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Natural Science Forum / Physics / Research / May 2008How can the Planck length be claimed to be the smallest length?
Many arxiv papers state that the Planck length is the smallest measureable length. On the other hand, the gravitational length L=2Gm/c^2 associated with every electron or proton is 19 or 22 orders of magnitude smaller than the Planck length. Nobody seems to doubt either of the two statements. What is the exact answer to this paradox? One can imagine at least 3 solutions: 1 - Lengths of objects can be smaller than L_Planck, but not positions. 2 - Lengths can be smaller than L_Planck if one makes many measurements and then makes a statistical average. 3 - There is an uncertainty relation between length L and position x: L x > L_Planck^2 There might be other answers. What is the canonical answer by researchers to this question? Thanks! John > Many arxiv papers state that the Planck length > is the smallest measureable length. [quoted text clipped - 28 lines] > > John The length scale you describe is the radius in which you would have to fit the Compton wavelength of the particle for quantum gravitational effects to come into play. You provided a formula that gave a length scale smaller than the Planck length. The question is, does this formula describe anything physical? For the electron this length scale is 10^-57 m For comparison, the electron's classical radius is 10^-15 m I do not believe any laboratory experiments have confined an electron to the degree you require. > Many arxiv papers state that the Planck length > is the smallest measureable length. Most of the time, this is merely a heuristic. The Planck length comes up as the scale beyond which quantum gravitational effects become non- negligible. This is shown using dimensional analysis, much in the same way as the Bohr radius, beyond which the full quantum mechanical description of the Hydrogen atom cannot be neglected. Note that the Bohr radius was derived before a modern quantum mechanical treatment of Hydrogen became available. A similar statement can be made about the Planck length. Current established theories neither require nor propose a minimal length. However, tentative models proposed for quantum gravity sometimes assume some kind of discreteness or granularity of space- time at the Planck scale. Thus, the nature of the Plank length as the smallest measurable one should be considered as one of the hypotheses assumed by these models. While there are arguments for the validity of this hypothesis, like all others, it must be subject to experimental verification. If you want a more detailed discussion of how to resolve your "paradox", you'll have to specify which model of quantum gravity you are assuming. Hope this helps. Igor > Many arxiv papers state that the Planck length > is the smallest measureable length. [quoted text clipped - 24 lines] > There might be other answers. What is the > canonical answer by researchers to this question? The Planck length comes from dimensional analysis. If we agree to have c = \hbar = G = 1 and dimensionless, then lengths must be expressed in multiples of the Planck length. By it's very nature dimensional analysis has nothing to say about numerical factors. Neither does it stipulate that these factors must be of order one. To go beyond dimensional analysis (and get the factors) you need a physical theory. One that explains for example why the electron mass is so small in terms of the Planck mass. Since this is sadly lacking we do not know whether or not statements like: gravitational length = order 10^{-20} Plancks do or do not have physical meaning. Purely from dimensional analysis there is nothing wrong with them, hence no paradox. Best, Jan > To go beyond dimensional analysis (and get the factors) > you need a physical theory. [quoted text clipped - 8 lines] > there is nothing wrong with them, > hence no paradox. String theory is often claimed to have a minimum length (the Planck length or near it), so does loop quantum gravity, so do many other approaches. The theory-independent approaches argue convincingly that lengths below the Planck length cannot be measured by any known procedure or device. The electron mass is also well known, and R=2GM/c^2 is not in doubt, as electrons have gravitational effects. It seems difficult to say that R=2GM/c^2 is wrong for electrons. So in practice there IS a paradox, because electrons are known experimentally to have a gravitational length smaller than the Planck length. Is there no hint of a way out? John > String theory is often claimed to have a minimum length > (the Planck length or near it), so does loop quantum gravity, > so do many other approaches. The theory-independent approaches > argue convincingly that lengths below the Planck length > cannot be measured by any known procedure or device. Unfortunately, neither string theory nor loop quantum gravity enjoys the status of an experimentally verified theory. The theory- independent approaches are precisely the ones that use dimensional analysis, as discussed previously by J. J. Lodder and myself. > The electron mass is also well known, and > R=2GM/c^2 is not in doubt, as electrons > have gravitational effects. It seems difficult to say > that R=2GM/c^2 is wrong > for electrons. What would it mean for the above paragraph to be wrong? You've defined a length scale R. You can write down any length you want and call it R, no-one will argue with you solely on that basis. > So in practice there IS a paradox, because > electrons are known experimentally to > have a gravitational length smaller than the > Planck length. I see. And what is this experiment that has measured the gravitational length of the electron? None of the modern particle physics experiments have probed lengths that are even withing a few orders of magnitude from the Planck length, not to mention beyond it. Hope this helps. Igor > > String theory is often claimed to have a minimum length > > (the Planck length or near it), so does loop quantum gravity, [quoted text clipped - 30 lines] > > Igor Hi Igor; Some photon wavelength equations relate the Planck length to the electron mass. L1/L2 = L2/L3 L1 = (L2)^2 (1/L3) = 2pi (3/2)^1/2 (Planck length) Where L2 is the photon wavelength with energy equal to the mass energy of one electron plus one positron and the wavelength L3 is (2pi)^2 (c) (one second). The L2 length is then equal to [(L1) (L3)]^1/2. When L2 is defined, the electron mass energy can have only one quantized value. Don Stevens On May 15, 5:51 pm, donjstev...@aol.com wrote: > > > String theory is often claimed to have a minimum length > > > (the Planck length or near it), so does loop quantum gravity, [quoted text clipped - 41 lines] > of one electron plus one positron > and the wavelength L3 is (2pi)^2 (c)(one second). so, if human beings decided to use a different unit of time than a second, L3 (and then L1) would come out to be a different physical value? i have trouble imagining that physical reality gives a rat's a.s what unit of time we humans happen to use. or the aliens on the planet Zog. r b-j > > To go beyond dimensional analysis (and get the factors) > > you need a physical theory. [quoted text clipped - 29 lines] > > John Paradoxes point out a flawed argument. The relationship reality -> mathematical structure is one to many. The problem here is an assumption that the relationship (a particular mathematical structure) -> reality is one to one. There in lies the rub. Perhaps there is no hard reality for a minimum length, or Oh No's divide it by c to dimensionaly come up with a time. That does not mean it can't have some value in a model based representation. Just do not confuse the model with the singular notion of reality. Rick > The Planck length comes from dimensional analysis. > If we agree to have c = \hbar = G = 1 and dimensionless, [quoted text clipped - 7 lines] > One that explains for example why the electron mass > is so small in terms of the Planck mass. because the Bohr radius is a_0 = ((4 pi \epsilon_0) \hbar^2)/(m_e e^2) which is a_0 = (m_P/m_e) (1/ \alpha) l_P where m_P is the Planck mass, l_P is the Planck length, and \alpha is the Fine-structure constant (which is not a particularly huge or tiny number). it's been said that the reason gravity is so weak is really because the masses of particles are so small ( m_e <<<< m_P ). but the reason that particle masses are so small is the same reason that the size of atoms are so big ( l_P <<<< a_0 ) which is another what of saying that the Planck length is so small (compared to the sizes of atoms or even particles contained therein). but, out of ignorance, i dunno why anyone says that it's the smallest length. it's a quantity, we can always define a length much smaller. but such a teeny length might not be in the ballpark of any physical thing. hell, maybe not even the Planck length is comparable to any physical thing. but i think the reason that the Planck length is tiny compared to the radius of an atom is the same reason the Planck mass is huge compared to that of an atom. r b-j On May 7, 10:31 pm, robert bristow-johnson <r...@audioimagination.com> wrote: > > The Planck length comes from dimensional analysis. > > If we agree to have c = \hbar = G = 1 and dimensionless, [quoted text clipped - 34 lines] > compared to the radius of an atom is the same reason the Planck mass > is huge compared to that of an atom. If gravity quantizes around the planck scale, then below that scale one does not have the convenience of a classical metric. Without a metric, how do you define length? > If gravity quantizes around the planck scale, then below that scale > one does not have the convenience of a classical metric. Without a > metric, how do you define length? If one takes 10^24 atoms of silicon in a single crystal (around 1 kg) it is undisputed that the whole object has a measureable gravitational length. The crystal bends space-time around it and attracts other masses; that is easy to measure. And there is a definite metric in our environment. It is also undisputed that all atom masses in the crystal essentially add up (the crystal binding energy can be neglected here). Since the gravitational length of the silicon crystal is defined as R=2GM/c^2, it is very hard to avoid saying that every silicon atom has a gravitational length given by the same formula, this time using the atomic mass. However, the gravitational length calculated in this way for one atom is much smaller than the Planck length. (about 10^18 times smaller). So it does seem that much smaller lengths than a Planck length have a physical meaning... John > If one takes 10^24 atoms of silicon in a single > crystal (around 1 kg) [quoted text clipped - 13 lines] > has a gravitational length given by the > same formula, this time using the atomic mass. It is true that a kilogram of silicon has a measurable gravitational field. This field corresponds to space-time curvature of order 1/L, where L is some length. We know for a fact that these gravitational effects are very weak, which in turn implies that L must be very large. When we measure gravitational effects due to this hunk of silicon, it is L that we measure, not the R that you've defined above. R would be the size of the hunk of silicon if it were dense enough to become a black hole. Since it is not a black hole, we have another demonstration that R is irrelevant to the physical situation. In short, your paradox is avoided because, no matter how small R is, it never comes up as an experimental measurement; only L does. And L is of regular macroscopic proportions. Hope this helps. Igor Thus spake JohnMS <john_m_stanton@yahoo.co.uk> >Many arxiv papers state that the Planck length >is the smallest measureable length. [quoted text clipped - 24 lines] >There might be other answers. What is the >canonical answer by researchers to this question? Many researchers in quantum gravity assume Planck length as a fundamental length scale. This should be regarded as a hypothesis for a model, not as an established fact. It is well below the scale of measurable lengths. My own research hypothesises that L=2Gm/c^3 is a fundamental time. Of course, I think that is more promising. :-) Regards  SignatureCharles Francis moderator sci.physics.foundations. charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and braces) http://www.teleconnection.info/rqg/MainIndex > Many arxiv papers state that the Planck length > is the smallest measureable length. [quoted text clipped - 28 lines] > > John Hello John; Let us suppose that the gravitational length (radius) of the electron is L: where L = 3Gm/c^2. This is the photon orbit radius for the electron mass. Next, suppose that the shortest meaningful distance is Planck length times the square root of (3/2). The circumference is 2pi (radius). A circumference is (2pi) (Planck length) (3/2)^1/2. This value is (3pi h G/c^3)^1/2. When a photon with energy equal to the mass energy of one electron plus one positron is gravitationally blue shifted to the wavelength (3pi h G/c^3)^1/2, the size reduction factor is (L/L)^2 rather than (L/L). This is because distance is shortened to match time dilation. The observable length will then be equal to the photon orbit circumference, 2pi (3Gm/c^2), while the radius is 3Gm/c^2. This is discussed in "Talk:Black hole electron", Wikipedia. Don Stevens |
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