Two Events in Special Relativity with on Time Dialation

[zellwwf]

Homework Statement

A Nova is a sudden, brief brightening of a star. Suppose Earth astronomers see two novas occur simultaneously, one in the constellation Orion and the other in the constellation Lyra. Both nova are the same distance from Earth, 2.5 E 3 c.y (lightyear) and are in exactly opposite directions from Earth. Observers on board an aircraft flying at 1000 km/h on a line from Orion towards Lyra see the same novas, but note that they aren't simultaneous.

A) For the observers on the Aircraft, how much time separates the novas?
B) Which one occurs first?

Homework Equations

x' = $\gamma$(x -vt)
t' = $\gamma$(t - xv/c^2)

The Attempt at a Solution

Attached.
Just please tell me if i am correct, else tell me where i went wrong and give me a hint to correct. Thank you

 

[zellwwf]

I was revisiting this question and i found a calculation error, my calculator gave me gamma to be 1, so i thought that it would be a little (very little) less that 1, which is why i put down 0.99999999999...
it turns out it is the other way around it is around 1.0000000000##..
will turning gamma into 1 or 1.00000047 provide the correct answer?
is gamma my only mistake there?

 

[voko]

You computation of Δx is wrong: it is not 5 light years, it is 5 thousand light years. But note you don't really need that value in meters, because in the formula you have (vΔx)/c²; so you just need (Δx/c), which is the number of seconds in 5K years, and you need to multiply that by v/c.

For γ, use 1. The aircraft's speed is truly infinitesimal compared to the speed of light.

The last part of your solution (about what happens first) is confusing because you did not clearly define what Δx and Δt are. You should clearly label the events as (x1, t1) for Lyra and (x2, t2) for Orion (or the other way around) and then let Δx = x1 - x2 and Δt = t1 - t2. Then when you get the result, its sign will tell you clearly what happens first. Positive Δt' will mean that the event labeled as (x1, t1) was after (x2, t2) in the S' frame, negative Δt' will mean the opposite of that.

 

[zellwwf]

Alright, i will report back to you with these :) thanks
can you (please) visit my other posts?
i just need to review before my upcoming problem set

 

[zellwwf]

OK, @Voko: here is my new attempt, think i covered all my mistakes
and here is my solution. again, i would be happy if you can just go over it, it's less that a page long. and i highlited the answers, but please check my arithmetic

 

[voko]

Here is my computation of Δx'.

v = 1000 km/h = 277.78 m/s
Δx/c = 5000 * 365.25 * 86400 = 157788000000 s.
vΔx/c^2 = 277.78 * 157788000000 / 299792458 = 146202.31 s

 

[zellwwf]

wow, ok , i calculated a "lightyear" extremely long... :P
nice to see you checkin in on me

 

[Chestermiller]

I have a different perspective on this problem that I would like to offer. Suppose that the two flashes arrive at the observer on Earth at time t = 0 in his frame of reference, and that the observer in the spaceship is just passing by adjacent to the Earth observer at that moment. His location in his frame of reference is x' =0, and his clock shows t' = 0. Since these observers are immediately adjacent to one another, they must be present at the same event. Therefore, the observer in the spaceship should also see that two flashes arrive simultaneously. I found this a very unexpected and relativistically counterintuitive result. So I did an analysis of the problem. I used the Lorentz Transformation to determine where and when the two flashes occurred in the S' frame of reference some ~ 5000 years ago, and then determined the arrival times of the flashes at the space ship's location x' =0 . The distances to the two flashes and the times that they originally occurred in S' were not the same as one another, but the arrival times at x' = 0 were. I am wondering whether any of this makes sense, and whether anyone would be willing to repeat my analysis.

Chet

 

[voko]

I am wondering whether any of this makes sense, and whether anyone would be willing to repeat my analysis.

Your analysis is quite correct. The observer at the aircraft detects the flashes at the same time. But because the distance to the novas are different in his frame, he concludes that the original events were not simultaneous. There was a similar discussion here just a few days ago: https://www.physicsforums.com/showthread.php?t=638659

 

[zellwwf]

I wish i can think that creatively by now :P