In diff geo terminology every smooth manifold has coordinate charts by definition. But a smooth manifold all by itself doesn't have a metric. So on that level you cannot tell if coordinates are curvilinear or not yet - so you cannot formally name them cartesian or spherical because they look...
Hmm, if the least thing you have is diffeomorphism between two manifolds they are at least topologically equivalent. If you are looking for something stronger then that, yet weaker then isometry there are a few relation in between. For example conformality is a quite a lot weaker then isometry...
Sorry, you feel that way. But of course you are right, physicists and mathematicians are very different and focus on have different focus of their interests, which is perfectly fine. Truth be told, the issue is that there is simply not enough opportunities to get a mathematical view on all these...
I agree, that for practical purposes it may seem to make little sense. Indeed, most of the time we just have one observer in the lab frame to worry about and the cases where we have multiple observers with their own setups between which we need to translate is specific to a few thought...
I wouldn't have to ask for this example if it i could find it everywhere in the books. Yes, what's discussed is what all the means for equations but that is straight forward as they don't need to distinguish between time and space anyway. But that does help so very little when discussing actual...
Yeah, i used exactly that approach to get the acoustic wave equation to conform to the same math as SRT - i.e. i derived the math framework from acoustic geodesics. Given a seemingly analog derivation it wasn't clear to me where the differences stem from which is why i tried to figure that out...
yeah, well i'd still think one should lead explaining relativity with this concept first. On that level it's not even really about relativity at all but rather treating the time coordinate the same as any other allowing it to mix. That concept is just not what one would intuitively think of -...
Let's same I have an observer A and B that initially occupy the same point at ##t=0## but they have a relative velocity to each other.
Now let's assume there is an object C that moves in a circular motion around some point from A's frame. The initial condition/position is given (in A's frame)...
Thank you very much for the effort and yes you are perfectly right. Sorry, i was somewhat occupied with work so didn't find the time to check this forum.
The "equation" i wrote down is just the one from my original post but corrected - and a mistake added. It represents the age relation of the...
Yeah, that's what ##\Delta t_{A,B}## is supposed to be and i get that it's a contradiction. My idea to use the grids and the ##\Delta t_{I,J}## is just so i can at least mathematically treat the twins frames as equivalent even though the experimental twin setups is not. Only in the close world...
Well, the event of A1,B1,C1 meeting is not used in determining any of the ##\Delta t_{I,J}##. However, I wrongly assumed that B2 will meet the C4 at the point where A3 is. Length contraction will indeed offset that meeting to happen before B2 arrives at A3 (in the scenario of symmetric grids)...
Thanks for the constructive responses. Finally we have at least established the basics of my original post and seem to agree on this very simple 2 grid scenario.
But the twin paradox is called so for a reason - sure i know it's not really a paradox, but the intent of it is to stress test the...
Fine, basic question then: in the following setup in a flat open world
... ------ A1 ------ A2 ------ A3 ------ ...
... -----> B1 -----> B2 -----> B3 -----> ...
which grid-vertex ages more between two meetings with the others grid vertices? those in A or in B?
A2 between meetings with B2 and...
The twins become indistinguishable if the access to certain information is restricted or rather the twins are forbidden from conducting some other experiments.
Unfortunately math and logic is always quite a bit more tricky when done correctly. Thus to put this into more rigorous mathematical...