an accelerometer) during the whole round-trip. The Cauchy–Schwarz inequality can be used to show that the inequality follows from the previous expression: Using the Dirac delta function to model the infinite acceleration phase in the standard case of the traveller having constant speed v during the outbound and the inbound trip, the formula produces the known result: In the case where the accelerated observer K' departs from K with zero initial velocity, the general equation reduces to the simpler form: which, in the smooth version of the twin paradox where the traveller has constant proper acceleration phases, successively given by a, −a, −a, a, results in where the convention c = 1 is used, in accordance with the above expression with acceleration phases and inertial (coasting) phases A rotational version Twins Bob and Alice inhabit a space station in circular orbit around a massive body in space. Bob suits up and exits the station. While Alice remains inside the station, continuing to orbit with it as before, Bob uses a rocket propulsion system to cease orbiting and hover where he was. When the station completes an orbit and returns to Bob, he rejoins Alice. Alice is now younger than Bob. In addition to rotational acceleration, Bob must decelerate to become stationary and then accelerate again to match the orbital speed of the space station. No twin paradox in an absolute frame of reference Einstein's conclusion of an actual difference in registered clock times (or aging) between reunited parties caused Paul Langevin to posit an actual, albeit experimentally undetectable, absolute frame of reference: In 1911, Langevin wrote: "A uniform translation in the aether has no experimental sense. But because of this it should not be concluded, as has sometimes happened prematurely, that the concept of aether must be abandoned, that the aether is non-existent and inaccessible to experiment. Only a uniform velocity relative to it cannot be detected, but any change of velocity .. has an absolute sense." In 1913, Henri Poincaré's posthumous Last Essays were published and there he had restated his position: "Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one."<ref>Poincaré, Henri. (1913), Mathematics and science: last essays (Dernières pensées).</ref> In the relativity of Poincaré