Something Strange Happens When You Follow Einstein's Math

- You can never see anything enter a black hole.

(bell dings)

Imagine you trap your nemesis in a rocket ship

and blast him off towards a black hole.

He looks back at you shaking his fist at a constant rate.

As he zooms in, gravity gets stronger,

so you would expect him to speed up,

but that is not what you see.

Instead, the rocket ship appears to be slowing down.

Not only that, he also appears

to be shaking his fist slower and slower.

That's because from your perspective,

his time is slowing down

at the very instant when he should cross the event horizon,

the point beyond which not even light can escape,

he and his rocket ship do not disappear,

instead, they seem to stop frozen in time.

The light from the spaceship gets dimmer and redder

until it completely fades from view.

This is how any object would look

crossing the event horizon.

Light is still coming from the point where he crossed,

it's just too redshifted to see,

but if you could see that light,

then in theory you would see everything

that has ever fallen into the black hole

frozen on its horizon, including the star that formed it,

but in practice, photons are emitted at discreet intervals,

so there will be a last photon emitted outside the horizon,

and therefore these images will fade after some time.

- This is just one of the strange results

that comes outta the general theory of relativity,

our current best theory of gravity.

The first solution of Einstein's equations

predicted not only black holes,

but also their opposite, white holes.

It also implied the existence of parallel universes

and even possibly a way to travel between them.

This is a video about the real science of black holes,

white holes, and wormholes.

- The general theory of relativity

arose at least in part due to a fundamental flaw

in Newtonian gravity.

In the 1600s Isaac Newton

contemplated how an apple falls to the ground,

how the moon orbits the earth and earth orbits the sun

and he concluded that every object with mass

must attract every other,

but Newton was troubled by his own theory.

How could masses separated by such vast distances

apply a force on each other?

He wrote, "That one body may act upon another at a distance

through a vacuum without the mediation of anything else

is to me, so great and absurdity that I believe no man

who has a competent faculty of thinking

could ever fall into it."

One man who definitely had a competent faculty of thinking,

was Albert Einstein and over 200 years later,

he figured out how gravity is mediated.

Bodies do not exert forces on each other directly.

Instead, a mass like the sun curves the spacetime

in its immediate vicinity.

This, then curves the spacetime around it

and so on all the way to the earth.

So the earth orbits the sun, because the spacetime

earth is passing through is curved.

Masses are affected by the local curvature

of spacetime, so no action at a distance is required.

Mathematically, this is described

by Einstein's field equations.

Can you write down the Einstein field equation?

- This was the the result of Einstein's decade of hard work

after special relativity

and essentially what we've got in the field equations

on one side it says,

tell me about the distribution of matter and energy.

The other side tells you what the resultant curvature

of spacetime is from that distribution

of matter and energy and it's a single line.

It looks like, oh, this is a simple equation, right?

But it's not really one equation.

It's a family of equations and to make life more difficult,

they're coupled equations, so they depend upon each other

and they are differential equations,

so it means that there are integrals

that have to be done, da, da da.

So there's a whole bunch of steps that you need to do

to solve the field equations.

To see what a solution to these equations would look like,

we need a tool to understand spacetime.

So imagine your floating around in empty space.

A flash of light goes off above your head

and spreads out in all directions.

Now your entire future, anything that can

and will ever happen to you will occur within this bubble

because the only way to get out of it

would be to travel faster than light.

In two dimensions, this bubble is just a growing circle.

If we allow time to run up the screen

and take snapshots at regular intervals,

then this light bubble traces out a cone,

your future light cone.

By convention, the axes are scaled so that light rays

always travel at 45 degrees.

This cone reveals the only region of spacetime

that you can ever hope to explore and influence.

Now imagine that instead of a flash of light

above your head, those photons were actually traveling in

from all corners of the universe

and they met at that instant

and then continued traveling on

in their separate directions.

Well, in that case then into the past,

these photons also reveal a light cone,

your past light cone.

Only events that happened inside this cone

could have affected you up to the present moment.

We can simplify this diagram even further

by plotting just one spatial and one time dimension.

This is the spacetime diagram of empty space.

If you want to measure how far apart

two events are in spacetime, you use something called

the spacetime interval.

The interval squared is equal to minus dt squared,

plus dx squared, since spacetime is flat,

the geometry is the same everywhere

and so this formula holds throughout the entire diagram,

which makes it really easy to measure the separation

between any two events, but around a mass,

spacetime is curved and therefore you need to modify

the equation to take into account the geometry.

This is what solutions to Einstein's equations are like.

They tell you how spacetime curves

and how to measure the separation between two events

in that curved geometry.

Einstein published his equations in 1915

during the First World War,

but he couldn't find an exact solution.

Luckily, a copy of his paper made its way

to the eastern front where Germany was fighting Russia,

stationed there was one of the best astrophysicists

of the time, Karl Schwarzschild.

Despite being 41 years old, he had volunteered

to calculate artillery trajectories for the German army.

At least until a greater challenge caught his attention,

how to solve Einstein's field equations.

Schwarzschild did the standard physicist thing

and imagined the simplest possible scenario,

an eternal static universe with nothing in it

except a single spherically symmetric point mass.

This mass was electrically neutral and not rotating.

Since this was the only feature of his universe,

he measured everything using spherical coordinates

relative to this center of this mass.

So r is the radius and theta and phi give the angles.

For his time coordinate, he chose time as being measured

by someone far away from the mass,

where spacetime is essentially flat.

Using this approach, Schwarzschild found the first

non-trivial solution to Einstein's equations,

which nowadays we write like this.

This Schwarzschild metric describes how spacetime curves

outside of the mass.

It's pretty simple and makes intuitive sense,

far away from the mass spacetime is nearly flat,

but as you get closer and closer to it,

spacetime becomes more and more curved,

it attracts objects in and time runs slower.

(gunshots firing)

Schwarzschild sent his solution to Einstein,

concluding with, "The war treated me kindly enough

in spite of the heavy gunfire

to allow me to get away from it all

and take this walk in the land of your ideas."

Einstein replied, "I have read your paper

with the utmost interest, I had not expected

that one could formulate the exact solution to the problem

in such a simple way."

But what seemed at first quite simple,

soon became more complicated.

Shortly after Schwarzschild solution was published,

people noticed two problem spots.

At the center of the mass, at r equals zero,

this term is divided by zero, so it blows up to infinity

and therefore this equation breaks down

and it can no longer describe what's physically happening.

This is what's called a singularity.

Maybe that point could be excused,

because it's in the middle of the mass,

but there's another problem spot outside of it

at a special distance from the center

known as the Schwarzschild radius, this term blows up.

So there is a second singularity. What is going on here?

Well, at the Schwarzschild radius,

the spacetime curvature becomes so steep

that the escape velocity, the speed that anything would need

to leave there is the speed of light

and that would mean that inside the Schwarzschild radius,

nothing, not even light would be able to escape.

So you'd have this dark object

that swallows up matter and light,

a black hole, if you will,

but most scientists doubted that such an object could exist,

because it would require a lot of mass

to collapse down into a tiny space.

How could that possibly ever happen?

(thrilling music)

Astronomers at the time were studying

what happens at the end of a star's life.

During its lifetime the inward force of gravity is balanced

by the outward radiation pressure

created by the energy released through nuclear fusion,

but when the fuel runs out, the radiation pressure drops.

So gravity pulls all the star material inwards, but how far?

Most astronomers believed some physical process

would hold it up and in 1926,

Ralph Fowler came up with a possible mechanism.

Pauli's exclusion principles states that,

"Fermions like electrons cannot occupy the same state,

so as matter gets pushed closer and closer together,

the electrons each occupy their own tiny volumes,"

but Heisenberg's uncertainty principle says that,

"You can't know the position and momentum of a particle

with absolute certainty, so as the particles become

more and more constrained in space,

the uncertainty in their momentum,

and hence their velocity must go up."

So the more a star is compressed,

the faster electrons will wiggle around

and that creates an outward pressure.

This electron degeneracy pressure would prevent the star

from collapsing completely.

Instead, it would form a white dwarf

with the density much higher than a normal star

and remarkably enough astronomers had observed stars

that fit this description.

One of them was Sirius B.

But the relief from this discovery was short-lived.

Four years later, 19-year-old Subrahmanyan Chandrasekhar

traveled by boat to England to study with Fowler

and Arthur Eddington, one of the most revered scientists

of the time.

During his voyage, Chandrasekhar realized

that electron degeneracy pressure has its limits.

Electrons can wiggle faster and faster,

but only up to the speed of light.

That means this effect can only support stars

up to a certain mass, the Chandrasekhar limit.

Beyond this, Chandrasekhar believed,

not even electron de degeneracy pressure

could prevent a star from collapsing,

but Eddington was not impressed.

He publicly blasted Chandrasekhar saying,

"There should be a law of nature

to prevent a star from behaving in this absurd way"

and indeed scientists did discover a way

that stars heavier than the Chandrasekhar limit

could support themselves.

When a star collapses beyond a white dwarf,

electrons and protons fuse together

to form neutrinos and neutrons.

These neutrons are also fermions,

but with nearly 2000 times the mass an electron,

their degeneracy pressure is even stronger.

So this is what holds up neutron stars.

There was this conviction among scientists

that even if we didn't know the mechanism,

something would prevent a star from collapsing

into a single point and forming a black hole,

because black holes were just too preposterous to be real.

The big blow to this belief came in the late 1930s

when Jay Robert Oppenheimer and George Volkoff

found that neutron stars also have a maximum mass.

Shortly after Oppenheimer and Hartland Snyder

showed that for the heaviest stars,

there is nothing left to save them when their fuel runs out,

they wrote, "This contraction will continue indefinitely,"

but Einstein still couldn't believe it.

Oppenheimer was saying that stars can collapse indefinitely,

but when Einstein looked at the math,

he found that time freezes on the horizon.

So it seemed like nothing could ever enter,

which suggested that either

there's something we don't understand

or that black holes can't exist,

(star explodes)

but Oppenheimer offered a solution to the problem.

He said to an outside observer,

you could never see anything go in,

but if you were traveling across the event horizon,

you wouldn't notice anything unusual

and you'd go right past it without even knowing it.

So how is this possible?

We need a spacetime diagram of a black hole.

On the left is the singularity at r equals zero.

The dotted line at r equals 2M is the event horizon.

Since the black hole doesn't move,

these lines go straight up in time.

Now let's see how ingoing and outgoing light ray travel

in this curved geometry.

When you're really far away,

the future light cones are at the usual 45 degrees,

but as you get closer to the horizon,

the light cones get narrower and narrower,

until right at the event horizon,

they're so narrow that they point straight up

and inside the horizon, the light cones tip to the left,

but something strange happens with ingoing light rays.

- They fall in, but they don't get to r equals 2M,

they actually asymptote to that value

as time goes to infinity,

but they don't end at infinity, right?

Mathematically they are connected and come back in

and they're traveling in this direction

and this bothered a lot of people,

this bothered people like Einstein,

because he looked at these equations and went,

"well, if nothing can cross this sort of boundary,

then how could there be black holes?

How could black holes even form?"

- So what is going on here?

Well, what's important to recognize

is that this diagram is a projection.

It's basically a 2D map

of four dimensional curved spacetime.

It's just like projecting the 3D Earth onto a 2D map.

When you do that, you always get distortions.

There is no perfectly accurate way

to map the earth onto a 2D surface,

but different maps can be useful for different purposes.

For example, if you wanna keep angles and shapes the same,

like if you're sailing across the ocean

and you need to find your bearings,

you can use the Mercator projection,

that's the one Google Maps uses.

A downside is that it misrepresent sizes.

Africa and Greenland look about the same size,

but Africa is actually around 14 times larger.

The Gall-Peters projection keeps relative sizes accurate,

but as a result, angles and shapes are distorted.

In a similar way, we can make different projections

of 4D spacetime to study different properties of it.

Physical reality doesn't change,

but the way the map describes it does.

- He had chosen to put a particular coordinate system

of a space and have a time coordinate, and off you go.

It's the most sensible thing to do, right?

- [Derek] People realize that if you choose

a different coordinate system

by doing a coordinate substitution, then the singularity

at the event horizon disappears.

- It goes away.

That problem goes away and things can actually cross

into the black hole.

- What this tells us is that there is

no real physical singularity at the event horizon.

It just resulted from a poor choice of coordinate system.

Another way to visualize what's going on

is by describing space as flowing in towards the black hole,

like a waterfall.

As you get closer, space starts flowing in

faster and faster.

Photons emitted by the spaceship have to swim

against this flow, and this becomes harder and harder

the closer you get.

Photons emitted just outside the horizon

can barely make it out, but it takes longer and longer.

At the horizon, space falls in

as fast as the photons are swimming.

So if the horizon had a finite width,

then photons would get stuck here,

photons from everything that ever fell in,

but the horizon is infinitely thin.

So in reality, photons either eventually escape or fall in.

Inside the horizon, space falls faster

than the speed of light,

and so everything falls into the singularity.

So Oppenheimer was right.

Someone outside a black hole can never see anything enter

because the last photons they can see

will always be from just outside the horizon,

but if you yourself go,

you will fall right across the event horizon

and into the singularity.

Now you can extend the waterfall model

to cover all three spatial dimensions,

and that gives you this, a real simulation

of space flowing into a static black hole

made by my friend Alessandro from ScienceClic.

Later we'll use this model to see what it's like

falling into a rotating black hole.

Now, I've never been sucked into a black hole,

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I wanna thank Incogni for sponsoring this part of the video

and now back to spacetime maps.

If you take this map and transform it

so that incoming and outgoing light ray

all travel at 45 degrees like we're used to,

then something fascinating happens.

The black hole singularity on the left

transforms into a curved line at the top

and since the future always points up in this map,

it tells us that the singularity is not actually

a place in space, instead, it's a moment in time,

the very last moment in time for anything

that enters a black hole.

The map we've just created is a Kruskal-Szekers diagram,

but this only represents a portion of the universe,

the part inside the black holes event horizon

and the part of the universe closest to it,

but what we can do is contract the whole universe,

the infinite past, infinite distance, and infinite future,

and morph it into a single map.

It's like using the universe's best fish eye lens.

That gives us this penrose diagram.

Again, light rays still always go at 45 degrees.

So the future always points up.

The infinite past is in the bottom of the diagram.

The infinite future at the top

and the sides on the right are infinitely far away.

The black hole singularity is now a straight line

at the top, a final moment in time.

These lines are all at the same distance

from the black hole.

So the singularity is at r equals zero,

the horizon is at r equals 2M,

this line is at r equals 4M,

and this is infinitely far away.

All of these lines are at the same time.

What's great about this map is that it's very easy to see

where you can still go and what could have affected you.

For example, when you're here, you've got a lot of freedom.

You can enter the black hole or fly off to infinity,

and you can see and receive information from this area,

but if you go beyond the horizon,

your only possible future is to meet the singularity.

You can still, however, see and receive information

from the universe.

You just can't send any back out.

Now think about being at this point in the map.

This is at the event horizon,

and now your entire future is within the black hole,

but what is the past of this moment?

Well, you can draw the past light cone

and it reveals this new region.

If you're inside this region,

you can send signals to the universe,

but no matter where you are in the universe,

nothing can ever enter this region

because it will never be inside your light comb.

So things can come out, but never go in.

This is the opposite of a black hole, a white hole.

What color is a white hole?

(Geraint exhales)

(Derek laughs)

- I mean, it's gonna be the,

it's not gonna have a color, right?

It's gonna be whatever's being spat out of it.

It depends what's in there and gets thrown out,

that's what you are going to see.

So if it's got light in there, it's got mass in there,

it's all gonna be ejected.

So the white hole kind of picture

is the time reverse picture of a black hole,

instead of things falling in, things get expelled outwards

and so whilst a black hole has a membrane,

the Schwarzschild horizon, which once you cross,

you can't get back out, the white hole has the opposite.

If you're inside the event horizon, you have to be ejected,

so it kicks you out kind of thing, right?

Relativity doesn't tell you which way time flows.

There's nothing in there that says that, that is the future

and that is the past.

When you are doing your mathematics

and you're working out the behavior of objects,

you make a choice about which direction is the future,

but mathematically, you could have chosen

the other way, right?

You could have had time point in the opposite direction.

Any solution that you find in relativity,

mathematically, you can just flip it

and get a time reverse solution

and that's also a solution to the equations.

- [Derek] Now, we've been showing things

being ejected to the right, but they could just as well

be ejected to the left.

So what's over there?

This line is not at infinity,

so there should be something beyond it.

If we eject things in this direction,

you find that they enter a whole new universe,

one parallel to our own.

- [Geraint] We can fall into this black hole,

and somebody in this universe here

could fall into this black hole in their universe,

and we would find ourselves in the same black hole.

(Derek chuckles)

- The only downside is that

we'd both soon end up in the singularity.

I guess I'm just trying to understand

where that universe appears

in the mathematical part of the solution.

Like, can you point to the part of the equation and be like,

so that's our universe, and then these terms here,

that's the other universe, or do you know what I mean?

Like- - Yeah,

well, it's coordinates, right?

Imagine somebody, right, came up with a coordinate system

for the earth, but only the northern hemisphere

and you looked at that coordinate system, right?

And you looked at it and you said,

"Ah, I can see the coordinate system, it looks fine,

but mathematically latitudes can be negative, right?

You've only got positive latitudes in your solution.

What about the negative ones?"

And they said to you, (scoffs) "Negative ones?

No southern hemisphere, right?"

And you've gotta go, "Well, the mathematics says that

you can have negative latitudes.

Maybe we should go and look over the equator

to see if there is something down there"

and I know that's a kind of extreme example,

because we know we live on a globe,

but we don't know the full geometry

of what's going on here in the sense that

Schwarzschild laid down coordinates

over part of the solution.

It was like him only laying down coordinates

on the northern hemisphere

and other people have come along and said,

"Hey, there's a southern hemisphere"

and more than that, there's two earths.

That's why it's called maximal extension.

It's like, if I have this mathematical structure,

then what is the extent of the coordinates

that I can consider?

And with the Schwarzschild black hole,

you get a second universe

that has its own independent set of coordinates

from our universe.

I want to emphasize right, this is the simplest solution

to the Einstein field equations,

and it already contains a black hole,

white hole and two universes.

- [Derek] That's what you get

when you push this map to its limits

so that every edge ends at a singularity or infinity.

- And in fact, there's another little feature in here,

which is that, that little point there where they cross,

that is an Einstein Rosen Bridge.

- To see it, we need to change coordinates.

Now this line is at constant crustal time

and it connects the space of both universes.

You can see what the spacetime is like

by following this line from right to left.

Far away from the event horizon,

spacetime is basically flat,

but as you get closer to the event horizon,

spacetime starts to curve more and more.

At this cross, you are at the event horizon,

and if you go beyond it, you end up in the parallel universe

that gives you a wormhole that looks like this.

- So that is hypothetically how we could use a black hole

to travel from one universe to another.

- Hypothetically, because these wormholes

aren't actually stable in time.

- It's a bit like a bridge, but it's a bridge that is long

and then becomes shorter and then becomes long again

and if you try to traverse this bridge,

at some point, the bridge is only very short, right?

And you say, "Oh, well, let me just cross this bridge."

But as you start crossing the bridge and start running,

your speed is finite, right?

The speed of light roughly and then the bridge starts,

becoming stretching and you never come out the other side.

- [Derek] This pinching off always happens too fast

for anything to travel through.

You can also see this if you look at the Penrose diagram,

because when you're inside one universe,

there isn't a light cone that can take you

to the other universe.

The only way to do that

would be to travel faster than light,

but there might be another way.

Schwarzschild solution describes a black hole

that doesn't rotate.

Yet, every star does rotate

and since angular momentum must be conserved,

every black hole must also be rotating.

While Schwarzschild found his solution within weeks

after Einstein published his equations,

solving them for a spinning mass

turned out to be much harder.