An analytical treatment of the clock paradox in the framework of the special and general theories of relativity

摘要

In this paper we treat the so called clock paradox in an analytical way by assuming that a constant and uniform force F of finite magnitude acts continuously on the moving clock along the direction of its motion assumed to be rectilinear (in space). No inertial motion steps are considered. The rest clock is denoted as (1), the to-and-fro moving clock is (2), the inertial frame in which (1) is at rest in its origin and (2) is seen moving is I and, finally, the accelerated frame in which (2) is at rest in its origin and (1) moves forward and backward is A. We deal with the following questions: (1) What is the effect of the finite force acting on (2) on the proper time interval Deltatau((2)) measured by the two clocks when they reunite? Does a differential aging between the two clocks occur, as it happens when inertial motion and infinite values of the accelerating force is considered? The special theory of relativity is used in order to describe the hyperbolic (in spacetime) motion of (2) in the frame I. (II) Is this effect an absolute one, i.e., does the accelerated observer A comoving with (2) obtain the same results as-that obtained by the observer in 1, both qualitatively and quantitatively, as it is expected? We use the general theory of relativity in order to answer this question. It turns out that Deltatau(I) = Deltatau(A) for both the clocks, Deltatau((2)) does depend on g = F/m, and Deltatau((2)) /Deltatau((1)) = (root1-beta(2)atanhbeta)/beta < 1. In it β = V/c and V is the velocity acquired by (2) when the force is inverted.