#### kaotak

Here's the situation of the problem, but not the question: There is an observer on the ground and a particle flies through the atmosphere with constant velocity and then decays.

I checked my answer to the problem by checking that the spacetime intervals (t^2 - x^2) in both the observer's frame and the particle's rest frame are the same. The spacetime interval I refer to is the spacetime interval between the event of the particle's creation and the event of the particle's decay.

I did not, however, get the same spacetime interval for each. Yet I thought that the spacetime interval was supposed to be invariant in all frames of reference. This would mean that the answer I got is wrong.

However, I think I may be mistaken. I looked over my notes and it says that the spacetime interval is invariant under rotation. Perhaps it is not invariant in this situation? For surely, the origins of the observer's frame and the particle's rest frame frames do not coincide. Thus the observer's frame cannot be just a rotation of the particle's frame.

So my question is: should the spacetime intervals in both the observer's frame and the particle's rest frame be the same? And to extend the question to other situations: are the spacetime intervals only invariant under rotation? If the origins of the frames do not coincide, are they still invariant? Is the spacetime interval always invariant, regardless of the setup of any two inertial frames? I would imagine they should be, just thinking about it visually, since intervals don't change length when you shift the graph.

Thanks in advance.