TWIN PARADOX REAL LIFE SIMPLE EXPLANATION

 Twin Paradox Real Life Simple Explanation 

As the postulates of special relativity lead to result which contradict 'common sense' a number of interesting paradoxes have been floated. We shall describe one of the most famous paradoxes of relativity- the twin paradox. consider the twins Ram and Balram living happily on the earth. Ram decides to make a trip to a distant planet p, which is at rest with respect to the earth, and come back. He boards a spaceship S1, going toward the planet with a uniform velocity. When he reaches the planet, he jumps from the spaceship S1 to another spaceship S1 which is going toward the earth. When he reaches the earth, he jump out and meets his brother Balram. 

As Ram returns from his trip and stands next to Balram, do they have equal age ? Or is Ram younger than Balram or is he older than Balram ?

To keep the calculation simple, let us assume the following date:

Distance between the earth and the planet = 8 light-years,

speed of S1 with respect to earth = 0.8c, and speed of S2 with respect to earth = 0.8c.

When we said that the distance between the earth and the planet P is 8 light-years, was it clear to you that this length is the length as measured from the earth frame ?

First, let us analyse the events from the point of view of Balram who is on the earth. For him, both the spaceships move at a speed 0.8c. So,

γ = 1/√1-v^2/c^2 = 1/0.6 . 

When Ram is on S1, he is moving and all his clocks run slower because of time dilation. His heartbeat, pulse beat, etc., represent clocks in themselves and they all run slower. Balram calculates that Ram will take 8 light-year/0.8c = 10 years to reach the planet P. But during all these 10 years, time is passing slowly on S1 and the clock will read only 10 yearsｘ 0.6 = 6 years in this period. The number of breaths taken by Ram corresponds to 6 years only.

Ram jumped into S2 for the return journey. This spaceship is also moving at 0.8c and for Balram , time passes slowly on S2 as well. Although 10 years passed on the earth during Ram's return journey, on the spaceship the journey was clocked at 6 years. Thus, Ram has aged only 12 years whereas Balram has aged 20 years during this expedition. Ram has become younger than Balram by 8 years. This difference in aging is real in the sense that Ram shows lesser signs of aging like he has lesser white hairs than his brother.

The observation of Balram is quite consistent with the special theory of relativity. Such experiment are indeed performed in laboratories with radioactive particles. particles are accelerated to large speed and are kept at these speed for quite some time by magnetic fields. These particles with large speeds have longer lives than their counterparts kept at rest in the laboratory.

The paradox arises when we analyse the events from the point of veiw of Ram. When he is in the spaceship S1, to him the distance between the earth and the planet is not 8 light-years. The earth and the planet P  are moving with respect to Ram and hence he is measuring contracted length. The separation is, therefore, 8 light-years ｘ0.6 = 4.8 light-years. As the planet is approaching Ram at 0.8c, the taken by the planet to reach Ram is 4.8 light-year/0.8c = 6 years. So according to Ram's clock, he jumped from S1 to S2 6 years after getting into S1. Once he is on S2, the earth and the planet are again moving with the same speed 0.8 Again, the earth is 4.8 light-years from the planet and is approaching at 0.8c. It takes 6 years for the earth to reach Ram. Thus, according to Ram's clock, he was out for 12 years from the earth, the same result as Balram had expected.

But how about Ram's calculation of Balram's age ? When Ram is on S1, the earth is going away from him with a speed 0.8c. Ram will find that the time on the earth is passing slower by a factor of 0.6 so that Balram is again slower than he is. The same is true when he is on S2. During this period also,  Balram is moving (toward Ram) with a speed 0.8c and hence time is passing slowly for Balram. As 12 years passes on Ram's clock, he calculates that Balram's clocks have advanced only by 12 years ｘ0.6 = 7.2 years in this period, According to this analysis, Ram should find that Balram is 12 - 7.2 = 4.8 years younger than him. 

This is the paradox. According to Ram, Balram's clocks are running slow and according to Balram, Ram's clock are running slow. Each thinks the other is younger. Where lies the fallacy ?

The fallacy lies in the fact that Ram has changed frames whereas Balram has stayed in an inertial frame. Thus, the roles of the twins are not symmetrical. The ordering of events are different in different frames and Ram must take that into account when he changes frames. Suppose Ram gets into the spaceship S1 when his clock reads zero. So does Balram's clock. What is the reading of the planet's clock at this instant ? According to Balram, it is zero because both the earth and the planet are at rest and the clock are synchronized in his frame. But that is not so in S1. As Ram gets into S1, he may have the following conversation with the captain of the ship. 

Captain: Welcome aboard S1. I saw you on the earth, coming toward us. Your jump to board this ship was perfect. Where are you going ?

Ram: Thank you. I am going to the planet P . How far is it from here and how long will it take for the planet to come to us ?

Captain: Planet P is 4.8 light-years from  us at the moment. It is coming toward us at a speed of 0.8c so it will take 4.8 light-years/0.8c= 6 years for the planet P to reach us.

Ram: Well the clock on the earth and the planet are running a bit slower than ours. I have been taught that moving clock run slow by a factor of γ. This factor is 1/0.6 for these clocks. So they will advance by 6 years ｘ0.6 = 3.6 years by the time the planet reaches us.

Captain: Yes, both the clock will advance by 3.6 years by the time you jump on the planet P.

Ram: The earth-clock was reading t = 0 as we passed the earth. This means when I jump on the planet P the clocks on the earth and the planet will be reading 3.6 years.

Captain: Here you are mistaken. Don't you remember that the planet's clock is not synchronized with the earth's clock ? The planet's clock is at the rear end, and hence is running 6.4 years ahead of the earth's clock. At the instant the earth's clock was reading zero, the planet's clock was reading 6.4 years. As the planet reaches us, both the clocks will advance by 3.6 years. So when you jump out of S1, the earth's clock will be reading 3.6 years but the planet's clock will be reading 10 years.

Ram understands the logic. In the earth's frame the two clocks read zero simultaneously. But in S1-frame, the event ''planet's clock reading zero'' occurred several years before '' earth's clock reading zero''. Six years pass in S1 and Ram finds that the planet P has reached him. He finds another spaceship S2 which is heading towards the earth. Ram jumps onto S2. In the process he looks at the planet's clock and finds that it is reading 10 years as calculated by him on S1. On S2, he starts talking to the commander of the ship.

Commander: Welcome to S2. How long will you be with us ? 

Ram: Thank you. I am going to Earth. Earth is at present 4.8 light-years from here and is coming toward us with a speed of 0.8c. So I will be with you for 6 years. The captain of S1 told me that the earth's clock is reading 3.6 years at this moment whereas the planet's clock reads 10 years. There is a difference of 6.4 years in the reading because the two clock are not synchronized. Also....

Commander: Sorry for interrupting you, but you are mistaken. It is true that the earth's clock and the planet's clock are not synchronized as they are moving past us. Also the difference in the reading of the two clocks is 6.4 years. But the planet's clock is at the front and the earth's clock is at the rear. It is the earth's clock that is leading by 6.4 years. At the moment the planet's clock reads 10 years and hence the earth's clock must be reading 16.4 years.

Ram: hmm... you are right. In S1, the earth was at the front and its clock lagged behind the planet's clock. But in S2 it is the other way round. Indeed the earth's clock reads 16.4 years whereas the planet's clock reads 10 years.

Commander: That's right. The earth's clock is reading 16.4 years at present. It will advance by another by another 3.6 years during the 6 years you will be with us. So it will be reading 20 years when the earth reaches you.

We see that the paradox is resolved.