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Exact stationary axially symmetric solutions of the 4D Einstein equationswith corotating pressureless perfect fluid sources are studied. A particularsolution with approximately flat rotation curve is discussed in some detail. Wefind that simple Newtonian arguments overestimate the amount of matter neededto explain these curves by more than 30%. The crucial insight gained by thismodel is that the Newtonian approximation breaks down globally, even though itis valid locally everywhere. No conflict with solar system tests arises.Comment: 14 pp, 4 eps figures; v3: considerably extended version with added  explanations and references, changed title, invited contribution to IJMP

NON-NEWTONIAN BEHAVIOR IN WEAK FIELD GENERAL RELATIVITY FOR EXTENDED ROTATING SOURCES