   Chapter 12, Problem 7C

Chapter
Section
Textbook Problem

One interesting (unbelievable?) implication of time dilation is contained in what is called the twin paradox. Imagine identical twins who decide to perform an experiment to test the accuracy of Einstein’s predictions about time dilation. After synchronizing their (identical) clocks, one of the twins enters a spaceship and travels to a distant star at a speed 95 percent that of light. Upon arrival, the traveling twin immediately turns around and heads home at the same speed. According to the twin on Earth, the clock aboard the spaceship runs slower than the one on Earth, SO that the traveling twin ages less than the Earth-hound sibling. But according to the space-faring twin, it is the clock on Earth that is running too slowly, making the stay-at-home twin the younger of the two. This is the paradox: each twin argues that the other will be younger at the end of the trip. How can this paradox be resolved?Which of the two twins is correct in their analysis, and why? If the trip requires a total time of 5 years to make as measured by the clock on Earth, what will be the time recorded for the trip by the clock on board the spaceship?

To determine

The contradiction of twin paradox of two persons in which one is at the earth system and other is moving on the spaceship and also determine the time recorded by the person who is on the spaceship when spaceship is moving with speed of 0.95c.

Explanation

Given info:

Speed of astronomer = v=0.95c

Time measured in earth frame = Δt=5years.

Formula used:

Formula for time dilation is defined as,

Δt=Δt'1 v2 c2Δt= time interval measured by frame that is at restΔt'= time interval measured by frame that is moving.

Explanation of twin paradox- Lorentz transformation equation is defined for the inertial frame of reference (the frame that is moving with constant speed with respect to other frame) and time dilation equation is defined under the Lorentz transformation. So, According to this theory, we can apply the time dilation equation when both observer moves with constant velocity. But in our case, astronomer starts his/her motion with rest and accelerate and, in returning situation, he/she decelerate the spaceship, comes to rest and finally moves backward by accelerating spaceship again. Thus, here, we can see that spaceship is not inertial frame (due to acceleration or deceleration). So, equation of time dilation is not valid for astronomer’s frame but we can apply the equation of time dilation for earth frame (however earth is also not an inertial frame completely but here we are ignoring its rotation, revolution and assume that earth is at rest.). Thus, both observers will not observe dilated time with respect to each other. Observer on the earth measured the larger time comparison to the time measured in astronomer in spaceship

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