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Is motion of a participant with constant proper acceleration, in a flat region, necessarily straight, hyperbolic motion (with respect to members of any inertial system, in that region)?

Or is for instance a participant who is moving at constant speed along a circular trajectory wrt. members of an inertial system said to have moved with constant proper acceleration, too?

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For motion along a circular trajectory the acceleration changes its direction, and is not constant.

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    $\begingroup$ Jerrold Franklin: "For motion along a circular trajectory the acceleration changes its direction" -- That's certainly true of the acceleration vector evaluated in reference to any inertal system (since the members of an inertial system, in a flat region, are not rotating). But if we're considering proper acceleration, i.e. the acceleration of the participant under consideration, at some instant, wrt. the corresponding instantaneously comoving inertial system then ... the notion of "(constancy, or change of) direction" may be more difficult; perhaps just a matter of convention. $\endgroup$
    – user12262
    Commented Sep 3, 2015 at 20:15

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