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This episode is supported by The Great Courses Plus.
At the event horizon of the black hole,
space and time are fundamentally changed.
Even professional physicists disagree on what
we expect to happen there.
But there is a powerful tool in physics
that can give us real intuition into the true nature
of the event horizon.
Its time you learned it.
Black holes, objects with densities
so high that there's this region, the event
horizon, where the escape velocity reaches
the speed of light.
Nothing that falls below the event horizon
can never escape and is lost to the universe
forever while we see falling objects freeze as time
stands still at the horizon.
And anything that happens below the event horizon
stays below the event horizon.
That's the official sanitized public version.
It's not entirely inaccurate, but the reality
is, of course, a good deal more complex and interesting.
Even ignoring the complications of Hawking radiation
or black hole rotational growth, the simplest black hole
of Einstein's general theory of relativity-- purely
gravitational, static, and eternal--
is a subtle and misunderstood beast.
But we can come to a powerful and intuitive understanding
of the beast.
Today I want to teach you how to use the same tool
that physicists use.
It's a tool that will let us easily
and so the most common questions about black holes.
For example, are objects falling through the event horizon
really physically frozen there from the point of view
of the outside universe?
Would you see the entire future history
of the universe playing fast forward at the instant
that you crossed the event horizon?
And do you see anything at all once you're
inside the black hole?
The tool that will answer these questions
is called the Penrose diagram, sometimes also
the Carter Penrose diagram.
It's a special type of space-time diagram designed
to clarify the nature of horizons.
But first, a quick refresher on basic space-time diagrams.
By graphing time versus just one dimension in space,
we can look at the limits of our access
to the universe due to its absolute speed
limit, the speed of light.
With the right choice of space and time units,
the speed of light becomes a diagonal line
on the space-time diagram.
The area encompassed by the so-called light-like paths
defines all future events or space-time locations
that we could potentially travel to or influence constrained
by the cosmic speed limit.
That's our forward light cone.
Our past light cone defines the region
of the past universe that could potentially have influenced us.
Let's drop a black hole onto our space-time diagram.
It lives at x=0 on the space axis,
but exists through all the times on the graph.
It has a point of infinite density, the singularity,
and an event horizon a bit further out.
The mass of the black hole stretches space and time
so that light rays appear to crawl out
of the vicinity of the event horizon
before escaping to flat space-time,
no longer following 45 degree paths.
Now let's throw a monkey into the black hole.
As it approaches the event horizon,
its future light cone bends towards the black hole
as fewer and fewer of its possible trajectories
lead away.
Below the event horizon, all possible trajectories
lead towards the singularity.
The problem with the regular space-time diagram
is that the path of light and the shape of the light cone
changes as space-time becomes warped.
That makes it difficult to figure out
what parts of the past and future universe
the monkey can witness or escape to.
And this is where the Penrose diagram comes in.
It looks like this.
It transforms the regular space-time diagram
to give it two powerful features.
It crunches together, or compactifies,
the grid lines to fit infinite space-time on one graph-- very
useful for black holes.
It also curves the lines of constant time
and constant space in what we call a conformal transformation
so that light always follows a 45 degree path.
That means light cones always have the same orientation
everywhere.
Super handy for understanding monkey trajectories.
This is the Penrose diagram for flat space-time
with no black holes.
Same as with a regular space-time diagram,
blue verticalish lines represent fixed locations
in one dimension of space and red horizontalish lines
are fixed moments in time.
Now, those lines get closer and closer together
towards the edge of the plot to encompass
more and more space-time.
They're extremely finely separated at the edges
so that any tiny stretch on the graph
represents vast distances and/or times.
The lines also converge together towards the corners
so that light travels a 45 degree path everywhere
on the diagram.
So a light ray starting from really, really far away
and coming towards us hugs the edge of the diagram
and crosses an enormous number of time and space steps,
only reaching us in our very distant future.
OK.
Let's drop a black hole into this space-time.
Nice and safely far off to the left.
And because we only have one dimension of space,
and any motion to the left brings
us closer to the black hole.
Its event horizon becomes the end
of the line in that direction.
The future cosmic horizon on the Penrose diagram
is replaced with a plunge into a black hole.
The compactified grid lines there
now represent the stretched space-time
near the event horizon.
An entirely new Penrose region represents the interior
of the black hole.
Weirdly, the lines of constant position and constant time
switch.
Space flows at greater than the speed of light inwards,
towards the central singularity.
It becomes uni-directional, flowing inexorably downwards,
just as time flowed inexorably forward
in the outside universe.
All paths lead to the inevitable singularity.
Once you're beneath the horizon, your future light cone
still represents all possible paths that you could take.
All of them end up at that singularity.
The only way to escape back to the outside universe
would be to widen your light cone
by traveling faster than light.
So you're out of luck.
Now that we've nailed the Penrose diagram,
we can use it to do some serious black hole monkey physics.
Our space-faring simian begins its journey
and emits a regular light signal that we
observe from a safe distance.
As it approaches the black hole, these light rays
have further and further to travel through increasingly
curved space-time and so the interval
between receiving signals also increases.
The progress of the monkey appears
to slow to a halt very close to the event horizon,
and the final signal at the moment of crossing never
reaches us.
It's trying to travel at the speed of light
against light speed cascade of space-time.
With this picture, we can start to answer
some very serious questions.
First, what would happen if the monkey remembered
to fire its jet pack at the last instant
before reaching the event horizon?
Well, it could still escape.
Its future light cone still includes a tiny sliver
of the outside universe.
It had better be a good jet pack because it's
going need to follow a very long near light speed path away.
It will, nonetheless, have experienced far less time
than us when it emerges into flat space-time
in our far future.
Assuming no jet packs, the monkey
is probably doomed to a graceful reverse swan dive
through the event horizon, watching
the entire future history of the universe
play out above it at that last instant.
Yet actually, no.
It doesn't see that at all.
The monkey's last view of the outside universe
is defined by its past light cone
that encompasses all of the light that will catch up to it
and that light is stuck following these diagonal lines
because it has to contend with the same stretched space-time
as the monkey.
There's no future universe spoiler promo.
If it could instead hover above the event horizon,
then it would see the universe in fast forward,
although that view would be compressed into a small circle
directly overhead.
Watching the monkey frozen on the event horizon
is going to make us feel a bit guilty, after a while.
Could we change our minds and launch a daring monkey rescue
mission?
Sadly, no.
Even if we do travel at the speed of light,
after a certain point there's no catching the monkey.
We would see it suspended above the horizon
as we're racing to meet it, but it will always
appear to be just a little further ahead
no matter how close to that horizon we dare to go.
Remember, the monkey isn't actually
above the horizon for infinite time,
it only appears that way to us because as long
as we're outside the event horizon, no times
that we can witness correspond to the monkey crossing
that horizon.
In order to see that crossing, we
would have to cross the event horizon ourselves.
Once inside the black hole, we could potentially
see the monkey below us.
All space-time within the black hole
is flowing toward the singularity faster
than the speed of light.
The two neighboring radial layers
aren't traveling faster than light relative to each other.
That means that the monkey's signal can still reach us,
although it might be more accurate to say
that we catch up to the monkey's outgoing signal.
But even that so-called outgoing light is still
moving downwards, doomed to hit the singularity along
with the monkey and our rescue mission.
All of this describes a non-rotating, uncharged black
hole, a Schwarzschild black hole.
Even this simple case is a good deal more complicated
than I let on.
For example, I only showed you half of the Penrose diagram.
The complete mathematical solution
for a Schwarzschild black hole has two additional regions,
one corresponding to a parallel universe
on the other side of untraversable wormhole,
the Einstein-Rosen Bridge.
And down here we have what we call a white hole.
These are strange mathematical entities
and probably aren't real, but we'll certainly
come back to them.
We'll also come back to what happens
if we set the black hole spinning
or add some electric charge.
Then our Penrose diagram blooms outwards
to include potentially infinite parallel regions of space-time.
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Hey, guys.
A couple of quick shout-outs.
First, a huge thank you to our first Patreon supporter
at the Big Bang level.
Antonio Park, you rock and I'm really looking forward
to our hangouts.
Your support is going to make a big difference, as is
the support at all levels.
So thanks again to everyone who's contributed on Patreon.
And a quick announcement of a couple of events
I'll be at early next year.
First, South by Southwest, March 10 to 19 in Austin, Texas.
A few months ago I asked you guys
to head to the South by Southwest panel picker
to vote for myself, Astrophysicist Katie Mack,
and It's OK to be Smart's Joe Hanson to be
picked for a panel titled, "We are All Scientists."
Well, thanks to you guys, we're scheduled.
We're going to talk about the value of and the threat
to critical thinking and scientific reasoning
and why these skills need to be seen
as accessible and important to everyone
and how we can act on this idea.
Also, for those of you thinking of attending grad school
in physics, I'll be talking about studying physics
at a professional level at the City University of New York's
Student Research Day, April 7 at the CUNY Graduate
Center in New York City.
More details closer to then, but for now I'll
put a link in the description and feel free to reach out
to me if you're interested.
OK.
Onto the comments from last week's episode
on De Broglie-Bohm theory.
Wow.
This is the closest I've seen a YouTube comment section come
to looking like a Q&A session after a professional physics
seminar.
Now, a lot of you wondered why I never mentioned the EM drive
when talking about pilot wave theory.
The answer is simple, there was nothing
useful to say on that connection.
In the recent paper out of Eagle Works Labs,
Herald White and collaborators present some results
on the thrust produced by their EM drive
and then go on to talk about how pilot wave theory might
explain the apparent conservation
of momentum-breaking results.
I might get into the details in an upcoming episode,
but for the sake of explaining pilot wave theory
this paper isn't relevant.
The connection is extremely speculative,
and honestly I wondered whether pilot wave theory was
chosen partly because the internet happens
to love it at the moment.
DinosaurFromtheFuture asks how it
can be that pilot wave theory predicts different particle
trajectories, given that the particles supposedly all start
at exactly the same point.
Well, the simple answer is that the particles don't start
at exactly the same points.
We just can't know exactly their location at the beginning.
See, pilot wave theory states that the particle riding
the wave does have a definite position at all times
and that position defines its future directory.
So if you know the position perfectly
and you know the wave function, you can perfectly
predict future locations.
However, you can't perfectly measure a particle position
without changing it slightly in ways
that themselves aren't perfectly predictable.
As a result, you never know exactly where a particle is.
This uncertainty leads to the range
of potential future trajectories,
including trajectories through one slit or the other.
More generally, it allows pilot wave theory
to agree with Heisenberg's uncertainty principle.
In the Copenhagen interpretation,
the uncertainty principle describes
the intrinsic randomness of the quantum world.
De Broglie-Bohm pilot wave theory
states that this uncertainty just
arises from our imperfect knowledge
and that the universe itself knows exactly
where all these particles are.
Vacuum Diagrams correctly points out
that to know the future trajectory of a particle,
you only need position, not velocity, as I had stated.
That velocity information is in the guiding wave.
Thanks for the correction and thanks also
for pointing out those extremely interesting papers that
detail certain failings of the pilot wave interpretation.
I'll link those and a couple of others
that take different sides in the description of this video,
as well as in the pilot wave episode.
In fact, there was some really heated and fascinating
discussion both for and against the pilot wave interpretation
and some of it was from people who
know a good deal more than I do, like Vacuum Diagrams.
Something I took from this is that Bohmian mechanics
is, on its own, very unlikely to be the full picture, even
ignoring the whole relativity issue.
That doesn't necessarily mean, though, that it's not useful.
I'll get back to why.
But first, as was pointed out to me in a nice email by physicist
and science writer Adam Becker, I wasn't entirely accurate
when I said that De Broglie, the founder of pilot wave theory,
remained convinced by Niels Bohr and his Copenhagen camp,
even after Bohm's work.
More accurately, De Broglie remained convinced
of the objections raised against his idea,
even after some of them were addressed in Bohmian mechanics.
To quote De Broglie from his 1956 book,
he, Bohm, assumes that the [INAUDIBLE] wave
is a physical reality, even the [INAUDIBLE]
wave in configuration space.
I have already stated why such a hypothesis appeared
absolutely untenable to me.
In fact, De Broglie was never a huge fan
even of his own simplistic particle
carried by a wave idea.
That formulation was a simplified version
of what was to be a much more intricate double solution
theory in which the so-called particle was actually a matter
wave itself embedded in and carried by the sine
wave, represented by the wave function.
He was unable to pull the math together
in time for the fated Solvay Conference,
and so derived the simpler description in which
the particle is point-like.
De Broglie never completed his full double solution theory,
but did work on it intermittently
throughout his life and was inspired
to return to it by Bohm's publication,
even if he didn't buy Bohmian mechanics.
The fact is we just don't know whether the reality that
drives the strange results of quantum experiments
is actually deterministic in the way
that we understand determinism.
But De Broglie-Bohm pilot wave theory
is a great example of how a deterministic theory can
at least go some way towards predicting the results
of quantum experiments.
Personally, I'm agnostic towards the relative truth
behind the Copenhagen, many-worlds,
pilot wave, or the other interpretations of quantum
mechanics.
I like the idea of a deterministic theory,
but the universe has often demonstrated
that it couldn't care less about our pet theories.
However, it's also shown itself to be
vulnerable to experimentation even for questions
that we previously thought untestable.
We'll figure this out, and until then it's
OK to like one theory over another
but belief should wait on the evidence.
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