. Obviously, Newtonian gravity has to be modified to make it relativistically correct. The same is true for the electrostatic Coulomb force. But, for electric charges there’s also magnetism, and that helps save the day. Clearly, what’s needed for gravity is an analog of magnetism. To create such an analog, Einstein appealed to geometry.