If one admit that the particle has the "real objective(observer independent ?)" trajectory x(t), then if it is "observed" from point O, then, the observed trajectory is x'(t')...One could simplify by taking t'=t, because it is not here an observer transformation. Then, there should exist a relationship of the form x'(t)=x(t-x1(t1)/c)....The problem is that I don't understand anymore the physical intuition that able to say if x1=x or x1=x' and which value t1(=t(?)) should have. Thanks for any simple explanation.
You mean it has an objective trajectory that can be parametrized as x(t) relative to a particular coordinate system? What do you mean by "point O". As long as two observers are using the same coordinate system x,t, it shouldn't matter where they are located in space, they will both have the same x(t) for the path of an object. Do you mean switching to a different coordinate system O which uses coordinates x',t'? By "observer transformation", do you mean the same thing as transforming into a different reference frame/coordinate system, or do you mean something else? You've lost me here...what do x1 and t1 represent? And again, what does x' represent?
Math<->Phys I just mean that when the first trajectory is specified x(t), then it's just a mathematical description that does not include the speed of light needed for the observer placed at the origin to "measure/see" the particle describing it's trajectory. But it is not even clear in my mind how I should formalize this...
What does your x(t) represent? Is it supposed to be the apparent position of a given object as seen by a given observer, with t measured on his own local clock? Usually when you write something like x(t) in relativity, you mean the path of the object in a given reference frame's coordinate system, not the apparent path--if I am at position x=0, and at time t=5 years I observe an event at position x=2 light-years, then I factor out the light-signal delay to conclude that the event's coordinates are x=2 light-years, t=3 years in my reference frame.
No, it's not here the observer dependent "reality". it's not the einsteinian observer dependent philosophy..it makes the assumption that there exists a reality independent of any observer..hence it is not "as seen"...the "as seen" has to be computed out of this one...but this is just a mind game again...
Einstein's theory is no more "observer dependent" than Newtonian physics (after all, velocity depends on your reference frame in Newtonian physics too). Relativity is fundamentally about quantities that do not vary by reference frame, like the proper time along a given path. You're not really making any sense to me, relativity certainly does say there is a reality indpendent of any observer, and I can't interpret the rest of what you say here. If you want to have a discussion, can you explain what you're talking about at greater length, maybe with a specific example as an illustration?
In special relativity it is not clear to me if there is such an observer independent reality, except all only on c=celerity of light based quantities of course..for Galileo it was t=time is observer independent makeable quantity...Infact if you have an observer independent reality, then you can deduce the observer transformation laws (which is basically a school homework),since SR is in fact just basing it's idea on Lorentz transform....So I take the same exercise : let's take an observer independent physical reality, but instead of c or ds^2..let's take a whole function X_reality(t) and let's find how two observers of this "weird observer independent reality" see it....is this is more understandeable for you ?
As I said, the proper time [tex]d \tau[/tex] (or spacetime interval ds) between any two points in spacetime is observer-independent, as is the proper time along a particular worldline. In the galilei transformation it's true that the time interval between two events is the same in all reference frames, while this isn't true in relativity. By observer transformation laws, do you mean the coordinate transformation between different reference frames? I don't get what you mean by "a whole function X_reality(t)"--can you explain this in more detail? Is "X_reality" a variable of some kind, and if so what does it measure--position along someone's X-axis, or something different? Again, a concrete example would probably help. It might also help if you would explain what X_reality(t) would mean in Newtonian mechanics, for comparison.
Well, yes by observer transformation rules, I mean how do coordinates of events (in relativity), but it could be other physical quantities, when you change your observer. For example mass tranformation laws....I have two remarks : 1) precise domain : how do you deduce mass tranformation between different observer via Lorentz tranformation. 2) generally (meta dialog about this method) : don't forget that when you make transformation laws between frame of reference, you pass under silence (because we don't know how to do it i suppose), that the law-maker itself is in a frame of reference).. Then, I just want to compute the following : the "lawmaker" trajectory is x_reality(t)...how does the two observer see that trajectory, so that i can deduce the transformation rules between their frame of reference... Your question about Newton is well done, because I asked once in this forum if somebody could reexplain me Newtons time concept in view of Einstein's review...for me it is as if nowadays, Newton't Time concept is a global concept valid for every observer, and hence in conflict with the local based time of einstein...
You don't really need to use relativistic mass, you're free to just stick with the concept of rest mass, which never varies. I'm not sure about how to derive the amount of force needed to accelerate an object that's already moving in your frame, which is what the concept of relativistic mass is based on. I don't understand this. What do you mean by "the law-maker"? I don't understand the meaning of "lawmaker", or what "x_reality" represents. Again, is this supposed to be a position variable in some coordinate system? Yes, the two theories say different things about time, but the Lorentz transformation does reduce to the Newtonian "Galilei transformation" in the limit as the relative velocity of different observers approaches an arbitrarily small fraction of c.
Right...the other stuff with the x(t) instead of ds as invariant is even for me not clear...but it is some kind of generalization of "observer independent" reality...if you want on a more human cognition based approach, if i say "black on white"...we could have several interpration of these words...(it's some kind of physical semiotic if you want..but in fact i don't even know what semiotic exactly means...)
What "other stuff"--something I said earlier, or something else? It's often hard to understand what you're saying because you introduce words and terms without explaining them. x(t) is not an invariant, even in Newtonian physics--for example, if I'm at rest with respect to a particular object, x(t) for the object will be constant in my frame, but if I'm moving relative to it, x will vary as t changes. But if you're talking about the point I was making in my first post, I was just saying that the position of an observer, and the light-speed delays he experiences, are not relevant, because an observer must take those delays into account when assigning coordinates to an event. So, two observers who are at rest with respect to each other but in totally different positions in space will assign an object the same x(t) in their common coordinate system--it's only when dealing with observers in motion relative to each other that you'll get different functions for x(t). Again, I'm not understanding what you're saying here--what does ambiguity in ordinary language have to do with the well-defined mathematical procedures for assigning coordinates to events in physics?
I think we really don't get along, because I told you from the beginning on, x_reality(t) was taken, as a starting hypothesis, as being an invariant quantity. Do you have medication for schizophrenia ??? I know they can cause connection problem in the brain....or maybe you just want to fool around ??? I think you use psychiatrist like approach to tickle the "unstable" candidate to push it in the hole....?? Just by "normal curiousity"....Or maybe it's just you don't want to enter the game proposed...free to you to remain on your battle-side, it's often like that in coop-working, nobody want to be the project depositer (because it's dumb), neither the worker (because there is calculation to do)
But you never explained what "x_reality" is supposed to mean! I asked if it was supposed to be the position coordinate in some coordinate system, but you didn't answer. Again, you're talking in your own private language here, the meaning of x_reality is not so obvious that other people are going to be able to understand it even if you don't provide a definition. I'm pretty sure my lack of ability to understand your invented terminology isn't a sign of a mental disorder. Try asking other people on this board if they understand clearly what you are trying to say, I think they'd all have the same problems understanding you that I'm having.
Well I think you just are too much meta-thinking...we don't care about what x_reality(t) in the phyiscal reality is, we just make a math games...if i remember well from the first message, i wrote something like : observer independent trajectory...which is of course understandble physically by nobody, because we all are observers, so if you are an observer you cannot observe, nor understand what that is because you need to be not an observer..(which is different than : equals for all observer...or am I mixing myself in literal problems ?)....does this is clearer that in this framework, even I cannot exlain you what x_reality(t) is, in this math exercise ?? Yes I understood, I should have written : observer free...which is almost the same difference between timeless and everlasting/eternal I see 2 ways of doing the case : either you have no observer, which is quite weird, or you say : it's valid for every observer, but then I wonder if it is the same as finding a "law" depending on every observer (if you know all the cases, then the global theory is applicable to everyone of them).... I think in physics there is a parallel between the two aspects of that science : phenomemology : you take all the experimental cases (passive theory making)...and then you put them together in the hope to find a law..and then you apply the laws (active theory applying)...which with social laws leads to some (funny) problems.
But I don't understand what "x_reality" means in any sense, not just "in the physical reality"! OK, can you explain in more detail what you mean by "observer independent"? The usual meaning of "observer independent" in physics would something like be "doesn't depend on what coordinate system you use to describe it, or on how you measure it, or whether you measure it at all." Are you using it to mean something different? If not, wouldn't the answer just be that trajectories cannot be observer-independent in either Newtonian physics or relativity, since different coordinate systems describe them differently? Or maybe one could say the trajectory itself is observer-independent even if the coordinates used to describe it are different, just like a curve drawn on a piece of paper is the same curve even though the function y(x) you use to describe the curve will depend on where you draw your x and y axes. In any case, is there anything about the problem you're trying to describe that's specific to relativity, or does it apply to all of physicist's attempts to describe the world using math? If the latter, maybe it would make more sense to move this discussion to the general philosophy, metaphysics & epistemology or philosophy of science, math & logic forum?
Yes, you're right, the only thing that was specific to relativity, is the maximal speed limit, inducing a change in the trajectory observed from a different standpoint than the one where it is described. Which puts forward the problem : what is it meant by giving a trajectory x(t) in a given frame ? Does it comprises the speed of light delay ? I think it would be a better idea to put in epistemology of science, it's a too basic notion to put it her....I don't how to move a complete thread...