On Friday, January 17, 2025 at 5:21:58 PM UTC-7 Alan Grayson wrote:
On Friday, January 17, 2025 at 4:00:56 PM UTC-7 Jesse Mazer wrote: On Fri, Jan 17, 2025 at 5:18 PM Alan Grayson <agrays...@gmail.com> wrote: On Friday, January 17, 2025 at 2:05:42 PM UTC-7 Jesse Mazer wrote: On Fri, Jan 17, 2025 at 1:37 PM Alan Grayson <agrays...@gmail.com> wrote: On Friday, January 17, 2025 at 11:25:54 AM UTC-7 Jesse Mazer wrote: On Fri, Jan 17, 2025 at 12:31 PM Alan Grayson <agrays...@gmail.com> wrote: On Friday, January 17, 2025 at 7:46:06 AM UTC-7 Jesse Mazer wrote: On Fri, Jan 17, 2025 at 9:38 AM Alan Grayson <agrays...@gmail.com> wrote: On Friday, January 17, 2025 at 7:29:19 AM UTC-7 Jesse Mazer wrote: On Fri, Jan 17, 2025 at 7:51 AM Alan Grayson <agrays...@gmail.com> wrote: On Thursday, January 16, 2025 at 5:52:52 PM UTC-7 Jesse Mazer wrote: On Thu, Jan 16, 2025 at 7:33 PM Alan Grayson <agrays...@gmail.com> wrote: On Thursday, January 16, 2025 at 2:39:55 PM UTC-7 Jesse Mazer wrote: On Thu, Jan 16, 2025 at 2:43 PM Alan Grayson <agrays...@gmail.com> wrote: On Thursday, January 16, 2025 at 11:36:48 AM UTC-7 Jesse Mazer wrote: On Tue, Jan 14, 2025 at 12:02 AM Alan Grayson <agrays...@gmail.com> wrote: Using the LT, we have the following transformations of Length, Time, and Mass, that is, x --->x', t ---> t', m ---> m' The length contraction equation is not part of the Lorentz transformation equations, the x --> x' equation in the LT is just about the position coordinate assigned to a *single* event in each frame. The length contraction equation can be derived from the LT but only by considering worldlines of the front and back of an object, and looking at *pairs* of events (one on each of the two worldlines) which are simultaneous in each frame--length in a given frame is just defined as the difference in position coordinate between the front and back of an object at a single time-coordinate in that frame, so it requires looking at a pair of events that are simultaneous in that frame. The result is that for any inertial object, it has its maximum length L in the frame where the object is at rest (the object's own 'rest frame'), and a shorter length L*sqrt(1 - v^2/c^2) in a different frame where the object has nonzero velocity v. The t ---> t' equation is likewise not the same as the time dilation equation, it's just about the time coordinate assigned to a single event in each frame, although it has a simpler relation to time dilation since you can consider an event on the worldline that passes through the origin where both t and t' are equal to 0, and then the time coordinates t and t' assigned to some other event E on this worldline tell you the time elapsed in each frame between the origin and E. And the LT don't include any mass transformation equation. Jesse You're right of course. TY. I see the LT as giving appearances because, say for length contraction, the reduced length is not measured in the primed frame, but that is the length measurement from the pov of the unprimed or stationary frame. In relativity one does not normally designate any particular frame to be the "stationary frame", since all concepts of motion and rest are defined in purely relative way; if one has two objects A and B in relative motion, one could talk about the frame where A is stationary (A's 'rest frame') or the frame where B is stationary (B's rest frame), but that's all. I'm not sure what you mean by "the reduced length is not measured in the primed frame"--which object's length are you talking about? If A's rest frame is the unprimed frame and B's rest frame is the primed frame, then the length of object A in the primed frame is reduced relative to its length in its own rest frame, i.e. the unprimed frame. *Let's consider a concrete example of a traveler moving at near light speed to Andromeda. From the traveler's frame, the distance to Andromeda is hugely reduced from its length of 2.5 MLY from the pov of a non-traveling observer. This seems to imply that the reduced length is only measured from the pov of the traveler, but not from the pov of the non-traveler, because of which I describe the measurement from the pov of the traveler as APPARENT. Do you agree that the traveler's measurement is apparent because the non-traveler measures the distance to Andromeda as unchanged? TY, AG * I don't know what you mean by "apparent", but there is no asymmetry in the way Lorentz contraction works in each frame--if we assume there is a frame A where Milky Way and Andromeda are both at rest (ignoring the fact that in reality they have some motion relative to one another), and another frame B where the rocket ship of the traveler is at rest, then in frame B the Milky Way/Andromeda distance is shortened relative to the distance in their rest frame, and the rocket has its maximum length; in frame A the the rocket's length is shortened relative to its length in its rest frame, and the Milky Way/Andromeda distance has its maximum value. The only asymmetry here is in the choice of the two things to measure the length of (the distance between the Milky Way and Andromeda in their rest frame is obviously huge compared to the rest length of a rocket moving between them), the symmetry might be easier to see if we consider two rockets traveling towards each other (their noses facing each other), and each wants to know the distance it must traverse to get from the nose of the other rocket to its tail. Then for example if each rocket is 10 meters long in its rest frame, and the two rockets have a relative velocity of 0.8c, each will measure only a 6 meter distance between the nose and tail of the other rocket, and the time they each measure to cross that distance is just (6 meters)/(0.8c). Jesse *By apparent I just mean that the measurement the LT gives in this case, is not what is actually measured in the target frame. Moreover, this is differnt from the situation in the Twin Paradox as discussed in another recent post on this thread. A*G What do you mean by target frame? If the unprimed frame is the frame where Milky Way/Andromeda are at rest and the primed frame is the frame where the rocket is at rest, are you saying the primed frame does not actually measure a shorter distance from Milky Way to Andromeda if we use the LT starting from the coordinates of everything in the unprimed frame? Or are you arguing something different? Are you using primed or unprimed as the "target frame"? Jesse *The target frame is the primed frame, the result of the LT. The unprimed frame is the traveler's frame moving at some speed toward Andromeda. It's often claimed that the result of applying the LT will yield the actual measurement in the primed frame, but this isn't the case in this example. AG* OK, so you want the unprimed frame to be the frame where the rocket is at rest and the Milky Way/Andromeda are moving? In that case the unprimed frame will be the one where the distance between Milky Way/Andromeda is contracted according to the length contraction equation, since they are moving in that frame and at rest in the primed frame. And as I told you, the LT is not the same as the length contraction equation, if you apply the LT to the coordinates of the worldlines of Milky Way/Andromeda in the unprimed frame, you will get the correct answer that in the primed frame these worldlines have zero velocity (constant position as a function of time) and a greater coordinate distance between them than they did in the unprimed frame. Jesse *Firstly, copied below is what I posted earlier today. Because I wanted to contrast the Andromeda case with the TP, I used an SR solution for the latter, and you will note that the line segment paths of the inscribed polygon are inertial paths, and by infinitely refining the partition, I get the circular motion for the return path of the traveling twin. You will also note that the Earth-bound twin is at rest, and is analogous to the rest bound observer in the Andromeda case. In the TP, the Earth bound twin measures the traveling twin's clock running slower than his own clock, using the LT, but this effect is real for the traveling twin; otherwise he wouldn't see himself younger than his twin when they are juxtaposed upon his return. In contrast, you have the Andromeda traveler also at rest, and measuring the contracting distance in his frame, while the resting twin measures time dilation in his frame. However, in the frame of the moving rod representing the distance from Earth to Andromeda, according to your analysis the observer in that frame does NOT measure his length contracted;. only the rest frame measures the length contraction. * First, there's a habit in your writings that I find ambiguous: your use of the word "rest" and "moving" in an unqualified way, not clearly specifying that you are just speaking in a relative way about the rest frame of a particular object/observer in your thought-experiment. Sometimes this actually leaves me unclear on which frame you are actually talking about, at other times I can infer which one you're talking about but I wonder if you're trying to implicitly suggest that your thought-experiment shows that we must accept some notion of an objective truth about which observer is "really at rest", as opposed to the standard physicist's understanding that rest and motion can only be defined in relative terms. So, can I ask that if you are just using these terms in a relative way, would you please always phrase it in a way that specifies whose rest frame you're talking about? (eg "the rocket's rest frame", "the Earth's rest frame" etc.) And second, if you *do* mean to make some argument that one of your thought-experiments suggests a concept of objective/absolute rest, in that case could you be explicit that you are making such an argument by talking about "absolute rest" or some similar term? Please let me know if you are willing to make this change to your way of writing before addressing my more specific questions below. Now, when you say "the frame of the moving rod representing the distance from Earth to Andromeda" and "only the rest frame measures the length contraction", do you mean to introduce a rod at rest relative to Earth/Andromeda into the thought-experiment, and when you call this a "moving rod" you are talking about the perspective of the rest frame of the rocket which is in motion relative to Earth/Andromeda, and the rocket's rest frame is what you mean by "the rest frame"? If so, it is of course true that the rod would be contracted in the rocket's rest frame but not in the rod's own rest frame, and similarly true that the rocket would be contracted in the rod's rest frame but not in the rocket's own rest frame. This symmetry is similar to time dilation in that if you have two clocks A and B in inertial relative motion, in the rest frame of clock A it will be clock B that's running slow while clock A is running normally, and in the rest frame of clock B it will be clock A that's running slow while clock B is running normally. Also when you say "You will also note that the Earth-bound twin is at rest" are you suggesting that the conclusion of this thought experiment is that the frame where Earth is at rest is more "correct" in some absolute sense, or just saying that this is what's true by convention if we analyze the problem from the perspective of the Earth rest frame? Would you agree or disagree that we could analyze the whole problem from the perspective of any other specific inertial frame, like a frame where the Earth is moving at 0.99c the whole time and the traveling twin is sometimes moving faster and sometimes moving slower during different sections of its non-inertial path, and we would get exactly the same answer to the question of how much each twin has aged at the moment they reunite at the same location (a 'local physical fact' in the sense I discussed before)? Jesse *When I refer to the non-traveling twin in the TP, do you find this ambiguous? Do you find the traveling twin's frame ambiguous? If you do, I don't how I can be more specific. * It's usually understood that "non-traveling twin" just means the twin that moves inertially between the two meetings, and "traveling twin" means the one that changed velocities at least once on its path between the meetings. *The non-traveling twin is at rest on the Earth throughout. I never heard of any other concept of the non-traveling twin. The traveling twin is moving with respect to the Earth. I never heard of any other concept of the traveling twin. AG* All that's important to the twin paradox is that one twin is inertial and the other is not. For example if twin A is moving away from Earth inertially the whole time, and twin B is at rest relative to Earth for a while and then accelerates to catch up with twin A, then twin A is the inertial twin and twin B is the non-inertial one during the time between their meetings, and so it's guaranteed that twin B will have aged less than twin A when they reunite. *On and off over many years I've read about the TP. Never, not once, have I read it described as you do. Moreover, for the moving observer to leave and return, it's impossible for that observer to be totally inertial. If you model any observer leaving Earth, that observer cannot be inertial. AG* If you mean something different, like the idea that there is some objective/absolute sense that the "non-traveling twin" is at rest rather than moving, then I would object to that. But as stated, those phrases don't involve the word "moving" or "at rest" without qualification, which is what I was objecting to in my comments above. Will you agree in future to specify what object/observer "rest" and "moving" are relative to if you mean them in a relative way (which can easily be specified with a phrase like 'moving relative to [some object]' or 'at rest relative to [some object]'), or else to explicitly use some phrase like "absolute rest" and "absolute movement" if you mean them in a non-relative way? *No absolute anything. All motion is relative to something. AG * OK, then are you willing to alter your way of writing about these things to prevent ambiguity, to always use phrases like "moving relative to [object]" or "at rest relative to [object]" to specify the relative motion/rest you are thinking of? *Concerning the Andromeda frames, there's a frame with a moving rod, representing the distance between the Earth and Andromeda,* "Moving" in an absolute or relative sense? If in a relative sense, moving relative to what? Are you talking about a rod which is at rest relative to Earth and Andromeda, and moving relative to the rocket? *The observer in Andromeda case is traveling, moving with respect to the Earth. Then we can assume this observer is at rest, relative to a moving rod which represents the distance from Earth to Andromeda. AG * * and the frame of an observer using this frame to determine the length contraction.* Are you talking about an observer on the rocket which is moving relative to Earth/Andromeda, *Yes. AG* using his own rest frame to determine the length contraction of the rod which is at rest relative to Earth/Andromeda? *The rod is moving, the observer is stationary in his rest frame, from which he calculates the length contraction. AG * The rod is moving relative to this observer A, and is thus contracted in A's frame. *But not, at that time, also contracted in B frame. This is different from the TP where time is dilated in frame B, the frame of traveling twin. IOW, using the LT in the TP, what is measured from the Earth or stationary frame, is what's actually measured in the moving frame. Not so in Andromeda problem. AG* *Let me clarify the problem I'm trying to resolve; notice the T in LT. It stands for Transformation, presumably from one frame to another. It's claimed that the LT will produce the measured result in the target frame, based on parameters of the source frame for the transformation. And this seems to be the case in the TP; from the source frame, the frame at rest on the Earth, the LT tells us what will be measured in the traveling frame. And it seems to do just that, since the clock in the traveling frame actually ticks slower as predicted. If it didn't, the traveling observer would not age slower than the stationary observer. But when we consider the Andromeda problem, the LT seems NOT to predict what the frame of the moving rod will measure. Maybe it does, as you indicated, but only when a measurement is taken, unlike the case of the TP, where the result seems inherent, and not requiring a measurement. AG * And likewise if we assume the observer A is on board a rocket as I suggested, the rocket is moving relative to an observer B who is at rest relative to Andromeda, and so the rocket is contracted in the frame of observer B. So length contraction is completely symmetric between inertial frames, as is time dilation--are you saying otherwise? Jesse *IOW, the cases are similar except for the fact that one involves time dilation and other involves length contraction, but what is measured in the target frames is hugely different. This is the puzzle I am struggling with; namely, why is time dilation a measurable result for the traveling twin, but length contraction is NOT a measurable result for the frame of the moving rod in the Andromeda case, even though the frames doing the measuring in both cases have measurable results in their frames, but not in their respective moving frames? AG* *PS for Clark; I am halfway through the video you posted. IMO, there can be several ways to solve a problem and acceleration is one legitimate way because there IS accelation for the traveling twin, and acceleration IS equivalent to gravity, and gravity DOES slow clocks. I will finish the video out of curiosity for the author's supposed solution, but it seems obvious that acceleration is one possible solution. AG* *"Houston, we have a problem!" Now let's consider time dilation using SR in the Twin Paradox. Imagine the traveling twin moving in a circle and returning to Earth, and imagine the circle contains a polygon consisting of straight paths, which will later be infinitely partitioned, whose limit will be that circle. As measured by the stationary twin, the traveling twin's clock is dilated along each segment, so when the twins are juxtaposed, the traveling twin's elapsed time is LESS than clock readings for the stationary twin. If this is correct, it demostrates that what the stationary twin measures, is actually what the traveling twin's clock reads. IOW, what happens to time dilation in this case is OPPOSITE to what happens to the frames for the trip to Andromeda! Do you understand what I am alleging -- that length contraction acts in an opposite manner compared to time dilation, when I would expect them to behave similarly? AG* About mass, since the measured mass grows exponentially to infinity as v --> c, isn't this derivable from the LT, but in which frame? AG The notion of a variable relativistic mass is just an alternate way of talking about relativistic momentum, often modern textbooks talk solely about the latter and the only mass concept they use is the rest mass. For example the page at https://courses.lumenlearning.com/suny-physics/chapter/28-5-relativistic-momentum has a box titled "Misconception alert: relativistic mass and momentum" which says the following (note that they are using u to denote velocity): "The relativistically correct definition of momentum as p = γmu is sometimes taken to imply that mass varies with velocity: m_var = γm, particularly in older textbooks. However, note that m is the mass of the object as measured by a person at rest relative to the object. Thus, m is defined to be the rest mass, which could be measured at rest, perhaps using gravity. When a mass is moving relative to an observer, the only way that its mass can be determined is through collisions or other means in which momentum is involved. Since the mass of a moving object cannot be determined independently of momentum, the only meaningful mass is rest mass. Thus, when we use the term mass, assume it to be identical to rest mass." I'd say there's nothing strictly incorrect about defining a variable relativistic mass, it's just a cosmetically different formalism, but it may be that part of the reason it was mostly abandoned is because for people learning relativity it can lead to misconceptions that there is more to the concept than just a difference in how momentum is calculated, whereas in fact there is no application of relativistic mass that does not involve relativistic momentum. Momentum is needed for situations like collisions or particle creation/annihilation where there's a change in which objects have which individual momenta, but total momentum must be conserved. It's also used in the more general form of the relation of energy to rest mass m and relativistic momentum p, given by the equation E^2 = (mc^2)^2 + (pc)^2, which reduces to the more well-known E=mc^2 in the special case where p=0. By the way, since relativistic momentum is given by p=mv/sqrt(1 - v^2/c^2), you can substitute this into the above equation to get E^2 = (m^2)(c^4) + (m^2)(v^2)(c^2)/(1 - v^2/c^2), and then if you take the first term on the right hand side, (m^2)(c^4), and multiply it by (1 - v^2/c^2)/(1 - v^2/c^2) and gather terms, you get E^2 = [(m^2)(c^4) - (m^2)(v^2)(c^2) + (m^2)(v^2)(c^2)]/(1 - v^2/c^2), and two terms cancel each other out so this simplifies to E^2 = (m^2)(c^4)/(1 - v^2/c^2), and then if you take the square root of both sides you get E = γmc^2. So the original equation for energy as a function fo rest mass m and relativistic momentum p can be rewritten as E=Mc^2 where M is the relativistic mass defined as M = γm, again showing that relativistic mass is only useful for rewriting equations involving relativistic momentum. 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 everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/9eba4b44-e843-41e1-9a40-d1b761d9699bn%40googlegroups.com.
