Worlds Enough And Time Reversed

Cathleen unmutes her mic. “Thanks, Kareem. Our next Crazy Theory presentation is from one of my Cosmology students, Jim.”

“Thanks, Cathleen. Y’all have probably heard about how Relativity Theory and Quantum Mechanics don’t play well together. Unfortunately, people have mixed the two of them together with Cosmology to spawn lots of Crazy Theories about parallel universes. I’m going to give you a quick look at a couple of them. Fasten your seat belt, you’ll need it.

“The first theory depends on the idea that the Universe is infinitely large and we can only see part of it. Everything we can see — stars, galaxies, the Cosmic Microwave Background — they all live in this sphere that’s 93 billion lightyears across. We call it our Observable Universe. Are there stars and galaxies beyond the sphere? Almost certainly, but their light hasn’t been in flight long enough to reach us. By the same token, light from the Milky Way hasn’t traveled far enough to reach anyone outside our sphere.

“Now suppose there’s an alien astronomer circling a star that’s 93 billion lightyears away from us. It’s in the middle of its observable universe just like we’re in the middle of ours. And maybe there’s another observable universe 93 billion lightyears beyond that, and so on to infinity. Oh, by the way, it’s the same in every direction so there could be an infinite number of locally-observable universes. They’re all in the same space, the same laws of physics rule everywhere, it’s just that they’re too far apart to see each other.

“The next step is a leap. With an infinite number of observable universes all following the same physical laws, probability says that each observable universe has to have twins virtually identical to it except for location. There could be many other people exactly like you, out there billions of lightyears away in various directions, sitting in front of their screens or jogging or whatever. Anything you might do, somewhere out there there’s at least one of you doing that. Or maybe a mirror image of you. Lots of yous in lots of parallel observable universes.”

“I don’t like that theory, on two grounds. First, there’s no way to test it so it’s not science. Second, I think it plays fast and loose with the notion of infinity. There’s a big difference between ‘the Universe is large beyond anything we can measure‘ and ‘the Universe is infinite‘. If you’ve been reading Sy Moire’s stuff you’ve probably seen his axiom that if your theory contains an infinity, you’ve left out physics that would stop that. Right, Cathleen?”

Cathleen unmutes her mic. “That quote’s good, Jim.”

“Thanks, so’s the axiom. So that’s one parallel universe theory. OK, here’s another one and it doesn’t depend on infinities. The pop‑science press blared excitement about time‑reversal evidence from the ANITA experiment in Antarctica. Unfortunately, the evidence isn’t anywhere as exciting as the reporting has been.

“The story starts with neutrinos, those nearly massless particles that are emitted during many sub‑atomic reactions. ANITA is one kind of neutrino detector. It’s an array of radio receivers dangling from a helium‑filled balloon 23 miles up. The receivers are designed to pick up the radio waves created when a high‑energy neutrino interacts with glacier ice, which doesn’t happen often. Most of the neutrinos come in from outer space and tell us about solar and stellar activity. However, ANITA detected two events, so‑called ‘anomalies,’ that the scientists can’t yet explain and that’s where things went nuts.

“Almost as soon as the ANITA team sent out word of the anomalies, over three dozen papers were published with hypotheses to account for them. One paper said maybe the anomalies could be interpreted as a clue to one of Cosmology’s long‑standing questions — why aren’t there as many antiprotons as protons? A whole gang of hypotheses suggest ways that maybe something in the Big Bang directed protons into our Universe and antiprotons into a mirror universe just like ours except charges and spacetime are inverted with time running backwards. There’s a tall stack of maybes in there but the New York Post and its pop‑sci allies went straight for the Bizarro parallel universe conclusion. Me, I’m waiting for more data.”

~~ Rich Olcott

A Little Summertime Monkey Business

Surely you’ve heard of The Infinite Monkey Theorem.  You probably don’t believe it.  No way could that monkey accidentally type out anything meaningful, much less the complete works of Shakespeare.  Well…

Home libraryIn several of his Discworld books, author Terry Pratchett featured something called Library-space, L-space for short.  It’s defined as “a dimension that connects every library and book depository in the universe.  L-Space is portrayed as a natural outgrowth of the fact that knowledge = power = energy = matter = mass and mass warps space, and therefore, libraries in the Discworld universe are a very dangerous place indeed for the unprepared”.

Somewhere, Pratchett wrote that L-space contains all the books that have been written, all those that will be written, and all those that would have been written but the author thought better of it.  Well, how big is L-space?

To over-estimate, suppose L-space contains a billion (109) books, each book is 500 pages long, each page contains 4000 characters, and the characters are chosen from an “alphabet” of 500 marks (upper- and lower-case letters, numbers and punctuation marks, all in normal, bold and italic forms in a several different fonts).  One book would then contain two million marks.

Now, how many possible books are there, including ‘impossible’ character combinations like “zqzqzqzq”?  We can construct a “possible” book by choosing some random one of the 500 marks as the first character, the same or a different one as the second character (500×500 = 5002 = 250,000 possibilities so far) and so on, until we’ve built (or our monkey has typed) a two-million-character book.  It could be a book that contains nothing but a string of a million copies of “zq” — but that’s OK, it’s still a possible book.  So is the book that contains all the works of Shakespeare and so is a typo version that inconsistently misspells “Romeo.”

On this basis there are some 5002,000,000 = 105,397,940 different possible books.  L-space with only a billion books is thus very small indeed compared to the number of possible books.  Put another way, the set of all possible books (which we can call B-space) could hold 105,397,931 versions of the L-space that initially seemed so immense.

Note that there are two distinct operations involved in the Monkey Theorem’s process

  1. Generate a string of characters, and
  2. Identify a meaningful substring within that.

The monkey* has no clue what it’s typing.  Any given random string might or might not be intelligible to someone who reads English, or German, or Cherokee.  The string might be a computer program in FORTRAN or JavaScript, or maybe a sequence of DNA icons for a gene mutation that makes green hair — or it (probably) would have no valid interpretation in any context.

The monkey doesn’t care, it’s just typing.

In the  second step of the process someone has to recognize Macbeth or The Tempest buried in all the nonsense.  If we were walking through the stacks of B-space and pulled a book off the shelf, what are the odds that the book we grabbed belongs to L-space?

The answer is one in 105,397,931.  That’s a very small probability, BUT IT’S NOT ZERO.  By construction, we’re guaranteed that all the L-space books are in B-space – but we have a vanishingly small chance of finding one of them.

Now for our extremely patient monkey who has been typing for a really, really long time.  It’s been at it long enough to produce many, many copies of B-space.  After all, even 105,397,940 is a very small number compared to infinity.

The core of the Infinite Monkey Theorem is that with so much opportunity for duplication, we are guaranteed that there exists at least one complete and perfect copy of B-space and so at least one good copy of L-space and so at least one good copy of all the works of Shakespeare.  Also there’s at least one copy of “zqzqzqzq”.

The challenge is in laying hands on that one good copy.  From a physicist’s perspective, it’s such a low-probability event that it can be ignored.  On the other hand, the probability of Life arising on Earth was pretty low, too, but I’m glad it happened.

~~ Rich Olcott

* – I had a great “Monkeys typing” graphic, but they were chimps.  Pratchett’s Diskworld Librarian would object, quite firmly, because apes aren’t monkeys.

** – I also had a pretty good “feature image,” a collage of many different monkey faces, but it seems at least one of them has a copyright lawyer.  Now the feature image is a picture of my library prior to the down-sizing.