On Fri, Dec 20, 2024 at 2:09 AM Alan Grayson <[email protected]> wrote:
> *Pedagogical" means what? * > > > Relating to how the subject is taught, in this case specifically which > concepts any teacher would see as important for students to understand. If > a student doesn't understand that different frames agree on all local > events, then they basically don't understand the first thing about how > relativity works. > > > *If car fits in one frame and not in another, isn't that what we would > expect, and yet in my prior post I wrote that this seems contradictory? Why > do you expect the frames must agree about this kind of local event? To > avoid a contradiction? AG* > > > As long as the laws of physics are Lorentz-invariant, that guarantees that > when different inertial frames apply the same equations (including length > contraction) they will get locally identical predictions, assuming they > both are using initial conditions which are equivalent under the Lorentz > transformation. > > > *Presumably, in this problem, the laws of physics are Lorentz-invariant, > but contrary to what you claim, they don't result in the same locally > identical predictions. Maybe I don't understand what you mean by "same > locally identical predictions". In fact, the results are diametically > opposite, about whether the car fits in garage. AG* > "The car fits" or "the car fits" are not statements about local events, i.e. statements about things that happen at a single spacetime point in one of Brent's diagrams. But the back of the car does pass the front of the garage at a single point in spacetime in this problem, so if there was a clock #1 attached to the back of the car and a clock #2 attached to the front of the garage, all frames would have to agree in their predictions about what each clock reads at the moment they pass through that one point in spacetime. Likewise if a clock #3 is attached to the front of the car and a clock #4 is attached to the back of the garage, those clocks would cross paths at a single point in spacetime so both frames would have to agree in their predictions about what they each read at the meeting, which they do. You can also imagine there is a ruler Rg at rest relative to the garage running along its length, and another ruler Rc at rest relative to the car and running along the same axis, so the two rulers are moving alongside each other at 0.8c. In this case, for any of the types of events I mentioned above like clock #1 passing clock #2, both frames also must agree about what marking on Rg this event coincides with in space, and what marking on Rc it coincides with. These are all facts about things that are happening at individual points in spacetime, not facts which require talking about a range of positions of times, like whether the car "fits". In Brent's scenario, assume clocks #1 and #3 at the back and front of the car were synchronized in the car's rest frame by the Einstein synchronization procedure, and clocks #2 and #4 at front and back of the garage were synchronized in the garage's rest frame using the synchronization procedure. Also assume the localized event of the back of the car passing the front of the garage coincided with both clock #1 and clock #2 there reading t=0 and t'=0 respectively, and that this happened right next to the x=0 mark on ruler Rc and the x'=0 mark on ruler Rg. All frames agree on these facts, which are exclusively about what happened at a single point in spacetime, namely the point where the back of the car passed the front of the garage. Given these assumptions, according to relativity they will *also* agree in all their predictions about a second event, the event of the front of the car reaching the back of the garage. Specifically they will agree that at the same point in spacetime as this second event, all the following are true: --Clock #3 at the front of the car read t = -7.5 --Clock #4 at the back of the garage read t' = 3.5 --this event of the front of the car reaching the back of the garage coincided with the x=12 mark on ruler Rc --this event of the front of the car reaching the back of the garage coincided with the x'=10 mark on ruler Rg There is no disagreement on any of these local facts. The only disagreement is that each observer adopts a different *convention* about which ruler and clocks to treat as canonical for the sake of assigning coordinates--the car rest frame defines time-coordinates by the clocks at rest in the car frame (clocks #1 and #3) and the ruler at rest in the car frame (Rc), while the the garage frame defines time-coordinates by the clocks at rest in the garage frame (clocks #2 and #4) and the ruler at rest in the garage frame (Rg). Based on these conventions, the car observer says the event of the back of the car passing the front of the garage happened AFTER the event of the front of the car reaching the back of the garage, therefore the car never "fit", while the garage observer says the event of the back of the car passing the front of the garage happened BEFORE the event of the front of the car reaching the back of the garage, therefore the car "did" fit. But this is not a disagreement about any of the local facts I mentioned. (BTW I earlier derived these numbers as the coordinates assigned to the event in each frame at https://groups.google.com/g/everything-list/c/gbOE5B-7a6g/m/43aKXeEUAQAJ but here I'm just emphasizing that coordinate judgments can be grounded in local readings on physical clocks and rulers, something I also talked about at https://groups.google.com/g/everything-list/c/gbOE5B-7a6g/m/BvxSA-b3AAAJ ) > > But are you asking a different question about what is the motive for > demanding that any claims about how things work in different frames needs > to pass the test of giving identical local predictions, in order to qualify > as good physics? If so just consider that there are all sorts of local > interactions in physics, like collisions, that cause changes that different > frames couldn't disagree about without being obviously inconsistent. For > example, say you have a clock that's wired to a small bomb that will cause > a localized explosion, which will be triggered when it reads 100 seconds. > And say you have another object in motion relative to the clock/bomb, say a > glass of water, which is going in the opposite direction so they will cross > paths. Imagine different frames could disagree in their prediction about > whether the event of the clock/bomb crossing paths with the glass of water > coincided was at the same local point in space and time as the clock > reaching 100 seconds--like, one frame predicts the clock reads 90 seconds > when they cross paths, a second frame predicts the clock reads 100 seconds > when it crosses paths with the glass of water. In this case, the second > frame would predict the glass of water was right next to the bomb when it > exploded, and so predicts that the glass will be broken up after the > encounter. Meanwhile the first frame would predict the glass of water has > already put some distance between it and the bomb by the time the bomb > exploded, so the glass would be intact after the explosion. This is a clear > physical contradiction, no? They can't both be right, and you could easily > falsify one frame's prediction just by looking at the glass afterwards. > > On the other hand, if all frames agree in all their predictions about > local events as in relativity (assuming Lorentz-invariant laws of nature), > then you don't get any contradictory predictions about such localized > physical interactions which affect the state of objects later. You may find > it counter-intuitive that they still differ in some kind of non-local > bird's-eye account of what happened, but you can't point to any differences > they will see on any measuring-instruments (since instrument readings are > also local events), like what a clock mounted on the back of the car reads > as it passes by the front of the garage. > > > *You keep asserting that the frames agree in all their predictions, when > in this problem they surely don't! So, I don't think we agree on this, if I > understand what you mean. AG * > See above about what I mean by localized events. > > > Do you disagree with my point that if different frames *didn't* have > differing definitions of simultaneity, it would be impossible for the two > frames to disagree about whether the car or garage was shorter without this > leading to conflicting predictions about local events, like what the clocks > mounted to front and back of the car will read at the instant they pass > clocks attached to the front and back of the garage?' > > > *I don't see how simultaneity or not helps in this situation. It seems > impossible for the car to fit when in motion. AG * > > > It helps by showing how the car can fit in the garage's frame without > leading the garage frame and the car frame to disagree in a single > prediction about local events. Does your "seems impossible" just mean you > find it counter-intuitive, not that you have a concrete argument about why > you think it *would* lead to disagreements in predictions about local > events? > > > *Well, in this case, using length contraction, the facts speak for > themselves. What could be counter-intuitive is that there's only one real > car, so how can Lorentz-invariant physics give us frame dependent results? > This seems to be not only a weak point in your analysis, but seriously > mistaken. AG * > There is no frame-dependence in predictions about localized events, and according to relativity these are the only real physical facts in the problem, everything else is a matter of conventions about how you *label* these events with position and time coordinates, no more problematic than a classical physics scenario . > > > > And in a later post, I elaborated on why differences in simultaneity are > critical to avoiding contradictory predictions about localized physical > events: > > 'In an imaginary alternative physics where different frames had no > disagreement about simultaneity but different observers still all believed > the length contraction equation should apply in their frame, then this > would be a genuine paradox/physical contradiction, because different frames > would end up making different predictions about local events. Think about > it this way--if there were no disagreement about simultaneity, there could > be no disagreement about the *order* of any two events (this would be the > case even if observers predicted moving clocks run slow like in > relativity). But if observer #1 thinks the car is shorter than the garage, > he will predict the event A (the back of the car passing the front of the > garage) happens before event B (the front of the car reaches the back of > the garage), and if observer #2 thinks the car is longer than the garage, > he will predict B happens before A. If there were no disagreement about > simultaneity this would lead them to different predictions about readings > on synchronized clocks at the front and back of the car/garage at the > moment of those events, specifically whether the clock at A would show a > greater or lesser time than the clock at B.' > > Jesse > > > *Jesse; in the near future I will try to address each of the issues you've > raised,* > > > OK, please prioritize answering the question about whether you understand > the basics of how position vs. time plots work in classical mechanics, > because that really is a crucial prerequisite if you want to hope to > understand anything about spacetime diagrams in relativity. If you don't > understand it I'm sure I could find a site that lays out the essentials. > And as a follow-up, did you ever study the basics of algebraic geometry? > Like if you had to plot a function like y = 4x + 5 on a graph with x and y > axes would you know how to do it? Likewise would you know the algebra > needed to figure out where that function intercepts with another one like y > = 2x +10? > > > *Sure, I have advanced degrees in math and physics. I'd solve for x, by > setting 4x + 5 = 2x + 10, and then solve for y to get the point of > intersection. (I sure hope I got that right!) I've seen spactime diagrams > before, but I'm more comfortable with explanatory text.* > OK, in a word problem if I say that in a classical problem, at t=0 seconds a spaceship is initially at position x=7 meters away from the origin, and it's moving in the +x direction at 12 meters/second, would you know how to write down the equation for its position as a function of time x(t), and plot this as a line on a graph with position in meters on the horizontal axis and time in seconds on the vertical? If so, that's really all that a "worldline" is. Likewise, if we have various such worldlines for different objects, and we want to know the position of each object at a particular time like t=5, do you understand why this would just be a matter of plotting a horizontal line that goes through the t=5 mark on the vertical axis (a classical line of simultaneity), and seeing the point it intersects each worldline? > *Tell me this if you can; in Brent's spacetime diagrams, he often has a > stretched car. Since there's nothing in the problem to indicate an > elogation of the car, what's Brent trying to illustrate? AG* > He's trying to illustrate a slanted line of simultaneity that connects two events that are simultaneous in the car's frame, as graphed in the garage frame. But the visual length of this line in the diagram is not meant to correspond to an elongated length in either frame, it just looks longer because it's being translated from relativistic (Minkowski) geometry where the length of a spacelike line segment is given by sqrt(x^2 - t^2) into a diagram in a 2D euclidean space (your computer monitor) where if we label the two spatial axes x and y, then the length of any slanted line segment is given by sqrt(x^2 + y^2). In Minkowski geometry the length of a slanted segment should be *less* than the distance along the x-axis between its endpoints, but in Euclidean geometry it's greater because of that switch from a minus to a plus, and we are only capable of intuitively visualizing Euclidean geometry so that's what we use for our imperfect diagrams. That's why the diagram has to show the car as longer here even though according to the relativistic math the proper length of that line segment should really be shorter. > > > * but for now let me just say I don't understand how to resolve this > issue, and my tentative pov is that relativity just isn't correct. Listen; > we start in a rest frame of a car which is longer than a garage. and have > no problem asserting that it won't fit. And that's how things seem from > both entities with physical observers. So far so good. Now we imagine the > car in motion and apply length contraction in both frames and we get > opposite results; namely, that in the car's frame, it won't fit in the > garage, but in the garage frame it does fit, and the fits gets easier as > the car's velocity increases. If I imagine a real car and a real garage, > from one frame it doesn't fit, the car's frame, and from the other frame, > the garage, it does fit. So, if intially the car doesn't fit, from the pov > of both physical entities should I expect contrary results when the car is > in motion? Maybe so. But I still can't wrap my head around the alleged > claim, that the observed reality will be frame dependent. I mean, how can > two observers in different frames, looking at a real car, disagree on what > they see?* > > > What do you mean "see"? Are you talking about what they see visually, in > terms of when light from different events reaches their eyes? If so, do you > understand that when we talk about "simultaneous" events in any frame, we > are *not* talking about events that are seen simultaneously in a visual > sense by an observer at rest in that frame, unless the observer happens to > be positioned equidistant from both events? > > > *If we imagine observers in each frame, humans seeing or instruments > measuring, how do you expect them to observe the same thing, when the final > results differ hugely? The car fits when observed from garage frame, but > not when observed from car frame! AG * > All they see is the sum of light from multiple events which are individually localized in space and time. Imagine for example that they are watching an image of the car and garage on a screen (it makes no difference to the problem), such that every bit of light they see was emitted by a specific pixel at a particular moment in time. In this case, even in a classical problem where there are no disagreements about simultaneity or distance in terms of the coordinates each observer assigns, as long as the light takes a finite speed to get from the pixel that emitted it to an observer's eye, different observers may visually *see* events at different times and in different orders. Even in this purely classical scenario you could have *visual* disagreements about whether the car fits (i.e. one observer sees the light from the event of the back of the car passing the front of the garage BEFORE seeing the light from the event of the front of car reaching the back of the garage, a different observer sees it AFTER), even though classically they won't disagree once they correct for light transit times in order to assign time-coordinates to these events. > > This was another point I made in an earlier post (at > https://www.mail-archive.com/[email protected]/msg97741.html > <https://www.mail-archive.com/[email protected]/msg97741.html> > ) which you didn't respond to: > > 'Note that when we talk about what happens in a given frame this is not > what any observer sees with their eyes, it's about when they judge various > events to have happened once they factor out delays due to light transit > time, or what times they assign events using local readings on synchronized > clocks that were at the same position as the events when they occurred. > > > *It could be both. I'm just asserting there is some objective reality > about whether the car fits or not, and from this I conclude a paradox > exists since results using contraction give opposite results. How do you > fail to reach this same conclusion? AG* > Do you definitely deny what I said about all observers agreeing about all local events, now that I've clarified a little what I mean by "local events"? Or are you saying that *even if* they agree about all local events, you still think there must be a separate objective truth about the question of whether the car fits, a fact of the matter that is somehow more than the sum total of all the facts about localized events (including all local readings on measuring-instruments)? > > > For example, if in 2025 I see light from an event 5 light years away, and > then on the same day and time in 2030 I see light from an event 10 light > years away, I will say that in my frame both events happened simultaneously > in 2020, even though I did not see them simultaneously in a visual sense. > And if I had a set of clocks throughout space that were synchronized in my > frame, when looking through my telescope I'd see that the clocks next to > both events showed the same date and time in 2000 when the events happened.' > > > * Incidentally, I just noticed that in one of Brent's recent posts with > two diagrams, he says there is a disagreement about simultanaeity, but I am > not sure if he's referring to comparing the two frames, and when I > interpreted this as his comparison, he got angry, denying my > interpretation. My bias is that the frames should agree (on what a bird's > eye observer would see?), but does that require disagreement about > simultaneity? AG* > > > What does "bird's eye observer" mean, if it's supposed to be something > more than just the sum total of all local events? > > > *Not a precise scientific term, so just forget it. It could be how God > sees everything, the ultimate observer so to speak, and finds your > conclusion baffling. AG * > Do you think someone with such a God's-eye perspective would find it baffling that different coordinate systems may disagree about which of two events has a greater x-coordinate, and that there is no objective truth about the matter independent of how we choose to orient our spatial x-y-z axes for the sake of assigning position coordinates? If you're OK with there being no objective truth about this, why are you suddenly *not* OK with the fact that there might similarly be no objective truth about which of two events has a greater t-coordinate, independent of our conventions about how to define coordinate systems? Jesse -- 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]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/CAPCWU3LF4UVdejjmHbB7jQMQgGnbZ2N7aNAf2GMejfjy%3DQrw8g%40mail.gmail.com.
