I have heard both that Planck length is the smallest length that there is in the universe (whatever this means) and that it is the smallest thing that can be observed because if we wanted to observe something smaller, it would require so much energy that would create a black hole (or our physics break down). So what is it, if there is a difference at all.
Spacetime – Is the Planck Length the Smallest Length That Exists in the Universe or the Smallest Observable Length?
discretephysical constantsquantum-gravityspacetime
Related Solutions
There is quite a lot to discuss around this topic, but I'll do my best to keep it brief. While we can't really say what energy structures at that scale "look like," we have some sense of what may be happening at that scale, and how those energy structures may be arranged.
First of all, let's get the scale straight. The Planck length has some interesting properties when it comes to energy. At this length, the energy of a single wavelength of light obeys the Schwarzschild condition, meaning that if we take the Compton wavelength of a Planck particle or Planck scale object (even a Photon), at this scale this wavelength is equal to the Schwarzschild radius. This means that it obeys the conditions of a tiny black hole. This in turn makes it's mass the Planck mass.1,2
While this might sound ridiculous, remember that what we're dealing with are the quantum dynamics at the smallest scale we currently theorize about, and what we know of the quantum vacuum at this scale is that it should be full of incredible amounts of energy. While calculated to be gargantuan (10113 joules per cubic meter),3,4 it seems that somehow this energy is so well balanced that it only shows up in what quantum electrodynamics calls "virtual particles" popping in and out of existence.
So let's get to the point. How is the energy so well structured that it seems like empty space?
Buckminster Fuller gives us a clue. His studies in architectural and energetic tensegrity showed that energy that is triangulated or tetrahedrally arranged can establish a perfect equilibrium. This means that regardless of the energy level, the energy is stable when in coherence and geometric alignment.
This means that if there are immense energies active at this scale, they must be tetrahedrally aligned in 3D, or triangulated in 2D to maintain equilibrium. Buckminster himself left us a very interesting quote foreshadowing this connection:
"Gravitational Field: Omnidirectional geodesic spheres consisting exclusively of three-way interacting great circles are realizations of gravitational field patterns... The gravitational field will ultimately be disclosed as ultra high-frequency tensegrity geodesic spheres. Nothing else." -RBF Definitions5
In my research, I found that you could easily begin to map gravitational topography of the spacetime fabric using this means, simply by following the rules of Bucky's domes. Where there is an energy point missing (radiated or compressed) from the hexagonal structure created by triangulation, there is a curvature formed (usually around pentagonal structures initially).
Loop Quantum Gravity also initially used triangulation to map quantum gravitational structures, though now it tends more towards "spin foam."6
Quantum Gravity and the Holographic Mass7 is an incredible peer-reviewed paper that applies these geometries to linking gravitational properties in cosmological scales to the energetic properties of the proton, and provides an elegant geometric solution for quantum gravity.
So what do things looks like at this scale? Well, if you took that fractal image and added a geometric lattice inside it, that might be close, because we see self-similar geometries all the way up to the macrocosm. Looking a 2D version of a lattice of gravitational curvature, it might look more like this:
If we want to see it in 3D, maybe something like this:
(source: netdna-cdn.com)
or like this if we're dealing with Planck Spherical Unit intersections:
(it won't let me post more images or links so here's the URL: https://s-media-cache-ak0.pinimg.com/236x/7e/44/e3/7e44e3e970c51cc539c6981623d87f41.jpg )
Great question, and I hope this gets you started in further exploration and research! There is certainly a need for better illustrations of the theories and principles at this scale.
Highest Regards,
Adam Apollo
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References:
Michel M. Deza; Elena Deza. Encyclopedia of Distances. Springer; 1 June 2009. ISBN 978-3-642-00233-5. p. 433.
"Light element synthesis in Planck fireballs" - SpringerLink
Peter W. Milonni - "The Quantum Vacuum"
de la Pena and Cetto "The Quantum Dice: An Introduction to Stochastic Electrodynamics"
Synergetics Dictionary Cards - RW Gray Projects
Loop Quantum Gravity (PDF Article) - IGPG Gravity PSU - Carlo Rovelli
Quantum Gravity and the Holographic Mass (PDF Paper) - SDI - Nassim Haramein
The Standard Model and General Relativity are both successful in appropriate limits, but they cannot be consistently combined for scales below $\sim\ell_P:=\sqrt{\dfrac{G\hbar}{c^3}}$ for various reasons. (By $\sim$, I mean "give or take a multiplicative constant that's besides the point here and may be hard to compute".) For example, what happens if you try to probe such length scales with a photon? How will its wavelength compare to its Schwarzschild radius?
When you ask about the physical meaning or significance of such short length scales, that's where it gets contentious. I'll try to summarise the range of views on this, but I'll probably fudge or simplify a few details:
- String theory says spacetime is infinitely divisible, but particles have a size $\sim\ell_P$. They therefore have worldsheets instead of worldlines, which smears Feynman diagram vertices as thus. This smearing removes the troublesome infinities from the treatment of gravity.
- Loop quantum gravity, in a sense, says the opposite: particles aren't posited to have size, but spacetime is quantised. Particles live at distinct lattice points. The area and volume of an object have operators in the Hilbert space, and these operators have discrete eigenvalues $\sim\ell_P^{2\,\mathrm{or}\,3}$.
- There have been attempts to combine ST with LQG (motivated by their respective pros and cons and their obtaining similar results from very different precepts, e.g. logarithmic corrections to the Hawking-Bekenstein entropy of black holes), but these are in their infancy. For now, it suffices to say such a union would introduce both deviations from the "point particles in infinitely divisible spacetime" idea that causes SM+GR problems.
- Another proposal is that $[\hat{x}_\mu,\,\hat{x}_\nu]=i\ell_P^2 \theta_{\mu\nu}$ is a non-vanishing antisymmetric tensor. This is far from developing into a full theory of quantum gravity, but it's an idea that's been explored in such attempts. Just as quantum mechanics says $[\hat{x}_j,\,\hat{p}_k]=i\hbar\delta_{jk}$ without the eigenvalues becoming discrete, the above use of noncommutative geometry requires only that $\sigma_{x_\mu}\sigma_{x_\nu}\ge\frac{1}{2}\ell_P^2|\theta_{\mu\nu}|$, not that eigenvalues of $x_\mu$ cannot differ by arbitrarily small fractions of $\ell_P$.
Best Answer
Short answer: nobody knows, but the Planck length is more numerology than physics at this point
Long answer: Suppose you are a theoretical physicist. Your work doesn't involve units, just math--you never use the fact that $c = 3 \times 10^8 m/s$, but you probably have $c$ pop up in a few different places. Since you never work with actual physical measurements, you decide to work in units with $c = 1$, and then you figure when you get to the end of the equations you'll multiply by/divide by $c$ until you get the right units. So you're doing relativity, you write $E = m$, and when you find that the speed of an object is .5 you realize it must be $.5 c$, etc. You realize that $c$ is in some sense a "natural scale" for lengths, times, speeds, etc. Fast forward, and you start noticing there are a few constants like this that give natural scales for the universe. For instance, $\hbar$ tends to characterize when quantum effects start mattering--often people say that the classical limit is the limit where $\hbar \to 0$, although it can be more subtle than that.
So, anyway, you start figuring out how to construct fundamental units this way. The speed of light gives a speed scale, but how can you get a length scale? Turns out you need to squash it together with a few other fundamental constants, and you get: $$ \ell_p = \sqrt{ \frac{\hbar G}{c^3}} $$ I encourage you to work it out; it has units of length. So that's cool! Maybe it means something important? It's REALLY small, after all--$\approx 10^{-35} m$. Maybe it's the smallest thing there is!
But let's calm down a second. What if I did this for mass, to find the "Planck mass"? I get: $$ m_p = \sqrt{\frac{\hbar c}{G}} \approx 21 \mu g $$
Ok, well, micrograms ain't huge, but to a particle physicist they're enormous. But this is hardly any sort of fundamental limit to anything. It isn't the world's smallest mass. Wikipedia claims that if a charged object had a mass this large, it would collapse--but charged point particles don't have even close to this mass, so that's kind of irrelevant.
It's not that these things are pointless--they do make math easier in a lot of cases, and they tell you how to work in these arbitrary theorists' units. But right now, there isn't a good reason in experiment or in most modern theory to believe that it means very much besides providing a scale.