Twins Paradox

The supposition that time differences between events are not absolute in reality was first expressed by Einstein in his famous work of 1905. He deduced from the equation for the dilation of time that the relation between the time data δτ, δt of clocks situated at the equator and at the pole, respectively, will be the result of the rotation of the earth with the circumferential velocity u ≈ 0.465 km/s (c ≈ 300,000 km/s is the speed of light):

Thus a clock at the equator should be delayed approximately 8/100 microseconds per day with respect to the clock at the pole.

Equation (1) determines the proper time of a moving clock in the inertial system of reference. In the case of time-dependent velocity u(t) it is necessary to take it as a differential equation and integrate it. Equation (1) represents a special case of the [Page 1383]conclusion of the relativity theory that the actual time is (apart from the constant multiplying factor) equal to the length of a section of the world line between the events (spacetime points) P and Q:

The clock paradox in its broadest sense means the existence of a difference between the actual times measured by the clocks moved in different directions between two meetings. from the geometrical point of view, this means different lengths of appropriate parts of world lines in spacetime. In the case of long and rapid motions, this difference will be reflected also in the biological age of organisms (according to A. Eddington, our faces represent clocks indicating passing years). Thus it is especially impressive when speaking about the twins paradox

The most common and most often discussed form of the paradox is the following: One of two identical twins will remain on the earth while the second makes a cosmic flight proceeding with the constant velocity u close to the speed of light. In the vicinity of a distant star the wandering twin will change the direction of his flight in a very short time interval compared to the total duration of the flight, and he will return to Earth. When the twins meet, the difference in their respective ages, corresponding to equation (1), will be immediately apparent.

The word paradox as used here refers primarily to the strangeness of the conclusion, whose theoretical and experimental legitimacy is beyond doubt. Nevertheless, even now (and sometimes even in physics publications) opinions are offered that the contradiction is real. They are based on the relativity of time. from the point of view of the wandering twin, motionless in his own system ∼, it is the earth-bound twin (at rest in the system S) who recedes and then again approaches with the velocity u. Could not the comparison of times give the opposite result for him?

The standard answer claims that the systems S and σ are not equivalent. Only system S is inertial for the whole time (no regard is taken of negligible effects connected with the motions and gravity of the earth and the sun). In order to return to the earth, the wandering twin changed the direction of his flight and felt great inertial forces in connection with this. His system is not permanently inertial and the use of equation (1) is consequently justified only in system S.

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