Relativity Series-5: Time and Special Theory of Relativity – Twin Paradox

Twin Paradox

Suppose there are two twin brothers, one of them remains on earth while the other goes for a space travel at a very high speed (comparable to the speed of light). The astronaut comes back from the voyage after certain number of years to see that his brother has aged more. 200 years back this would pass for a story in science fiction. But in the post-theory-of-relativity era, it’s as comprehensible as sun rising in the east. This phenomenon is called as twin paradox. This concept is also covered during our Physics tuition classes for Singapore-Cambridge H3 syllabus students.

The paradox just suggests that the phenomenon is seemingly paradoxical although it’s not a paradox as such. As said in the previous article, it is one of the consequences of special theory of relativity. One of the postulates of special theory of relativity is that the speed of light (in vacuum) remains constant irrespective of observer’s motion. Let’s take a hypothetical example where Mr. A was moving at a speed of ‘0.5 c’ (‘c’ is the speed of light in vacuum) and Mr. B is at rest with respect to some reference. Now when Mr. B sees light, it appears to move at speed ‘c’. This makes sense and does not violate our notions. As per the discussion in the article Relativity Series-1, when Mr. A sees light, it should be moving at 0.5 c. This is because, as per conventional principles of relative motion when both Mr. A and the car (in the example cited in the first article) were travelling at same speed with respect to some reference, Mr. A sees the car to be stationary. And similar should be the case with light. But the postulate of special theory of relativity and the observation of Michelson and Morley state otherwise. As per relativistic consequences, Mr. A should also see that the light appears to be moving at speed ‘c’. But how can both Mr. A and Mr. B, who are in relative motion with each other, see light to be moving at the same speed? The answer is time dilation.

When Mr. A is travelling at a very high speed, his clock ticks slowly which ensures that the speed of light remains constant for both Mr. A and Mr. B. So when light travels 3 x 108 meters with respect to Mr. B he notes it as one second. But for Mr. A, who himself is travelling at 1.5 x 108 m/s (with respect to Mr. B), the light appears to have moved only 1.5 x 108 meters. Speed of light should remain constant, given by c = 3 x 10m/s. We know that speed = distance travelled / time, since the distance travelled by light is 1.5  x 108 m w.r.t Mr. A, the time light takes to travel that distance is 0.5 seconds and hence when Mr. B’s clock ticks 1 second, Mr. A’s clock ticks only 0.5 seconds. This happened because of the motion of Mr. A with respect to Mr.B. So when one of the twin brother’s went for a space trip and other was dwelling on earth, due to the relative velocity of the astronaut brother with respect to his twin, his clock ticks slow and hence he aged less.