John and Joe: Twins in Space-Time

[Kuiper83]

Homework Statement

Two twin astronauts, John and Joe, leave the Earth at the same time for a trip to far away stars. According to their mother, who stays on earth, John reaches a star that is 4 light years away after 5 light years and Joe reaches a star that is 6 light years away after 10 light years. After reaching their destination they return along the same path with the same velocities as their outbound trip. When both twins are back home who will be younger?

Homework Equations

Δs^2=(Δct)^2-(Δx)^2
ct'=γ(ct-xv)

The Attempt at a Solution

I used the lorentz transformation in the form of
ct'=γct-βγx
with γ=5/3 and 5/4
and β =3/5 and 4/5
which gave me times of ≈2.6s and 4.45s which means John is younger.

However, while I got the correct temporal relation between the twins I got no credit. Is this because I should have used the proper time/spacetime interval? This was an exam question and I feel like I should have gotten some credit, and so I just wanted to make sure that I had some basis for these claims.

 

[anathema]

Dear fellow scientist, it seems like you have approached the problem correctly by using the Lorentz transformation to calculate the time dilation experienced by each twin. However, it is important to also consider the proper time and spacetime interval in your calculations. The proper time is the time measured by an observer who is at rest with respect to the events being measured, and the spacetime interval is the distance in spacetime between two events. In this case, the proper time for each twin would be the time experienced by their mother on Earth, and the spacetime interval would be the distance between the Earth and the stars they traveled to.

By taking into account the proper time and spacetime interval, you will be able to accurately calculate the time dilation and determine which twin is younger when they return home. I encourage you to review your calculations and see if you can incorporate these concepts to get a more accurate result. Keep up the good work in your studies!