Lattices and Lorentz invariance

Discussion in 'Physics & Math' started by Farsight, Oct 22, 2011.

  1. rpenner Fully Wired Staff Member

    Messages:
    4,833
    QED explanation for pair production:
    \(\begin{eqnarray} F_{\mu\nu}& = & {\partial A_\nu\over\partial x_\mu}-{\partial A_\mu\over\partial x_\nu} \\ {\cal L}_{\textrm{QED}}& = & {\cal L}_{\textrm{fermion}} + {\cal L}_{\textrm{boson}} + {\cal L}_{\textrm{interaction}} \\ {\cal L}_{\textrm{fermion}}& = &- \bar{\psi}\left(m - \gamma^\mu\frac{\partial}{\partial x_\mu} \right)\psi \\ {\cal L}_{\textrm{boson}}& = &-\frac{1}{4}F_{\mu\nu}F_{\mu\nu} \\ {\cal L}_{\textrm{interaction}}& = &-ie {\bar{\psi}\gamma^\mu A_\mu\psi} \end{eqnarray}\)
    The last term shows that the fermion field couples pair-wise with the vector boson field. With time-like fermions, pair production, annihilation, electrostatics and magnetism are just part-and-parcel of treating space and time symmetrically.
     
  2. Google AdSense Guest Advertisement



    to hide all adverts.
  3. Farsight Valued Senior Member

    Messages:
    3,475
    Come on now rpenner. That's no explanation. That's just a paltry attempt to hide behind mathematics. You don't even inform the reader what the various terms are. Everybody reading this thread can see that you can't explain what happens in pair production, and that you're employing sophistry to try to conceal the fact. A word of advice: remember the saying: You do not really understand something unless you can explain it to your grandmother. If you don't understand it, or alternatively if you have difficulty explaining it in layman's terms, just say so. People will hold you in much higher regard for that than they will if they perceive you to be trying to pull the wool over their eyes.
     
  4. Google AdSense Guest Advertisement



    to hide all adverts.
  5. Farsight Valued Senior Member

    Messages:
    3,475
    You wish.

    There's nothing philosophical about pointing to the hard scientific evidence and making it clear that there ain't no time flowing in that parallel-mirror light clock, just light moving. Through space. Not through spacetime.

    You might think it's useful, but it isn't. It's wrong. And it's all downhill from there.

    I'm in a position to comment because I'm well-read on experimental physics and original material by the likes of Maxwell, Einstein, and Minkowski, and because I've analysed the terms in mathematical expressions.

    This really is an attempt at an argument from authority. If you don't like being challenged, then all you have to do is point out the flaws and explain where the challenge falls down. Shouldn't be too difficult. Only it is, isn't it? That's why you end up making bizarre statements that contradict experimental physics, and yourself. And totally losing it. It is not a pretty sight, Alphanumeric.

    Oh stop whining. Stick to the physics.

    I'm the one pointing to the experimental evidence here.

    Like pair production, the Einstein-de Haas effect, magnetic moment, electron diffraction. Like I said, I'm the one supporting my argument with scientific evidence here. You're the one dismissing it all.

    I've learned a great deal. That's why I can beat the intellectual snot out of you every time.

    I enjoy talking to pryzk. I've learned plenty from him. However all I've learned from you is how not to behave.

    If you've got nothing to fear just stick to the physics and explain carefully why I'm wrong. If you can't do this and if instead you continue with all the outraged abuse, people will draw their own conclusions. They aren't stupid you know.

    No. I've already said I don't understand the details.

    I can. I passed A-level Maths, and did more maths related to electromagnetism and logic within my Computer Science degree course. I've even given paid-for maths tutoring to A-level candidates.

    Let's make this one the same then, huh?

    I accepted, on the condition that you stowed the abuse. Your very next post was full of your usual ad-hominem garbage, so I withdrew. Because you cannot be trusted.

    I noticed rpenner's post. I'll take a look at it next.

    You've always displayed your intellectual arrogance and your insecurity.

    I've addressed lots of questions. Everybody who can read can see this. And they can also see that your assertion that I've addressed nobody's questions is simply bizarre. Take care, people will begin to conclude that you've got a "beautiful mind", if you catch my drift.

    Just doing my bit for physics, Alphanumeric. And don't forget, this isn't a journal, this is a discussion forum. And you're supposed to be a qualified physicist. A professional physicist. So start acting like one.

    I was threatened. It's there for all to see.

    No I can't. But your inference doesn't hold, because string theory is divorced from scientific evidence, whilst "my work" is not.

    Now stop wasting my time. Stick to the physics.
     
  6. Google AdSense Guest Advertisement



    to hide all adverts.
  7. Guest254 Valued Senior Member

    Messages:
    1,056
    I think QED is all about invisible turtles, and it's invisible turtles sitting on top of invisible turtles. I'm willing to bet you can't prove me wrong. The fact you can't prove me wrong is evidence that it's all about turtles.

    Farsight, your behaviour in this thread has made me realise why you stopped responding in the academia thread. I'm sorry to say it, but all talk and no trousers seems like it describes you pretty well.
     
  8. Farsight Valued Senior Member

    Messages:
    3,475
    But despite all that, the Einstein-de Haas effect is real. Here's an easy-reading description of it on the urbana website. I have to say that it sounds like you're complaining that I'm arguing from the authority of scientific evidence here, rpenner.

    I referred to a historic version of the article because the non-sequiteur justifying instrinsic spin was removed about three months after I started pointing it out. I didn't pick a particular version, just one old enough to still contain the spinning-like-a-planet text. I have no known connection or contact with wikipedia contributors.

    The fine-structure constant is a "running" constant, varying down to 1/128. See NIST and note the comment "Thus the famous number 1/137 is not unique or especially fundamental".

    It was just an old version selected at random. All I was looking for was the non-sequiteur concerning intrinsic spin. I'll repeat it for convenience:

    "Electrons are spin-1⁄2 particles. These have only two possible spin angular momentum values, called spin-up and spin-down. The exact value in the z direction is +ħ/2 or −ħ/2. If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast. Even if the electron radius were as large as 14 nm (classical electron radius) then it would have to be rotating at 2.3×1011 m/s. The speed of rotation would be in excess of the speed of light, 2.998×10^8 m/s, and is thus impossible.[2] Thus, the spin angular momentum has nothing to do with rotation and is a purely quantum mechanical phenomenon. That is why it is sometimes known as the "intrinsic angular momentum."

    I take issue with "weird".

    It's no straw man. Instead it's relevant to the given explanation for why electron spin is "intrinsic", which is that if the electron was spinning like a planet, it would be spinning at more than c. This is then used not to say that it isn't spinning like a planet, but that it isn't spinning. This IMHO is a catastrophic error of logic, and history will not be kind to it.

    With respect you don't understand mass, rpenner. Inertia is merely the flip side of momentum. Have a read of Light is heavy by van der Mark and 't Hooft. If you trap a massless photon in a mirror-box, you add mass to that system. Open the box to release the photon, and the system loses mass, just like the radiating body in Einstein's E=mc² paper. Then consider the electron in terms of a photon in a box of its own making. Don't hesitate to ask me questions. When I've answered those questions to your satisfaction, I'm confident you will gain an appreciation that does not require you to abandon the QED spin ½ particle. You merely reinterpret "the electron field" as "the electron field component", et cetera.

    I'm quietly confident that an enhanced QED which does not confront human intuition will be the winner.

    It interacts as if it's a point-like object, but so does the photon, which has a wavelength and can be similarly diffracted. The evidence is ambiguous, and QED employs virtual photons which are usually interpreted as not being part of the electron structure.

    I have no issue with "elementary particle". l do however have an issue with "fundamental particle".

    The analogy I gave was It's like probing for a cannonball in a whirlpool, and then declaring that the electron can be no bigger than a ping-pong ball because you'd have otherwise felt it. It's an imperfect analogy, a better analogy might employ a vector field with reference to Maxwell's Theory of Molecular Vortices and gravitomagnetism along with a vector field:

    Please Register or Log in to view the hidden image!


    GNU Free Documentation License image by AllenMcC, shown on the English Wikipedia vector field article.

    There is no central body in the centre. The electron is created from an electromagnetic wave in pair production, think of it as a standing wave going at c without getting anywhere, remember that an electromagnetic wave is a field variation, and thus a standing wave is a field. So it's all field. There is no pointlike body at the centre.

    To be continued. Thank you for your professionalism and civility.
     
  9. Farsight Valued Senior Member

    Messages:
    3,475
    Get lost, kid. The grown-ups are talking physics here.
     
  10. Guest254 Valued Senior Member

    Messages:
    1,056
    That reaction doesn't instill confidence in me. With regard the tone of your response, I'm fairly sure (though not positive) that I'm the most senior academic on the forum.
     
  11. Farsight Valued Senior Member

    Messages:
    3,475
    But this is a physics discussion, and as you said here, you're not a physicist. Przyk and rpenner are, and they've given considered sincere responses. They eclipse you, and your "seniority" is academic.

    Now, where was I?
     
  12. rpenner Fully Wired Staff Member

    Messages:
    4,833
    I'm a physicist? How so? I think you mean AlphaNumeric, since he's got that nice doctoral thesis and articles.

    I believe Guest's reply was to your partition of the forum into "grown-ups" and others that you attempt to exclude from the discussion, and Guest is pointing out that he regularly has grown-up discussions with physicists and other people capable of rational discussion.
     
  13. AlphaNumeric Fully ionized Moderator

    Messages:
    6,697
    Dirac, Stoke, Green, Hawking, Penrose and of course Newton were not physicists by degree but to claim they could then be ignored would be ridiculous. I consider myself a mathematician but I have.a doctorate from a physics department and now I apply my capabilities to specific problems in physics.

    Farsight, you don't really grasp how physics and maths blend together significantly, as illustrated by the time you asked to be given physics vector calculus, not just mathematical stuff, as if such a division exits. An example of how flawed that thinking is is complex analysis. Holomorphic functions make up a truly vast area of pure mathematics. Some of the most abstract beautiful pure mathematics pertains to it. And yet every holomorphic function is interpretable as a particular type of fluid flow!

    This division you believe exists doesn't but you lack the experience to realise that.
     
  14. Farsight Valued Senior Member

    Messages:
    3,475
    Noted.

    Because I didn't read the paper, and didn't realize that the inference that nothing rotates was contained within it.

    What you said was So all three problems are well-addressed by the hypothesis that the electron is well-modeled by a Poincaré-invariant quantum field theory of a finite mass spin-1/2 particle coupled to the massless helicity-1 field of photons. There's a lot of heavy lifting to get from that to numerical comparison of experiment with theory, but it's a simple hypothesis that is far more precise (and useful) than "matter is made of waves." And I said Run that by me again: what's matter made of? You have declined to give an answer, and have instead attempted to deflect attention from your inability to do so by claiming that I don't care about precision and usefulness. It fools nobody. Lift your game.

    Not so. We have no evidence for the existence of the human soul. But we do have evidence for the existence of electrons. And we do have evidence of pair production. Lift your game.

    I'm glad we agree on that. Pop-science is often disparaged, but it does often accurately reflect the prevailing view. Don't think however that I'm a fan of all pop-science. There's pop-science, and there's pop science.

    The errors concerning nanometres and femtometres are not important. The non-sequiteur is. Here is is again:

    "Electrons are spin-1⁄2 particles. These have only two possible spin angular momentum values, called spin-up and spin-down. The exact value in the z direction is +ħ/2 or −ħ/2. If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast. Even if the electron radius were as large as 14 nm (classical electron radius) then it would have to be rotating at 2.3×1011 m/s. The speed of rotation would be in excess of the speed of light, 2.998×10^8 m/s, and is thus impossible.[2] Thus, the spin angular momentum has nothing to do with rotation and is a purely quantum mechanical phenomenon. That is why it is sometimes known as the "intrinsic angular momentum."[/b]

    You're no fool rpenner, think this through. The given argument is that if the electron was spinning like a planet, it would be spinning at more than c. This isn't used to assert that it isn't spinning like a planet, but that it isn't spinning. It's a juvenile error of logic. You surely must be able to see this. Even a child can see this logical fallacy.

    I have some sympathy with this view. In fact I'd go further, and venture to say that woo features heavily in certain pop-science books. Like I said, there's pop-science, and there's pop science.

    No. I'd trust the scientific evidence.

    You need concern yourself when challenged with scientific evidence. If you cannot respond to it, then all your professional science first class cuts no ice. You cannot explain how pair production works. And the first class smoke-and-mirrors you attempted does not conceal that fact. Nobody is taken in by "mathematical incantations". Do not use them. Just come clean and say I don't know, and people will hold you in higher regard for it.

    Sorry. No. My bad.

    It'll be a pleasure rprenner. Though I doubt that it'll be my publication. I'm just leaning on the tiller.

    Forget them. Just get your calculator out and work out λ = 4π / c^1½. Don't worry about dimensionality. It isn't a trick. Or perhaps I should say that taking the dimensionality out of it is the trick. Walk round in a tight circle banking your arms. Think of h and h-bar. Then maybe you'll understand how it does rotate. And why we don't call it a photon any more.

    Nice talking to you rpenner. Always a pleasure.
     
  15. Farsight Valued Senior Member

    Messages:
    3,475
    No, not him. He's just a string theorist. Sorry rpenner, I gathered you were both a mathematician and a physicist. My mistake? If so I'll bear that in mind in our ongoing conversations. I'll cut you some slack.

    Could have fooled me, rpenner. Take a look at his posts. Ouch. An awful lot of them are abusive. Sneering, even. Not much sign of rational discussion there. Insecurity I guess, what with guys like you and przyk eclipsing him an' all. I mean, even a guy like me straight off the street leaves him in the shade. Mind you, I am the intellectual equivalent of a cage fighter. He wouldn't dare take me on. By the way, I'm pleased you summoned up the courage to step up to the plate. Much better than sniping from the sides. Good man.

    And so to bed.
     
  16. przyk squishy Valued Senior Member

    Messages:
    3,151
    Farsight, this is getting silly. AlphaNumeric is easily one of the most experienced contributors in the physics sub forum here, and he is certainly more familiar with many areas of physics, including quantum field theory, than I am. I'm not even mentioning string theory here: I'm judging Alpha by what he posts, and string theory is not something he posts very frequently about, if at all.

    Guest is a mathematician and I don't know how extensive his knowledge of physics is, but at the very least his mathematical background has given him excellent insights in special and general relativity. Guest may playfully poke certain posters with a stick a lot but when he decides to contribute to a thread with his experience, his responses tend to be spot on. His contributions to CptBork's recent relativity thread were exactly things CptBork needed to know for instance. In general I only see people with a certain minimum level of experience in mathematics and physics that are really able to appreciate Guest's contributions. When he says he suspects he's one of the most senior academics on the forum, it's very believable.
     
  17. rpenner Fully Wired Staff Member

    Messages:
    4,833
    Indeed. But why do you think I haven't thought this through?

    In SI units, the classical electron radius is: \(r = \frac{1}{4 \pi \varepsilon_0} \frac{e^2}{m c^2} \; \approx \; 2.8 \, \textrm{fm}\)

    http://www.google.com/search?q=what... pi * electric constant * c^2 * electron mass)

    If it was orbital angular momentum associated with a that radius, we have:
    \(\frac{\hbar}{2} = L = r p \quad \Rightarrow \quad p = \frac{h \varepsilon_0 m c^2}{e^2} \; \approx \; 19 \times 10^{-21} \, \textrm{kg} \cdot \textrm{m} \cdot \textrm{s}^{-1} \)
    In Newtonian physics, we have \(p = m v\) whereby \(v \approx 68.5 c\). Bing bang boom, your pop-physics authors stop there. Some might want to use a less efficient arrangement like a solid sphere of constant density, in which case the equator moves faster. But this model doesn't begin to explain the gyromagnetic ratio.

    In relativistic classical physics, we have to have a more complicated model with components and a binding energy.
    \(U = E - mc^2 \\ E^2 = m_0^2c^4 + p^2c^2\).
    If we assume that the components are massless, \(m_0 = 0\), then we have \(E = \frac{h \varepsilon_0 m c^3}{e^2} \; \approx \; 5.6 \textrm{nJ} \; \approx \; 35 \, \textrm{MeV}\) most of which represents an unknown binding energy, U. If the components are not massless, U is even larger. If the components are any number of photons, the model dies because no such photon-photon coupling is observed, the Pauli exclusion principle is not explained, electron-photon coupling isn't explained and the electron gyromagnetic ratio is not explained. Moreover, we have slammed electrons together at much more than 35 MeV center-of-mass energies, and no side-effects related to this have been seen. Experiment rules this out.

    In relativistic quantum physics, this doesn't work at all since there is no quantum wave function associated with non-integer spins orbital angular momentum.

    But in relativistic quantum physics, we don't need electrons to be other than point particles with intrinsic angular momentum, as Wigner showed in 1939 was not in violation of any law of the universe articulated.

    Y S Kim, "Wigner's last papers on spacetime symmetries", in Proceedings of the IV Wigner Symposium, Guadalajara, 1995 (River Edge, NJ, 1996), 1-10.
    as quoted in http://www.gap-system.org/~history/Biographies/Wigner.html

    Another fundamental theorem of relativistic quantum mechanics is the spin-statisics theory which in our space-time of 3+1 dimensions you get spin-1/2 particles acting like fermions for free.

    BTW, all of this comes of the top of my head except for the Wigner biography and quote. If I pooh-pooh my physics credentials, it is because I am conscious of how high the skill ladder goes and don't want to fall afoul of the Dunning–Kruger effect. Physicists do physics and can't not do it. Professionals get paid to practice a profession. Experts are sought after for their expertise. So a physicist needs about 3 seconds to figure out the from the definition of the classical electron radius that Newton would require the electron to be moving at about 137/2 times the speed of light. Real physicists are aware of models of composite electrons, like technicolor and some of them can quote to you the current experimental bounds. But they also know Wigner and the spin-statistics theorem and that describing a symmetry is a laudable goal of understanding the universe, since a description of a perfect symmetry is perfect understanding, and a description of an experimentally perfect symmetry is precise to exactly the limits of human information.
     
    Last edited: Dec 3, 2011
  18. Guest254 Valued Senior Member

    Messages:
    1,056
    Precisely.

    But Farsight, you are neither a physicist nor a mathematician, and yet here you are, waxing lyrical. If we are to use you as a benchmark, I would seem to be overqualified!

    But I don't want to dwell on this point, I'm happy for you to carry on. The salient part of my post seems to have been overlooked. I mentioned invisible turtles and the fact you couldn't prove me wrong - I was hoping you might draw a comparison...
     
    Last edited: Dec 2, 2011
  19. Guest254 Valued Senior Member

    Messages:
    1,056
    And this is why I'm keen for you to continue to post. Gold.
     
  20. rpenner Fully Wired Staff Member

    Messages:
    4,833
    I think it is a trick, like the ones used by kidnappers to lure children into vans.

    Dimensionality matters since physics quantities aren't numbers but quantities.

    The speed of light isn't 299792458 and isn't 18737028625/100584. It's certainly not both since 299792458 >> 18737028625/100584.
    But units let us make sense about this.
    c = 299792458 m/s = 18737028625/100584 miles/second

    I think what you are trying to say that if A = 1 meter and B = 1 meter/second (values which have no physical significance), then
    \(\lambda = 4 \pi A \left( \frac{B}{c} \right)^{\tiny \frac{3}{2}} \approx 2.421 \, \textrm{pm} \) and \(\lambda_{\tiny \textrm{Compton}} = \frac{h}{m_{\tiny \textrm{electron}} c} = 2.4263102389 \pm 0.0000000016 \, \textrm{pm}\) which is not only unphysical numerology, but is uninteresting numerology in that it just gets three significant digits correct.

    http://physics.nist.gov/cgi-bin/cuu/Value?ecomwl

    The relative error is \(\frac{\lambda - \lambda_{\tiny \textrm{Compton}} }{\lambda_{\tiny \textrm{Compton}} } \approx -0.2%\)

    --
    Approximating c as 300000000 m/s or 1,800,000,000,000 furlongs/fortnight is more accurate than this approximates the Compton wavelength of the electron.
     
    Last edited: Dec 2, 2011
  21. Farsight Valued Senior Member

    Messages:
    3,475
    I didn't claim that.

    That's absolutely not a problem. And I really do not have an issue with mathematicians or mathematics. But I do take exception to a mathematician who is so afraid to enter into a physics discussion that he attempts to pour scorn instead.

    I do grasp it. As I've said before, mathematics is a vital tool for physics, we can't do physics without it. I know I don't use much in my "analysis of mathematical terms like E m c and t", but don't think I'm hostile to it.

    In all seriousness I'd say that there is an issue wherein what you'd call mathematical physicists don't always pay sufficient regard to scientific evidence, and sometimes put too much emphasis on mathematical abstraction.

    I'm sure that all readers have noted that you still haven't entered into any discussion of the physics here, and have drawn their own conclusions.

    It is, przyk. Mine was a cheap shot, in retaliation to abuse. Let's have no more of it, let's stick to the physics discussion.

    All points noted przyk. Let's move on.

    rpenner: I've got to go out now, but I've just skimmed your last post. There's a small binding energy adjustment that I omitted, related to the g-factor. The expression c^½ / 3π isn't a trick either. Your posts look interesting, I'll get back to you properly later.
     
  22. AlphaNumeric Fully ionized Moderator

    Messages:
    6,697
    Farsight attempts to dismiss me because I did a phd in string theory, thus somehow invalidating any physics related stuff I might have done. Never mind I had to learn and be competent in classical dynamics, electromagnetism, special and general relativity, quantum mechanics, quantum field theory, symmetries, Hamiltonian mechanics, statistical physics and an arseload of mathematical methods courses which all of physics is written in. Never mind I taught relativity, electromagnetism, dynamics and quantum mechanics. Never mind my research related to an understanding of cosmology and also QCD. Never mind my research since leaving academia has spanned the aerospace, defence, energy and telecommunications industries. Never mind my job involves solving real world problems. Never mind by every single measure I am superior to him in every single relevant area of physics. No, that doesn't count because Farsight just knows how things work and it isn't inline with what competent well read people have observed about reality.

    I'll reply to your quote by quote post Farsight later, I dont like using iPads to type long elaborate posts.
     
  23. rpenner Fully Wired Staff Member

    Messages:
    4,833
    From post 25:
    From Post 33:
    From Post 48:
    From post 171:
    From post 178:
    Again regarding numerology:
    If we define physics quantities A (a length) and B (a speed) by
    \(A = \frac{27 \pi^2 h }{4 m_{\tiny \textrm{electron}} c} \left( \frac{m_{\tiny \textrm{proton}}}{m_{\tiny \textrm{electron}}} \right)^3 \; \approx \; 1.000636 \, \textrm{m}\)
    \(B = \left( \frac{m_{\tiny \textrm{electron}}}{m_{\tiny \textrm{proton}}} \right)^2 \frac{c}{9\pi^2} \; \approx \; 1.001062 \, \textrm{m/s}\)

    Then we have trivially:
    \(4 \pi A \left( \frac{B}{c} \right) ^{\tiny \frac{3}{2} } = 4 \pi \frac{27 \pi^2 h }{4 m_{\tiny \textrm{electron}} c} \left( \frac{m_{\tiny \textrm{proton}}}{m_{\tiny \textrm{electron}}} \right)^3 \left( \left( \frac{m_{\tiny \textrm{electron}}}{m_{\tiny \textrm{proton}}} \right)^2 \frac{c}{9\pi^2} \, \frac{1}{c} \right) ^{\tiny \frac{3}{2} } = \frac{h}{m_{\tiny \textrm{electron}} c} \)
    and
    \(\frac{1}{3\pi} \sqrt{ \frac{c}{B} } = \frac{1}{3\pi} \sqrt{ c \, \left( \frac{m_{\tiny \textrm{proton}}}{m_{\tiny \textrm{electron}}} \right)^2 \frac{9 \pi^2}{c} } = \frac{m_{\tiny \textrm{proton}}}{m_{\tiny \textrm{electron}}}\)

    So the trick is finding A and B with those specially chosen factors which render them close to 1 m and 1 m/s respectively. Then, if one knocks out the scaffolding, hiding A and B and the units involved, to the untrained eye it looks like physics experimental quantities are arising by simple arithmetic operations. But this is a trick to dress up pseudo-physics and numerology as reliable information about the universe.

    It is the soul and substance of intellectual dishonesty. A rape of the minds of the ignorant. A scurrilous blight upon our shores. A betrayal of trust placed in soi-disant authorities.

    Unwinding the trick requires but a first year undergraduate's understanding of physics and a smattering of high school algebra.

    The expressions \(4 \pi \left( \frac{1}{c} \right) ^{\tiny \frac{3}{2} }\) and \(\frac{1}{3\pi} \sqrt{ c } \) are physically meaningless when it comes to describing the universe. The first is not a length and the second is not a mass ratio. The approximations \( \left( 4 \pi \, \textrm{m} \right) \left( \frac{1 \textrm{m/s} }{c} \right) ^{\tiny \frac{3}{2} }\) and \(\frac{1}{3\pi} \sqrt{ \frac{c}{1 \textrm{m/s}} } \) throw away reliable information about the universe in pursuit of a magic trick.
     
    Last edited: Dec 3, 2011

Share This Page