August 12, 2008

Googol and Googolplex

My kids (from 3 and 5 years old) use the large number word Googolplex frequently in their conversations. I mistakenly explained a Googolplex as a 1 with 100 zero's (which I now know is only a Googol). To set the record and their education straight I thought it would be nice to actually look-up the origin and explanation of this intriguing word which baffles anyone trying to name a large number (just try to name a larger number from the top of your mind...). [Toine]

Googol and Googolplex

Some time before 1940, mathematician Edward Kasner asked his (at the time) nine-year-old nephew to name the number 10100, and the nephew invented the name googol. Kasner then named 10googol with the name googolplex.

googol is a class-2 number, and is reasonably easy to comprehend. You can even write it out:


The number of particles (protons, electrons, photons, etc.) in the universe has been estimated as 1080, so while a googol is an awful lot bigger, it's still roughly the same idea.

Googolplex on the other hand, is 10googol, or:


and that's a number far too big to relate to anything physical — or is it?

This description comes from Don Page, and is highly paraphrased:

In quantum physics one talks about distinct and indistinct states, overlapping wave functions and many other things, and one of the important principles that comes out of it the concept of how much time it takes for a system with many possible states to spontaneously assume a certain, given state. For simplicity it is imagined that the system is inside an impermeable, rigid box (which thus prevents anything from going in or out and thus isolates the system). If you're outside the box you have no idea what's inside, and any state can spontaneously occur (even a higher-energy state). For example, if the system is a proton and a neutron, the possible states include: A hydrogen atom, a hydrogen atom with the electron in a higher "shell", a separate proton and electron, a neutron, or many other things. The amount of time you might have to wait for one of these states to occur depends on the number of possible states, which depends on the number of particles inside.

Cosmologists and physicists often use these concepts to discuss what might happen inside black holes. The inside of a black hole is isolated from us (nothing can get out) and so one can talk about all the possible states that might exist inside the black hole and how long it might take for a certain state to arise. In theory, if left inside a rigid nonpermeable box for long enough (so that nothing new can go into the black hole) the contents of the box will at some point in the future cease to be a black hole and instead be real matter.

The formula for the number of possible states is a really big exponential function with Planck constants in it and other such things. I don't have the exact formula but we do know that if the black hole had a mass equal to the Milky Way Galaxy, plus the Andromeda galaxy, and Magellanic clouds, and a few other similar globular clusters in our region, then the number of states, and the amount of time you'd have to wait, is somewhere around a googolplex. Its such a big number that there's no way to distinguish what units should be used. It could be a googolplex seconds or a googolplex millenia. Whatever the units, thats the amount of time you'd have to wait for the contents of an impermeable, rigid box containing a black hole with the mass of the local galactic neighborhood to spontaneously become identical to the local galactic neighborhood as we know it.

The original (and much better) description is below.


Edward Kasner and James Newman, Mathematics and the Imagination, Penguin, 1940.
[Source: ]

How to Get A Googolplex
So I finally convinced you that you cannot imagine something close to Googolplex? This was my belief, too, until I was taught a lesson by Don Page,, a gravitational physicist. But beware, it's deep into physics.

Stephen Hawking, Nature 248 (1974), pp. 30-31, and Communications of Mathematical Physics 43 (1975), p. 199ff, found that the dimensionless entropy (i.e., the entropy after dividing by Boltzmann's constant k) of a black hole is 1/4 its area in Planck units, or 4*pi times its mass squared in Planck units for a Schwarzschild (i.e., uncharged and nonrotating) black hole.

Planck units are those one gets when one multiplies the appropriate powers of Planck's reduced constant h-bar=h/(2*pi), the speed of light c, and Newton's gravitational constant G to get a quantity of whatever dimension one wants.

E.g., the Planck unit of area, or simply the Planck area, is (h-bar*G)/(c^3)=2.612x10^{-70} square meters, the Planck length is the square root of this, or 1.616x10^{-35} meters, the Planck time is the Planck length divided by the speed of light, or Sqrt[h-bar*G/c^5]=5.391x10^{-44} seconds, and the Planck mass is Sqrt[h-bar*c/G]=2.177x10^{-8} kilograms.) Thus one would need a Schwarzschild black hole of mass Sqrt[10^{100}/(4*pi)]=2.821x10^{49} Planck masses, or 6.140x10^{41} kg, to have an entropy of one Googol. Since the sun has a mass of about 1.989x10^{30} kg or 9.137x10^{37} Planck masses, one needs about 3.087x10^{11} solar masses to give a black hole with an entropy of one Googol.

If the entropy of a black hole represents the (natural) logarithm of the number of states of similar macroscopic configurations (as it does for other thermal systems, and as I shall assume is also true for black holes, though there is some controversy about this), then a black hole with an entropy of one Googol would have e^{googol}=10^{googol/(ln10)} states. To get a Googolplex of states, the entropy needs to be larger than a Googol by a factor of ln10, and so the mass of the black hole needs to be larger by the square root of ln10, or about 4.685x10^{11} solar masses to give a Googolplex of macroscopically similar states. This is a bit more than three times the 1.4x10^{11} solar masses that C. W. Allen, Astrophysical Quantities, Third Edition (The Athlone Press, University of London, 1976), p. 282, attributes to our Galaxy, but Allen, p. 287, attributes 10^{11.5} solar masses to the Andromeda Galaxy (M31), so if one adds that, and perhaps one or more smaller galaxies in our Local Group, such the Large Magellanic Cloud, the Small Magellanic Cloud, M32, etc. (each of which are a few percent of the mass of our Galaxy), one should get to a mass such that the corresponding black hole would have a Googolplex of states with similar macroscopic configurations. Certainly adding the Maffei Galaxy, with a mass of 10^{11.3} solar masses according to Allen, would put one over what is needed for this, unless Allen's numbers are significantly overestimated. (I would guess that dark matter would make the actual masses rather larger than the estimates Allen quotes, but I don't know the latest figures on the masses of components of our Local Group.)

Thus if one takes some (probably moderately large) fraction of the mass of our Local Group of galaxies, puts it into a black hole, and asks how many states there are with a similar macroscopic appearance, one would get a Googolplex.

Obviously, Mr. Page is into big numbers, too. "The number of states in a black hole with a mass roughly equivalent to the Andromeda Galaxy". And you've got a Googolplex. Couldn't be simpler, could it?

However, after a user's complaint that it is a little too technical even for an advanced physics student,

Huh? I took physics for two years, and calculus for two years, but your
explanation that there is indeed enough matter in the
universe to comprise a Googolplex floored me.
Mr. Page decided to add another paragraph.

This result, plus the assumption that a black hole is a nearly ideal mixer of macroscopic information, would imply that if one put this black hole into a hypothetical rigid nonpermeable box, a few million light years in size, and looked at the contents once a year, it would look like our present Local Group for the first time again after roughly a Googolplex of years. In other words, a Googolplex is a quantitative measure of an extension (to a much larger system) of the temporal periods expressed by such colloquial phrases as "until the cows come home," which would give recurrence times that would presumably be long in human terms but much shorter than a Googolplex if these smaller systems (e.g., of cows and their home) would actually last long enough.

You might be amused to note that in Information Loss in Black Holes and/or Conscious Beings? to be published in Heat Kernel Techniques and Quantum Gravity, edited by S. A. Fulling (Discourses in Mathematics and Its Applications, No. 4, Texas A&M University Department of Mathematics, College Station, Texas, 1995) (University of Alberta report Alberta-Thy-36-94, Nov. 25, 1994), hep-th/9411193, I estimated a quantum Poincare recurrence time for the quantum state of an extremely hypothetical rigid nonpermeable box containing a black hole with the mass of what may be the entire universe in one of Andrei Linde's stachastic inflationary models and got 10^(10^{10^[10^(10^1.1)]}) Planck times, millenia, or whatever. As I wrote in the following line, "So far as I know, these are the longest finite times that have so far been explicitly calculated by any physicist."

Don Page

Dont mistake Googolplex with Googleplex which is searchengine Google headquarters in Mountain View, California, (USA). I assume they liked the alliteration and reference to this large number when naming the complex. [Toine Fennis]

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