On Tuesday, December 10, 2024 at 1:18:12 PM UTC-7 Jesse Mazer wrote:
On Tue, Dec 10, 2024 at 5:41 AM Alan Grayson <[email protected]> wrote: On Tuesday, December 10, 2024 at 3:20:20 AM UTC-7 Jesse Mazer wrote: On Mon, Dec 9, 2024 at 11:55 PM Alan Grayson <[email protected]> wrote: On Monday, December 9, 2024 at 9:08:33 PM UTC-7 Jesse Mazer wrote: On Mon, Dec 9, 2024 at 8:28 PM Alan Grayson <[email protected]> wrote: On Monday, December 9, 2024 at 4:54:34 PM UTC-7 Brent Meeker wrote: On 12/9/2024 3:24 PM, Alan Grayson wrote: On Monday, December 9, 2024 at 2:01:28 PM UTC-7 Brent Meeker wrote:> > Nothing odd about dilation and contraction when you know its cause. > But what is odd is the fact that each frame sees the result > differently -- that the car fits in one frame, but not in the other -- > and you see nothing odd about that, that there's no objective reality > despite the symmetry. AG The facts are events in spacetime. There's an event F at which the front of the car is even with the exit of the garage and there's an event R at which the rear of the car is even with the entrance to the garage. If R is before F we say the car fitted in the garage. If R is after F we say the car did not fit. But if F and R are spacelike, then there is no fact of the matter about their time order. The time order will depend on the state of motion. Brent Since the car's length can be assumed to be arbitrarily small from the pov of the garage, why worry about fitting the car in garage perfectly, and then appealing to difference in spontaneity to prove no direct contradiction between the frames? It seems like a foolish effort to avoid a contradition, when one clearly exists. AG What's the contradiction? *ISTM that the car can, or cannot fit in garage given the initial condition that in the rest frame, the car is longer than the garage; in other words there is an objective reality, but the frames differ on whether the car fits or not.* Can you define what "fits in the garage" would mean in objective terms, i.e. a definition that does not depend on simultaneity or choice of frame? If you can't even define that phrase in any objective terms, why do you believe there is any objective reality here? Is it just some kind of strong intuitive hunch that there "should" be some kind of objective reality attached to such a notion? *"Fits" means the car's length is equal to garage's length, or less.* Do you define the "length" of each in the standard relativistic frame-dependent way? * If car exactly fits, this is ambiguous from the pov of garage frame due to lack of simultaneity, and this is the consensus solution to an alleged paradox. But what happens if the car's velocity is increased, so car fits with room to spare? This is the case I have been posting about. AG * * If one avoids the issue of simultaneity, by not requiring the car to perfectly fit in the garage, we get opposite conclusions from the frames. AG* The paradox does not depend on the assumption that the car is the same length as the garage in either the car frame or the garage frame (or that they have the same rest length), *I don't make this assumption. AG* Then what did you mean by "not requiring the car to perfectly fit in the garage"? What does perfect vs. imperfect fit mean here, if "perfect" does not refer to them being exactly the same length? if that's what you mean by "perfectly fit in the garage". Even if the car's rest length is much shorter than the garage's rest length, so it fits easily in the garage frame, *No, this isn't the initial assumed car length. Its length is assumed larger than the garage, and the question is whether it can fit due to its motion which causes length contraction. AG* The paradox can refer to any scenario where the car fits in the garage frame but not in the car frame, this can happen regardless of whether the car's rest length is larger or smaller than the garage's rest length. *If we assume as an initial condition that the car's length is smaller than the garage's length, then for some velocity and greater, the car will not fit in the garage from the car's frame, so we're in the same situation as when we assumed as an initial condition that the car's length is greater than the garage's length. The contradition persists and differences in simultaneity does not come to the rescue, since it's measured the car's frame and couldn't be used to show the car fits, exactly, or loosely, since it doesn't fit. AG* Difference in simultaneity does come to the rescue by showing that it's a difference in coordinate-based descriptions of the situation, and that there is no objective coordinate-independent way to define the question of whether the car fits. you could always pick a sufficiently large relative velocity such that the car would be longer than the garage in the car's rest frame, *No, the car's length decreases in the garage frame only due to its motion. In the car's frame, the car's length doesn't change. AG* I didn't say the car's length changed in the car's frame, I just said that if you pick a sufficiently high relative velocity, then in the car's rest frame the car will be longer than the garage (in this case due to the garage's length being shortened in that frame). *I don't follow. If the garage's length is shorten, the length of the car remains unchanged. AG * "the car will be longer than the garage" is just a relative comparison, it doesn't imply anything about the length of the car changing. If the garage's length shrinks to a length shorter than the car in the car's frame, then that means the car is longer than the garage in that frame. and thus it would not fit in that frame, so the paradox remains. Not sure what "opposite conclusions from the frames" could mean if you don't have a specific way to define "fits in the garage" in a way that doesn't depend on picking some frame or another. Jesse *As I understand the problem, in the initial rest frame the car is assumed to be larger than the garage. Then the question is whether it can fit when the car is in motion due to length contraction. In the car's frame, the garage length decreases, so there is no possibility of the car fitting. OTOH, from the pov of the garage frame, the car's length shrinks, so there is some velocity where it fits perfectly. If the velocity continues to increase, the car fits with room to spare. So, I have shown that the frames differ in concluding whether the car fits, or not, and the question is whether this is a paradox. If you conclude it is not, then you deny there's an objective reality such that the car fits, or doesn't fit. And "fits" just means the car's contracted length is EQUAL TO or LESS than the garage's length. AG* Yes, I've said before that there's no objective frame-independent reality about whether the car fits, that's part of the standard answer to this paradox. Do you accept that this is a valid way of resolving it? You seemed to be objecting in your earlier comment when you said "ISTM that the car can, or cannot fit in garage given the initial condition that in the rest frame, the car is longer than the garage; in other words there is an objective reality, but the frames differ on whether the car fits or not", but maybe I misunderstood? Jesse *My conclusion is that there's a contradiction in evidence, since the situation of the car fitting and not fitting makes no sense. AG * Why does it make no sense? Did you read my earlier analogy of the purely geometric scenario with two cylinders in 3D space that are at an angle relative to each other and cross each other in something like an X-shape, so there is some region where their interiors overlap? If you define a Cartesian x-y-z coordinate system and call the z-dimension the "vertical" one so that a "horizontal cross-section" of each cylinder would be where they intersect with a 2D plane of fixed z-coordinate, then the question "is there any horizontal cross-section of cylinder #1 that fits wholly inside the horizontal cross-section of cylinder #2" can depend on your choice of how the x-y-z axes are oriented, i.e. it's a coordinate-dependent question. Do you follow this (if not I can draw a diagram), and if so do you agree that "is there any horizontal cross-section of cylinder #1 that fits wholly inside the horizontal cross-section of cylinder #2" is a question with no objective coordinate-independent answer? If this makes sense to you, why is the question about the car and the garage any different? Instead of 2D cross-sections of 3D cylinders, this is basically just a question about 3D cross-sections of 4D world-tubes, and the answer similarly depends on the angle of the cross-section. Jesse I don't see how your complicated geometry has any relevance. In the situation at hand, we essentially have a one-dimensional coordinate system consisting of a straight line, the path along the garage. AG -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. 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