First Year Relativity Problem from Exam

[Heresy]

Homework Statement

The Earth and the Planet of the Apes are in a reference frame where they are stationary relative to one another, and are measured to be 2 light years apart. The observers on Earth send a TV signal to the Planet of the Apes with a picture of a banana. 0.6 years later, the observers on Earth receive a TV signal from the Planet of the Apes saying "Give us the banana!".

Question 1: What is the minimum speed that the TV signals must travel to fulfill the above conditions? (This was not the original wording but feel free to ignore this question, I don't quite remember it anyway)
Question 2: What is the spacetime interval between the two events, as measured by an observer on Earth?
Question 3: What is the spacetime interval between the two events as measured by an observer on a rocket traveling from Earth to PoA who finds that the two events (Earth and PoA sending their respective TV signals) are simultaneous?
Question 4: According to the observer, what is the distance between the two planets?
Question 5: At what speed is the observer moving relative to Earth and the PoA?

The Attempt at a Solution

Event 1: Earth sends TV signal to PoA
Event 2: PoA sends TV signal to Earth

Most of the people in my class (including me) assumed that one event caused the other, but is that possible if an observer in a reference frame measures them to be simultaneous?

If I take Event 2 to happen at t = -1.4 years and Event 1 to happen at t = 0 years (for the signal from PoA to reach the Earth at t = 0.6 years), the spacetime interval turns out to be a negative number. Answer to Question 4 turns out to be 1.4 x 10^16 meters, and the answer to Question 5 turns out to be 0.63c

Was I on the right track?

 

[Doc Al]

Most of the people in my class (including me) assumed that one event caused the other, but is that possible if an observer in a reference frame measures them to be simultaneous?

How can these events be causally related? The signal from 1 won't even reach PoA until long after the the signal from 2 has arrived at earth. (But you're correct that if are simultaneous in some frame, then they can't be causally related.)

If I take Event 2 to happen at t = -1.4 years and Event 1 to happen at t = 0 years (for the signal from PoA to reach the Earth at t = 0.6 years), the spacetime interval turns out to be a negative number. Answer to Question 4 turns out to be 1.4 x 10^16 meters, and the answer to Question 5 turns out to be 0.63c

Show how you arrived at these results.

(Planet of the Apes is the earth! )

 

[Heresy]

Most people assumed a causal relation because Question 1 implied that a signal went to the PoA and back within a time interval of 0.6 years... as I said earlier I don't remember the exact wording of this question so I'll just let it go.

Operating under the Earth's reference frame here
-Distance between Earth and PoA is 2 light years, so for the signal from PoA asking for the banana to reach the Earth at t = 0.6 years, PoA must have emitted their TV signal at t = -1.4 years. Earth emitted their signal at t = 0 years

-Spacetime interval squared = c^2 (deltaT)^2 - (deltaX)^2
-Converted light years to metres and years to seconds
-Setting deltaT (1.4 years or 44150400 seconds) and deltaX (1.89 x 10^16 m) yields a result of s^2 = -1.83 x 10^32 metres squared. (Answer for question 2 and 3)

For the observer to measure them as simultaneous, I used the invariant spacetime interveral by setting deltaT = 0 and solving for deltaX, which came out to be 1.35 x 10^16 metres (Answer for question 4).

To find the speed of the observer flying on the rocket, I used the formula for length contraction

1.35 x 10^16 = sqrt(1-v^2/c^2)1.89 x 10^16

With a little bit of algebra, v turns out to be 0.70c (answer for question 5)

Um... I guess the minor difference in numbers came about when I was fiddling with the number of digits I was handling with - but is my method correct?

NB: I didn't write this for my exam on Wednesday morning, but it doesn't even matter to me anymore - I just want to know how to do it properly.

 

[Doc Al]

Um... I guess the minor difference in numbers came about when I was fiddling with the number of digits I was handling with - but is my method correct?

Perfectly correct.

 

[Heresy]

Alright, thanks a lot!