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#91
Jan1513, 05:45 PM

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I object to giving any physical meaning to simultaneous space. Simultaneity is a convention. For inertial observers (or in an inertial frame used to analyze some overall scenario), there is a standard convention any reasonable person would use; how far it makes sense to extend it (for an observer) depends on how long they have been inertial. For noninertial observers there is no preferred convention except 'locally'. A noninertial observer is analogous to the GR situation  only local frames (with standard simultaneity convention reasonably preferred sufficiently locally in time and space).
I believe this is how Einstein viewed it, but that is neither here nor there. [There is also the sense of relatively arbitrarily chosen simultaneity surfaces used to construct coordinates useful for some problem. Obviously, I don't consider coordinates a feature of physical reality.] 


#92
Jan1513, 08:12 PM

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If you had said things like "I find that looking at hyperplanes of simultaneity helps me to make sense of what is going on" (which is pretty much what LastOneStanding said right before you entered the thread to support what he was saying), I doubt we would have had any sidebars. But you insist on saying things like "hyperplanes of simultaneity are fundamental to relativity", which implies (incorrectly) that they are necessary to *any* understanding of relativity, and then claiming that Einstein said so too, which is a strained (at best) interpretation of what he said. 


#93
Jan1513, 09:37 PM

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By the way, you did a very excellent job of explaining the doppler approach. I've read a number of accounts of this, most recently Paul Davies's discussion, and yours is as good as any and better than mostparticularly with your use of the diagrams. 


#94
Jan1513, 09:43 PM

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#95
Jan1513, 10:18 PM

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#96
Jan1513, 10:24 PM

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However, coordinates are a really really convenient calculating tool in many problems... So we use them a lot. 


#97
Jan1513, 10:50 PM

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Since we need to do that globally anyway, we can also do it locally. We then clearly see that the laws of physics don't care one bit what coordinate systems we use, and the actual laws of physics are expressed entirely in terms of invariant quantities. 


#98
Jan1513, 10:51 PM

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Then, in post #38, you made the statement that I first responded to: 


#99
Jan1513, 11:53 PM

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Because the way it is usually defined i.e. some quantity that does not change under coordinate transformation, is confusing as it is defining invariants using concept of coordinates and consequently coordinate dependant quantities that we are using to construct coordinates. So coordinate dependant quantities are more basic than invariants. 


#100
Jan1513, 11:54 PM

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Your claim about some preferred meaning to your chosen 'simultaneity space' is equivalent to insisting that only cartesian coordinates are valid on a plane. Even more, that if we draw some arbitrary curve on a plane, and then want treat it as a coordinate axis, we must use lines perpendicular to its tangent at each point. 


#101
Jan1613, 07:51 AM

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I was specifically thinking of falsifiability and Occham's razor, both of which can be derived from Bayesian inference, which is the mathematical framework for inductive reasoning. So falsifiability is not a counterexample and I stand by my previous assertion. For example, proper time can be defined physically as the time measured by a clock. It can also be defined geometrically as the integral of the spacetime interval along a timelike path. Neither of these definitions require coordinates. Similarly with the other invariant quantities used in physics. You can define all of your physical theories in terms of these invariant quantities without reference to coordinates. Then, once you add coordinates, you can note that all of the quantities that show up in your physical theories are invariants, and you can refer to them collectively as "invariants" without at all implying that they are less basic than coordinates and coordinatedependent quantities. 


#102
Jan1613, 10:24 AM

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#103
Jan1613, 11:08 AM

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Hence we suddenly had a need for the concept of translating between measurement outcomes, and one way to do that is via invariants. But the invariant is more than just a mathematical trick for doing the translation it is the thing that the measurements are referring to, in the sense that it is the thing that is objective. So to me, the main lesson of relativity is that measurements are only "objective" if we keep track of the state of the measurer, whereas the invariants we construct from the measurements are objective in the true sense of being the same for all observers. That's also why a special relativistic invariant is indeed just a kind of mathematical trick, a means to an end, whereas a general relativistic invariant is actually what the laws of physics must refer to (at least, insofar as general relativity is a good theory of physics). That was, after all, Einstein's primary motivation for GR he never liked singling out the inertial observers, and I imagine that's because it didn't seem very objective to do so. I think an analogy can help us see deeper into what objectivity means. Imagine a "chick flick" that is being reviewed. Let's rampantly overgeneralize and say that women like this movie and men find it boring. Now imagine a male reviewer who pans the movie and a female reviewer who says it's oscarworthy. Are either of those reviews making objective claims about the movie? No, the objective claim, and the best review, are simply the statement that women will like this movie and men will hate it (again ignore the absurdity of such sweeping generalizations about movies). Can we say if the movie is good or not? No, we cannot, there is no objective way to do that all that is objective is to account for how each person will experience the movie. And how can we tell how each person will experience the movie? By considering what is invariant about that movie what aspects can men and women both agree this movie has? So even though we might thus say that experiencing a movie is something subjective, we can still say that accounting for that experience is objective. It is the latter, not the former, that underpins science, and hence the need for invariants. That's what relativity is trying to tell us, and it was completely new to science at the time, but then quantum mechanics came along and gave us additional reasons to track what the observer is doing. Personally, I'd say the main lesson of physics of the 20th century is that we can never again imagine that physical reality has an existence completely separate from how we perceive it. But then again, Einstein never liked quantum mechanics! And on the matter of the "reality" of the concept of relativity of simultaneity, I agree completely with DaleSpam. What's more, I'd say the wellknown "Andromeda paradox" that bobc2 is talking about makes pretty clear the unreality of the entire concept of global simultaneity. We should have learned from relativity that simultaneity is a strictly local concept whose usefulness gets diluted with larger and larger (invariant) separation between the events. Maybe this lesson will someday prove false, but it's all we have to go on at the present moment (pun intended). 


#104
Jan1613, 11:15 AM

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#105
Jan1613, 11:26 AM

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(nitpick: 4 vectors are also invariants, as are the tensors of GR. Also, the laws built from those objects are themselves invariants, as are the invariant parameters embedded in those laws like e and c. But measurements are always scalars, so the positivist might further restrict the "real" invariants to just the scalars, whereas a more rationalistic philosopher might allow all the classes of tensorial invariants, and the parameters of the theory, to be considered "objectively real." Personally, I hold that no quantity that has units is something real, but here we are just talking about what can be considered an invariant. So perhaps we must allow zonde that even the concept of an "invariant" contains some troubling elements, a suggestion that we have not penetrated the mystery completely!)



#106
Jan1613, 11:30 AM

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#107
Jan1613, 11:34 AM

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For example: I say this stuff you are telling me is a crap. And my statement is invariant. Someone else can say: "zonde says this stuff Ken G told him is a crap." But it's says nothing what others will think about it. 


#108
Jan1613, 11:42 AM

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You are right, the point is just that we must admit some ambiguity exists even in the meaning of that which is "invariant." Objectivity is the underpinning concept, yet we can be troubled by these various different forms of "things that are the same for all observers." Does a law have to be the same for all observers in the same way that a proper time along a path does, or the charge of the electron? In relativity, there is no need to distinguish these flavors of invariance, as the theory is built from all of them, but future theories that relax the postulates of relativity might need to navigate those differences. For example, a proper time over an infinitesmal interval has a perfectly good reason to be considered an invariant in relativity, as it results from a metric inner product over that interval, but what justification do we have that the charge of the electron is the same in all reference frames? It's not really part of the structure of the theory, it is simply Occam's razor.



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