Wikipedia:Reference desk/Archives/Science/2021 August 10
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August 10
editProving the relativistic formulas (that use Gamma factor), of momentum, or of mass:
editI wonder if it's possible to prove the well known relativistic formulas: m=γmo, p=γmoV, without relying on the concept of energy as used in Relativity theory, and mainly without relying - on the equivalence of energy and mass - nor on the assumption about the four-vector of energy and momentum.
I do allow, though, to use all Newton's laws accepted by Special Relativity, along with the two basic assumptions of Special Relativity about the invariance - of light speed - and of laws of Physics (by "laws of Physics" I mean Newton's laws mentioned above along with the two basic assumptions of Special Relativity mentioned above). I may permit additional assumptions as well, provided that they are really "intuitive" or at least not as "heavy" as the assumptions mentioned above about energy. 185.120.124.50 (talk) 08:17, 10 August 2021 (UTC)
- The only real alternative to energy as a concept is action, and if you thought energy was non-intuitive and "heavy", oh boy. But you asked for it... The equivalent of "conservation of energy" using action is the principle of least action and general relativity can be derived using the concept of Einstein–Hilbert action. --Jayron32 10:51, 10 August 2021 (UTC)
- I suspect we don't understand each other. I haven't rejected the idea of using principles about energy, nor of using the conservation of energy - as it's acknowledged in Newton's Mechanics as well. I have only rejected the idea of using the concept of energy as used in Relativity theory, like the equivalence of energy and mass, and like the four-vector of energy and momentum, and like the energy- momentum relation. 185.120.124.50 (talk) 11:12, 10 August 2021 (UTC)
- I'm not sure why you've rejected those ideas. It's not like you get to pick and choose which parts of well-established scientific principles you get to "accept" and then expect everything that depends on them to work. --Jayron32 12:08, 10 August 2021 (UTC)
- Oh. I don't reject anything on an ideological basis. It's only a logical game: I'm only asking if it's possible to logically prove the well known relativistic formulas (that use Gamma factor) of momentum or of mass, while only using - both Newton's laws (including the conservation of energy/momentum as used in Newton's Mechanics) and the two basic assumptions of Special Relativity about the invariance - of light speed - and of laws of Physics. It's also fine to use other intuitive assumptions, as long as they may be considered to be sufficiently "intuitive" by Newton. 84.228.238.42 (talk) 16:58, 10 August 2021 (UTC)
- It seems like you're asking if General Relativity could be established as long as one never considers Special Relativity; for various reasons, I'm not sure that's possible. General Relativity starts with Special Relativity, and it generalizes it from inertial frames to accelerating frames. Mass-energy equivalence and the four vector are inextricable parts of SR. The important part of GR for our purposes is the Equivalence principle between gravitational and inertial mass; another key point is that while pre-Special Relativity physics had established that the speed of light was invariant, GR establishes that the spacetime path of light is also invariant (it follows a geodesic in curved space time.) In order to start to get at what much of General Relativity concludes, you really need those parts of Special Relativity you are discarding as a thought exercise. I'm not as strong in the mathematical end of this as most other people here, so it may be possible to do what you are asking, it just may require some really complicated work-arounds. But I doubt even that. --Jayron32 17:28, 10 August 2021 (UTC)
- I'm quite amazed at what you ascribe to my question: Actually, it has nothing to do with General relativity: By "mass" I'm only referring to inertial mass. Further, as opposed to what you claim, I did (and still do) allow to use - at least the two basic assumptions of Special relativity, i.e. the invariance - of light speed - and of laws of Physics. 84.228.238.42 (talk) 18:49, 10 August 2021 (UTC)
- But mass-energy equivalence is both an inseparable part of Special Relativity and a law of physics. You can't just pull at threads and not expect the entire blanket to not just unravel like that. --Jayron32 12:43, 11 August 2021 (UTC)
- by "laws of Physics" I mean what Einstein meant, i.e.: All Newton's laws accepted by Special Relativity, along with the two basic assumptions of Special Relativity about the invariance - of light speed - and of "laws of Physics".
- Actually, my question is about a game in Logic/Mathematics only. Just as I'm allowed to ask if - from a logical (rather than mathematical) point of view - one can prove the existence of negative numbers while only assuming the existence of natural numbers (the answer being "No": You can't prove it), so I'm asking here if - from a logical/mathematical (rather than physical) point of view - one can prove the formula m=γmo while only assuming all Newton's laws accepted by Special Relativity along with the two basic assumptions of Special Relativity about the invariance - of light speed - and of laws of Physics. 84.228.238.42 (talk) 08:47, 13 August 2021 (UTC)
- But mass-energy equivalence is both an inseparable part of Special Relativity and a law of physics. You can't just pull at threads and not expect the entire blanket to not just unravel like that. --Jayron32 12:43, 11 August 2021 (UTC)
- I'm quite amazed at what you ascribe to my question: Actually, it has nothing to do with General relativity: By "mass" I'm only referring to inertial mass. Further, as opposed to what you claim, I did (and still do) allow to use - at least the two basic assumptions of Special relativity, i.e. the invariance - of light speed - and of laws of Physics. 84.228.238.42 (talk) 18:49, 10 August 2021 (UTC)
- It seems like you're asking if General Relativity could be established as long as one never considers Special Relativity; for various reasons, I'm not sure that's possible. General Relativity starts with Special Relativity, and it generalizes it from inertial frames to accelerating frames. Mass-energy equivalence and the four vector are inextricable parts of SR. The important part of GR for our purposes is the Equivalence principle between gravitational and inertial mass; another key point is that while pre-Special Relativity physics had established that the speed of light was invariant, GR establishes that the spacetime path of light is also invariant (it follows a geodesic in curved space time.) In order to start to get at what much of General Relativity concludes, you really need those parts of Special Relativity you are discarding as a thought exercise. I'm not as strong in the mathematical end of this as most other people here, so it may be possible to do what you are asking, it just may require some really complicated work-arounds. But I doubt even that. --Jayron32 17:28, 10 August 2021 (UTC)
- Oh. I don't reject anything on an ideological basis. It's only a logical game: I'm only asking if it's possible to logically prove the well known relativistic formulas (that use Gamma factor) of momentum or of mass, while only using - both Newton's laws (including the conservation of energy/momentum as used in Newton's Mechanics) and the two basic assumptions of Special Relativity about the invariance - of light speed - and of laws of Physics. It's also fine to use other intuitive assumptions, as long as they may be considered to be sufficiently "intuitive" by Newton. 84.228.238.42 (talk) 16:58, 10 August 2021 (UTC)
- I'm not sure why you've rejected those ideas. It's not like you get to pick and choose which parts of well-established scientific principles you get to "accept" and then expect everything that depends on them to work. --Jayron32 12:08, 10 August 2021 (UTC)
- I suspect we don't understand each other. I haven't rejected the idea of using principles about energy, nor of using the conservation of energy - as it's acknowledged in Newton's Mechanics as well. I have only rejected the idea of using the concept of energy as used in Relativity theory, like the equivalence of energy and mass, and like the four-vector of energy and momentum, and like the energy- momentum relation. 185.120.124.50 (talk) 11:12, 10 August 2021 (UTC)
- The Lorentz factor was derived independently by FitzGerald and Lorentz in a conceptually Newtonian setting, as part of a theoretical explanation of the observed invariance of the speed of light. It appears in the formulas for momentum in special relativity, which extend the Lorentz transformations consistently to other physical quantities than just length and velocity. Presumably it is the simplest consistent extension, but perhaps it is even the only one. --Lambiam 18:00, 10 August 2021 (UTC)
- Yes, but the Lorentz factor itself comes after the Lorentz transformation, which deals with the four vector, one of the verboten concepts the OP wants to avoid invoking. No four-vector, no Lorentz transform, no Lorentz factor. --Jayron32 18:05, 10 August 2021 (UTC)
- The original Lorentz transformations concerned only length and time. It all began with the length contraction formula, first formulated by FitzGerald in 1889, which already involved (although FitzGerald did not use the symbol). The four vector came later, when it was realized that extensions were required for physical quantities involving other dimensions than length and time to have a consistent body of physical laws beyond the kinematics of bodies in free fall. What I do not know if there was a choice, or if all such extensions result in isomorphic sets of transformations. I suspect, though, that the choice was forced. --Lambiam 08:41, 11 August 2021 (UTC)
- Yes, but the Lorentz factor itself comes after the Lorentz transformation, which deals with the four vector, one of the verboten concepts the OP wants to avoid invoking. No four-vector, no Lorentz transform, no Lorentz factor. --Jayron32 18:05, 10 August 2021 (UTC)
- The Lorentz factor was derived independently by FitzGerald and Lorentz in a conceptually Newtonian setting, as part of a theoretical explanation of the observed invariance of the speed of light. It appears in the formulas for momentum in special relativity, which extend the Lorentz transformations consistently to other physical quantities than just length and velocity. Presumably it is the simplest consistent extension, but perhaps it is even the only one. --Lambiam 18:00, 10 August 2021 (UTC)
- You should be able to get there by considering a collision at low (nonrelativistic) velocities in the lab frame, boosting to a reference frame moving at relativistic velocity relative to the lab frame, and requiring momentum to be conserved in the moving frame. Any consideration of energy will only be in the lab frame, so you're not making assumptions about relativistic energy or mass, you're only requiring that they have the correct Newtonian limit at low velocity. You can find papers like [1] that do something along these lines. --Amble (talk) 20:54, 11 August 2021 (UTC)
- Do you mean it's impossible to prove m=γmo (under my conditions mentioned in the beginning of this thread), if the velocities of the colliding balls are arbitrarily large? 84.228.238.42 (talk) 08:47, 13 August 2021 (UTC)
- That equation is a definition of m and as such cannot be proved. Moreover, it's a useless and often misleading definition. The mass of an object is always its "rest mass", it's a property of the object and does not change through a change of reference system (as described by γ). The term "relativistic mass" should be reserved for composite objects, where it changes due to a change in internal energy, but never through centre-of-mass motion. --Wrongfilter (talk) 09:15, 13 August 2021 (UTC)
- Your argument, it's a useless and often misleading definition. The mass of an object is always its "rest mass", it's a property of the object and does not change through a change of reference system (as described by γ), is controversial. Sometimes Einstein thought like you, sometimes not.
- Einstein: "the law of constancy of mass applies to a single physical system, only when its energy remains constant". 84.228.238.42 (talk) 12:19, 13 August 2021 (UTC)
- Just the opposite -- I meant that you can derive the relativistic momentum and mass formulas for arbitrary velocities, "without relying on the concept of energy as used in Relativity theory", by considering a collision where the lab-frame velocities are low enough to work in the Newtonian limit. The paper I linked takes a slightly different approach that's probably an even better answer for you: it uses an elastic collision where you don't need to rely on energy beyond some very basic symmetry considerations. That treatment allows the velocities to be relativistic even in the lab frame. To derive the mass formula, of course, you will need to decide beforehand what kind of mass you are interested in, since there are several different mass concepts that only agree in the Newtonian limit. --Amble (talk) 16:36, 13 August 2021 (UTC)
- That equation is a definition of m and as such cannot be proved. Moreover, it's a useless and often misleading definition. The mass of an object is always its "rest mass", it's a property of the object and does not change through a change of reference system (as described by γ). The term "relativistic mass" should be reserved for composite objects, where it changes due to a change in internal energy, but never through centre-of-mass motion. --Wrongfilter (talk) 09:15, 13 August 2021 (UTC)
- Do you mean it's impossible to prove m=γmo (under my conditions mentioned in the beginning of this thread), if the velocities of the colliding balls are arbitrarily large? 84.228.238.42 (talk) 08:47, 13 August 2021 (UTC)