If the universe goes on forever, does that
mean there are infinite versions of you? And does that mean that one of them didn't do that really embarrassing thing that one time? The cosmological equations that so beautifully
describe our universe make an uncomfortable prediction: interpreting them in the most
straightforward way, they tell us that the universe may be infinite. Or not; it could turn out that the universe
contains enough matter and energy to close in on itself and be finite, or perhaps the
simplest interpretation of the cosmological equations is too simple. But according to our best theoretical understanding,
an infinite universe seems at least possible - and some would say likely. If so this raises an even more crazy possibility. An infinite universe may literally contain
every possible thing allowable by the laws of physics - each in infinite multitude. And that includes infinite versions of you. Today I’m going to try to convince THIS
version of you that all those other versions are also real. To start with we need though we need some monkeys. In fact, we need infinite monkeys. You may have heard the old thought experiment
- if an infinite number of monkeys tap randomly on an infinite number of typewriters, eventually
one of them will accidentally type out the complete works of Shakespeare. This proposition is the infinite monkey theorem. The reasoning is simple enough: assume each
monkey is tapping at all keys randomly. There’s a chance that a given monkey will
tap the first character of the first play: a capital “A”. Let’s say there’s around a one in 100
chance on a typical typewriter assuming capital "A" is as likely as small "a." Then the second character - a space - 1 in 100 again and so on. “A tempestuous noise of thunder and lightning heard...” the chance
of that is 1/100^50 or 10^-100 - and that’s 100 billion billion times less likely than
winning a raffle draw in which there are as many raffle tickets as there are particles
in the universe. Probably that monkey will mess up on the very
next character ... but given enough monkeys one will eventually get through the Tempest,
move on to the Two Gentlemen of Verona, Merry Wives, etc, with 99 out of a hundred monkeys
dropping out at each subsequent character. And 99 out of every hundred monkeys that make
it to the last character of the last play will mess up there ... stupid monkeys. But one of those monkeys will correctly complete
Cymbeline and all of Shakespeare’s First Folio with a probability of 1/100^3.75 million. That’s right, I counted all the characters
in all Shakespeare's plays. Don’t say I never do anything for you. This is an unthinkably small probability. But it’s not zero probability, and the Infinite
Monkey Theorem tells us that any non-zero probability event will definitely happen given
a sufficiently large number of trials. Perhaps you can start to see how this applies
to there being infinite yous in an infinite universe. In a perfectly deterministic universe, the
starting conditions in any given region - like positions, velocities, etc of all particles
- perfectly determines the future history of any point with that region. That is, until the disrupting influence of
external regions has time to reach that point. The properties of each region are effectively
random - set in the beginning of the universe by quantum processes. If they’re set just right then the future
history of that region will be identical to our part of the universe, leading to the formation
of the Milky Way, the Earth, William Shakespeare, and you. The probability of getting every particle
just right is unthinkably smaller than the already unthinkably small probability of a
monkey typing out Shakespeare. But it’s not a zero probability, and so
with infinite regions of the universe - infinite trials, the infinite monkey theorem tells
us it’s got to happen somewhere. In fact, it’s got to happen infinite times
and with infinite variations. So infinite yous and infinite versions of
you. That is, IF our assumptions are right. To decide if this is really the right conclusion
there are a few things we need to get straight. First up, in such an infinite universe we
wouldn’t find everything - only everything that COULD happen from the range of initial
conditions developing under the laws of physics. Infinite monkeys with standard English typewriters
will eventually reproduce all of Shakespeare and every Harry Potter, but lacking an inverted
question mark could never do the original spanish version of Borge’s Library of Babel. Similarly, there’s no version of you out
there where you have Captain Marvel superpowers. Though I guess there could be one where you’re
Batman. Or NOT Batman, if, you know, you’re already
Batman in this one. OK, so refining our question: in an infinite
universe, does every POSSIBLE thing happen infinite times? Well still, not necessarily. If there are infinite possible starting configurations
for any one region of the universe, then there can be infinite regions without any doubling
up. But if there are finite starting points then
at least SOME of those starting configurations have to be repeated infinite times. Technically we could imagine that OUR part
of the universe is unique and it’s just other regions that are duplicated, but we
have no good reason to assign that sort of specialness to our region. So if there really are finite possible starting
conditions then this region is probably repeated. So what does it mean for two regions of the
universe to have the same starting conditions? It means that every particle, or chunk of
quantum field, or whatever elementary pixel of reality - has matching properties between
the two regions. How closely matching would they need to be? Close enough that the subsequent 13 and a
half billion years of bouncing around leads to indistinguishable results - and to you
existing now. In such a chaotic system, even tiny differences
in the starting conditions will lead to massive divergences in that future history. But in principle there is an allowable level
of ridiculously tiny deviation in the starting conditions between two regions that would
still lead near-identical 13.5 billion year histories. We can’t tune the starting conditions to
an infinite degree and still get different results - and that’s the key point. It seems there must be finite possible starting
configurations. That means at least some regions must be repeated
infinitely. Therefore you’re Batman. I should also note that it’s not just the
particles that define starting conditions, there’s also the laws of physics themselves. The constants of nature or the energy of the
vacuum may well be different between regions. But in most cases we expect these values to
repeat themselves infinite times in an infinite universe - eventually leading to an exact-enough
repetition of both the laws of physics AND the arrangement of particles. Repetitions of the initial conditions are
inevitable as long as the properties defining those conditions can take on a finite range
of values. If one or more properties of the universe
can take on values over an infinite range then no repetitions would be necessary. For things like the mass or charge or other
properties of individual particles, an infinite range means going to very, very high values
for these properties - and that tends to be prohibited by a combination of conservation
laws and the formation of black holes. It IS technically possible for the energy
of the vacuum itself to have any value, even enormously large. But universes with large vacuum energies will
exponentially expand, until that vacuum energy decays to lower values in smaller regions
within. In fact that’s essentially the eternal inflation
picture - one of the most popular ways to produce infinite universes. I simplified this argument a bit, but elaborating
doesn’t change the conclusion. For example, you could argue that fundamental
quantum randomness will cause even identical starting configurations to produce different
results. But that still doesn’t give us an infinite
number of distinguishable histories because the number of possible configurations of particles
at every instant is still finite. In fact, quantum randomness could allow different
starting conditions to evolve into a universe that looks like this one. There’s actually a much simpler proof that
there’s a finite number of different possible configurations of any region of the universe. Jacob Bekenstein and Steven Hawking showed
that the maximum amount of entropy in any region of space is proportional to the number
of tiny Planck areas covering its surface. We can interpret that Bekenstein bound as
the maximum number bits of hidden information. And 2 to the power of which is the number of
different possible configurations for that volume. For our 46-billion light year observable universe there are definitely no more than 2^10^123 possible unique configurations. As long as the greater universe has more than
than that many observable universe-sized regions, one of them should be identical to this one. And actually, the Bekenstein bound is the
hard upper limit - for non-black holes the number of possible configurations is much,
much lower. OK, so unless there’s something weird hiding
in the laws of physics that we don’t understand, an infinite universe probably does duplicates
its parts infinite times. We covered most of the important assumptions
in this crazy hypothesis - except the main one. That the universe is actually infinite at
all. Now this may never be testable - we can measure
the observable part of the universe to greater and greater precision and see if it has an
open or closed geometry... but even then we still have to hope that it’s reasonable to extrapolate
our cosmological equations forever. On the other hand, if we somehow verify the
eternal inflation picture then it’s very hard to avoid an infinite universe, or at
least one that’s infinite-enough to duplicate itself. On the other-other hand, many good physicists are
very uncomfortable allowing infinities into any models describing the physical universe
- and their intuition shouldn’t be entirely discounted. But let’s get back to the real question:
in an infinite universe, is it inevitable that somewhere a monkey will type out Shakespeare? Well actually, the experiment has been done - researchers at the University of Plymouth gave six Celebes crested macaques a computer for one month. Turns out monkeys don’t hit all keys randomly. In fact mostly these ones hit the letter S
and pooped on the machine. But who knows - perhaps there’s a near duplicate
region of the universe where Shakespeare’s plays are just long strings of S’s underscored
with fecal smears - and where the infinite monkey theorem has been experimentally verified. As for whether there are infinite versions
and variations of you. Honestly, I was dismissive of the idea before
I started writing this script. Now I think it’s hard to avoid that conclusion
... assuming an infinite universe. And at risk of over quoting whatshisname: I could
be bounded in a nutshell, and count my self a King of infinite spacetime. Today we're catching up on comments for the last
two weeks - first up we got the three-body problem and then we'll do our episode on quantum
hacking with the s-matrix theory. Ilavenya rightly points out that no nice analytical
solutions to the 3-body problem exist in general relativity - Einstein's modern theory of gravity. The solutions only exist in special cases for Newtonian
gravity. And in fact it's worse than that - forget
three bodies, general relativity gives no perfect analytical solutions for TWO bodies
in orbit around each other. That's because objects in orbit are all moving,
so the curvature of the background spacetime is constantly changing. It's hard to write down a simple equation
of motion when the coordinate system is changing. That said, approximate solutions work very
well in some cases - for example if one of the masses is much heavier than the other
then the Schwarzschild solution gives simple equations for orbits which work pretty well
as long as the bodies aren't too close together. Jordan Miller suggests that the Euler and
Lagrange solutions to the three body problem are just 2 body solutions in disguise. So to remind you - Euler and Lagrange came up
with a set of 3-body configurations that are now known as the Lagrange points. In any 2-body system, there are these 5 places
that a third body can be placed and will stay in stable orbit. If the 3rd body has a low mass relative to
the others then the original 2-body system is undisturbed, and in that case Jordan is correct. It's a two body solution. But actually, if you're willing to adjust
the original 2-body system then a 3rd body of any mass can be added - either with 3 bodies
in a permanent straight line or in an equalateral triangle. These would no longer be called "Lagrange
points", but they are true stable or metastable 3-body orbits. RobTheImpure would like to know what is meant
when we talk about "spaceless timeless particle scattering" in the context of the s-matrix. This is indeed a weird idea. How can particles scatter off each other without
some notion of cause and effect or location? Well to start with, there is a causal order for
the incoming and outgoing particles - the former cause the latter, and so they must come first. But in s-matrix theory and quantum field
theory, time and space in the interaction region are fuzzy. I'll say more on this in followup episodes,
but to give you a simple example. Imagine an interaction where an electron emits
a virtual photon which then deflects another particles - say, a proton. That's exactly the same interaction as if
the proton emitted the photon to deflect the electron - in other words, the direction of
the flow of time is irrelevant from the perspective of the virtual photon - what matters is the
structure of the interaction. Another example is the s-channel - t-channel
duality, in which you get identical scattering results when space and time switch places. And on that note I do have a couple of corrections
from the last two episodes.I got the labeling of the t and s channels backwards in the S-Matrix episode. The t channel is where 2 particles scatter
off each other by exchanging a virtual particle, while the s-channel is where the particles
annihilate each other into a virtual particle, which then creates 2 new particles. I said them the other way around. Also I mispronounced the name Newton's great
work - it's PrinKipia - hard K, not soft principia. Don't you hate it when you read a word a million
times before you hearing it said out loud, but by then it's stuck in your head that way
no matter how many times you get corrected you still say it that way. Anyway, usually I blame mispronunciations
on my dialect - as in "oh, yeah, that's how we say it in Australia". I don't think that works in this case. And finally, there was some interest in our
depiction of the proton being made of lego bricks as a viable theory - quantum lego dynamics,
as Steve Plegge puts it. VioletteOrdinaire prefers this offering from
Denmark it's much more straightforward than the other Copenhagen interpretation.
This view of Infinity as anything that has a non-zero probability has happened or will happen seems more accessible to my monkey brain.
But the universe can't be infinite.
The universe wasn't infinitely large at the time of the Big Bang, and It's been getting bigger ever since. Between the Big Bang and now, at no point could it suddenly become infinitely large.
It's like starting at zero and trying to count to infinity. You'd need an infinite amount of time to do it, and so far we've "only" had 13.8 billion years
Interesting hypothesis and well explained basics of chaos theory. My problem is that all of this assumes that universe is predeterministic to the point that starting configuration of our solar system (and the billions of light years of our immediate universe) already dictated whether Trump gets elected a second term and subsequently the end of human civilization ALREADY BILLIONS OF YEARS AGO. Down to the last sequence of letters in this comment, which are: KSCAWDRAWWMONKEYWRITING##FDE
Like many others, Matt jumps from the idea that the world splits for every quantum "measurement-like" interaction to the idea that it splits for every decision someone makes. This would be true only if every decision involved a measurement-like interaction. Clearly, there is lots of decoherence going on when someone makes a decision, but that doesn't mean that the outcome of the decision is affected. Most of our decisions are for reasons, and those reasons don't change due to quantum jiggle. (Even if the jiggle causes a hurricane in some worlds and a light breeze in others a week later.)