Consider a universe with Newtonian gravity (no general relativity), but with special relativity. (Is this even possible? Newtonian gravity might be fundamentally incompatible with special relativity.) We probably also want point masses. Does this combination yield interesting dynamics?

Consider an object in such an extremely eccentric orbit that it approaches the speed of light near periapsis. Does the fact that apparent mass increases at high speeds cause interesting effects? Perhaps periapsis precession, reminiscent of general relativity. Consider an even more eccentric orbit whose orbital speed would exceed the speed of light at periapsis. How would special relativity prevent that?

The Painleve Conjecture and "Off to infinity in finite time" showed that with just Newtonian gravity, point masses can get accelerated outward to infinite speed. How would special relativity prevent that?

Probably relevant: Ehrenfest (rotating disc) paradox. It might force general relativity as a consequence of special relativity, especially if we want the equivalence principle.

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