Calculating Time Period for Simultaneous Relativity in All Ref Frames

[AlexCdeP]

Homework Statement

Challenge: a rather eccentric group of astronomy students wanted to celebrate the impact of the Shoemaker-Levy comet on Jupiter by holding a party of sufficiently long duration that their celebrations were simultaneous with the impact of the comet in all inertial reference frames.
For how long do they need to party?
How might the party end?
[The distance from the Earth to Jupiter is 8E11m and you may neglect their relative motion]

Homework Equations

The Lorentz transformation formulae
x=γ(x'+vt')
t=γ(t'+vx'/c²)

where γ=(1/(1-v²/c²)^1/2 and is the gamma factor.

Where x is the distance in one frame and x' is the distance in another, and t is the time in one frame and t' is the time in the other. v is the velocity relative to one another.
At least I think these are the equations I need. Please correct me if I'm wrong, I may have a fundamental misunderstanding of the situation.

The Attempt at a Solution

OK, I have spent a fair while on this question and although initially I made the mistake of taking the party and Jupiter as being two separate reference frames like an idiot I finally realized that I can in fact consider the two events to be in the same reference frame. Calling x the distance in this reference frame x is now 8E11m and the time difference is of course 0 as they are simultaneous so t=0. However in order for these to be simultaneous in all reference frames I now have four unknowns as i don't know γ x' or t' or v, I can cope with three but not four. Which means that I'm missing a clue in the question here, any ideas?
While I was writing I suddenly thought maybe t is not 0 but t' is 0, is t in fact the length of time it takes light to travel from Jupiter? If that's the case then this is solved. I really like this question! :D Relativity = mind blown
Thanks in advance

 

[AlexCdeP]

Is it a trick question? I thought there could never be agreement on the simultaneity of events between reference frames.:( So they would have to party forever...I don't get it.

 

[vela]

Just because the events aren't simultaneous in all frames doesn't necessarily mean that the range of time difference for the two events goes from -infinity to +infinity. You want to show if that's indeed the case or not.

 

[AlexCdeP]

Thanks for the reply! And you're right of course I've found its not necessarily infinity. You helped me realize what I actually had to do.
OK so this is my final result. Would be great if an you or another could check that its a valid answer.
The length of time that the party continues on for is equal to the time difference between the party on Earth and the comet hitting Jupiter. The only way I can think about it is if the party is some form of theoretically continuous or even infinite event? Is this the right way to think about it? Anyway..

t'=γ(t-vx/c²)
Now if we make t'=Δt' or the difference in time between the two events as seen in any other reference frame which is the length of the party I'm looking for. t=Δt is just zero because the events are simultaneous with each other in the Earth and Jupiters inertial frame. And finally x=Δx is 8E11m.

Therefore: length of party=γ(v(8E11)/c²)
Hence the length of the party depends entirely on the velocity of the other reference frame relative to the Earth and Jupiter frame. This means that the party could theoretically be infinite because the value of gamma can be is infinite. I'm pretty certain there's not enough information given in the question to gain an actual numerical answer to this question... unless I've missed something
I'm still unsure as to what to write for how the party could end? Clearly it would end with the comet hitting Jupiter. Is this correct?
I understand this is a bit of an awkward question to help with because it doesn't seem to have a definite answer... so massive thanks to anybody who replies. :)

 

[vela]

This is an interesting but confusing question. What you're trying to find is how long the party must last in the partiers' rest frame so that the comet hitting Jupiter and some spacetime point in the world line of the party are simultaneous.

You've actually analyzed the opposite direction. If the party lasts a finite time in its rest frame, you can always find a moving frame in which the party lasts arbitrarily long. This should make sense because by moving fast enough, the observer will see the partiers' clock slow down enough to make the party last as long as they want in the observer's frame.

If you draw a spacetime diagram in the students' rest frame, the answer is pretty clear if you realize that the party and the collision are spacelike-separated in all frames. You can also find the answer by using the Lorentz transformation and a bit of simple reasoning.