Questions about Relativity time travel scenarios.

Question

 

a)

Is it possible for you to travel far enough and fast enough so that when you return home from a trip you are younger than your stay at home sister, who was born 5.0 yr after you?

 

b)

 

Suppose you fly on a rocket with a speed of v = 0.99c for 1 yr. according to the ship’s clocks and calendars. How much time elapses on earth during your 1-year trip?

 

 

 

c)

If you were 22 when you left home and your sister was 17, what are your ages when you return?

 

 

 

Explain your responses with at least 150 words for each question. 

Answer 1

(a). When you ask if it is possible to travel far enough and fast enough to arrive home younger than your sister, this depends on whether we are speaking theoretically or practically. It is entirely possible to achieve this if you travel close to the speed of light. If you travel at 0.99 of the speed of light and are gone for 0.8 years, a little over 5 years will have passed for your earthbound sibling. We can see this by using Einstein's time dilation formula:

Δt' = Δt/(√(1-v^2/c^2))

where t' is the time on Earth, t is the time as measured by a clock moving with the spaceship, v is the speed of the rocket, and c is the speed of light. Remember when using this formula to keep your units consistent. If you are measuring the speed of the rocket in m/s, the speed of light must be in m/s. Unfortunately, we've not managed to build a rocket that can travel at 0.99 of the speed of light. The fastest rocket, according to Guinness World Records, travels at 192,220 m/s. Plugging this into the above formula, we see that no amount of journey within a human lifetime would stretch time to a five year difference. If someone travelled at this speed for a hundred years, 100.1 years would have passed on Earth.

(b) This same formula helps answer your question about how much time passes on Earth if you travel at 0.99 the speed of light for one year. Because the rocket's speed is already represented as a fraction of the speed of light, it is super easy to plug in. Keeping the time in units of years for the sake of simplicity, we arrive at the following:

Δt' = 1/(√(1-0.99^2)) = 7.09 years.

You could arrive at the same answer by converting the years to seconds, but this saves the extra step of converting the seconds back to years. We can do this because when the speed of the rocket and the speed of light are divided, their units cancel, so we can put in whatever unit of time is the most convenient for us. Therefore, if you could somehow travel at 0.99 of the speed of light, one year would have passed for you, but approximately 7 years would have passed for your sibling on Earth. The closer you get to the speed of light, the greater the difference between the clocks on the spaceship and on Earth.

(c) Assuming we are retaining the speed of the journey and the time taken over the journey from the previous question, we can plug the amount of time we found in the previous question here. If you were 22 when you left home, you would be 23 when you returned to Earth. On the rocket ship, time would appear to pass at the same rate as normal. Your clocks would measure one year passing, so you would be one year older. Meanwhile, for your sister, a little over 7 years would have passed due to relativistic time dilation. As a result, if she had been 17 to begin with, she is now 24 years old. You left older than her, and you came home younger than her! A quick note is that the reason why you are the one who comes home younger rather than the other way around (in relativity, all motion is relative after all and moving and stationary perspectives can be flipped) is because you undergo periods of acceleration in your rocket ship while moving up to your velocity and then back down to rest when you get home, which marks you as the moving frame of reference.