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Twin Paradox in Relativity

Twin Paradox in Relativity
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A longer life but it will not seem longer.

The time dilation effect is reciprocal in nature. If an inertial frame S’ is moving with a velocity $ \displaystyle \vec{v}$ with respect to a frame S, then the S -frame observer feels that S’ -clock is running slow whearas S’ – frame observer feels that S -clock is running slow. Does any of the two clocks actually run slow?

Twin Paradox in Relativity

Suppose A and B are twin brothers born at the origin O of the co-ordinate system at a time t0. The brother B is immediately set in motion along the X-axis with a high speed $ \displaystyle v=\frac{{\sqrt{3}}}{2}c$. Suppose he travels for 15 years as measured by his own clock up to the point P and then returns back to O with the same speed and therefore in the same time. In the opinion of the brother B who is in the moving frame S’, he has been travelling for 30 years since his birth and therefore his age is 30 years. This is the proper time $ \displaystyle {{T}_{o}}$ as it has been measured by a clock in the inertial frame in which the event takes place, the event being the ageing of B.

The age of B or the corresponding observed time T as measured by his twin brother A who is staying at O’ in the stationary frame S with respect to which brother B is moving is given by

$ \displaystyle T=\frac{{{{T}_{o}}}}{{\sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}}}=\frac{{30}}{{\sqrt{{1-\frac{3}{4}}}}}=60\text{ years}$

Therefore, the age of B is 60 years from the point of view of the twin brother A who has stayed at O. But when B returns back to O he is only 30 years old.

Twin Paradox in Relativity
Image Credit: Gauri Sharma | Pinterest

If we examine the situation from the point of view of the twin brother B who is in the moving frame S’, the brother A who has stayed at O is moving with a velocity $ \displaystyle \frac{{\sqrt{3}}}{2}c$ in the opposite direction. Therefore we might expect A to be 30 years old upon the return of twin B (who is in the moving frame S’) who himself will be 60 years old at this time – the precise opposite of the result obtained in the preceding paragraph.

This is known as the twin paradox.

To resolve this paradox we must note that the twin B who is in the moving frame S’ is in an accelerated reference frame, because he undergoes an acceleration at various times in his journey, when he takes off, when he turns around, and when he finally comes to a stop. The result of special theory of relativity which hold only frames of reference in relative motion at constant velocity cannot be applied at all to the above situation. We must apply the formulas of general theory of relativity which hold for accelerated frame of reference. According to the principle of equivalence, in this theory a large acceleration produces effects similar to those produced by strong gravitational fields due to which the clock tick more slowly.

Hence, the brother B in the moving frame S’ is indeed younger on his return back to the point O, than his twin brother A in the stationary frame S. It should, however, be clearly noted that B’s life span has not been extended to him since however long his 30 years n the moving frame may have seemed to his brother A, it has been 30 years as far as B is concerned.

The nonsymmetric aging of the twins has been verified by experiments in which accurate clocks were taken on an airplane trip around the world and then compared with identical clocks that had been left behind. An observer who departs from an inertial system and then returns after moving relative to that system will always find his or her clocks slow compared with clocks that stayed in the system.

Reference: Wikipedia

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