Today I want to tell you why some scientists
believe that our universe is really a 3-dimensional
projection of a 2-dimensional space.
They call it the “holographic principle”
and the key idea is this.
Usually, the number of different things you
can imagine happening inside a part of space
increases with the volume.
Think of a bag of particles.
The larger the bag, the more particles, and
the more details you need to describe what
the particles do.
These details that you need to describe what
happens are what physicists call the “degrees
of freedom,” and the number of these degrees
of freedom is proportional to the number of
particles, which is proportional to the volume.
At least that’s how it normally works.
The holographic principle now says
that you can describe what happens inside
the bag by encoding it on the surface of that
bag, at the same resolution.
This may not sounds all that remarkable, but
it is.
Here is why.
Take a cube that’s made of smaller cubes,
each of which is either black or white.
You can think of each small cube as a single
bit of information.
How much information is in the large cube?
Well, that’s the number of the smaller cubes,
so 3 cube in this example.
Or, if you divide every side of the large
cube into N pieces instead of three, that’s
N cube.
But if you instead count the surface elements
of the cube, at the same resolution, you have
only 6 x N square.
This means that for large N, there are many
more volume bits than surface bits at the
same resolution.
The holographic principle now says that even
though there are so many fewer surface bits,
the surface bits are sufficient to describe
everything that happens in the volume.
This does not mean that the surface bits correspond
to certain regions of volume, it’s somewhat
more complicated.
It means instead that the surface bits describe certain correlations between the pieces of volume.
So if you think again of the particles in
the bag, these will not move entirely independently.
And that’s what is called the holographic
principle, that really you can encode the
events inside any volume on the surface of
the volume, at the same resolution.
But, you may say, how come we never notice
that particles in a bag are somehow constrained
in their freedom?
Good question.
The reason is that the stuff that we deal
with in every-day life, say, that bag of particles,
doesn’t remotely make use of the theoretically
available degrees of freedom.
Our present observations only test situations
well below the limit that the holographic
principle says should exist.
The limit from the holographic principle really
only matters if the degrees of freedom are
strongly compressed, as is the case, for example,
for stuff that collapses to a black hole.
Indeed, the physics of black holes is one
of the most important clues that physicists
have for the holographic principle.
That’s because we know that black holes
have an entropy that is proportional to the
area of the black hole horizon, not to its
volume.
That’s the important part: black hole entropy
is proportional to the area, not to the volume of the black hole.
Now, in thermodynamics entropy counts the
number of different microscopic configurations
that have the same macroscopic appearance.
So, the entropy basically counts how much
information you could stuff into a macroscopic
thing if you kept track of the microscopic
details.
Therefore, the area-scaling of the black hole
entropy tells you that the information content
of black holes is bounded by a quantity which is proportional
to the horizon area. This relation is the origin of
the holographic principle.
The other important clue for the holographic
principle comes from string theory.
That’s because string theorists like to
apply their mathematical methods in a space-time
with a negative cosmological constant, which
is called an Anti-de Sitter space.
Most of them believe, though it has strictly
speaking never been proved, that gravity in
an Anti-de Sitter space can be described by
a different theory that is entirely located
on the boundary of that space.
And while this idea came from string theory,
one does not actually need the strings for
this relation between the volume and the surface
to work.
More concretely, it uses a limit in which
the effects of the strings no longer appear.
So the holographic principle seems to be more
general than string theory.
I have to add though that we do not live in
an Anti-de Sitter space because, for all we
currently know, the cosmological constant
in our universe is positive.
Therefore it’s unclear how much the volume-surface
relation in Anti-De Sitter space tells us
about the real world.
And for what the black hole entropy is concerned,
the mathematics we currently have does not
actually tell us that it counts the information
that one can stuff into a black hole.
It may instead only count the information
that one loses by disconnecting the inside
and outside of the black hole.
This is called the “entanglement entropy”.
It scales with the surface for many systems
other than black holes and there is nothing
particularly holographic about it.
Whether or not you buy the motivations for
the holographic principle, you may want to
know whether we can test it.
The answer is definitely maybe.
Earlier this year, Erik Verlinde and Kathryn
Zurek proposed that we try to test the holographic
principle using gravitational wave interferometers.
The idea is that if the universe is holographic,
then the fluctuations in the two orthogonal
directions that the interferometer arms extend
into would be more strongly correlated than
one normally expects.
However, not everyone agrees that the particular
realization of holography which Verlinde and
Zurek use is the correct one.
Personally I think that the motivations for
the holographic principle are not particularly
strong and in any case we’ll not be able
to test this hypothesis in the coming centuries.
Therefore writing papers about it is a waste
of time.
But it’s an interesting idea and at least
you now know what physicists are talking about
when they say the universe is a hologram.
