A commonly accepted modern SR/Gr statement is "timelike geodesics are maxima of the proper time". I really shall not dispute with these experts, who are all professors. But this statement is clearly questionable. It bothered me for a long time. The H & T experiment and GPS all show the statement is questionable. As I mentioned in another thread, in a gravitational field, an orbiter's world line is at geodesic and a standing person's worldline is not a geodesic, but the clock of the orbiter is slower than the standing person. This contradicts that statement. My guess is that the statement shall be altered as "Timelike geeodesics are the maxima of the proper time compared with an interseting worldline that share the same local comoving inertial framet at the initial intersecting event point."