The Twin Paradox.

Today I want to talk about something fascinating. Something astray from my usual theme on this blog. Ever since I was a kid ( five I think? ) I have been obsessed by the huge concept of space and time. I have recently been reading A Brief History of Time by Stephen Hawking which has opened my mind to a plethora of topics. For instance, about an Irish philosopher ‘Bishop Berkeley’ who believed that all material objects and space and time are an illusion (seriously?). But today I’m not going to talk about ancient philosophers and their beliefs. Apart from that, I’ve been reading about Einstein’s special theory of relativity which is what I want to focus on today.

One cannot deny the fact that it has revolutionized our understanding of space, gravity, time, and our universe. It has revolutionized our way of thinking. Einstein’s general theory of relativity implies that the universe must have a beginning and, possibly, an end. As Hawking says ‘ The old idea of an essentially unchanging universe that could have existed, and could continue to exist, forever was replaced by the notion of a dynamic, expanding universe that seemed to have begun a finite time ago, and that might end at a finite time in the future.’ So basically it states that space and time are not independent of each other, rather they must be considered as the same object, space-time. One of the consequences of space-time mixing is time dilation. One of the most fascinating thought experiment, possibly, in special relativity is the twin paradox or clock paradox. But to understand that we would first have to delve deep into the concept of time dilation. So it goes like this: we do know that the time measured from such a frame where the two events occur at the same place is called proper time interval and the time interval measured by a frame where the events occur at different places is called improper time interval. Now, this is where the concept of time dilation comes in. The proper time interval between the occurrences of two events is smaller than the improper time interval by a factor Y. This phenomenon is called time dilation. Now that that’s sorted out let’s talk about the twin paradox. It’s going to get complicated so if you don’t want your brain to get all fuzzy you should probably leave now.

Still with me? That’s the spirit. Okay here goes. Imagine that there are two identical twins. Let’s name them S and H (to avoid complications). Let’s consider a hypothetical situation where H travels to another planet (let’s name it Vulcan) on a spaceship S1 whereas S is left on Earth. So H reaches the planet Vulcan and takes another spaceship S2 on his return journey to travel back to Earth. The question here is whether the ages of S and H would remain the same when they meet again?

Let’s access this situation now. Imagine that the distance between the two planets is 8 light-years. Then the speed of S1 with respect to Earth would be 0.8c. Here we consider the viewpoint of S (the one left behind). The speed of S1 and S2 can be given by the formula γ=1√1−v2c2 γ = 1 1 − v 2 c 2 = 1/0.6. According to this S calculates that the time taken for H to reach the planet Vulcan would be 10 years. But the time is passing slowly on S1 due to time dilation so the clock on S1 reads 10×0.6=6 years. Now H jumped to S2 on the return journey. So the speed of S2 according to S is 0.8 c. Now time passes slowly on S2 as well. Although 10 years have passed on the return journey for H only 6 years have passed. Thus H has aged only 12 years whereas S has aged 20 years. Now let’s consider the viewpoint of H. When H is on spaceship S1, the distance between Earth and Vulcan is 8 light-years×0.6 = 4.8 light-years. (The Earth and Vulcan are moving with respect to H and hence he is measuring contracted light.) So time taken by H to reach Vulcan will be 6 years( 4.8 light years/0.8 c = 6 years). So according to H, he jumped from S1 to S2 6 years after getting into S1. Once he is on S2, the Earth and Vulcan are moving again with the same speed 0.8c. Again the Earth is 4.8 light-years from Vulcan and is approaching at 0.8c. Again it takes 6 years for Earth to reach H. So according to H’s clock, he was away for 12 years from Earth. Now if H will calculate S’s age (one left on Earth): When H was on S1, Earth is going away from him with a speed 0.8c. So time on Earth is passing slower by a factor of 0.6. Then H concludes that S is aging slower than him. The same goes for S2, S is moving towards H so time is passing slowly for S. As 12 years pass on H’s clock, he calculates that S’s clock has advanced by only by 12 years×0.6 = 7.2 years in the period. According to this analysis, H should find that S is 12-7.2=4.8 years younger than him.

Then how to dissolve this paradox? You see this situation here is only a paradox if one has the idea of absolute time at the back of one’s mind. In the theory of relativity, there is no absolute unique time, but instead, each individual has his characteristic measure of time that depends on where he is and how he is moving. In S1 the Earth was at the front and its clock lagged behind the Vulcan’s clock. Here the Vulcan’s clock is at the rear end and hence is roaming 6.4 years ahead of the Earth’s clock. At the instant Earth’s clock was reading 0, the Vulcan’s clock was reading 6.4 years. As the Vulcan reaches H, both the clocks will advance by 3.6 years. So when H gets out of S1, Earth’s clock will be reading 3.6 years but Vulcan’s clock will be reading 10 years. However, in S2, this is reversed. The Vulcan’s clock is at the front and Earth’s clock is at the rear. It is Earth’s clock that is leading by 6.4 years. At that moment, Vulcan’s clock will read 10 years and hence Earth’s clock must be reading 16.4 years. Earth’s clock will advance by another 3.6 years during the 6 years of the return journey. So it will now be 20 years when the Earth reaches H. You see how this paradox is resolved. To sum it all up in Hawking’s words ‘No particular measurements are any more correct than any other observer’s but all measurements are related.’

2 thoughts on “The Twin Paradox.”

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