Wikipedia:Reference desk/Archives/Science/2024 March 27

= March 27 =

No-signaling and the infinite hat game
(posting in RDS instead of RDMath because the math is pretty clear; it's the alleged "physics" that is not making sense to me)

Recently someone added a note to our article on the axiom of choice a reference to an interesting paper that describes an allegedly counterintuitive consequence of AC. The first bit is fun but not super-surprising to a set theorist (which I am by academic training though not by current employment). The game involves a countably infinite collection of men (just to make the pronouns easy) lined up in the order of the natural numbers. They are given hats, either red or white. No one can see his own hat, but each can see the hats of every man in front of him. Each is supposed to guess the color of his own hat. They are allowed to agree on a strategy beforehand.

It turns out that if they choose and agree on a transversal of the equivalence relation of finite differences on sequences of red and white (that is, two sequences are equivalent if they differ in only finitely many places), then no matter how the hats are placed, all but finitely many of them can guess correctly. The proof of this is essentially trivial once you've understood the definitions. AC guarantees that such a transversal exists.

Now, the counterintuitive thing here is that no man can get any information whatsoever about the color of his own hat just from the colors of the hats in front of him, but nevertheless the strategy works perfectly after some point. I guess if you want you can take this as a point against AC, though from my perspective it doesn't really compete with the clear and simple intuition in favor of AC.

But where I think the paper might go off into the weeds a bit, or maybe I just haven't been clever enough to get what it's talking about, is when it tries to connect this with forms of the no-signaling principle. They identify a "probabilistic" and "functional" version of the NSP, and claim to show that, allegedly counterintuitively, the "functional" version is a stronger "resource".

This I cannot make heads or tails of. First, I don't see how either version of the NSP is a "resource", unless the word is being used in some way I'm not familiar with. As far as I can tell they are constraints, not resources. So my first clear question is, is there some physics-specific sense of the word "resource" that would make this make sense?

Then, in the "functional" version, they allow an oracle for a transversal of the equivalence relation. I don't think they allow that in the "probabilistic" version, though I haven't combed over every line of it, because if they did, you could just use it and ignore the probabilistic part. So isn't it the oracle that's the extra "resource" here? And then, second clear question, why should it be counterintuitive that you can do more with the oracle than without it? --Trovatore (talk) 05:19, 27 March 2024 (UTC) Pinging, who added the note. --Trovatore (talk) 06:35, 27 March 2024 (UTC)


 * Not relevant to your question, but I find this statement in the paper questionable: "After all, the whole of the theoretical apparatus of physics relies heavily on the use of set theory, thus on the AC." I doubt the theoretical apparatus of physics needs more than naive set theory; if formalized, I expect it can be accommodated equally comfortably in ZF+AD. --Lambiam 07:46, 27 March 2024 (UTC)
 * Well, physics does use functional analysis, which is most comfortably formulated in a context where AC is true. For example the Hahn–Banach theorem is inconsistent with AD. --Trovatore (talk) 16:16, 27 March 2024 (UTC)
 * The idea that the infinite hat game can tell us something about physics is IMO absurd. It should be completely obvious that the lined-up players cannot do better than one would expect from guessing in a purely Newtonian universe, so this has nothing to do with space-like separation. At any time, a player can only have information about a finite number of other hats, so good luck in forming the equivalence class. The equivalence class exists in a Platonic sense, but the player has no access to it. One might as well let player $$k$$ apply AC to the set $$\{\{c_k\}\},$$ in which $$c_i$$ is the hat colour of the $$i$$-th player. While the player cannot know this set of sets, it "exists", so in the logic of the puzzle one may apply AC. So if the player now does apply AC, we also apply some smoke, and boom!, paradox. --Lambiam 08:05, 27 March 2024 (UTC)
 * The authors understand that, I believe. That's not exactly the point of my question.  I still can't follow why they think it's especially counterintuitive that they get more with a "functional" no-signaling than an ordinary one, or even just what they mean by that.  As far as I can tell, probability is a fifth wheel in the whole argument. --Trovatore (talk) 16:19, 27 March 2024 (UTC)
 * Just a note to before this gets archived.  My assertion that "the authors understand that" refers to the bit about the scenario not being physically realizable.  The bit starting with "[o]ne might as well" is incorrect.  I'm not sure it was intended seriously.  In case it was, the error is that the set Lambiam posits depends on the actual arrangement of hats, whereas the strategy given by the transversal of the equivalence relation does not depend on that. --Trovatore (talk) 07:32, 2 April 2024 (UTC)

Here, in a book titled Bell Nonlocality, the author refers to "a hypothetical no-signaling resource called a PR-box (Popescu and Rohrlich, 1994)". The reference is to: Popescu, Sandu and Rohrlich, Daniel (1994). Quantum nonlocality as an axiom. Foundations of Physics, 24, 379–385. --Lambiam 08:45, 27 March 2024 (UTC)
 * Hmm, that does sound like somewhere I might figure out what they mean by "resource"; thanks. --Trovatore (talk) 16:21, 27 March 2024 (UTC)
 * Gah. And if you replace the Axiom of Choice with Axiom of determinacy which is supposed to make things simpler I think you'd probably still end up with this problem. NadVolum (talk) 10:07, 27 March 2024 (UTC)
 * It's not totally clear what you mean by "this problem", but I do think that AD should imply that the hat wearers almost surely can't do better (or indeed worse) than an asymptotic density of 1/2 correct, when the hats are placed with random colors. That's because AD implies that all the sets involved are measurable, and since each wearer has no information about the color of his own hat, his guess can be no better (and also no worse) than a coin flip, regardless of what (pure or mixed) strategy he plays.  I guess that might be what the authors are calling out as "unintuitive", given that AC implies the wearers have a pure strategy that gets them to asymptotic density 1 of correct guesses.  They seem to think that mixed strategies ought to be more powerful than pure strategies and they find it surprising that this is not true; I am not sure why they find that surprising.
 * On a side note, AD does not always make things "simpler". It does imply that sets of reals have nice regularity properties.  But it has some odd implications elsewhere (for example that the cofinality of $$\aleph_3$$ is $$\aleph_2$$). --Trovatore (talk) 16:00, 28 March 2024 (UTC)
 * I believe you're right okay, and have shown why the AC case isn't a problem either. For any particular finite number a random ordering of the men would have a zero chance of having got to the last difference between their's own sequence and the representative of the equivalence class by then - even if the probability is one at infinity. NadVolum (talk) 17:02, 28 March 2024 (UTC)
 * I had to puzzle over this response, because it sounds right, but it can't be: Probability is countably additive, so if it were really true that for any n the probability of all guesses past n being correct were zero, then the probability of eventual correctness would also be zero.
 * The problem turns out to be that the first probability isn't zero; it doesn't exist. If there were such a probability, it would equal the coin-flipping measure of the transversal.  But the transversal has inner measure zero, but outer measure greater than zero.  You can pick the outer measure to be anything you want in the interval (0, 1].  I came up with a nice little argument to show that you can reach outer measure 1.  It involves wellordering the possible sequences of hats in one list, and the possible closed sets of positive measure in another list, both in the minimum order type $$2^{\aleph_0}$$.  Then you go back and forth up the two lists.  You pick a sequence from the first list and throw it into the transversal, and then you strike out all sequences equivalent to it, both out of the first list, and out of all sets in the second list.  Then you pick a closed set of positive measure from the second list, and take a sequence that hasn't been struck out of it and put it in the transversal, and again strike out all sequences equivalent to it.  By cardinality considerations you can keep going on both sides.  When you exhaust all of both lists, you have a transversal that meets all closed sets of positive measure, so it must have full outer measure.) --Trovatore (talk) 07:40, 2 April 2024 (UTC)

A red photon and a blue photon are approaching each other in opposite directions. Is the whole system at rest?
On the one hand, it seems the whole system is apparently at rest, because they are approaching each other in opposite directions, by the same speed, while carrying identical masses (being zero). On the other hand, both photons have different momenta, so the whole system's momentum is not zero, so apparently the whole system can't be at rest. That's why I wonder, what's the correct answer to the question: Is the whole system at rest?

I guess it's not at rest, yet I'm not sure about what's the mistake in my first consideration, by which I've concluded the whole system is apparently at rest. 147.235.215.72 (talk) 14:03, 27 March 2024 (UTC)


 * At rest relative to what? There's no priviliged reference frame to automatically define "at rest" against.  If the two photons are approaching each other, you can CHOSE a reference frame where the net momentum of the system is zero.  PianoDan (talk) 14:23, 27 March 2024 (UTC)
 * Here is my answer to your question (in your first sentence): Relative to whoever sees the first photon as red and the second photon as blue. 147.235.215.72 (talk) 15:29, 27 March 2024 (UTC)
 * The mass $$m_\lambda$$ of a photon in empty space is given by $$m_\lambda=E_\lambda/c^2=(hc/\lambda)/c^2=(h/\lambda)/c.$$ The magnitude of its momentum, $$p_\lambda,$$ is given by $$p_\lambda=h/\lambda.$$ We see that the two are related by $$m_\lambda=cp_\lambda.$$ So if their momenta are not equal in magnitude, neither are their masses. --Lambiam 14:31, 27 March 2024 (UTC)
 * Actually, there is a dispute over whether light has mass. My question assumes light has no mass, yet I know my question could have a clear answer if we assumed light had mass. 147.235.215.72 (talk) 15:28, 27 March 2024 (UTC)
 * There is no serious dispute over whether light has (rest) mass. People have come up with modifications to Maxwell's equations that would apply if it did, but there's no evidence that it does, there are very tiny experimental bounds on what it can be, and there would be (as I understand it, though it's not my area) some thorny theoretical problems if it did.  Basically it can be taken as established (subject to revision, as science always is) that the rest mass of a photon is exactly zero.
 * Whether light should be considered as having nonzero relativistic mass is a different sort of "dispute", not really about facts but about terminology and bookkeeping. Most writers today consider it better to avoid the concept of relativistic mass. --Trovatore (talk) 16:12, 27 March 2024 (UTC)
 * Actually the sort of mass I referred to when I responded to Lambiam, was exactly the sort of mass they had referred to, i.e. relativistic mass. Wasn't it clear from the very beginning? Anyway, I agree with you that this dispute is about terminology, but my previous response still holds. 147.235.215.72 (talk) 16:35, 27 March 2024 (UTC) 16:35, 27 March 2024 (UTC)
 * Suppose that in our frame of reference $$F$$ the two photons approach each other head-on. Now in every frame of reference $$F'$$ moving relative to $$F$$ at a velocity parallel to that of the photons, the photons approach each other head-on with the speed of light too (although Doppler-shifted). So if that's sufficient to consider $$F$$ the rest frame of the system of two photons, then $$F'$$ must be one too, but as $$F'$$ moves relative to $$F$$, that doesn't work so well.
 * No, the rest frame of a system of photons would be the one where their total momentum is zero. Such a frame exists provided there are at least two photons not on parallel trajectories. PiusImpavidus (talk) 17:28, 27 March 2024 (UTC)
 * Regarding your last sentence:
 * (a) Please notice the two photons I'm asking about are moving in parallel trajectories (in opposite directions).
 * (b) Additionally, I can't understand why you had to add this stipulation about the non-parallel trajectories: In my view, it's quite easy to find an appropriate reference frame in which both photons have idetical (yet opposite) momenta, as follows: Since the original reference frame I'm asking about sees the first photon as red and the second photon as blue, so the new reference frame is supposed to see both photons as having the same color, indeed Doppler-shifted, but still the same one, probably yellow, because this is the medium color on the spectrum between red and blue. Don't you think it's quite easy to find such a new reference frame?
 * Regarding your answer: What you wrote, only supports my second consideration (in my original post), by which I concluded the whole system in the original reference frame was not at rest. Actually, your conclusion is exactly the conclusion in my second consideration, ibid. However, please notice I didn't ask for additional considerations (as your ones) that support my second consideration ibid., but rather for an explanation about what's wrong in my first consideration ibid., by which I concluded the whole system in the original reference frame was at rest. Do you think you can point at the exact stage (in my first consideration ibid.) that contains a mistake? 147.235.215.72 (talk) 18:42, 27 March 2024 (UTC)
 * The system is considered at rest if its total momentum is zero. In your situation the momentum is not zero. Therefore the system is not at rest. Ruslik_ Zero 20:09, 27 March 2024 (UTC)
 * Sources?
 * This is new to me. Given a multi-particle system, I've always thought that if the particles have identical masses, then the system's velocity should be calculated as the average of the velocities of the particles in the system. At least, this is how we could calculate the system's velocity if the particles having identical masses were massive ones. 147.235.215.72 (talk) 20:22, 27 March 2024 (UTC)
 * You obviously thought wrong. Photons have no mass and $$p=\gamma mv$$ simply does not apply. Do yourself (and ourselves) a favour, accept that and move on. --Wrongfilter (talk) 20:26, 27 March 2024 (UTC)
 * Sorry, but this is exactly what I've said to Lambiam: "My question assumes light has no mass". In other words, photons have zero mass. Hence they have "identical masses (being zero)", as I wrote in my original post.
 * But my question was not answered yet. Is the system at rest? 147.235.215.72 (talk) 20:43, 27 March 2024 (UTC)
 * Your question has been answered many times. The system is not at rest because the total momentum is not zero. --Wrongfilter (talk) 20:53, 27 March 2024 (UTC)
 * This explanation had only been given by Ruslik_Zero (before you repeated their answer), but the other editors gave me other explantions, e.g. that light has mass, while I didn't understand their explanations, because I assume photons have no mass, i.e. they have identical masses being zero.
 * As for Ruslik_Zero's explanation, which is now also your one, did you read my response to them? If you did, then what's the answer to what I asked them? 147.235.215.72 (talk) 21:09, 27 March 2024 (UTC)
 * What question? "Sources?"? Any decent textbook on special relativity. Listen, HOTmag, in a thread from a couple of days ago you said yourself that relativity defines momentum as $$p = \gamma m v$$ (and that would be the formal basis for your handwaving argument). Now plug in what you know about photons: $$m = 0$$, $$v = c$$. This gives $$\gamma = \infty$$ and $$p = \infty \cdot 0$$, which is of course undefined. You cannot compute the momentum of a photon that way. The momentum of a photon is given by the de Broglie relation, $$p = hc/\lambda$$. --Wrongfilter (talk) 21:23, 27 March 2024 (UTC)
 * I remember I mentioned the formula of relativistic momentum (By the way I don't remember HOTMAG has responded to my current question, yet I do remember they responsed to me on other occasions in this reference desk), but my question did not assume this relativistic formula. My question has only assumed that if the particles have identical masses (whether a zero mass or a non-zero mass), then the system's velocity can be calculated as the average of the velocities of the particles in the system. I didn't connect this calculation with any momentum. 147.235.215.72 (talk) 21:56, 27 March 2024 (UTC)
 * The average velocity equals $$\frac{m_1\mathbf{v}_1+m_2\mathbf{v}_2}{m_1+m_2}.$$ If $$m_1=m_2=0,$$ this comes out as $$\frac\mathbf{0}0.$$ --Lambiam 23:03, 27 March 2024 (UTC)
 * Again, you've used the concept of momentum. This is also the concept I used when I concluded in my second consideraion (of my first post) that the whole system was not at rest, yet I didn't have to devide by zero (as you had), because I didn't use the formula you've used, but rather the formula of the sum of momenta (when a given photon's momentum is received by e.g. the photon's wavelength) for getting the whole system's momentum.
 * However, in my first consideration (ibid.), as well as in my previous response, I bypassed the very concept of momentum, by using a totally different argument for concluding that the whole system was at rest...
 * ALL IN ALL, my question is not about what else we can claim (as you're claiming now) for proving that the whole system is not at rest, but rather about what's wrong in my first consideration (of my original post) or in my argument of my previous response, wherein I bypassed the very concept of momentum and concluded that the whole system was at rest. 147.235.215.72 (talk) 01:01, 28 March 2024 (UTC)
 * My reply used no other concepts than mass and velocity, taking the average velocity to be the weighted average, being the velocity of the centre of mass. --Lambiam 12:43, 28 March 2024 (UTC)
 * Indeed, you didn't utter the explicit word "momentum", yet you did use products of mass and velocity, which is actually the same as using momentum, even without saying explicitly "momentum". This is how I describe your method of calculation, as far as you are concerned. However as far as I'm concerned, my first consideration (of my original post) hasn't used momenta, nor products of velocities multiplied by anything else whatever it is. My first consideration (ibid.) has only used velocities, being actually the velocities of particles that have the same mass (while assuming light has no mass). Why doesn't my method work with photons (having identical masses), whereas it does work with massive bodies (having identical masses), while in both cases I bypass the concept of momentum, as well as any product of a velocity multiplied by anything else? 147.235.215.72 (talk) 16:41, 28 March 2024 (UTC)
 * I could have written it as $$\sum_i w_i\mathbf{v}_i,$$ were $$w_k=\frac{m_k}{\sum_i m_i}.$$ No velocities were harmed in this formula by multiplying them by velocities. --Lambiam 22:57, 28 March 2024 (UTC)
 * All right, so you managed to bypass the concept of momentum, you replacing it by the concept of $$w_k,$$ but what I've managed to do is to bypass any concept other than a velocity, at least as far as massive bodies are concerned, while my question was: Why does my method, which bypasses any concept other than a velocity, fail when working with photons (having identical masses), whereas it does work with massive bodies (having identical masses), while in both cases the method bypasses any concept other than a velocity? 147.235.215.72 (talk) 19:48, 1 April 2024 (UTC)
 * Rule A: In a system of particles with equal mass, the rest frame is the frame where the average velocity of the particles is zero.
 * Rule B: In a system of particles, the rest frame is the frame where the total momentum is zero.
 * Rule A follows from rule B, assuming that the mass isn't zero (and it isn't), but we have to be careful how to define that mass that must be equal. Rule A only follows if the particles have identical relativistic masses, $$\gamma m_0$$ for massive particles or $$h/(\lambda c)$$ for photons, the thing you get when dividing momentum by velocity. You, in your first consideration, used the rest mass of the particles. That works for non-relativistic particles, but not for photons. PiusImpavidus (talk) 11:30, 28 March 2024 (UTC)
 * If they move in opposite directions, I call them antiparallel. PiusImpavidus (talk) 09:55, 28 March 2024 (UTC)
 * In the first paragraph of my first answer, I told what was wrong with your first consideration: if the first consideration were true, it follows that there is a second rest frame, moving relative to the first. But two frames moving relative to each other cannot both be rest frames for the same system of particles, so the premise must be wrong. Therefore, the first consideration must be false. PiusImpavidus (talk) 10:12, 28 March 2024 (UTC)
 * The contradiction is rooted in relativity. Classically, the antiparallel photons' speeds should entail the velocity additions of the classical Galilean transformations, but according to relativity light speed is invariant (instead of invariant distances, time intervals and simultaneity). Had the OP been talking simply about two sound waves (red-like and blue-like) having a medium's speed of sound converging from opposite directions then their first consideration that they are at rest is accurate and it follows that the waves' inertial mass and momentum is moving with respect to them in the same direction as the red wave moves. Modocc (talk) 15:16, 28 March 2024 (UTC)
 * You claim my first consideration (of my original post) leads to a contradiction. But I have been aware of that, from the very beginning, in my oringinal post, where I presented also the second consideration which contradicts the first one, so you didn't have to remind me of what I've been aware of. i.e. that the first consideration leads to a contradiction.
 * When we encounter two apparently correct arguments contradicting each other, while we don't know yet which one is the wrong one, then for finding the wrong argument, we must point at the wrong stage of the argument, by declaring "This stage is unjustified because it's not based on any well established rule". I claim, that my first consideration (ibid.) is apparently well established, beacsue it uses the same method we could have used if the photons had been massive bodies. Just think, what would occur if two massive bodies, having the same masses, were approaching each other in opposite directions, but at the same speed: Wouldn't we agree, that the whole system were at rest, we bypassing the very concept of momentum? We would, so isn't this fact sufficient for using the same method for photons, we bypassing the very concept of momentum? The question is, why do we have to discriminate a pair of particles having zero mass, while we actually manage to bypass the very concept of momentum, just as we would've been allowed to bypass it - had we used the same method for massive particles? 147.235.215.72 (talk) 16:41, 28 March 2024 (UTC)
 * "...my first consideration (ibid.) is apparently well established,.." What do you mean by it is well established? If you mean photon's lightspeed is c, yes that has been. Neither the observer(s) or the net momentum of the photons are necessarily at rest except as you have referenced them. Modocc (talk) 17:15, 28 March 2024 (UTC)
 * I mean, that if two massive bodies, having the same masses, are approaching each other in opposite directions, but at the same speed, then all of us agree, that the whole system is at rest, we bypassing the very concept of momentum, and this is a well established fact, isn't it? The question is: Why doesn't this well established method work with photons (having identical masses), whereas it does work with massive bodies (having identical masses), while in both cases I bypass the concept of momentum, as well as any product of a velocity multiplied by anything else? 147.235.215.72 (talk) 19:48, 1 April 2024 (UTC)
 * Any system of energetic particles (such as those orbiting each other) is at rest only if its total momentum is zero. Modocc (talk) 21:33, 1 April 2024 (UTC)
 * Your claim is exactly my second consideration mentioned in my original post. But my question was about my first consideration ibid. which contradicts the second one ibid.
 * Anyway, the question mentioned in the last sentence of my previous response (you've just responded to), is not about my second consideration but rather about my first one, while your response only repeats my second consideration, instead of referring to my first one. 147.235.210.126 (talk) 10:40, 2 April 2024 (UTC)
 * You said: "...if two massive bodies, having the same masses, are approaching each other in opposite directions, but at the same speed, then all of us agree, that the whole system is at rest...". By "massive" include all the body's energy along with its KE; you cannot simply add their inertial masses. Those masses are at rest not because they are equal (they are not vectors), but because their momentum vectors are equal and opposite. The photon's are not, so only your second consideration that they are not at rest is correct (v>0). Modocc (talk) 12:54, 2 April 2024 (UTC)
 * by "massive", I mean: "having a positive restmass". That said, do you agree to the following rule: Any system composed of two bodies, which move at the same speed in opposite directions, and which carry identical positive masses, is at rest. If you agree to this rule, then I claim that this agreed rule has nothing to do with momenta, because it does not mention them, nor does mention any product of a velocity multiplied by a mass. 147.235.210.126 (talk) 13:50, 2 April 2024 (UTC)
 * You previously wrote "Anyway, contrary to what you ascribe to me, I didn't refer to the rest mass." So now you admit that you implicitly meant and want to consider only particle rest mass. By "positive masses" meaning equal rest masses then no no and no. With field(s), all real bodies carry energy and momentum. Modocc (talk) 14:14, 2 April 2024 (UTC)
 * Let's avoid confusing one issue with another one. In my original post I wrote "identical masses (being zero)", and later I made it clear that "I didn't refer to the rest mass", as far as my original post was concerned. But when I asked you my last question about the "massive" bodies I made it clear that "by massive, I mean: having a positive restmass", as far as my last question to you was concerned. They were two different clarifications referring to two different issues, and let's avoid confusing between them.
 * What do you mean by your first "no"? I want to be more specific about it: Do you agree to the following rule: "Any system composed of two bodies, which move at the same speed in opposite directions, and which carry identical positive masses, is at rest". 147.235.210.126 (talk) 16:57, 2 April 2024 (UTC)
 * You write "positive masses" which is ambiguous. Please rephrase. Modocc (talk) 17:29, 2 April 2024 (UTC)
 * Ok, I'll be more explicit: Do you agree to the following rule: "Any system composed of two bodies, which move at the same speed in opposite directions, and which carry identical positive rest-masses, is at rest"? 147.235.210.126 (talk) 18:26, 2 April 2024 (UTC)
 * No, because in general that cannot apply to the inertia of waves, irrespective of whether or not we invoke relativity and its systemic corrections. Consider The Poseidon Adventure (1972 film), a wave that capsized the ship. The ship's wake fans out and intersects the wave. To a stationary observer in the water the relative sizes of the wave to the wake matter. In other words, far more energy and momentum is traveling in the direction of the large wave than what the wake can add. However, if the energies/momenta carried by the wave and wake happen to be equal and opposite then the observer that is at rest sees that their total momentum is in fact zero; otherwise its nonzero in some direction. The photons are like these waves in that they carry different energy and momenta too. The fact their phase velocities are not ever at rest (so no mass, unless one is talking about group velocities) and travel at the same speed like the ocean waves doesn't matter. Modocc (talk) 19:05, 2 April 2024 (UTC)
 * Note my last question to you was not about waves nor about photons, but rather about bodies whose rest masses are positive, so let me repeat the question, now with bold letters emphasizing this detail:
 * So, do you agree to the following rule?
 * "Any system composed of two bodies, which move at the same speed in opposite directions, and which carry identical positive rest-masses, is at rest". 147.235.210.126 (talk) 20:39, 2 April 2024 (UTC)
 * I do not agree. Photons are a lot like electrons in that they are point-like particle-wave bodies that are known to carry energy and momentum. Modocc (talk) 20:52, 2 April 2024 (UTC)
 * So let me formulate it now as follows:
 * "Any system composed of two balls, which move at the same speed in opposite directions, and which carry identical rest-masses being 1 kg, is at rest".
 * Agree? 147.235.210.126 (talk) 21:30, 2 April 2024 (UTC)
 * Only if each ball carries equal momenta. However even baseballs with equal momenta might curve due to unequal effective masses because of their interactions with nearby matter. Modocc (talk) 21:45, 2 April 2024 (UTC)
 * What if the baseballs don't curve? Are your sure your first reservation, stating: "Only if each ball carries equal momenta", will still be needed? If you think it will still be needed, then you must think there will be cases, my claim - to which you added this reservation - may still be untrue without your reservation, but I wonder what cases they may be - if the baseballs don't curve, because AFAIK, my claim you added this resevation to is always true - if the baseballs don't curve - even without adding any reservation, so let me repeat my claim - now with the condition that those baseballs don't curve, and let me know if you still think there are cases this claim is untrue unless you add your reservation. So my claim is:
 * "Any system composed of two baseballs, which don't curve, and which move at the same speed in opposite directions, and which carry identical rest-masses being 1 kg, is at rest". Agree? 147.235.210.126 (talk) 08:33, 3 April 2024 (UTC)
 * The two baseballs' contributions to the momentum vector of the system adds to zero meaning it is at rest. Modocc (talk) 12:28, 3 April 2024 (UTC)
 * So do you think the rule mentioned in my previous response is true, without having to add any reservations to this rule, or you think it still needs some reservations for it to be true? 147.235.210.126 (talk) 15:46, 4 April 2024 (UTC)
 * I'm well aware that your first consideration and your second consideration contradict each other. That was clear from the beginning. What I'm telling you (for the third time now) is that the first consideration also contradicts itself, therefore can never be true.
 * If you want more details (which I already gave above): your argument depends on all particles having the same mass. But what mass is that? It's the relativistic mass, but you used rest mass. PiusImpavidus (talk) 00:16, 29 March 2024 (UTC)
 * It seems you didn't read Trovatore's response, nor my response to them. Anyway, contrary to what you ascribe to me, I didn't refer to the rest mass. 147.235.215.72 (talk) 19:48, 1 April 2024 (UTC)
 * The net momentum of any system of particles needs to be zero for it to be considered at rest in an inertial frame of reference and that includes all of its energy. So the OP did error in neglecting that. Modocc (talk) 20:53, 29 March 2024 (UTC)
 * Your claim is exactly my second consideration mentioned in my original post. But my question was about my first consideration ibid. which contradicts the second one. For more details, see my previous response to you above. 147.235.215.72 (talk) 19:48, 1 April 2024 (UTC)
 * If you want more details (which I already gave above): your argument depends on all particles having the same mass. But what mass is that? It's the relativistic mass, but you used rest mass. PiusImpavidus (talk) 00:16, 29 March 2024 (UTC)
 * It seems you didn't read Trovatore's response, nor my response to them. Anyway, contrary to what you ascribe to me, I didn't refer to the rest mass. 147.235.215.72 (talk) 19:48, 1 April 2024 (UTC)
 * The net momentum of any system of particles needs to be zero for it to be considered at rest in an inertial frame of reference and that includes all of its energy. So the OP did error in neglecting that. Modocc (talk) 20:53, 29 March 2024 (UTC)
 * Your claim is exactly my second consideration mentioned in my original post. But my question was about my first consideration ibid. which contradicts the second one. For more details, see my previous response to you above. 147.235.215.72 (talk) 19:48, 1 April 2024 (UTC)

Photon wavelengths are not fixed invariably at their emission. Neither an indelibly red nor an indelibly blue photon exists in timespace because the wavelength of any received photon depends on the relative motions of source and receiver. The scenario of "a red photon and a blue photon...approaching each other in opposite directions" may be merely the perception of a viewer who is moving fast enough along the line joining the two sources in the direction of the latter photon's source. Philvoids (talk) 18:28, 28 March 2024 (UTC)
 * Correct, and my question only refers to that viewer mentioned in your last sentence. 147.235.215.72 (talk) 19:48, 1 April 2024 (UTC)