(Re-Answered) Spacetime interval homework question?

[CCCP]

(This problem was previously asked but poorly answered. Here is a link to the previous locked thread. https://www.physicsforums.com/showthread.php?t=202855)

1. State the problem, all variables and given/known data
The star Alpha goes supernova. Ten years later and 100 ly away, as measured by astronomers in the galaxy, star Beta explodes.

An alien spacecraft passing through the galaxy finds that the distance between the two explosions is 120 ly. According to the aliens, what is the time between the explosions?


s=Spacetime interval
c=Speed of light≈3 x 10^8 m/s or 1 lightyear/year
Δt=Change in time from reference of astronomers in the galaxy=10 years
Δx=Distance in light years between Alpha and Beta's explosions from reference of astronomers in the galaxy= 100 light years
Δx'=Observed distance between explosions observed by Alien spacecraft

Δt'=Observed time between explosions from the reference of the Alien spacecraft =?

Homework Equations

s^2=c^2(Δt)^2 - (Δx)^2 =c^2(Δt')^2 - (Δx')^2

The Attempt at a Solution

Plug in known variables to equaiton (Unit abbreviations- ly=light year; y=year)
=> (1ly/y)^2 *(10y)^2 - (100 ly)^2 =(1ly/y)^2*(Δt')^2 - (120 ly)^2

Because the spacetime interval s between the two events is invariant, we can use algebra you can get rid of the c term as it appears on both sides. The equation is simplified to:

=> (10y)^2 - (100y)^2 = (Δt')^2 - (120 y)^2

=>-9900y^2 = (Δt')^2 - 14400 y^2

=>(Δt')^2=4500

=>Δt'=67.08 years

 

[anathema]

Therefore, according to the aliens, the time between the two explosions is approximately 67 years. This is significantly different from the 10 years measured by the astronomers in the galaxy, which can be attributed to the effects of time dilation caused by the relative motion between the two observers. This phenomenon is a key concept in the theory of relativity, where the passage of time is relative to the observer's frame of reference. It is important to note that the speed of light is constant and the spacetime interval between two events is always the same, regardless of the observer's frame of reference. This is a fundamental principle in understanding the behavior of the universe.