Q: What is the Planck length? What is its relevance?

Physicist: Physicists are among the laziest and most attractive people in the world, and as such don’t like to spend too much time doing real work.  In an effort to streamline equations “natural units” are used.  The idea behind natural units is to minimize the number of physical constants that you need to keep track of.

For example, Newton’s law of universal gravitation says that the gravitational force between two objects with masses m1 and m2, that are separated by a distance r, is F = \frac{Gm_1m_2}{r^2}, where G is the “gravitational constant“.  G can be expressed as a lot of different numbers depending on the units used.  For example, in terms of meters, kilograms, and seconds: G = 6.674\times 10^{-11}\frac{m^3}{kg\,s^2}.

In terms of miles, pounds, and years: G = 7.248\times 10^{-6}\frac{mi^3}{lb\,y^2}.

In terms of furlongs, femptograms, and  fortnights: G = 3.713\times 10^{-34}\frac{fl^3}{fg\,fn^2}.

Point is, by changing the units you change the value of G (this has no impact on the physics, just the units of measurement).  So, why not choose units so that G=1, and then ignore it?  The Planck units are set up so that G (the gravitational constant), c (the speed of light), \hbar (the reduced Planck constant), and kB (Boltzmann constant) are all equal to 1.  So for example, “E=mc2” becomes “E=m” (again, this doesn’t change things any more than, say, switching between miles and kilometers does).

The “Planck length” is the unit of length in Planck units, and it’s \ell_P = \sqrt{\frac{\hbar G}{c^3}} = 1.616\times 10^{-35} meters.  Which is small.  I don’t even have a remotely useful way of describing how small that is.  Think of anything at all: that’s way, way, way bigger.  A hydrogen atom is about 10 trillion trillion Planck lengths across (which, in the pantheon of worldly facts, ranks among the most useless).

Physicists primarily use the Planck length to talk about things that are ridiculously tiny.  Specifically; too tiny to matter.  By the time you get to (anywhere near) the Planck length it stops making much sense to talk about the difference between two points in any reasonable situation.  Basically, because of the uncertainty principle, there’s no useful (physically relevant) difference between the positions of things separated by small enough distances, and the Planck length certainly qualifies.  Nothing fundamentally changes at the Planck scale, and there’s nothing special about the physics there, it’s just that there’s no point trying to deal with things that small.  Part of why nobody bothers is that the smallest particle, the electron, is about 1020 times larger (that’s the difference between a single hair and a large galaxy).  Rather than being a specific scale, The Planck scale is just an easy to remember line-in-the-sand (the words “Planck length” are easier to remember than a number).

That all said (and what was said is: don’t worry about the Planck constant because it’s not important), there are some places on the bleeding edge of physics where the Planck length (or distances of approximately that size) do show up.  In particular, it shows up in the “Generalized Uncertainty Principle” (GUP) where it’s inserted basically as a patch to make physics work in some fairly obscure situations (quantum gravity and whatnot).  The GUP implies that at a small enough scale it is literally impossible, in all situations, to make a smaller-scale measurement.  In the right light this makes it look like maybe spacetime is discrete and comes in “smallest units”, and maybe the universe is like the image on a computer screen (made up of pixels).

How bleeding edge is the GUP?  So bleeding edge that there isn’t even a wikipedia article about it.  Like a lot of things in string theory (this is an opinion), these sort of patches may prove to be mistakes.  So, spacetime may come in discrete chunks, but the most we can say is that those chunks (if they exist) are very, very, very, very small.

You’d never notice (at least, the experiments designed to notice haven’t so far).

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39 Responses to Q: What is the Planck length? What is its relevance?

  1. john says:

    Awesome photo! Is that Marie Curie at the table by Albert? Oh what is give to be a fly on the wall at that meeting. To hear the various theories of the day, the ones that didn’t get further than friendly chatter at the whisky table.

  2. Jasper A. Visser says:

    Yes it is. There’s a list of all the attendants on the wiki:
    http://en.wikipedia.org/wiki/Solvay_Conference

    Planck is in there also, sitting at the table, 4th from the left.

  3. m. While these results provide some hints of Planck-scale gravity, neither of these experiments was designed as a tool to specifically test gravity, so Gharibyan warns that uncontrolled pieces of setups could mimic observed effects.

  4. Mehrdad says:

    Question: Why do some Planck units seemingly put a quantum limit on quantities, like P. length and P. time, while others like P. energy and P. mass are quite irrelevant?

  5. Mehrdad says:

    “How bleeding edge is the GUP? So bleeding edge that there isn’t even a wikipedia article about it.”
    No offense, Mr. Physicist, but I hate it when people act so subjectively towards Wikipedia. If it isn’t there, make a account, and create the page.

  6. The Physicist The Physicist says:

    They’re probably all irrelevant. Planck length just happens to have been added into a particular theory (in a very patched, ad hoc, kind of way).

  7. Pingback: Research Paper | Walk the Planck!

  8. Pingback: Research Paper by Bryce Estes | Walk the Planck!

  9. Gregor says:

    One visualization of the size of the Planck length I have come across is this:
    take a dot which is at the limit of being visible with the unaided eye: 0,1 mm in diameter. Enlarge this dot to the size of the now visible universe (10^27 meters) and the size of a 0,1 mm dot is about the Planck length in diameter.
    (Actually, I don’t get the computation it seems like the dot in the last case would be 1*10^-60 meters but what do I know…)

  10. James says:

    If you had two basket balls and one you kept halving and the other you kept doubling in size, what would you reach first. The plank scale or the edge of the universe.

  11. The Physicist The Physicist says:

    You’d reach the edge of the observable universe first.
    That actually made it impossible to come up with a useful comparison (the Planck scale is really small).

  12. James says:

    WHAT! By a factor of what?

  13. Anthony says:

    I’m really fascinated by the virtual reality theory of everything. It resolves virtually (no pun intended) every problem in physics.

    http://arxiv.org/ftp/arxiv/papers/0801/0801.0337.pdf

  14. David says:

    @James
    Lets say the basketball is 1m in dia. (we’ll do this approximately).
    The observable universe is, say, 13.7 billions years old, giving a 27.4 ly diameter.
    So put “convert 27.4 *10^9 light years to m” in google
    gives 2.59 × 10^26 meters (approx, use the tools you’ve got or do it longhand).

    Again, google “how long is a planck length” gives “1 planck length =
    1.62 × 10^-35 meters”

    Result: The planck length is smaller than the universe is larger (of a comfortable human dimension of one meter or a basketball) by 9 orders of magnitude. That’s really^9 small.

  15. Gregor says:

    According to the “Standard Model” the universe went through a period of ‘inflation’ where space inflated dramatically and became quite large. As a result the universe is probably quite a bit larger (in diameter) than you calculate, perhaps 90 billion light years across presently…

  16. James says:

    Thanks David It just blows me away that there is so much more inner space than outer space.

  17. Don Mart says:

    From 1.62 × 10^-35 meters, up to 2.59 × 10^26 meters, that’s a total of 62 orders of magnitude, and we are at the 36th order of magnitude.

    Interesting, because 36/62 = 1.7, and 2.59/1.62 = 1.6, the “golden ratio” comes to mind

  18. Alex says:

    Wikipedia has new information about the Planck length. See http://en.wikipedia.org/wiki/Planck_length (proofs). This information reveals the true meaning of the Planck length.

  19. Frank Weyl says:

    The limits to our reality are defined by quantum mechanics within the Planck constants.
    Distant,time,energy,frequency and mass.
    Apart from the Planck mass, there are no numbers below these when defining aspects of our reality.
    0/0 therefore is not an indeterminate and if you see this about to occur in any calculation you have to either rationalize it or realize that there is something radically wrong with your equations.
    As the universe has the shape of a Riemann-sphere (or Mobius sphere) with no externality,
    and started from what can only be termed a Planck-object of 10^-35cm in diameter, all (internal) numbers stop at these constants.
    What is the meaning of 0/0 = 0?
    The symbol 0 means ordinarily the absence of something. Zero credit in my account means I have no money in my account. Defining 0 that way is simple and I initially concluded that there should be no further problem with this definition. When given the instruction 3 – 3 the solution is simply 0, since 3 is taken off itself; nothing remains.
    However, reflecting further on the subject I realised, that things are not that simple.
    Let us describe something concrete. Subtract 3cm from itself that is 3cm – 3 cm. Normally one would conclude that 3cm minus itself gives also 0cm. But in order to know an object’s length one must first measure its length. So can one measure the lengths of objects with infinitesimal accuracy and precision or are there limits?
    Quantum mechanics imposes limits on length, time, energy and mass (etc.). It tells us that there is no smaller distance between two points which can be measured than Planck distance which is 1.616048616 × 10^-33 cm. One can measure 3cm + 1-33cm, but not 3 + 1-34cm.
    Implied is, that measuring the length of an object below Planck length is meaningless. When applying the above considerations to the calculation of 3cm – 3cm the solution is not zero anymore but Lp. If we state 3cm minus 3cm is 0 we must implicitly be capable to measure the length of an object below Planck distance to ensure, that no number defining some distance below that point occurs, and that we would find only zeros after the decimal point infinitesimally. This however is impossible. When subtracting 3cm minus itself one must therefore give the ‘limit of length which is Lp for the solution. I write:
    3cm – 3cm = Lp.
    Further implied is that we can replace here Lp for zero. This is in calculations which involve length measurements.
    Now back to the question if 0/0 has any significance.
    I write: 3cm = 3cm
    Applying algebra I get:
    3cm – 3cm = 0cm I solve the terms on the left to obtain:
    0cm = 0cm
    And finally I bring the 0cm on the right to the left and equal the equation with 0cm on the right side to attempt a true statement.
    0cm/0cm = 0cm.
    I highlighted the given solution which is zero in red, since it is not in accord with my previous insights; 3cm – 3cm = Lp.
    Note also, that in the equation 0cm/0cm = 0cm (mathematical statement) which is derived from 3cm – 3cm =0cm and 0cm = 0cm, all the zeros are highlighted red. It indicates that the derived numbers zero which occurred in the expression which is my final calculation 0/0 = 0 are all untrue/ incorrect. Therefore the whole statement 0cm/0cm = 0cm is not significant; meaningless.
    Let us do the above calculations again obeying the Planck limit of length.
    I write:
    3cm – 3cm = Lp.
    Lp = Lp.
    Note that I did not write Lp cm because Lp is (I assume) not a length in that sense. It is the limit of measuring length.
    The next step is to bring (for example) the right term to the left and calculate the terms on the left as part of an equation.
    Lp/ Lp =?
    Dividing Lp by Lp I do not however treat the same as 0/0. Lp is a definite value. So any value divided by itself is 1 (except 0/0 = 0; ordinary definition). I write:
    Lp/ Lp = 1
    Let us look at 1/0. My scientific calculator gives ‘error’, Microsoft Mathematics gives ‘indeterminate’.
    I replace zero for Lp
    1/ Lp = 6.1879326531 * 1032
    Now 0/1 is normally 0. But…
    Lp/1 = Lp
    Addenda: I mentioned that some zero’s where highlighted in red. The colour didn’t transfer from my copy/paste operation.
    This was collaboration between me and my mathematics student Joachim Delbeke. I gave him the question and the above; with slight addition, is his work.

  20. mktea says:

    Is Lp/2 not halve the size of Lp? Making it smaller than Lp?

  21. Paul says:

    No wonder I lose stuff.

  22. Frank Weyl says:

    mktea says:
    October 8, 2014 at 11:04 pm
    Is Lp/2 not halve the size of Lp? Making it smaller than Lp?
    Quote from my original response:
    “Implied is, that measuring the length of an object below Planck length is meaningless. When applying the above considerations to the calculation of 3cm – 3cm the solution is not zero anymore but Lp. If we state 3cm minus 3cm is 0 we must implicitly be capable to measure the length of an object below Planck distance to ensure, that no number defining some distance below that point occurs, and that we would find only zeros after the decimal point infinitesimally. This however is impossible. When subtracting 3cm minus itself one must therefore give the ‘limit of length which is Lp for the solution.”

  23. Hi, just a thought….speculative and entirely unproven.

    Let’s imagine that reality as we see and experience it, is created by the actions of our biological brain as it processes the information coming to it. The reality we thus perceive is virtual. We believe it to be real and indeed experience it as such, because the person or centre of being that we see ourselves to be is also created by that very same process of sensory re-creation that our brain so deftly performs. Life is thus a dream, one in which we are immersed fully and attentively within and one in which the very being that we believe ourselves to be, is also created.

    Given this, if we examine the apparent nature of this outer dream, and delve ever deeper into the finer constructs of reality, we eventually get to the point where measurements of space and time becomes chunky i.e. beyond which we can divide no further…the planck length. This has been suggested to show evidence that supports the virtual reality model of creation in that if creation were an entirely unconstructed form then it would be continuous. May I suggest that the observation of quanta represents the resolution of our brains ability or capacity (processing limits) to create and sustain the virtual reality that we perceive around us. The planck length and other quantifications of measurement thus offer an insight into the formation of reality, the world and everything that we believe to be in it. The difference to other theories of virtuality is that I suggest the creator of that reality is not some other external or other worldly entity or the workings of some future sophisticated computer program, but is in fact the manifestation of our own physical brain in cognition.

    Our brain creates an impression of reality based on the perceptions of its senses and it creates within that setting, an avatar of ourselves in which our centre of consciousness abides. We thus are created and set amongst the creation of our thoughts.

    What would be interesting, would be to discover if the limitations of the planck length ie the limits of resolution of this virtual model, correlate with the actual physical processes of the brain. To do that would be to show quite elegantly, the true relationship of the apparent material universe and it’s fundamental, created nature.

    If you care to read my book which is all, purely speculative, then please do. I am Brian Mccully, and the book is called Why Goldfish Never Die.

  24. Billy Brown says:

    Physicist, you are really funny (in a good way) and should definitely write a book, that is if you haven’t already!

  25. Alex says:

    The physical meaning of the Planck length, read by https://en.wikipedia.org/wiki/Virtual_black_hole

  26. Frank Weyl says:

    To moderator. I couldn’t edit my post, could you?
    Thank-you Alex,
    Perhaps this .pdf could act as an addendum to your: ‘https://en.wikipedia.org/wiki/Virtual_black_hole’
    Goto:
    http://arxiv.org/pdf/1307.2283.pdf

  27. Bryan says:

    Since the electron is 10^20 times larger than the Planck length,I cannot understand
    why “they” claim …”…all the matter in the Universe was the size of the head of a pin
    before the Big Bang.” I wish I would stop hearing that. “They” were not there when it happened and it is all speculation but presented as fact…Show me some kind of proof,not wild imaginations.

  28. Bryan says:

    P.S. Wild imaginations like that interfere with my own observations.Giovanni Schiaparelli viewed Mars and claimed there were canals. “Black holes” were not supposed to let light escape,yet observable “jets” are seen shooting from “them”.
    “Black holes” have not even been observed,Sgr A is the closest and does pose interesting questions about what is going on there…Please put a question mark on your “observations and ideas” if you have no direct proof.Getting to the point I cannot believe much at all what is written in books.”Facts” keep changing decade to decade.

  29. Bryan says:

    P.S.S. Nothing against you Physicist,its just some of what Ive observed happens in science. Messes up the whole data stream.

  30. Tor Snyder says:

    Can neutrinos be measured in planck units?

  31. PROF. Rob Frank says:

    You should ll watch the movie “Yesterday was Lie” for fun… it deals with Planks Constant in a artful and really deep way. Just sYING

  32. PROF. Rob Frank says:

    (forgive the misspellings – and pause at 21:54 to explain)

  33. Alex says:

    There is a new article about the Planck length http://vixra.org/abs/1507.0149

  34. Frank Weyl says:

    Alex: Thank you for this.

  35. Liam Murphy says:

    “The “Planck length” is the unit of length in Planck units, and it’s \ell_P = \sqrt{\frac{\hbar G}{c^3}} = 1.616\times 10^{-35} meters. Which is smallI don’t even have a remotely useful way of describing how small that is.”

    I do not know if this is useful but I came across a description recently that goes like this… If one puts a period on a page then expands that period to the size of the observable universe…then put down another period on that, then that second period is equivalent to the Planck length. Forgive any sloppy language I am not a scientist.

  36. Angel says:

    @Bryan:

    There is evidence that shows that the Big Bang happened. Not only does that evidence exist, but the Big Bang has been replicated in particle collisioners. It’s a thing. We know it happened. We just don’t know enough to make accurate conclusions about the Big bang itself. But it happened. That much we know.

  37. Angel says:

    @Frank Weyl:

    Your argument isn’t really valid because it is not applicable. Numbers are not based on reality. You cannot talk of numbers using physical properties of reality. Irrational numbers are not even possible in the physical universe, let alone complex numbers. However, inside mathematics, we have been able to work with infinitesimals and wheel algebra. The Planck length is the smallest length possible. However, inside mathematics, the very rules of algebra that you assume to be founded on reality despite not being so prohibit having such a thing as a smallest possible number. That is why there exists no greatest integer. That is why the reciprocal of zero does not make sense — or at least one of the reasons. On the other hand, 0/0 must be indeterminate. ox=0 for any x. Therefore, 0/0 must equal every single number that exists at the same time. That si the very definition of indeterminate. Every other indeterminate expression simplifies into 0/0 in some way, such as 0^0, for instance: 0^0=0^(1-1)=0^1/0^1=0/0. 0^0 is indeterminate because 0^0 is also indeterminate. We know infinite geometrical sums work. But adding a number of infinite things doesn’t work in the real physics.

  38. Dr. K. Padma Prasad says:

    the Page is interesting and informative

  39. Frank Weyl says:

    Angel,
    Thank-you for your observations above.
    Reality is.
    Mathematics is a language and evolving logically and, in some cases, esoterically.
    One can alter the logic of approach to a problem in mathematics but cannot alter the state of reality.
    Mathematics has and will continue to define complex physical phenomenon to a remarkably fine degree of resolution. But, even with these mathematical tools at our disposal, certain areas of physics,( i.e. QFT or QM) in which these mathematical tools and descriptions are used and appear, are only approximates and similes. We have yet to design a mathematical regime which defines precisely the underlining reality of quantum action.
    Hence our ‘tongue in cheek’ dissertation above.
    If we all keep along the same line of mathematical logic/reasoning without question, mathematical evolution will suffer and stifle innovation.
    As a physicist I am grateful to you mathematicians in giving us a mathematical structure/language in which we can define phenomena and I personally read this site and other mathematics sites to further my ‘vocabulary’, mathematical grammar and in some instances…philosophy of number.

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