On Thu, Feb 6, 2025 at 5:48 PM Jesse Mazer <[email protected]> wrote:
> > > On Thu, Feb 6, 2025 at 3:46 PM Alan Grayson <[email protected]> > wrote: > >> >> >> On Thursday, February 6, 2025 at 6:48:34 AM UTC-7 Jesse Mazer wrote: >> >> On Thu, Feb 6, 2025 at 4:48 AM Alan Grayson <[email protected]> wrote: >> >> On Monday, February 3, 2025 at 6:44:29 AM UTC-7 Jesse Mazer wrote: >> >> On Mon, Feb 3, 2025 at 3:16 AM Alan Grayson <[email protected]> wrote: >> >> On Sunday, February 2, 2025 at 11:37:48 PM UTC-7 Jesse Mazer wrote: >> >> On Mon, Feb 3, 2025 at 12:09 AM Alan Grayson <[email protected]> wrote: >> >> On Sunday, February 2, 2025 at 9:39:04 PM UTC-7 Jesse Mazer wrote: >> >> On Sun, Feb 2, 2025 at 11:04 PM Alan Grayson <[email protected]> wrote: >> >> *In mathematics, functions have domains and ranges, standard terminology. >> A function maps domain sets to range sets. The image of a function is the >> set containing its range. Again, standard mathematical terminology. The >> contraction formula, derived for the LT, is a function. With me so far? AG* >> >> >> In math the domain and range of a function have no physical >> interpretation, they are just sets of mathematical objects with no comment >> about what they "mean". For example, in purely mathematical terms, the >> length contraction formula l = L*sqrt(1 - v^2/c^2) has as its domain values >> of the variables L and v where L can be any member of the set of real >> numbers > 0, and v is any member of the set of real numbers larger than or >> equal to 0 but smaller than the value of c (in whatever units you're using >> like 299792458 meter/second), and the range is the value of l which can >> likewise be any member of the set of real numbers > 0. >> >> If you are speaking more metaphorically, basically just saying that any >> equation like length contraction can be thought of as a sort of machine >> that takes two values (a proper length L and a speed v) as input/"domain" >> and spits out another value (a length l in whatever frame measured the >> object to be moving at v) as output/"range", then I'm fine with that. >> >> >> *Now about the substance. If coordinate frame O2 is the domain of the >> formula function, the x values are elemments in its domain, and the x' >> values are elements in its range.* >> >> >> If you are talking about the LT function when applied to a problem where >> we start with coordinates in O2 as input and get coordinates in O1 as >> output, sure. If you're talking about the length contraction formula, no, >> the only inputs to that are a proper length and a velocity in a single >> frame, and the output is the length in that same frame. >> >> >> *Here's where we disagree. ISTM, there's a convention, that the image >> frame of the formula, is moving wrt the frame applying the contraction >> formula,* >> >> >> The length contraction formula is not exactly translating between >> different frames at all, at least not in the sense that the input >> exclusively consists of variables from one frame and the output a variable >> from a different frame, the way the LT does. In the length contraction >> formula, both the speed v which is used as input and the contracted length >> L' which is given as output are measured in the SAME frame, the frame of >> the observer who sees the object moving at speed v relative to their own >> frame. The proper length L, which is also used as input, can be thought of >> as length in a different frame, namely the object's own rest frame. >> >> Since there is a symmetry of motion where the speed of B relative to A is >> the same as speed A relative to B, I suppose the v in the length >> contraction formula *could* instead be defined as the speed of the observer >> as measured in the object's rest frame, rather than defining it as the >> speed of the object in the observer's frame as textbooks normally do. If >> you think of it in that alternate way where v is the speed of the observer >> in the object's frame, then you could consider both input variables L and v >> to be measured in one frame (the object's rest frame) and the output >> variable L' in another (the observer's frame). Even though v is not >> normally described this way, it'd make no difference mathematically if you >> did. But if we do think of it this way, I presume you'd then say the >> object's rest frame is the frame "applying the contraction formula", and >> the observer's frame is the "image frame"? >> >> >> >> * and L' is the contracted length in the frame in relative motion. If >> you claim the contraction occurs in the same frame from which the formula >> is applied, then won't we get no contraction? AG * >> >> >> See above, the normal way of describing the length contraction formula >> involves a mix of measurements in two different frames as input so there is >> no single 'frame from which the formula is applied', but if you want, you >> do have the option to think of the meaning of v in a different way such >> that both L and v are defined in terms of measurements in a single input >> frame (the rest frame of the object), in which case the contracted length >> L' would be in a different output frame (the rest frame of the observer). >> >> >> >> >> Also note that there is nothing in the LT formula that restricts which >> frame you take as input and which you get as output. You can just as easily >> start with the coordinates in O1 as input and use the formula to find the >> coordinates in O2 as output. I just said this in the post above and even >> offered to give you a numerical example of how this would work. >> >> >> *Of course, but why do that? AG * >> >> >> Because you asked how the LT could be used to predict a contracted >> length--in order to do that, the output of the LT formula has to be defined >> in a frame where the object has non-zero velocity. >> >> >> >> >> * The contraction formula is a mapping or correspondence from coordinate >> sets O2 to coordinate sets O1, the moving frame in relative motion wrt O2.* >> >> >> The contraction formula doesn't do that, the LT *can* be used to do that >> if you start with coordinates in O2 as input and then get coordinates in O1 >> as output, but as I say above it can just as easily map coordinates in O1 >> taken as input to coordinates in O2 given as output. >> >> >> * That is, the contraction formula maps O2, the frame with no rod, to O1, >> the frame with the rod.* >> >> >> I asked you several times in the last comment (and in a number before >> that) to clarify if by "no rod" you just mean the rod is not *at rest* in >> O2, or if you mean that O2 literally doesn't "see" the rod to assign it >> coordinates, or something else. >> >> >> *There is one rod is at rest in O1, and its frame in moving relatively >> wrt O2. AG* >> >> >> But you agree that the observer O2 can measure and assign coordinates to >> the rod in O2's own rest frame? >> >> >> >> >> * You say, and I now agree, that there's no contraction of the one and >> only rod in O1.* >> >> >> Yes. >> >> >> * So what happened to contraction?* >> >> >> The rod is contracted in other frames like O2, and as I said you're free >> to use the LT to start with the coordinates in O1 used as input and then >> use the LT formula to get the coordinates in O2 as output (once you know >> the coordinates of the front and back of the rod in O2 it's easy to get the >> length of the rod in O2 from that). And if you use the length contraction >> formula rather than the LT it's an even simpler matter to derive the rod's >> contraction in O2, you just need the rod's proper length L as well as its >> velocity v in O2, you enter that L and v into the length contraction >> formula as input and get the contracted length l in O2 as output. >> >> >> *When we map from O2 to O1, you agreed, no contraction, so if we map from >> O1 to O2, won't there also be no contraction?* >> >> >> No, since the rod is at rest in O1 and moving in O2, if you map from O1 >> to O2 using the LT you get a contracted length for the output. As I said I >> could give you a numerical example showing this if you want. >> >> >> *If rod is at rest in 01 and in relative motion wrt O2 (because O1 is the >> moving frame) and we map from 02 to 01 we get no contraction, but if we map >> in opposite direction, from O1 to O2, we get contraction? Could you explain >> how you reach these conclusions? AG * >> >> >> The quickest way to get this conclusion is just to know that the >> predictions of the LT about length will always match those of the length >> contraction equation which was derived from it, and if the rod has some >> known proper length like L=10 and some nonzero velocity like v = 0.6c >> relative to O2, then in O2 frame its length is 10*sqrt(1 - 0.6^2) = 8, but >> since it has a velocity of v = 0 relative to O1, then in the O1 frame it >> must have length 10*sqrt(1 - 0^2) = 10. The LT will agree with this so if >> you use the equations with O2 as input and O1 as output, you'll have a >> length of 8 as input and a length of 10 as output (no contraction in output >> frame). But if you use the LT equations with O1 as input and O2 as output, >> you'll have a length of 10 as input and a length of 8 as output >> (contraction in the output frame). >> >> If you don't want to just trust the principle of "the LT agrees with the >> length contraction equation", you can verify this by direct calculation >> using the LT equations. Let's call O2 the unprimed frame and O1 the primed >> frame. In the O2 frame we have the following equations for the position as >> a function of time of each end of the rod (i.e. the worldline of each end): >> >> Back of the rod: x = 0.6c*t >> Front of the rod: x = 8 + 0.6c*t >> >> These equations tell you that at t=0 the back of the rod is at x=0 and >> the front of the rod is at x=8, so the rod has a length of 8 in the O2 >> frame, and it's moving at 0.6c. >> >> Then in the primed O1 frame we have the following equations: >> >> Back of the rod: x' = 0 >> Front of the rod: x' = 10 >> >> These equations tell you that at any choice of time coordinate t' the >> back is always located at x'=0 and the front is always located at x'=10, >> i.e. the rod is at rest in this frame and has a length of 10 (its proper >> length). >> >> One way to use the LT is to directly plug the full equations of motion in >> one frame into the LT equations as input, and after a little algebra get >> out the equations of motion in the other frame as output. That's what I did >> in that post at >> https://groups.google.com/g/everything-list/c/ykkIYDL3mTg/m/giZVF9PpDQAJ >> going from the equations of motion in the frame where the rod was moving >> (here the O2 frame) to the equations of motion in the frame where the rod >> was at rest (O1). This does involve a little algebra though. A simpler way >> of just checking that the LT map between those equations of motion is just >> to pick the coordinates of some individual points that are along a given >> end's worldline in the coordinates of one frame, and see that they always >> map to coordinates of individual points that are along the same end's >> worldline in the coordinates of the other frame. >> >> For example in the O2 frame, the points (x=0, t=0) and (x=3, t=5) and >> (x=6, t=10) and (x=9, t=15) would all lie along the worldline of the BACK >> of the rod given by x = 0.6c*t in this frame (I'm assuming we're using >> units like light-seconds and seconds where c=1). If you take any of those >> points as input and use the x-->x' LT to find the corresponding coordinates >> of the point in the primed O1 frame as output, you will get (x'=0, t'=0) >> and (x'=0, t'=4) and (x'=0, t'=8) and (x'=0, t'=12) [note that the x-->x' >> equation in this case is x' = 1.25*(x - 0.6c*t) and the t-->t' equation is >> t' = 1.25*(t - 0.6*x)]. You can see that all of these points do lie along >> the line x'=0, the equation for the BACK of the rod in the O1 frame. >> >> Similarly, in the O2 frame, the points (x=8, t=0) and (x=11, t=5) and >> (x=14, t=10) and (x=17, t=15) all like along the worldline of the FRONT of >> the rod given by x = 8 + 0.6c*t in this frame. If you take any of those >> points as input and use the x-->x' LT to find the corresponding coordinates >> in the primed O1 frame as output, you will get (x'=10, t'=-6) and (x'=10, >> t'=-2) and (x'=10, t'=2) and (x'=10, t'=6). You can see all these points >> lie along the line x'=10, the equation for the FRONT of the rod in the O1 >> frame. >> >> Then if you want to go in reverse, starting with O1 coordinates as input >> and getting O2 coordinates as output, you can use the LT equations for >> x'-->x which in this example is x = 1.25*(x' + 0.6c*t'), and the one for >> t'-->t which is t = 1.25*(t' + 0.6*x'). You can verify that the reverse >> mapping here works too, for example if you take (x'=10, t'=2) as input (a >> point I listed above as being on the worldline of the front of the rod) you >> get back (x=14, t=10) as output. So you can verify this way that if you are >> given the equations x' = 0 and x' = 10 for the rod in the O1 frame >> (corresponding to a rod at rest with length 10 in the O1 frame), if you >> pick any point along those lines and map to the O2 frame with the LT, as >> output you always get points along x = 0.6c*t and x = 8 + 0.6c*t >> (corresponding to a rod of length of length 8 and velocity 0.6c in the O2 >> frame). So, here the input coordinates represent a rod with its proper >> length of 10, and the output coordinates represent a rod with a contracted >> length of 8. >> >> Jesse >> >> >> >> Let's start from the beginning to make sure we're on the same page. Using >> the contraction formula L' = L * sqrt ( 1 - (v/c)^2), where L is the rest >> length of rod in frame f1, moving at velocity v wrt frame f2, and L' is the >> contracted length of the rod as calculated from the pov of f2. Do you agree >> with my interpretations of these variables? Do you agree that the measured >> length of rod in f1 is never L', but always its rest length L? These are >> Yes or No questions. TY, AG >> >> >> As long as we are considering a case where f2 is different from f1 (which >> is normally the only situation where anyone would bother to use the >> formula), I agree with everything you just wrote. But the one caveat I'd >> add is that one is technically free to consider the special case where f2 = >> f1, i.e. you are imagining that the frame f2 of the observer who is using >> the equation is the same as f1, the rest frame of the rod, so in this case >> v=0 and L' = L. In my first paragraph above (the one starting with 'The >> quickest way to get this conclusion') I had L=10, and I considered both the >> case of an observer O2 moving at 0.6c relative to the rod who used L=10 and >> v=0.6c as input to the formula to output a length L' = 8 in his frame, and >> also the case of observer O1 who was at rest relative to the rod and used >> L=10 and v=0 as input to the formula to output a length of L' = 10 in his >> frame. Of course the latter is a trivial case and it's easier to just >> remember that rest length/proper length is what will be measured in the >> frame where the rod is at rest, but I included it to illustrate the point >> that either observer can apply the formula to get the correct length in >> their frame as the output. >> >> Jesse >> >> >> In the scenario posited, you agree that the observer in the moving frame, >> f1, where the rod is at rest, always measures the rod as having length L, >> whereas the observer applying the contraction formula, resident in frame >> f2, always measures the rod as having a contracted length as L', which is >> always less than L, >> > > I would not call f1 "the moving frame" as that is the sort of > non-comparative designation I don't like to use. And I also made clear it's > not *always* true that the observer in f2 measures a length less than L, > because you can consider the special case where f2 = f1, i.e. the observer > applying the length contraction formula is at rest relative to the rod > (observer O1 in the previous discussion)--in that case the rod has length > L' = sqrt(1 -0^2)*L = L according to the formula. So it would be more > accurate to say that the length L' which is the output of the length > contraction formula is always *greater than or equal to* the proper length > L in the rod's rest frame. > Sorry, above I should have written that the length L' which is the output of the length contraction formula is always *smaller* than or equal to the proper length L. > > >> why do you deny my claim (so it seems) that the LT, via the contraction >> formula, never predicts what the observer in f1 will actually measure? AG >> > > I don't understand what "the LT, via the contraction formula" is supposed > to mean here. If you are continuing to endorse the strange idea that the LT > would predict a contracted length in f1 (rather than the proper length L > which is actually measured), then it seems to be an obvious consequence of > your idea that the LT would have to *differ* in its predictions from the > length contraction formula. Please tell me if you agree/disagree with the > following four statements: > > 1. If observer O2 has a velocity of v=0.6c relative to the rod, and uses > the length contraction formula to predict the length in his own frame, then > the output of the formula is that the rod has contracted length L' = 0.8*L. > > 2. If the observer O1 has a velocity of v=0 relative to the rod, and uses > the length contraction formula to predict the length in her own frame, then > the output of the formula is that the rod has its non-contracted length L' > = L. > > 3. In order for the LT's predictions to agree with the predictions of the > length contraction formula when converting between O1's frame and O2's > frame in either direction, then regardless of which frame's coordinates you > use as input and which you use as output, the numerical predictions should > agree with statements #1 and #2 above. > > 4. So, if your input coordinates are those of O2 where the length is 0.8*L > and the velocity is 0.6c (as per statement #1 above), then if the LT agrees > with the length contraction formula, that means the output coordinates in > O1 should say that the length is L and the velocity is 0 (as per statement > #2 above). Similarly if your input coordinates are those of O1 where the > length is L and the velocity is 0 (as per statement #2 above), then if the > LT agrees with the length contraction formula, the output coordinates in O2 > should say the length is 0.8*L and the velocity is 0.6c (as per statement > #1 above). > > Agree/disagree? > > 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/CAPCWU3LZDj%2BNS7GjBTsGBojk68AAdLN1uNetWQKTgCDcdejDaA%40mail.gmail.com.
