The Twin Paradox

As the postulates of special relativity lead to reslts which contradtics our so called common sense, a number of interesting paradox have been found.

A paradox can be simply understood as a statement which is hard to understand as it contains two oppostie facts.

Discussing the Twin Paradox:
Consider 2 twins Ram and Sham living on Earth. Let say Ram decides to make a round trip to a distant planet P.

To understand the situation easily we will assume planet P to be at rest with respect to Earth. He boards a spaceship S₁ goind towards P with a constant velocity. When he reaches Planet P he jumps to another spaceship S₂ which is going to Earth. When Ram returns from his trip will he be older or younger than his twin?.

Lets calculate:
Lets first assume some data to make calculation easier.
Distance between planets = 8 light years.(From Earth frame)
Speed of both S₁ and S₂ = 0.8c (c = speed of light)

γ=1/[1-{(v^2)/(c^2)}]^2
γ= 1/0.6.

So for Sham spaceships move at 0.8c and Rams's journey will take 20 years. But for Ram everything will be 0.6 times slower due to time dilation. So even the number of breaths that Ram takes will be 0.6 times less than he should have. All of his clocks even his heart rate will decrease 0.6 times. So we can say that Ram is now 8 years younger than his twin now....or can we?

The paradox arrises when we look at things from Ram's frame.

As you see for Ram the planets are moving and hence he is measuring contracted length. The separation is 8x0.6 lightyears.=4.8 light years, the time taken by Ram = 4.8/0.8 years = 6 years.

So according to Ram he jumped from S₁ to S₂ 6 years after getting into S₁. Simmilarly time for his return journey will also be 6 years. Thus according to Ram he was out of Earth for 12 years, the same result as Sham expected.

But how about Ram's calculation of Sham's age? When Ram is on S₁, Earth is going away from him with a speed of 0.8c. Ram will find out that the time on Earth will be 0.6 times slower and hence Sham should age 0.6 times slower than he is. The same will also be true for return journey. Hence Ram should find that Sham will age is 12x0.6= 7.2 years.

This is the paradox. According to Ram Sham's clocks are running slow and for Sham Ram's clocks are running slow. Each thinks that other is younger. So who is wrong and who is right??

The problem lies in the fact that Ram is changing frames and also that the clocks on earth are not synchronized.

In Earth's frame both the clocks were at t=0 simultaniously. But in the frame of S₁ the t=0 of Planet P has occoured many years before.

On keeping this in mind we will find that Sham's observations were correct.