# Would one actually find their doppelgänger in a “Googolplex Universe”?

I've recently become a fan of Numberphile, and today I happened to watch their video regarding Googol and Googolplex. In the video, I found a rather confusing claim that I'm hoping someone here can help me sort out.

It's first stated in the introduction, and then explained in more detail at around 4:10. The claim can be summed up as this:

In a universe which is "a Googolplex meters across", if you would travel far enough, you would expect to eventually begin finding duplicates of yourself.

Getting deeper into the detail of this, Tony Padilla explains that this is because there is a finite number of possible quantum states which can represent the volume of space in which your body resides. That volume is given as roughly one cubic meter, and the number of possible states for that volume is estimated at $10^{10^{70}}$. This is obviously much less than the number of cubic meters within a "Googolplex Universe", and so the idea does make some sense.

But I believe the idea relies on a false premise. That premise would be that the universe as a whole is entirely comprised of random bits of matter. We can easily see that this is not true. The vast majority of our observed universe is occupied by the near-vacuum of space, and those volumes which are not empty are occupied by some fairly organized objects which all interact according to certain laws and patterns.

So, I'm curious to know two things:

1. Who originally proposed this idea? Is it something that perhaps Tony just came up with to include in the video, or is there a noted physicist or mathematician who actually wrote about this at some point?

2. Given the possibility that a universe similar to ours could exist and be one Googolplex across, would this actually be probable? Or, would the natural order and organization of the universe prevent this from being as likely as it might be in a more random universe?

A note regarding the size of the universe, in respect to this question. In the linked video, the following constraints are mentioned:

• The number of particles in the universe is $10^{80}$
• Stated at 1:38 in the video.
• The number of the grains of sand that could fit within the universe is $10^{90}$.
• Stated at 1:55 in the video.
• The number of Planck volumes in the universe is $10^{183}$
• Stated at 2:22 in the video.
• The size of the universe is $(10^{26}m.)^{3}$
• Stated at 6:36 in the video.

A few times in the video, Tony does qualify his statements by referring specifically to the observable universe. However, sometimes it's not quite so clear.

So, to simplify the problem for the purposes of this question, let's assume:

• Our universe is finite, and much less than a Googolplex in diameter.
• The "Googolplex Universe" suggested in the video, and in question here, is also finite.
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Regarding the source of claims such as those, I believe they are due to Max Tegmark, likely from this article. He has curious ideas, though discerning whether they are correct, useful or at least in the realm of possibility is far beyond my capacity. He is a strong proponent of the idea of a mathematical Universe. If I recall correctly, he goes to the point of stating that anything exists at all merely because mathematics/logic exists, and everything we perceive as reality is logical statements playing out in a marvelously complex fashion very far down the line. Why does it happen? Because it can, basically.

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I think this assertion would be more correct if it was something like:

There are more 1 cubic-meters of matter units in a googolplex-meter-wide universe than there are possible quantum states of 1 cubic-meters of matter units, so at least one of those quantum states of 1 cubic-meters of matter occurs in more than one 1 cubic-meters of matter somewhere.

In fact there are at least as many duplicates of some kind, to make up the difference between the number of possible states, and the number of total volumes.

But, this does not imply that any particular configuration is duplicated. Heaven forbid there be more of me out there. Maybe a rather simple configuration accounts for 99.99% of all the duplications.

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Exactly, he's implicitly assuming all configurations are equally probably. There is no reason to believe that. – Tobias Brandt Aug 1 '14 at 21:41

Well, the universe has evidently produced you at least once, so we can certainly say the probability of producing you is nonzero.

Of course, you're the end result of a very long chain of events, encompassing all sorts of cultural, biological, planetary, astrophysical, and cosmological processes. Ultimately (well, to the extent of our cosmological knowledge) your existence traces back to certain random quantum fluctuations during the inflationary era of the Big Bang, which acted as the seeds for galaxies to form, and so on.

If we imagine that the universe extends far beyond the part we can see (the observable universe), at least $10^{10^{100}}$ meters in each direction, but has evolved from similar initial conditions everywhere, then it isn't too difficult to imagine that somewhere in that immensity, some inflation fluctuations occurred that were sufficiently similar to ours, to generate a galaxy sufficiently similar to the Milky Way, to generate a planet sufficiently similar to Earth, to be inhabited by beings sufficiently similar to us, that one of them is very similar to you.

All of this would be done by exactly the same physical processes that produced you. It doesn't require that the universe be filled with a random distribution of all possible physical states; it only requires that whatever random processes contributed to your existence will randomly occur again, which they certainly will in a big enough universe. I won't attempt to quantify it, but if a googolplex meters is not enough, just make it bigger. :)

As for the history of this idea, it seems quite similar to the notion of Poincaré recurrence, which dates back to 1890. That idea concerns repetition of states over time, due to arbitrarily-unlikely fluctuations occurring if you wait long enough, but it's straightforward to think also of repetition in space due to arbitrarily-unlikely fluctuations occurring if you have enough space for them to occur in.

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Another such argument was made by Jaume Garriga and Alexander Vilenkin, see here:

A generic prediction of inflation is that the thermalized region we inhabit is spatially infinite. Thus, it contains an infinite number of regions of the same size as our observable universe, which we shall denote as O-regions. We argue that the number of possible histories which may take place inside of an O-region, from the time of recombination up to the present time, is finite. Hence, there are an infinite number of O-regions with identical histories up to the present, but which need not be identical in the future. Moreover, all histories which are not forbidden by conservation laws will occur in a finite fraction of all O-regions. The ensemble of O-regions is reminiscent of the ensemble of universes in the many-world picture of quantum mechanics. An important difference, however, is that other O-regions are unquestionably real.

But you can just consider all the useful information stored in your brain that you are aware of. Even if there exists someone who is not an exact physical copy, it can still be identical to you if whatever algorithm his/her brain is executing is the same as the one your brain is executing. This means that the local environment only needs to be approximately the same. A copy of me doesn't have to have the exact same number of hairs growing in his head, because I have never counted the exact number. The Earth's radius can be several meters more or less. The copy of the Milky Way galaxy does not need to contain the exact same number of stars. Only what I am aware of must be the same, and it's only the awareness that counts not if it is actually true, although the two things will be strongly correlated.

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So, what you're saying is, there will be some exact copies and some near-exact copies who won't even be able to realize that they aren't exact copies. – Iszi Sep 25 at 20:32
@Iszi, that's indeed what I'm saying. Also, I believe (but have to admit that it's just my belief), that it's essential to think of ourselves as ensembles of nearly identical states instead of a sharply defined state. If you think of yourself as a machine in some well defined state, you get into philosophical problems that can be addressed better if you replace the machine by a set of machines in slightly different states. This makes the algorithm executed by the machines well defined, at the expense of not knowing exactly what state it is in which it couldn't know anyway. – Count Iblis Sep 25 at 20:42

If you mass 120 kg (264 lbs) and the average atomic weight in your substance is, we'll be generous, 25, then you contain $(120,000/25) \times (6.022 \times 10^{23})$ atoms or $2.9x10^{27}$ atoms. They can be arranged in $(2.9x10^{27})!$ ways.

\begin{align} n! &\approx \sqrt{2\pi n}(n/e)^n = (10^{14})(10^{27})^{27} \end{align}

is a big number, dwarfing string theory's $10^{500}$ acceptable vacua.

How much time are you given to look? Then QM intrudes (Heisenberg compensators?).

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