Alice and Bob are hovering close to a
non-spinning black hole and Bob decides
to jump in. What happens to him? You
probably know from Bob's own
perspective, he does enter the black
hole. But the question is, what happens
from Alice's perspective? The most
common story I hear is as Bob falls
closer, Alice sees him slow down and
eventually freeze at the horizon. But,
and this is important, that's just what
she sees. In reality, Bob actually goes
in. She just can't see it.
But wait a second. Doesn't time really
slow down near a black hole and stop at
the event horizon from Alice's
perspective? So, shouldn't it mean that
it's not just what she sees, but he
actually takes an infinite time to enter
the hole? So, from her perspective,
nothing should ever enter a black hole,
right?
And if that is the case, how do black
holes form in the first place? Because
matter would take an infinite time to
collapse. Or shouldn't black holes
evaporate even before Bob enters in? And
would Bob actually see the entire
universe in fast forward as he falls
through? I never got a satisfactory
answer for all these questions until
now. For the past few months, I've been
going through this incredible book on
relativity. And my god, I think I
finally cracked it. So, in this video,
we're going to pretend we don't know
anything about black holes or event
horizons and step by step rediscover the
real physics behind them intuitively,
answering every single burning question
about event horizons. So, if you're
ready for this, let's begin.
We start by talking to Wolf Gang Rler,
the person who coined the term event
horizon. And he asks a seemingly random
question. Mahesh, can you outrun light?
What do you mean? Here's our friend
Alice, and there's a flashlight behind
her. As we turn on the flashlight, she
blasts her ship and starts accelerating.
The question is, with enough
acceleration, can she outrun that beam
of light or will the light eventually
catch up to her?
My immediate response is, Rler, that's
an easy question. I mean, look, no
matter how much Alice accelerates, she
will always be slower than Light. So,
the gap between them keeps shrinking,
and eventually, given enough time, Light
will always catch up. or will it?
Reindler reminds us that even if a
shuttle is at rest in space, it is still
moving through time. So in spacetime,
its world line is vertical. If it's
moving at a constant velocity through
space, its world line is slanted. The
faster it moves, the more it tilts with
the maximum tilt corresponding to that
of light. Now we will conveniently
choose units like light years for space
and years for time or light seconds and
seconds. That way this maximum angle
always turns out to be 45°. It's just
nice. So this means all physical world
lines must lie within this 45°.
Meaning your entire future is restricted
to this region, your future light cone.
You simply cannot reach events outside
it ever. Anyways, since our friend Alice
is accelerating, what does her world
line look like?
Well, when she's at rest, her world line
starts vertically. Then she accelerates,
so her velocity keeps increasing,
meaning her world line gets curved. But
remember, even if she were to accelerate
forever, according to relativity, it
will take her an infinite amount of time
to reach the speed of light. This means
her world line will curve closer and
closer to that 45°, but it will take an
infinite amount of time to actually
reach 45°.
In other words, it will get closer and
closer to this 45° line, but it will
only touch it at infinity. So, the world
line becomes a hyperola with this 45° as
meaning a line that touches it at
infinity. But wait, a 45° line is also
the world line of light. So this means
if a flash of light were to start
anywhere in this region, it will
eventually reach her. But if it starts
exactly here, that light would take an
infinite amount of time to reach her.
Which means if the flash starts anywhere
behind this line, it would never reach
her.
So think about what this means. As long
as she's constantly accelerating, even
though she will never reach the speed of
light, her speed will always be slower
than that of light, she can outrun it,
provided that light ray starts from far
back enough. But Rindler [clears throat]
says it's not just about light. It
reveals something much deeper. If Bob,
for example, was outside this region, he
can always catch up with Alice because
Alice's world line is part of his
future. But once he enters this region,
it'll be impossible for him to reach her
because it'll be impossible for him to
escape it. Holy smokes. This is
beginning to sound like the event
horizon of a black hole.
But Rener, that's crazy. This is flat
spacetime. There's no gravity. There's
no mass anywhere. And Rner says exactly.
Apparently, just the act of accelerating
can leave behind a patch of spaceime
which cannot influence you anymore.
Signals from there can never influence
you. He calls that patch of spaceime the
horizon. And of course, we will call it
the RNL's horizon. I absolutely love it
when we start with a seemingly simple
question and then dig deeper and end up
with something so profound. And that's
25% off for your first payment. All right, back to the video. Rener's
horizon sounds awesome. But I have a
question, Rener. This horizon only
exists for Alice because she is
accelerating, right? Yeah. And if she
stops accelerating, light from
everywhere can now reach her and so the
horizon disappears, right? Yes. Okay.
This horizon is starting to sound pretty
fake to me. Why are we even talking
about it? Because Alice isn't the only
one who experiences this particular
horizon. We can draw a family of
hyperbolic world lines all sharing the
same 45° as meaning they will all share
the same horizon. Just that the world
lines farther away are wider, meaning
they need to maintain a lower
acceleration and the world lines closer
need to maintain a higher acceleration.
And I look at this and I'm like, "Oh my
god, this is starting to look very
similar to a bunch of observers hovering
near a black hole."
"Exactly," says Rindler. Without looking
outside, Alice and her friends have no
way of telling whether they are
accelerating in an empty patch of space
or hovering near a black hole. In both
cases, there is a horizon behind them.
They experience a gravity-like force
which gets weaker with distance. So
locally the physics is identical. This
is the Einstein's equivalence principle.
Acceleration is indistinguishable from
gravity. Which means even though render
the horizons seem fake, they are pretty
intuitive. And more importantly, the
physics carries over to real black holes
in curved spacetime which is anything
but intuitive. This is the strategy.
This is how we can rediscover the entire
physics of black holes without having to
touch messy equations.
Rener, you beauty, take a bow.
But wait,
there is a problem.
If they were all hovering near a real
black hole, then Alice, the event
horizon, and all her friends would be at
fixed distances from each other, right?
Yeah. But Rimler, look here. We can
clearly see the horizon and all her
friends are getting closer and closer to
each other. So I'm sorry Rler the two
situations are clearly not identical
or are they?
See from our inertial perspective Rener
says I'm absolutely right. They're all
getting closer to each other. But what's
happening to the distances from their
perspective? That's important because
it's their horizon.
Well, isn't it the same? No, says Rler.
To answer this question, we need to
understand how their simultaneity works.
Wait, what's simultinity?
Somewhere millions of light years away,
aliens could have started their invasion
to Earth. But what's crazy is that even
though that invasion has started from my
perspective, from the perspective of my
wife who's just walking right here, it
wouldn't have for her. That invasion
would only happen a few months later.
For her, probably they're still
discussing right now. That sounds crazy.
The invasion should either start or not.
How can both be true? Well, there's a
surprisingly intuitive way to think
about it. Well, it's kind of like
driving and coming across this rock.
Right now, I'm just passing that rock.
It's right next to me. So, it's in my
present. Not the past, nor the future,
but the present. But if my wife were
driving at a slight angle, she would be
right next to me. But look, she wouldn't
have passed that rock yet. It's yet to
happen. It's in her future. So how can
the rock be both in the future and the
present at the same time?
Well, since we are moving at an angle,
our lines of sight which defines our now
are tilted with respect to each other.
We are basically considering different
slices of the space as now. That's
what's happening over here. Right? Now
guess what? The exact same thing happens
in spaceime. Because of relative motion,
our world lines are tilted with respect
to each other. And that means our
definitions of now what we call the
lines of simultinity are also tilted
with respect to each other. So look my
presence lies of spaceime is tilted with
respect to my wife's. So the alien
invasion could have started right now
for me. It's in my present. But for my
wife it's yet to happen for her right
now. They're probably still discussing
it. So the situation is analogous to two
cars.
But there is an important difference
when it comes to the cars. I can just
look besides me and say, "Aha, there's
that rock. I am passing it right now."
And of course, my wife would look at her
side and say, "Aha, I I see no rock over
here. That rock is ahead of me. I'm yet
to pass it." So we can both confirm
what, you know, our present is and our
future is. We can see it literally. But
when it comes to spaceime, we can't see
what's happening right now far away
because for that light needs to reach
our eyes. So when you extend your
simultaneity lines far away, it's not
about appearances. It is about your
reality. It tells you what really is
happening from your perspective far
away. Relative motion can truly make my
present my wife's future. But why don't
we notice this in our everyday life?
Well, because this picture is highly
exaggerated, our everyday speeds,
relative speeds are much much much
slower than the speed of light,
which means our world lines are almost
parallel, making our simultaneity also
almost parallel. And so for events close
by, we pretty much agree on what's
present, past, and future. And we tend
to think this extends to the entire
universe, but it doesn't. If you start
looking at events millions of light
years away, the differences begin to
show up. This is probably why Einstein
said, and I quote, "The distinction
between past, present, and future is
only a stubbornly persistent illusion."
So the mind-boggling takeaway here is
the idea of what's happening right now
only carries physical meaning locally
only then we'll mostly agree on past
present and future. But if you extend
the idea of simultaneity far away it's
necessary to do that to build space-time
maps and we'll do it. But if you extend
that and you start thinking of what's
happening right now far away, well, it
wouldn't have any physical meaning
because my present can be someone's
future.
Anyways, coming back, this means drawing
Alice's simultaneity is the key to
understand what's happening from her
perspective. But how do we do that? I
mean, her world line is curved, so it
sounds pretty complex. The spac-time
geometry is definitely non-intuitive,
but what helps me is that it's
incredibly analogous to regular space
geometry. Here's what I mean. If we go
back to the car analogy, in regular
space, rotations are circular, right?
So, look, the line of sides, the
simultaneities change as a tangent to
the circle. But in spacetime, rotations
are hyperbolic. So the simultaneity here
will change as a tangent to the
hyperola. See analogous, right? But
there's more.
When a car goes in a circle, the
simultaneity is drawn at every moment
meet at the center. Right? Similarly, if
a spacecraft takes a hyperbolic path in
spaceime, the simultaneity lines all
meet up at the vertex. But wait, we
already know Alice's world line is
hyperbolic. So this is what her
simultaneity slices look like. They all
meet at the vertex. And if this isn't
beautiful enough, get this. Just like
how all points on circle are equidistant
from the center, all points on the
hyperola are at the same proper distance
from its vertex. Now, it doesn't look
like that because we can't draw
hyperbolic geometry, but this situation
is analogous. Gosh, tell me this isn't
beautiful.
So, coming back, we now have Alice's
space-time map. How she slices her
spacetime every moment. But what exactly
does this mean? Well, if we bring back
the flashlight, its world line would be
vertical because it's at rest with
respect to us. And the light would be,
of course, at 45°.
So, according to our lines of
simultaneity, the front of the beam
comes closer and closer to her.
But according to her, at this moment,
the front of the beam is here. But a
little later at this moment, it's still
there at the same distance behind. Wait,
it doesn't look like it's at the same
distance. But remember, all distances
from the vertex to the hyperola are the
same from her perspective.
Which means no matter how much time
passes for her, that beam of light
doesn't progress at all. for her. It's
frozen there in time at a fixed location
behind her.
Ooh,
you can begin to see the whole infinite
time dilation at the horizon where it
comes from. Oh my god, this is
incredible. And of course, the exact
same analysis applies to the entire
family. That light beam is frozen in
time at a fixed distance behind each one
of them. Again, Rena reminds us they
don't see the light beam frozen because
remember seeing requires light to reach
your eyes. Simultaneities doesn't tell
you what you see. Your simultaneity
lines tell you what's happening right
now far away. So this is not about
appearances. This is about their
reality. from their perspective that
beam of light is just frozen there at a
fixed distance behind them.
So indeed the situation is locally
indistinguishable from hovering close to
a real black hole. So this means we can
use the same physics, the same
space-time map that we have discovered
for the event horizon of a real black
hole. So, all we have to do now is zoom
out and complete the map. There it is.
Isn't it beautiful?
But of course, I have a lot of
questions.
First of all, here each shuttle is now
hovering at a fixed distance, right?
Meaning they are all at rest with
respect to each other, right? Yeah. So,
a part of me is wondering, shouldn't
their world lines be vertical? So why
does it look curved? Why isn't there
world line vertical? Render says we
could have drawn them vertically like
this. And I immediately look at this and
say, hey, this feels so much more
intuitive. Then why aren't we drawing it
this way? Well, because if you actually
solve Einstein's field equations, then
in this particular space-time diagram,
you'll find that the path of light is
curved.
And in fact, look, it looks like nothing
can even enter the event horizon.
That led people like Einstein and
Edington to believe black holes might
not even exist.
But of course, eventually we realize
that what we're looking at is just a
map. And just like how every flat map of
curved earth is distorted,
maybe every space-time map that we draw
of a curved spacetime is also distorted.
Some maps preserve correct angles but
distort sizes massively and others
preserve sizes but they distort angles.
So this map of spacetime preserves
vertical world lines but the paths of
the light gets distorted. So physicists
started wondering if they could somehow
build a new map of spaceime where the 45
degree light path could be preserved and
that required a lot of math but we
eventually found it and guess what
that's the map that we have found it but
we did it way more intuitively
I mean in this map the world lines of
the light is always at 45°.
How do we know? Well, because we built
it from renler horizon, remember? And
Renler horizon comes from flat spacetime
where light travels at 45°. Since we
used the equivalence principle and
translated that into our curve spacetime
in our map, the 45° got preserved. So,
this was the reason why we did it all
this way. This allowed us to sidestep
the whole equations and coordinate
transformations, all the messy math and
intuitively
build the map. And now that we have it,
we can use this as a tool to answer all
our burning questions.
All right, let's say Alice is hovering
over here. So this is her world line.
And let's say she drops Bob somewhere
over here. What's going to happen to
Bob? To answer that question, we draw
Bob's future Litecoin. And we can
immediately see Bob has a future inside
the event horizon. So he can enter the
black hole, which we can immediately see
from here. But Bob also has a future
over here, which means he can avoid the
event horizon. As long as he uses his
rockets and, you know, fires it at the
right amount, he can definitely avoid
the event horizon. So that makes sense.
But of course, since Bob doesn't have a
rocket, let's assume that he takes a
path something like this. This is a
freef fall path and he enters the
horizon. We'll talk about what happens
inside a little bit later, says Rindler.
But now the big question is from Bob's
perspective, how do things look like?
Well, as he follows along this path,
notice more and more of his future now
lies inside the horizon. You can clearly
see that. Now just before reaching the
horizon if he changes his mind and if he
had fire you know if he had rockets
again no look look there is a small
window that he can exit but once he
enters the horizon that's it you can
clearly see there is no escape because
even if it travels at the speed of light
he cannot escape he's now trapped
but let's say he realizes there's no way
back and he sends one last transmission
message he says goodbye universe
Of course, even that message will never
come out because it'll also be trapped
inside the black hole. But from this, we
can immediately see that these events
happen for Bob. And now we understand
why we say from Bob's perspective, he
does enter the horizon. We can clearly
see he enters the horizon in finite time
in his clock and he can do stuff inside
the horizon. But now comes the big
question. What is the story from Alice's
perspective?
Okay, first let's think about what she
sees. All we have to do is draw a 45°ree
line at periodic intervals. Here it
goes.
And now notice when Alice is somewhere
over here, she gets the signals pretty
periodically. But as time goes on,
notice the delay increases and
eventually the last signal takes an
infinite time for her to reach over
there. which means indeed she sees him
slow down. The best thing about this
picture, I don't have to talk about time
dilation anymore. I can draw the path of
light and I can talk about what she
sees. Now comes the main question. What
happens from her perspective? Not just
what she sees, but what actually happens
from her perspective. For that, we have
to use her lines of simultaneity.
So each line of simultaneity represents
a moment in Alice's perspective. So
let's call this as t= 0. t= 1 2 3 4 5
and t= infinity would be somewhere over
here. So at t=0 she's over here and from
her perspective Bob is over here.
Remember she doesn't see Bob over there
but this is where Bob is from her
perspective.
Okay at one a little later uh Bob is
over here. A little later Bob is over
here. A little later Bob's over here.
Ooh.
Ooh. You can begin to see that Bob
actually does slow down. Notice he's
slowing down from Alice's perspective.
This is not about w this is not about
appearances. From her perspective, Bob
will always be always be outside the
horizon. Remember
this particular 45°ree line represents
the infinite time which means Alice
actually has to wait an infinite amount
of time for Bob to enter the horizon. So
even that story is true. It's not just
about appearances.
Bob really takes an infinite amount of
time to enter the horizon from Alice's
perspective.
So, Renler asks the most important
question from Alice's perspective. Did
Bob ever say goodbye, universe?
Well, my immediate response is no,
because from Alice's perspective, it
already takes an infinite amount of time
to just enter the horizon. So, clearly
nothing ever can happen beyond the
horizon. So, no. Nothing ever happens
beyond the horizon from Alice's
perspective.
Or does it? Here's a table and there's
an ant. What's the coordinate of this
ant right now? And I say, well, for that
we need a grid. Okay, let's put a grid.
What's the coordinate right now? Okay,
it's um x= 3 and y = 2. What if we
change the grid? What if we use a
different kind of coordinate system in
which the x= 1 line is over here? The
next line x= 2 is halfway in between and
x= 3 would be halfway in between this
and x= 4 will be here. x= 5 will be here
and so on and so forth and so forth and
eventually x= infinity will be somewhere
over here. Okay. Now according to this
coordinate system, what's the coordinate
of this ant?
Well, I say you we can't assign a
coordinate anymore. So Rler asked, does
that mean the ant does not exist
according to this system? I say that's
absurd. Of course, the ant exists. And
Miller says, "No, no, no, no. I mean,
think about it. Even if I go through
infinite values of x, I can't reach the
ant. So obviously that ant does not
exist as far as this coordinate system
is concerned." Right? And I say, "Come
on, Reinda. What are you talking about?
Whether the ant exists on the table or
not does not depend upon what kind of
grid you put on it.
And that's exactly the point. If we go
back over here, Bob, Alice, and
everybody in this universe shares the
exact same spacetime. And events on that
spacetime either happens or it doesn't.
It's not subjective. It's not relative.
and it's independent of the coordinate
system that you put on it. Now clearly
we can see that this event of Bob saying
goodbye happens and so it happens for
everybody. It happens for Bob, it
happens for Alice as well because she
also shares the same spaceime.
So if you think of it that way, these
events exist and so by definition they
happen. So Bob does enter the horizon.
Bob does say goodbye. And everything
that Bob does inside happen even from
Alice's perspective because they exist
from Alice's perspective.
But wait, Rener, if that event happens,
when does it happen according to Alice?
And Rindler says never.
What? Never in her timeline.
Okay. Oh, okay. I think I finally
understand the source of all the
confusion. Hear me out. Okay. In
everyday life, we have only two choices.
Either an event exists and when it does,
it's always a part of our past, present,
or future. We can always talk about when
it happens. Okay? And in case it's not a
part of our timeline, then those events
never happen. They don't exist at all.
This is what we have intuition for. This
is all that we usually experience in
everyday life. But black holes force us
to accept a third possibility. It's
possible for events to exist
but not be a part of our past, present
or future. It's possible for event to
happen but not be a part of our
timeline.
And this are all these these are all the
events inside the horizon.
Now if somebody had just told me this I
wouldn't have accepted it. I mean like
it sounds cool but come on. But now we
are seeing it right in front of our
eyes. These events exist because we all
share same spaceime. It exists for
everyone. And yet we cannot assign a
coordinate to it because Alice's map
doesn't extend to it. But wait, I still
can't reconcile this with the fact that
it takes an infinite amount of time for
Bob to enter the horizon. I mean, that's
what the map tells us, right? Alice
literally has to take an infinite amount
of time for Bob to enter the horizon.
So, how how can I reconcile with the
fact that events do happen? Like I am
not able to put it all together. And
Brindler says actually Mahesh, it's
quite easy. Remember what we learn about
simultaneity.
The idea of what's happening now far
away has no physical significance. Which
means even though from Alice's
perspective it takes infinite time for
Bob to enter the horizon,
it it carries no real meaning.
Okay, I find that hard to accept. And
Rler says, "All right, suppose you truly
believe that Bob will never enter the
horizon, that Bob is always outside the
horizon as far as Alice is concerned."
Then that would also mean that we can go
and rescue Bob anytime in the future,
right? Like that's the literal meaning
of that statement. He's always outside
the horizon. And I say, "Yeah, that
makes sense." Rner says, "Let's test it
out. Let's see if Alice can always save
Bob anytime in the future because it's
always outside the horizon. So, let's go
back over here. How do we check this?
Well, let's say Alice is over here. Can
she save Bob now from entering the
horizon? To answer that question, all we
have to make sure is whether Bob's world
line outside the horizon intersects with
Alice's future. Is it a part of Alice's
future light cone? So let's just
concentrate on this world line and let's
look at Alice's future light cone and we
immediately see nope this is not a part
of her future. So she cannot rescue Bob
anymore even though her sim simultaneity
tells her that Bob is outside she can't
rescue him.
In fact, we cannot draw a 45 degree line
from here and realize that this is the
threshold. As long as Alice is past this
particular point, she can never save
him. But if Alice is below that
threshold, then look, she can save him
because she can reach to him before he
enters the horizon. So, this really
represents the last chance to rescue
Bob.
This totally convinced me. I mean, if
Bob was truly out there always, then we
should have been able to rescue him
always. But notice, we can't. So, that
clearly tells me, and I am able to
accept it, that that's that carries no
physical meaning. So, we shouldn't we
should stop taking it literally.
I love this map.
Okay, on to some other confusing
questions. If things take infinite time
to enter the horizon, how does a black
hole form in the first place? Shouldn't
all the material be stuck near the
horizon forever? Well, again, let's go
back to the map. Let's say we have a
star. Let's think about that star before
it forms a black hole. Now if that star
is stable then the surface also is a is
also the world line of that surface is
also a hyperbola. It's very similar to
hovering.
And now let's say that here is where it
starts collapsing. Then it starts going
inwards and then eventually it becomes
smaller than its own shore child radius.
the critical radius below which it
starts becoming a black hole and that's
where it becomes and then the event
horizon is formed and again if we draw
Bob somewhere over here we can clearly
see the Bob you know what are the Bob's
op um options he can always avoid the
whole thing and just you know hover he
can go and hit that star over here but
if he takes a path like this he will
enter the horizon Again, from Alice's
perspective, if you look at our
simultaneity lines, it says the event
horizon takes an infinite amount of time
to form that all the stuff is always out
there. It never reaches the horizon. The
horizon never forms according to her.
But that statement carries no physical
meaning because remember what's
happening right now far away carries no
physical meaning. So if she drops Bob,
Bob can certainly enter the horizon and
she too can certainly enter the horizon
and check for herself, the horizon will
exist.
What about black hole evaporation?
Because another confusion we have is if
it takes infinite amount of time for Bob
to fall through the horizon and we know
according to our current models that
black holes can evaporate in finite time
horrendous amount of finite time but
nevertheless finite amount of time
shouldn't the black hole evaporate
before Bob enters
let's see suppose the black hole
evaporation event happens somewhere over
here now honestly I have no idea what's
going to happen beyond this point to the
map. The horizon disappears or it's it
goes away but it doesn't matter because
look the horizon exists from here to
here and Bob can certainly enter the
horizon. So if Bob took this trajectory
then Bob would have entered the horizon
he would be inside the black hole. But
again from Alice's perspective it takes
an infinite time for the horizon to
form. It takes an infinite time for Bob
to enter. It takes an infinite time for
a black hole to even evaporate. What
does it mean? Nothing.
Events far I think you get it. But I'm
going to say it one more time. Thinking
about what's happening now far away
carries no physical meaning. That I
think is the moral of the story over
here.
What really happens to Bob inside the
horizon? Why can't he hover there? Or
why can't he orbit the center like we
can do outside? We know he can't do that
because we've heard of the stories. But
can the map help us understand that? All
right, let's do that, says Rindler. So,
we now need to map the insides. How do
we do that? Here's the way to think
about it. If you look on the outside,
each hyperola represents a fixed
distance from the black hole, right? So,
we can give each hyperola an R
coordinate. So, the R equals 1, let's
say, represents the event horizon 1.2 2
1.4 these are different distances at
which you can hover. Now the question is
can you hover inside the event horizon
to do that I need to draw another
hyperola
with
um with the same as 45°ree line and
notice I can't do that physically this
makes sense because as I go closer my
acceleration needs to increase to
maintain this hyperola and eventually
when I reach the event horizon the
acceleration needs to be infinite
because the hyperola converts into a 45
degree V and that is the world of light
and that takes an infinite acceleration
if you want to maintain that beyond that
you cannot think about a hyperola with
the same asytote opening to the right
that's why you can't hover but we can
draw a family of hyperboly inside
opening upwards so I can draw a hyperbol
that looks like this so geometrically
speaking once I go inside the lower
values of r must look like this because
remember the idea is straightforward. We
draw hyperboly that are uh that have the
same asytote.
And so this is what a lower value of r
would look like. An even lower would
look like this. And eventually the
center would look somewhat like this.
And so this is the map of the inside of
the black hole.
So now let's look at what happens when
you go inside. Notice that inside your
future
lies towards the lower values of R.
Meaning wherever you go, it's impossible
now to avoid these different values of R
compared to what happens outside.
Outside you can stick to a particular
hyperbol over here. You can hover over
there. You can stick to a value of R and
you can stay. If you choose, you can go
here or if you choose you can go here.
All of that is a part of your future.
That's how space works. You are free to
move anywhere you want. But notice
inside you are no longer free to move
any values of R. You have to go to this
R and then after that you have to go to
this R and then eventually you hit the
center. It's unavoidable.
So in that sense space behaves like
time. Oh this is why we say space
behaves like time. This is why we say
your future lies at the center. You
cannot avoid it just like how you cannot
avoid next Saturday.
That's what's happening and that's why
you cannot hover. You cannot orbit.
Whatever you do you have to reach the
center. But what happens as somebody
something reaches the center of the
black hole? Well, turns out that the in
the curvature tends to go to infinity.
It's technically called the remon
curvature tensor that goes to infinity.
Um tidal forces that means go to
infinity and that will certainly
spaghettify anything and shred it apart.
But what happens at a quantum level? If
you had a quantum object that went
towards the center, we don't know what
happens. And the center is called the
singularity because you have infinity
over there and we we don't know what
would happen at a quantum level. So
that's an unknown. That's an open
question for physics as of today.
However, what about the time
coordinates? Now we can draw time
coordinates just like we drew over here.
And notice time coordinates behave the
opposite way outside. If you look at the
time coordinates, you have no choice but
to go from time t= 1 to 2 and then from
2 to 3. You cannot hover at a particular
time coordinate. That's not allowed. You
have to go. You cannot avoid your
future. But look inside. You can stick
to a particular time coordinate or you
can jump to another time coordinate or
you can jump to this time coordinate. So
in that sense time behaves like space.
Space behaves like time.
But what does that physically mean? Rler
Rler says nothing. It's just
coordinates. It doesn't mean anything.
What's more important is that for Bob,
[snorts] the future lies towards the
center. If Bob were to look back as he
falls through, would he see the entire
future of the universe in fastforward
motion? Let's go back to the map. Let's
just focus on how Bob sees Alice as he
falls through. For that, he just we just
have to draw 45°ree light signals from
Alice all the way over here. And when we
do that, we notice he does not see the
entire future of Alice. In fact, if you
can clearly see over here, this is the
last signal he will ever get from Alice.
Beyond that, he doesn't receive any
signal. And same will be the case for
the rest of the universe, which means
no, he does not see the entire future of
the universe in fastforward motion.
[laughter]
Last scenario. So we already know that
there is a threshold beyond which Alice
cannot save Bob. And let's say Alice has
crossed that threshold as you can see
right now. And Alice realizes she loves
Bob. She cannot live without him. So she
would rather plunge into the black hole
herself and meet up with Bob one last
time before moving towards singularity
together. So can she always do that or
is there a window for that as well?
Again, we can go that go back to our
map. Let's say Bob's world line ends
over here. It cannot continue beyond
that because this is the center. And so,
we can now draw a 45° line from here.
And now, this would represent the last
chance to even meet Bob. So if she is
somewhere between here and if she
decides to plunge into the black hole
inside the horizon, she can meet up with
Bob and be together for the rest of
their lives. But if she misses that
window, she will never be able to meet
him even inside the black hole. And that
would really be a sad story. But of
course our final question could be what
does this part represent?
And what if this was a rotating black
hole? Well then Bob could have a
completely different future and an
exciting future altogether. In fact, he
could even meet a different Alice in a
different universe altogether. But
that's a story for another day. See you.