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.