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can you fit all of the information in
the universe into a region smaller than
the universe
let's find out
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there's quite a bit of stuff in the
universe to put it mildly hundreds of
billions of galaxies each with hundreds
of billions of stars each with
rather a lot of particles in them and
then there's all the stuff that isn't
stars the dark matter black holes
planets and the particles and radiation
in between the stars and galaxies not to
mention space itself with its
fluctuating quantum fields dark energy
blah blah stuff everywhere
but is the universe actually made of
stuff an increasing number of physicists
view the universe view reality as
informational at its most fundamental
level and its evolution through time can
be thought of as a computation
and then there's the simulation
hypothesis in which that computation is
engineered by
who knows what or who
how big a memory bank would you even
need to compute a universe
seriously let's figure it out how much
information does it take to describe the
entire observable universe after we're
done with that i'm going to have an even
cooler challenge question for you
i casually mentioned in the last episode
that our 3d universe may just be a
projection of information imprinted on
its two-dimensional boundary no biggie
that's the holographic principle and
we've talked a lot about some ideas
leading up to it don't worry the full
holographic principle episode is still
coming
but our recent episode on black hole
entropy and some of the lead-ups to that
might be helpful here you can also watch
this video as a standalone and go back
to those earlier ones later if you feel
like it but the main point the really
weird surprising point is that the
maximum amount of information that can
fit in a volume of space is not
proportional to that volume is
proportional to the surface area of that
region of space
jacob beckenstein figured this out by
realizing that the entropy of a black
hole is proportional to the surface area
of its event horizon
but entropy is just a measure of hidden
information so the bekenstein bound is
equally a limit on how much information
you can fit into any region of space
we'll come back to the beckenstein bound
in a sec but for now let's think about
why this dependence on surface area is
surprising well common sense would
suggest that the maximum information
content depends on volume not surface
area i mean a pile of thumb drives has a
total storage capacity that depends on
its volume but instead of storage
capacity let's think about the
information needed to perfectly describe
a patch of space
you think that to fully describe say the
universe you need to know what's going
on in every tiniest possible 3d chunk
that smallest element is roughly a cube
one planck length on a side where the
plunk length is the smallest meaningful
measure of distance at around 1.6 times
10 to the negative 35 meters
it's the smallest possible chunk of
space so let's say it can contain the
smallest amount of information one bit
per plunk volume
that's kind of like saying we can
describe the universe completely if we
go through all of its quantum voxels and
answer the yes no question of whether
it's full or empty
this probably way underestimates how
much info you really need to describe
the universe but let's start with this
anyway
so how many planck volumes are there in
the universe well the radius of the
universe is something like 47 billion
light years which is a few times 10 to
the power of 61 plunk lengths
4 on 3 pi r cubed so the universe
contains 10 to the power of 183
plunk volumes
you'll see estimates that the radius of
the universe is a mere 10 to the 60
plunk lengths rather than 10 to the 61
and that its volume is 10 to the power
of 180 units and that's because
cosmologists tend to round down at every
step and so drop a bunch of orders of
magnitude astrophysicists are by
comparison highly accurate we only drop
factors of 2 but what's a few orders of
magnitude between friends
you might argue that more than one bit
can fit at each grid point in the
universe
you might be right particles have more
information than just their position it
will be better to use the number of grid
points in quantum phase space which
includes position but also other degrees
of freedom like momentum spin direction
etc in other words we should count all
possible quantum states in the universe
anyway the real number is going to be
way higher than our estimate of 10 to
the 180 but as we'll see we're already
way way higher than the actual
information limit of the universe
okay so 10 to the power of 180 or so
bits is the minimum if you want to
describe every 3d quantum voxel
completely independently
but the beckenstein bound tells us that
the information content of any volume
isn't the number of these plunk volumes
but rather the number of plunk areas on
the surface
the observable universe has a surface
area of 10 to the power of 120 to 10 to
the power of 124 plunk areas depending
on whether you're rounding like a
cosmologist or an astrophysicist so the
storage capacity of the universe is
around 10 to the 60 lower than the
number of volume elements it contains
so how do you encode a whole universe in
a space far smaller than the universe
itself
there must be some amazing compression
algorithm
maybe middle out
in fact in a sense the holographic
principle is a compression algorithm see
you don't really need one bit per volume
element of the universe they'll be like
having separate completely independent
memory elements for every empty pixel in
an image file
most of space is indeed empty let's say
instead you only need a single bit of
information for every element in phase
space that's occupied in other words one
bit per elementary particle the
observable universe contains something
like 10 to the power of 80 protons each
proton has three quarks and there are a
similar number of electrons most other
particles are much rarer so we're still
in the realm of 10 to the 80 to 10 to
the 81.
neutrinos and photons formed in the big
bang are probably a billion times more
abundant than protons
that's verified experimentally in the
case of photons the cosmic microwave
background has around 10 to the power of
89 photons across the observable
universe so almost all of the
information and for that matter the
entropy in particles is in neutrinos and
in the cosmic microwave background
photons
the situation with dark matter is
unclear so let's just round up to 10 to
the power of 90 bits of information in
particles in our universe that sounds
like a lot but happily it's way less
than the 10 to the power of 120 ish
limit of the beckenstein bound but
there's one more source of information
black holes as i mentioned last time
black holes contain most of the entropy
in the universe the relationship between
black hole entropy and information
deserves some thought
black hole entropy in terms of number of
bits tells you the information you'd
need to describe all possible initial
states that could have possibly formed
the same black hole
of which there are many
two to the power of the number of bits
of entropy
and for black holes that entropy is the
bekenstein bound the number of plank
areas on its event horizon because the
information about the black hole's
previous state is lost to fully describe
it you need to fully describe its event
horizon you need its full beckenstein
bound in information
how much information is that
let's take a super massive black hole as
an example
sagittarius a star in the center of the
milky way which has a mass of four
million suns its event horizon is around
12 billion meters giving it a surface
area of 10 to the power of 90 to 10 to
the power of 91 plunk units
so the milky way's black hole has as
much entropy and hidden information
as all of the matter and radiation in
the entire rest of the universe
and there are some hundreds of billions
of galaxies in the universe each with
its own supermassive black hole we're
talking something like 10 to the power
of 101 to 10 to the power of 102 bits of
entropy or information black holes
contain by far most of the entropy in
the universe and require most
information to fully describe
but again we're still below the
bekenstein bound for the whole universe
i know you're as relieved as i am we are
nowhere near the universe's memory limit
the universe can keep having particles
and you can leave your horribly bloated
email inbox alone
but what would actually happen if the
universe contained too much information
say in the form of too many particles
what if we started to fill up those
empty plunk size cubes of space
throughout the universe until it
contained more information than the
bekenstein bound allowed
well the answer is straightforward
enough at the moment the universe
reached its informational limit it would
immediately become a black hole with an
event horizon as big as the current
cosmic horizon
it would be the end of space time
the beckenstein bound does apply equally
to engineered information storage as it
does to black holes and universes
it gives the limit of storage capacity
within a given volume before the
interior collapses into a black hole
that suggests a nice challenge question
so answer me this
imagine a super advanced civilization
built a computer with memory storage
that operates at the beckenstein bound
essentially they're computing on the
surface of a black hole event horizon
how large a black hole computer would
you need in mass and radius to contain
enough data to simulate the entire
observable universe
let's ignore really high entropy stuff
like black holes the cosmic background
radiation and neutrinos also ignore dark
matter just regular matter like protons
electrons etc assume one bit per
elementary particle and i also have an
extra credit question
go and read the paper computational
capacity of the universe by seth lloyd
link in the description and answered
this
how long would that black hole computer
take to simulate the entire universe
you need to make a bunch of assumptions
and there isn't a single perfect answer
but i'm really curious to see what you
come up with
for both questions i'll post detailed
wording and perhaps some hints on
twitter and on my own instagram links in
the description
by the way the latter is very new and
i'm using it among other things to give
you a bit of a behind the scenes on
spacetime stuff especially graphics as
well as the extra details on challenge
questions
submit your answers to the challenge
question within two weeks of release of
this episode email them to pbsspacetime
gmail.com and use the subject line
simulated universe challenge
use exactly this subject line and check
your spelling because we filter by
subject line please make sure your
answers are neat and show complete
working
and use full english sentences to
explain your thinking you may know that
you're a genius but it's important to be
able to communicate that fact to others
okay assuming no one switches off the
simulation i'll see you next week for a
new episode of spacetime
so we're a bit behind on comment
responses due to my travels we're going
to catch up on responses to the life on
mars and the end of the universe
episodes today and next week we'll get
to responses to black hole entropy as
well as today's episode starting with
our episode on the history of life on
mars stushdu says cool the show is
current enough to acknowledge the recent
discovery of liquid water on mars that
is
yeah for a show that's about a 13.8
billion year old universe we do try to
keep it on human time scales at least
most of the time in fact this episode
was even more up to date than we'd
planned we'd already filmed it including
the bit about the underground lake when
the giant dust storm hit mars and the
opportunity rover went silent
we re-recorded and tried to pretend that
we intended it all along to be a homage
to opportunity
and speaking of opportunity this is
probably a good opportunity to update
you further about opportunity
uh no word there yet i'm afraid
opportunity is still silent the dust
storm is nearly over and nasa will keep
trying to make contact at least for
several weeks
patrick dunn offered a very reasonable
clarification of a point that i rushed
over regarding the viking landers
labelled release experiment which had an
initial positive result for possible
biotic activity which then couldn't be
replicated patrick points out that what
i should have said was that they
couldn't induce a second peak of gas
release from the same samples the key is
that they were the same samples opening
the possibility that anything living in
that soil could have been harmed by the
disturbance caused by the experiment
resulting in the second null result not
saying there was life i have to look
deeper into that but this highlights the
danger in simple interpretation of
second-hand reports i.e you gotta read
the original paper and now on to our
episode on the end of the universe
spencer tweedy is amused at my use of
the word prompt for the time scale of
evaporation by hawking radiation
yeah that's the weird thing about time
scales of the end of the universe
they're so ridiculously long that even
other ridiculously long time scales look
short in comparison
orders of magnitude are wacky
but scientific intuition is served very
well if you can get your head around
comparative orders of magnitude
particularly in astrophysics it's really
useful to be able to ignore
insignificant amounts of space or time
or energy in the presence of much larger
quantities regarding the fact that the
hypothetical proton decay time is 10 to
the power of 40 years vinay k asks if a
proton created in say 10 to the power of
36 years from now would still decay in
10 to the power of 40 years or would it
decay in 10 to the power of 40 years
after its birth well
neither actually that 10 to the 40 years
is a crude estimate of the proton's
half-life assuming it decays at all
which it might not
half-lives are more commonly used for
radioactive elements but can be used for
anything that has a constant probability
of decaying in a given amount of time so
the half-life is the amount of time it
takes for something to have a 50 chance
of decaying at some point during that
time
the actual amount of time it takes to
decay is random and there's a constant
probability of it happening at every set
interval so if these decays happen at
all then after 10 to the power of 40
years every proton in the universe will
have had a 50 chance of decaying
that means around 50 of them will have
decayed and many of those will have
decayed much sooner than 10 to the 40
years
after another 10 to 40 years 50 of the
remaining protons will have decayed and
so on that halving of the number of
protons will happen 10 times by 10 to
the power of 41 years so the fraction of
protons left will be 0.5 to the power of
10 or around 1 1000th
to reliably eliminate all of the 10 to
the power of 80 protons in the
observable universe you need around 265
half-lives or 10 to the power of 42 to
43 years marco dala gasparina points out
that the nothing can be defeated with
the help of the luck dragon
is that a metaphor for quantum
fluctuations in impossibly distant
futures spontaneously generating a new
big bang by pure chance
or do you mean that while we're waiting
through the infinite future for a new
big bang quantum fluctuations will
produce all sorts of weird things first
like boltzmann brains and falcore
valid point maybe our tax will come back
too
i hope so
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you