Thank you to Babel for supporting PBS. Don't panic, but it's possible that an
event horizon is forming right behind you right now. Don't bother looking. You won't see it. It
definitely won't look like the classic picture of the black hole that you're probably familiar
with. You won't even know that it's growing to envelop you until it's way too late.
for the universe's ultimate fate. Now onto the episode. By now, we all have an idea of what black
holes are supposed to look like. Discs of absolute darkness on the sky, surrounded by gravity warped
starfields, and the final electromagnetic screams of stuff about to fall in. We think of the
black hole as this roughly spherical surface of lightlessness. And the name we give that surface
is the event horizon. If I were to ask where is the black hole, you might point to the event
horizon. And that would be fair because a black hole is really defined as something with an event
horizon. The surface of no escape surrounding a region so packed with mass and energy that
even light becomes trapped. The event horizon is as good a wear as anything for the black hole.
And if I were to ask what is the black hole, it seems fair to answer anything inside that horizon.
Well, it turns out these answers are somewhere between inadequate and just plain wrong. Even the
question, when does the horizon exist isn't that simple. Event horizons are actually much harder
to pin down both in space and time. And when it comes to black holes, a little bit of ambiguity
can get you into a lot of trouble. So for safety sake, let's get clear about the event horizon. If
we define the event horizon as that surface from which light cannot escape, we need to ask escape
to where? One pretty reasonable answer would be to require that in order to be considered above
the event horizon, light has to be able to truly escape, which means it must be able to get far
enough away from the black hole that it no longer is affected by its gravity. This is actually close
to the formal definition. Light must be able to reach infinite distance which is physics ees for
very very far away. But this definition is already at odds with how we tend to imagine the event
horizon as the scary black disc or sphere. Let's approach that disc then to investigate. We'll
travel to the very near vicinity of what we think of as an event horizon. Now, after collecting
some Hawking radiation or whatever, we switch on our ridiculously powerful rockets to move away.
Because we carefully stayed above what looks to us like the horizon, we're able to make some headway.
But then, misfortune strikes. A giant asteroid or something falls into the black hole from above
us and the black hole grows in mass and the event horizon expands and we're enveloped. But according
to our earlier definition, if we or if light cannot escape to an extreme distance, then we
were already beneath the event horizon and we're already doomed. Yes, something that black surface
did expand to envelop us, but it wasn't the true event horizon at all. So, let's dig a little
deeper into the formal definition of the event horizon because it completely changes the what,
where, and even the when of our understanding of a black hole. The textbook definition, the one that
Steven Hawking and George Ellis sit down in their book, The Large Scale Structure of Spacetime,
is as follows. An event horizon is the boundary of the causal past of future null infinity. The
causal past of future null infinity. Okay, super helpful. Well, we can break it down maybe a little
further. We'll use a Penrose diagram. Longtime viewers will remember these and it'll make
things a little clearer, hopefully. Before we complicate things by adding black holes, here's
a Penrose diagram of an infinite flat universe. Time increases upwards and one dimension of space
is traced horizontally, but the contours of space and time are curved so that the path of light is
always 45°. And those lines are also increasingly squished together towards the edges so that
infinite space fits in this finite map. The points at the top and bottom represent our infinite
future and past. And the left and right points are infinite spatial boundaries. But it's the diagonal
edges of the diagram that we need to pay attention to. These are the null infinities that Hawking and
Ellis are talking about. In particular, the future null infinity is where light rays end
in the infinite future. And the past null infinity is where they appear to come from if we trace them
back in time forever. Now we can add a black hole to the Penrose diagram like this. This
is an eternal black hole. It was always there and always will be there. It's just sitting out there
to our left and any motion in that direction will eventually reach it. And that means we can replace
one future null infinity by the event horizon. Below the event horizon, space and time grid lines
actually switch place. Moving up on this diagram is equivalent to falling deeper in the black hole
towards the inevitable destination of the black hole center. the singularity which occupies all
future times inside the black hole. We have previous episodes that explain all of this in
more detail. With this sort of diagram, we can see that nothing can escape the black hole. In order
to escape, something would need to take a path shallower than 45° and that means going faster
than light. Even light gets stuck forever at the event horizon if it starts there and inevitably
at the singularity even if it was trying to move outwards. Hawking and Ellis's definition
is clearer in the context of this diagram. They define an event horizon as the boundary of the
causal past of future null infinity. This is the future null infinity and its causal past is
anything that can get to it traveling no greater than light speed. That means the entire universe
except the region below the event horizon. Nothing behind that boundary can affect future infinity or
indeed anything else in the universe that isn't itself behind the same event horizon. Okay, so
it feels like we're saying the same thing that we already knew but with more words and more lines.
But crucially, we we've now been precise enough about black holes as a causal structure. There's
something that determines what events are allowed and forbidden in the universe. In doing this,
we remove a lot of ambiguity that we ran into earlier. But we also discover something really
strange. The event horizon definition is no longer local. It doesn't refer to a single location in
space or point in time. It's the boundary below which neither you nor light rays can escape to a
great distance, aka reach future null infinity, even if the black hole that eventually traps
you hasn't started forming yet. So the event horizon is a property of the entire spaceime
reaching all the way into infinite future. It's not just a statement about what is possible now,
but what will be possible based on everything that will happen to this patch of spaceime. Physicists
sometimes describe the event horizon as being teological from the ancient Greek term telos
meaning final purpose. It tells us about the final state of the universe, the paths that make it out
to infinity and those that eventually end up stuck inside the black hole. That's what I mean when I
say that an event horizon might have just passed you right now and you wouldn't even realize. Let's
do another thought experiment to see why. We'll start back on Earth. Relatively flat space free of
black holes. However, for unfortunate reasons that we don't need to get into, but may involve vogons,
there's a spherical shell of radiation of light collapsing towards us. This is an example of the
Va spacetime devised by Indian physicist Prahalad Vaja as a toy model to explore spaceimes and event
horizons of a collapsing star without all of the messy matter. In this thought experiment,
it feels like there's no event horizon as the shell of radiation converges on us. We don't
even know that it's coming because no signal can travel faster than the radiation zone collapse.
It's ultimately going to form a black hole with a radius governed by the amount of energy in that
radiation. But according to our formal definition, the event horizon forms long before the black
hole. In fact, it grows from the center of the collapsing sphere at the speed of light, reaching
its maximum size when it intercepts the infalling radiation. That's when the black hole truly forms,
but the horizon was already there. To visualize this, let's say that you're at the center of the
collapsing shell and there's a lastditch effort to escape. You shoot a lighteed signal out into
space to see if any passing spaceships will beam you up. If your signal makes it beyond
the ultimate size of the event horizon before the incoming radiation crosses that same point,
then it escapes and maybe you hit your way out of there. But if not, then it and you remain stuck
behind the horizon. If you send your signal too late, then it never reaches future null infinity.
So by our definition, it was always behind the true event horizon, trailing it as it expands
to become a black hole. Let's see what the Penrose diagram looks like for a Vega spacetime. It starts
like our black hole-free diagram, which we'll cut in half because who cares about this entire
side of the universe? The sphere of collapsing radiation looks like this single line in our one
dimension of space. It eventually reaches us, but the black hole it forms comes fully into existence
when it collapses below that certain radius. The event horizon is then a diagonal line upwards from
that point. Just as in our black hole containing Penrose diagram, the spaceime within that event
horizon is terminated by a singularity. Now, it's easy to see which light rays escape the forming
event horizon and which don't. Ones that we send out early enough to reach future null infinity
are the ones that escape. Those that depart too late fail to pass the involing radiation before
the black hole forms. But these rays were doomed from the very start. And so we need to trace the
true event horizon backwards to encompass this entire region. We have a part of the universe
that is trapped even though it doesn't know that it's in a black hole. So yeah, a nasonent event
horizon might be forming behind your couch as we speak. You might want to check that out.
Although you'll have to wait until the end of the universe before you can be absolutely certain. A
secret event horizon could be due to the collapse of something huge, a giant region of the galaxy
or even the universe billions of years from now. All of this is a weird way to think about event
horizons, but does anything really change? Who cares if we can't find the horizon? Well, the
most obvious problem is that an event horizon is used to define the boundary of a black hole. So,
it makes it hard to talk about where a black hole is. It contradicts our astrophysical intuition.
We know that there is a black hole at the center of our galaxy called Sagittarius A star and that
stars will get eaten if they get within a certain distance. But this local description feels at odds
with the teological definition. This is even more apparent when we try to simulate the merges of
black holes on a computer. Because Einstein's equations are difficult to work with in these
highly dynamical regimes, we rely on simulations to tell us what the spaceime will look like, we
set up Einstein's equations to describe two black holes in spiraling, then evolve the simulation
forward in time until the black holes merge, and we're left with a single black hole. But where
and with what shape are the event horizons of the merging black holes and of the final result? If
a horizon is teological, then it's defined across the entirety of the simulation to future infinity.
We could get a single time slice out to see how spacetime has evolved, but it won't tell us where
the horizon is. We could possibly trace light rays back in time from the end of the simulation and
look for the boundary of a region from which no light gets out alive. And that's very much in the
spirit of Hawking and Ellis's definition. But this is very awkward as a way to find black holes and
not necessarily very instructive if we care about the local shape of spaceime during the simulation.
And besides, we still want to know what to call that black surface that we think we're escaping
but are really doomed because the real event horizon is already above us. In the special
case of a black hole that's done forming and will never absorb new stuff, that black surface really
is the event horizon in the technical sense that we just defined. But in the real universe where
black holes tend to keep growing, we need another name for the current snapshot of the black holes
boundary. And that name is the apparent horizon where the formal event horizon represents the
global inescapable boundary integrated across all future time. The apparent horizon is a more local
beast. It's the locally inescapable boundary, the thing that you can't exit even right now,
regardless of whether you can make an ultimate escape. The formal definition of the apparent
horizon is also in terms of what light rays can do. But now it's not whether they can reach future
null infinity. It's whether they can make any progress whatsoever in the outward direction.
Technically, we say that the apparent horizon is the outermost surface from which all outward-
facing null geodysics are converging inwards. Having a local definition of the horizon is nice,
especially because it lets us understand what things look like if we're not at null infinity
or otherwise really far away. When we see these awesome simulations of merging black holes,
what we're seeing is the evolution of the apparent horizon. So why bother with the first definition
of the event horizon at all? Well, it's because the apparent horizon is relative. Every observer
will agree on the definition of the true event horizon, the ultimately inescapable region. But
the definition of the apparent horizon varies not just over time, but also with frame of reference.
In relativity, your sense of now varies with your velocity and the curvature of spaceime. Each sense
of the present is a 3D spatial slice out of the four-dimensional spaceime. And that slicing
changes with perspective. The apparent horizon will also change between these slicings. And it's
even possible to find slicings or definitions of a specially extended now that include a black hole,
but in which the apparent horizon disappears. But that's not true of the true event horizon.
If you're behind that, it doesn't matter how you slice spacetime, you are stuck. In fact,
every possible apparent horizon is guaranteed to always lie inside or on the actual event horizon.
In this sense, the apparent horizon gives a conservative estimate about the extent of the
black hole. It also gives a much more useful way to study the behavior of dynamical black holes.
black holes that are currently feeding like in quazars or in merging black holes. In that case,
the apparent horizon does something quite unexpected. You have these apparent horizons for
the inspiraling black holes that warp as they get close, but the apparent horizon of the final isn't
just a smooth merging of the two. That larger joint apparent horizon forms only in the instant
that the event horizons merge. So, the apparent horizon is far from a substitute for the event
horizon. It's a genuinely different object with its own interesting properties. So, what, where,
and when is an event horizon? And is there one forming behind you right now? Well, if you care
about what's happening close to a black hole at a particular time, then you need to get creative.
You need to choose reference frames and ask where for that observer light appears to turn inwards at
the apparent horizon. But if you want the most clear definition, the definition that matters
if you want to escape a black hole, then the true event horizon is not a property of local space or
immediate time. It's a thing defined by the past and the future and exists as a causal boundary
cutting its way through spaceime. Thank you to Bubble for supporting PBS. If you spend any time
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