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What does infinity mean to a physicist

  1. Mar 11, 2010 #1
    Infinity has always been a problem for me, apart from the concept I do not believe it exists. numbers go on to infinity and even negative infinity but they don't really exist so they don't count (hehe). some people say the universe is infinite in size. well apart from infinite in the same way as a spheres curved surface is infinite then no the universe cannot be infinite.

    I raised this with a colleague, he said from a maths point of view (and presumably physics) that infinite just means immeasurably large but finite.

    Is that true?

    If so please call it something else other than infinite may I suggest [tex]\infty[/tex]-1 (lol)

    If its not true how can infinite be used within equations to describe 'real' things because its impossible.
  2. jcsd
  3. Mar 11, 2010 #2


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    I don't think we've ever discovered infinity in nature. I tend to think it doesn't exist either. In physics, whenever we say infinity, we really mean "a huge number". So big, as it were, that it causes terms to go to zero in the approximation.

    This simplifies mathematics and allows for series approximations. I don't think anything is really going to infinity though.
  4. Mar 11, 2010 #3
    We do use infinity in nature. For example, a gravitational singularity which is found at the center of every black hole. We know that there is no degenerate matter that can maintain equilibrium with the force of gravity because it is too strong. There is no known force in the universe that can stop this incredible warping of space.

    The result is an astronomically large quantity of matter being crushed into a point in space with zero volume and infinite density.

    P = M / V
    ∞ = M / 0
    Last edited: Mar 11, 2010
  5. Mar 11, 2010 #4
    Infinity doesn't exist in the real, physical world. However it does exist in mathematics, a world we have invented.
    We use mathematics to model the real world, and it works just fine until we get an infinity. Then we know that our mathematical model of the real world has broken down at that point.
  6. Mar 11, 2010 #5
    Infinite usually means that something is wrong with the current equations. In a singularity there seems to be infinite acceleration of gravity as well (which doesn't make any sense):


    Plug in the 0 for r:
  7. Mar 11, 2010 #6
    Actually, that makes perfect sense. It only has an infinite acceleration at zero volume. There is no such thing in space where there is zero volume, so I don't see the problem.
  8. Mar 11, 2010 #7
    The problem is there can be no infinite. so there can be no infinite acceleration of gravity. if the singularity is the sum of all things that exist today (laymans interpretation) then the sum of all things today must equal the singularity.

    So if there is no infinite gravity today then there wasn't in the singularity.

    So in my most humble opinion the singularity cannot be modelled by the math or there was no singularity because it is an impossible state.

  9. Mar 11, 2010 #8
    Well the problem is that GR predicts all matter in a black hole to be crushed into a singularity with zero volume. So yes, infinite acceleration due to gravity is a problem.
  10. Mar 11, 2010 #9


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    Actually, GR insists on a hole; filling that hole with a point of space-time is a violation of GR.

    (And trying to use Newton's formula for gravitational "force" is absolute nonsense in this context)
  11. Mar 11, 2010 #10
    Well, its only infinite at the very singularity. Even a 10^-99 meters apart from the singularity, the acceleration becomes finite.
  12. Mar 12, 2010 #11
    Even the "singularity" at the center of a black hole has a finite, albiet small, volume. A singularity would imply that it has an exact position, which violates the uncertainty principle.

    Infinity can be thought of as a tool, but I don't think it really exists in the conventional sense (except maybe in an infinite multiverse thingy, the universe itself is presumably finite. I say this with a B.S. degree in Layman's Physics).
  13. Mar 12, 2010 #12

    What type of degenerate matter can stop gravity from crushing it into zero volume?
  14. Mar 12, 2010 #13


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    Mathematically, infinity must exist because there must exist the infinitesimal.

    For example, there are an infinite number of real numbers between 1 and 2 (technically the more appropriate term might be uncountable, but by its definition, still infinite).

    One might argue that any practical number, such as the [tex] \sqrt{2} [/tex] which represents the diagonal of a square, might trail off with all zeros eventually, if one were to calculate out the decimal places far enough. But if that were true, it would be possible to represent [tex] \sqrt{2} [/tex] as a ratio of integers, albeit very, very large integers.

    But here is rough proof that this is not possible. Suppose [tex] \sqrt{2} [/tex] can be represented by a ratio of integers, M/N (where M and N might be very large, yet still finite).

    Suppose that we reduce M/N such that M and N are the smallest units possible. Now there are 3 possibilities regarding M and N.
    (a) M and N are both odd.
    (b) M is odd and N is even.
    (c) M is even and N is odd.
    Note that M and N cannot both be even, because then we could simply divide both by 2, and retain the same ratio.

    Given the above, we could say,

    [tex] \sqrt{2} = \frac{M}{N}. [/tex]

    Squaring both sides gives us

    [tex] 2 = \frac{M^2}{N^2} [/tex]

    which is,

    [tex] 2N^2 = M^2. [/tex]

    Here we can be confident that M is even because 2N is even: The the square of an odd integer gives an odd number. 2 times any integer is even thus 2N is even. Thus if M2 is even, M must be even. That also implies that N must be odd (because M and N cannot both be even).

    Since M is even, let's make a substitution that 2L = M.

    [tex] 2N^2 = {2L}^2, [/tex]

    [tex] 2N^2 = 4L^2, [/tex]

    [tex] N^2 = 2L^2. [/tex]

    But this means that N must also be even, since multiplying any integer by 2 creates an even number. making both M and N even, which contradicts the original proposition. Thus [tex] \sqrt{2} [/tex] cannot be represented by a ratio of integers. [tex] \sqrt{2} [/tex] belongs in some different class of numbers, that cannot be represented by a ratio of finite integers (namely irrational numbers).

    One way to interpret this is [tex] \sqrt{2} [/tex] has a truly infinite number of decimal places to represent it -- they do not trail off to all zeros eventually. Another way to interpret this is that [tex] \sqrt{2} [/tex] cannot be represented by a ratio of integers -- at least not finite integers -- at least not unless both M and N are infinite.

    So if one believes that squares exist, and if one believes that squares have diagonals (i.e. one believes that isosceles right triangles exist), arguably one must believe that infinity exists (at least mathematically speaking). This logic might be debatable, but it does present a very strong argument.
  15. Mar 12, 2010 #14
    Despite the fact that QED is a really nice, really accurate theory, there's no way it can be used to predict exactly how degenerate matter would behave at the core of a black hole. With that said however, it may indeed be packed into a small space, but it can't possibly have a zero volume due to the uncertainty principle. Degenerate matter is still matter, and matter takes up space. Just because relativity is flawed and suggests that an infinitely small singularity is possible, doesnt mean that it IS possible, especially in light of the fact that it's incomplete (non-quantum).
  16. Mar 12, 2010 #15
    There's no such thing as infinity. It's a fabrication of the human imagination. The important thing about "infinity" is that it shows up mathematics as an imperfect model of reality (I'm sure mathematicians/physicists hate to hear this, but it's TRUE).
    Last edited by a moderator: Mar 12, 2010
  17. Mar 12, 2010 #16

    Unimaginably small is still finite. any thing smaller than that doesn't exist. The discussion is about reality not mathematical modeling.

    Proof of infinity in math well thats easy, [tex]\frac{1}{0}[/tex] = [tex]\infty[/tex]. it exists in the mathematical model and that is true.

    Just like zero, a very important number in maths it does not relate to a real thing. I am holding zero rocks in my hands means there's no rocks there. I could quite easily say I am holding zero universes in my hand and be 100% correct. If I said I am holding an infinitesimally small universe in my hand, I'd be lying but no-one could PROVE IT. If I found it, then it must be finite. so whatever real thing I found at that scale it must be smaller again MUCH smaller, so no-one could prove I am lying.

    infinitesimally small is as impossible in the real universe as infinitely big.


    BTW: each time you add a decimal point thats not zero it gets bigger not smaller so infinitesimally small must be by definition All Zero's
  18. Mar 12, 2010 #17
    Hey, this is my first post on the Physics Forums. I just find the idea of infinity interesting, so I though I'd comment. Zeno's Paradox comes to mind. and this might be a good example of how math is an imperfect representation of the world. the distance from one point to another can be separated into pieces, each 1/2 the length of the last (.5D, .25D, .125D, etc...). using summation notation, you could technically add up all the parts to equal the distance, D. but also, theoretically I think (as Zeno thought), one can infinitely divided D into an infinite number of pieces. thus, you would have to add an infinite number of increasingly infinitesimal distances. applying this to the real world, if one crossed one section of the distance at a time, then you could never get to the other point, because you would cross and increasingly infinitesimal distance, thus crossing and infinite number of distances. and infinite number of real numbers exist, both outside of and inside of 1. so if one actually applied this to the real world, because matter has volume and thus a dimension of length, there could theoretically be an infinite number of pieces of matter, increasingly infinitesimal. so you could make the conclusion, perhaps, that infinity exists in matter because the space around (and of) us is infinitely "deep" (is filled with an infinite number of infinitesimal particles of matter). which also brings to mind the oddity that since one does appear to go from one point and reach another, one can traverse infinity, just as one traverses infinity in math by counting from 1 to 2. but since math is really not a perfect model of the world, this likely isn't true. And although I don't really understand a singularity, is supposed to be an infinitely dense/small thing with extreme gravity? because one could make the argument that it has an infinite pull because it is nothing. nothing is technically infinity. I think of infinity as having no point in it because it has no beginning and no end. the only other thing I can think of that has no start or end is 0-nothing. if a singularity is nothing, then technically it is infinitely deep and pulls in matter infinitely. you can't measure it because it is infinite because it is nothing. I don't know. this is all really hypothetical. I'm not a real scientist, I'm just interested in physics, so I joined the forums.
  19. Mar 12, 2010 #18


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    Sigh, allow me to correct some errors.

    The only infinitessimal real number is zero.

    1/0 is nonsensical, when viewed as arithmetic of real numbers.

    Whether or not a mathematical structure has an element called "infinity" is entirely up to the mathematical structure.

    I don't know of any (non-contrived) mathematical structures that:
    • contain a non-zero infinitessimal number
    • that number has an (infinite) reciprocal
    • the reciprocal is called "infinity"

    All math is, more or less, a "fabrication of the human imagination".
  20. Mar 12, 2010 #19


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    Yes, I suppose you're right. :tongue2:

    Leopold Kronecker has a famous quote saying,

    "Die ganzen Zahlen hat der liebe Gatt gemacht, alles andere ist Menschenwerk.
    (The dear God has made the integers, all the rest is man's work.)"
    -Leopold Kronecker


    Anyway, slightly off topic (or maybe not!) here is fun game to play. I've actually stumped many mathematically inclined people with this riddle (many for well over 10 or 20 minutes).

    Proof that 2 = 1
    Find the flaw! (game/riddle)

    We start with the postulate that a = x.

    [tex] (1) \ \ \ \ \ a = x [/tex]

    Add a to both sides.

    [tex] (2) \ \ \ \ \ a + a = x + a [/tex]

    Simplify left side,

    [tex] (3) \ \ \ \ \ 2a = x+a [/tex]

    Subtract 2x from both sides,

    [tex] (4) \ \ \ \ \ 2a - 2x = x + a -2x [/tex]

    Factor the left side,

    [tex] (5) \ \ \ \ \ 2(a - x) = x + a -2x [/tex]

    Simplify the right side

    [tex] (6) \ \ \ \ \ 2(a - x) = a - x [/tex]

    Divide both sides by (a - x)

    [tex] (7) \ \ \ \ \ 2\frac{(a - x)}{(a-x)} = \frac{a - x}{a-x} [/tex]


    [tex] (8) \ \ \ \ \ 2 = 1 [/tex]

    Therefore 2 = 1. QED.

    Obviously 2 is not equal to 1, so there must be a flaw in the above logic. Specify the step which has the (first) flaw and the reason it is flawed.

    If you already know the answer, or if you figure it out, you might wish to keep it to yourself for awhile (on this thread) lest spoil it for the rest. :rofl:
  21. Mar 13, 2010 #20


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    Although there are some advanced branches of mathematics where an entity called "infinity" exists, in this post I refer to standard real numbers and complex numbers, which are what most physicists would use.

    In the standard real number system there is no such number as infinity. Mathematicians do use the words "infinity" or "infinite", but only a shorthand for some other way of expressing it. When being rigorous, you should never write [itex]x = \infty[/itex], but you can write [itex]x \rightarrow \infty[/itex], which has a specific technical meaning.

    To give an example, you might say that the energy of a particle tends to infinity as the particle's velocity tends to the speed of light ([itex]E \rightarrow \infty[/itex] as [itex]v \rightarrow c[/itex]). But this means the same thing as "no matter how much energy you supply to a particle, it will never reach the speed of light". You specify an energy, as large as you like, and the particle can exceed that energy by going fast enough (but still slower than light). Similarly, any sentence containing the word "infinity" or "infinite" can be rephrased in terms of finite quantities.

    If an equation of physics seems to give you an answer of infinity, that means you've misused the formula and it doesn't apply in that circumstance.
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