**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,

Mr. Page decided to add another paragraph.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.

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,don@page.phys.ualberta.ca

CIAR Cosmology Programme

Theoretical Physics Institute

University of Alberta

Edmonton, Alberta, Canada

Frank Pilhofer <fp -AT- fpx.de> Back to the Homepage Last modified: Sun Jun 3 14:44:31 2001