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...
Yes, the topologies are clearly different. "Unrolling the cylinder" onto a plane repeatedly by creating a copy of its contents obviously creates a different topology. Take the point of where one twin and its clone is. In the closed world the clone is actually the twin itself and therefore these...
Well, two similar cases actually: yes, the closed world example is one but the grid example is another.
See, I wanted to carry the closed world example to the open world scenario an make them as indistinguishable as possible. So instead of going around the world and coming back from the other...
Sure: apparently in a closed world each frame has a characteristic quantity: the delay between two signals sent out in opposing directions coming back around the world to their source. Unlike the closed/open SoS this one distinguishes each frame in a much finer fashion. Show that no such...
except that every twin is inertial in my example and the ##\Delta t_{I,J}## treat each as such so by your words finding the oldest would already get quite tricky.
Do you even know what the word "proof" means?
So this is indeed considered correct. I understood the original explanation but I was not sure if it was really correct given the loss of equivalence.
Can you point me to a proof of that is issue is specific to a closed universe? I mean the closed world makes the asymmetry quite obvious, but...
Originally i just wanted to look at how much analogy can be made between light and sound waves using all that math has to offer to depict them in most similar framework possible - just so as to have a different perspective to understand some things better. Anyhow, no matter how well one tries to...
That said, isn't the isotropy of light actually a convention and not a hypothesis? I asking this because i find that trying to make a self-consistent hypothetical model which assumes otherwise always ends up in a logical contradiction when attempting to model an experiment measuring the...
Thank you! this is exactly what i have in mind. But is there a general name for these type of transformations? As you can see i am still lacking the terminology to properly express my questions, hence i easily create misunderstands by using a wrong formulation. There is also the issue that i...
To be honest i am not entirely sure how this type of transformation of an exchange of metrics is to called properly and I am indeed not sure if applying a pullback via a non-isometric map achieves what i have in mind. Generally i am thinking of something akin to Weyl-transformations but to...
Oh, no no no. I'm definitively not talking about an induced metric. Okay maybe i call my map ##\phi## instead of ##id## to point out it maps between different metric spaces. Furthermore i don't want to pullback the original metric, therefore the image of the map is not an embedded space.
Think...
The simples case of what i have in mind would be a Weyl transformation. This also uses the ##id## for the pullback. But in this scenario ##id## is neither an automorphism (no self mapping) nor an isometry - as it maps between two actually different metric spaces. It should only be...
It is clear that all the information contained within the geometry cannot simply get lost via a transformation, so it has to transform into something else. My thinking was not about trying to choose what should be a force and what geometry - but merely chose a different metric and see what...
i don't exactly understand your premise here. You can define various physical objects or length which do change relative to the SI-meter depending on where they are (and in a well defined way). My thinking was that one could equally set these things to have constant length instead thus rendering...
Hmm, okay. Well, there are different types of manifolds i guess and we are not talking about the same here. Consider the identity function as acting only on the bare set of ##M## and forget about it knowing about any other structures defined on it maybe.
Perhaps i better make a concrete...
Yeah, this is an aspect i was thinking a lot about lately. In particular about the assumption that a rigid object, i.e. the "meterstick" length is unchanged. The structure of Riemann geometry makes me think this postulate is not physical in nature, given the circular definition of any real...