Twin Paradox

What is Twin Paradox

Firstly we try to understand that, what is paradoxical statement? A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.

 As the postulates of special relativity leads to results which contradict "common sense", a number of interesting paradoxes have been floated. We shall must describe the most famous paradoxes of relativity - THE TWIN PARADOX. Consider the twins Jack and John living happily on earth. Jack decided to make a trip to distant planet P, which is at rest with respect to earth and come back. He boards on the spaceship S1, going towards the planet with uniform velocity. When he reaches the planet he jumps from spaceship S1 to the spaceship S2 which is going towards the earth. When he reaches the earth, he jumps out and meet his brother john.

As Jack returns from his trip and stand next to his brother John, do they have equal age? Or is Jack younger than John or is he older than John.

To keep the calculation simple we assume this data:

Distance between the earth and the planet = 8 light years

Speed of S1 with respect to earth = 0.8c

Speed of S2 with respect to earth = 0.8c.

We will calculate time dilation factor which known as (γ). It will come out to be 0.6.

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

 Jack jumped into S2 for the return journey. This spaceship is also moving at 0.8c and for John, time passes slowly on S2 as well. Although 10 years passed on earth during Jack's return journey on the spaceship the journey was clocked at 6 years. Thus, Jack is aged 12 years whereas John has aged 20 years during this expedition. Jack has become 8 years younger than John. 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 observations of John is quite consistent with the special theory of relativity. Such experiments are indeed performed in laboratories with radioactive particles. Particles are accelerated at high speed and kept at these speeds at quite some time by magnetic fields. These particles with large speed 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 Jack. When he is in the spaceship S1, to him the distance between the earth and the planet is not 8 lightyears. The earth and the planet P are moving with respect to Jack and hence he is measuring contracted length. The separation is therefore,  8 light-years x 0.6 = 4.8 light-years. As the planet is approaching Jack at 0.8c, the time taken by the planet to reach Jack is 4.8 light-years/0.8c = 6 years. So according to Jack's clock, he jumped from S1 to S2 6 years after getting into S1. Once he is on S2, the earth are the planet again moving with the same speed 0.8c. 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 Jack. Thus, according to Jack's clook, he was out 12 yers from the earth, the same result as John expected.

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

This is the paradox. According ton Jack, John's clock is running slow and according to John. Jack's clock is running slow. Each thinks the other is younger. Where lies the fallacy?