Something Strange Happens When You Follow Einstein's Math

Veritasium
29 Apr 202437:02

Summary

TLDRThe video script explores the fascinating science behind black holes, white holes, and wormholes, as predicted by Einstein's general theory of relativity. It explains how black holes, formed from the collapse of massive stars, create a gravitational pull so strong that not even light can escape. The script delves into the concept of the event horizon, where time appears to slow down and objects seem to freeze. It also touches upon white holes, which are the theoretical opposite of black holes, ejecting matter and light. The concept of wormholes, as shortcuts through spacetime, is introduced, suggesting the possibility of travel between parallel universes. However, the script highlights the theoretical nature of these phenomena, the mathematical complexities involved, and the current consensus that traversable wormholes and parallel universes are unlikely to exist based on our understanding of physics.

Takeaways

  • 🌌 **Black Holes and Visibility**: You can't see anything enter a black hole due to the extreme gravitational pull that slows down and eventually freezes light from escaping.
  • 🚀 **Time Dilation**: As an object approaches a black hole, it appears to slow down and time appears to dilate from the perspective of an outside observer.
  • 🔮 **Redshift**: Light from an object near a black hole gets redshifted to the point of invisibility as it loses energy trying to escape the black hole's gravity.
  • ⚫ **General Theory of Relativity**: Einstein's theory describes gravity not as a force between masses but as a curvature of spacetime caused by mass.
  • 📘 **Einstein's Field Equations**: These complex equations describe how the distribution of matter and energy in the universe curves spacetime.
  • ⭕ **Schwarzschild Metric**: The first non-trivial solution to Einstein's equations, describing the spacetime curvature outside a spherically symmetric, non-rotating mass.
  • ⚠ **Singularity**: A point in space where the curvature of spacetime becomes infinite, such as at the center of a black hole.
  • 🌟 **Stellar Collapse**: When a massive star runs out of fuel, it can collapse under its own gravity, potentially forming a black hole if it exceeds the Chandrasekhar limit.
  • 🌀 **Rotating Black Holes**: Unlike non-rotating black holes, rotating ones have a more complex structure with an ergosphere and inner horizon, allowing for theoretical traversal of the singularity.
  • 🌐 **Multiverse Theory**: Some solutions to Einstein's equations suggest the existence of white holes and parallel universes, although these are speculative and not proven.
  • ⛓ **Wormholes**: Hypothetical tunnels in spacetime that could, in theory, provide shortcuts between different points in space or even connect to other universes, but their existence is unconfirmed.

Q & A

  • Why can't we see anything enter a black hole?

    -As an object approaches the event horizon of a black hole, the gravitational pull causes time to slow down as observed from a distance. This effect, known as gravitational time dilation, makes the object appear to slow down and eventually freeze at the event horizon. Additionally, the light emitted by the object gets redshifted to the point where it becomes too dim to be seen.

  • What is the Einstein field equation?

    -The Einstein field equation is a set of ten interrelated differential equations that describe the fundamental interaction of gravitation as a result of the distribution of mass and energy within spacetime. It is represented as a single line equation involving the metric tensor, the Ricci tensor, and the energy-momentum tensor, but it is actually a system of coupled, non-linear partial differential equations.

  • How does the Schwarzschild solution describe the spacetime around a black hole?

    -The Schwarzschild solution provides the first non-trivial solution to Einstein's field equations, describing how spacetime curves outside of a spherically symmetric, non-rotating mass. It indicates that spacetime is nearly flat far from the mass, but becomes increasingly curved as one approaches it, which slows down time and attracts objects.

  • What is a singularity and why is it a problem in the context of the Schwarzschild solution?

    -A singularity is a point in spacetime where certain physical quantities, such as density or curvature, become infinite. In the context of the Schwarzschild solution, there are two problematic singularities: one at the center of the mass (r=0) and another at the event horizon (r=2M). These singularities indicate a breakdown of the solution and our understanding of physics at these points.

  • How did Chandrasekhar's work contribute to our understanding of stellar collapse?

    -Subrahmanyan Chandrasekhar showed that electron degeneracy pressure, which supports white dwarfs against gravitational collapse, has a limit—the Chandrasekhar limit. Stars with a mass greater than this limit cannot be supported by electron degeneracy pressure and will continue to collapse, potentially leading to the formation of a neutron star or a black hole.

  • What is the Penrose diagram and how does it help visualize the universe?

    -The Penrose diagram is a graphical representation of spacetime that shows the entire history of the universe in a single map. It is a highly simplified 'fish-eye' view that contracts the infinite past, infinite distance, and infinite future into a comprehensible diagram. It helps visualize the structure of spacetime, including the behavior near black holes and the concept of a white hole.

  • What is an Einstein-Rosen Bridge and how is it related to wormholes?

    -An Einstein-Rosen Bridge, also known as a wormhole, is a hypothetical structure in spacetime that would allow for direct passage between two distant points in the universe or even between two separate universes. It is a solution to the equations of general relativity, but its existence and stability are subjects of ongoing debate and research.

  • How does the Kerr solution differ from the Schwarzschild solution?

    -The Kerr solution is a solution to Einstein's field equations that describes a rotating black hole, unlike the Schwarzschild solution which describes a non-rotating black hole. The Kerr solution accounts for the effects of rotation, leading to a more complex structure with an ergosphere and an inner event horizon, and a ring-shaped singularity instead of a point singularity.

  • What is the role of exotic matter in the theory of wormholes?

    -Exotic matter, which has a negative energy density, is theorized to be necessary to keep wormholes open and stable. It would counteract the natural tendency of wormholes to collapse under their own gravity. However, the existence of such exotic matter is speculative and not supported by current experimental evidence.

  • Why do scientists believe that traversable wormholes and parallel universes might not exist?

    -While solutions to Einstein's field equations suggest the possibility of traversable wormholes and parallel universes, the physical realization of these phenomena is doubted due to the requirement of exotic matter with negative energy density, which is not observed in nature. Additionally, the stability and consistency of such structures with the known laws of physics are subjects of debate.

  • What is the significance of the event horizon in the context of a black hole?

    -The event horizon is the boundary around a black hole beyond which nothing, not even light, can escape the gravitational pull of the black hole. It is the point of no return for any matter or radiation approaching the black hole. From an outside observer's perspective, objects appear to freeze at the event horizon due to extreme gravitational time dilation.

  • How does the concept of a white hole relate to that of a black hole?

    -A white hole is a theoretical concept that is essentially the reverse of a black hole. While a black hole is characterized by the property of not allowing anything to escape its gravitational pull, a white hole is thought to expel matter and energy. It is derived from a time-reversed solution of the black hole equations, suggesting that if matter falls into a black hole in one direction of time, it could be ejected in the opposite direction of time.

Outlines

00:00

🌌 The Enigma of Black Holes

This paragraph introduces the concept of black holes, explaining that nothing, including light, can escape once it crosses the event horizon. It discusses the perspective of an observer watching an object fall into a black hole, noting how time appears to slow and eventually stop at the event horizon. The paragraph also touches on the implications of the general theory of relativity, which predicts not just black holes but also white holes and the possibility of parallel universes. It highlights the historical context, leading up to Einstein's formulation of the theory of general relativity, which describes gravity as the curvature of spacetime caused by mass.

05:01

📐 Spacetime and the Schwarzschild Solution

The second paragraph delves into the mathematics of general relativity, specifically Einstein's field equations, which describe how the distribution of matter and energy curves spacetime. It explains the complexity of these equations and the need for advanced mathematical tools to solve them. The narrative then shifts to the historical context of Karl Schwarzschild, who found the first non-trivial solution to Einstein's equations, known as the Schwarzschild metric. This solution describes the curvature of spacetime outside a spherically symmetric, non-rotating mass. The paragraph also discusses the concept of the spacetime interval and how it is affected by the presence of mass, leading to the discovery of singularities at the center of the mass and at the Schwarzschild radius, which is the critical point where escape velocity equals the speed of light.

10:01

đŸ’„ Stellar Collapse and the Birth of Black Holes

The third paragraph explores the end of a star's life cycle and the conditions under which a black hole might form. It describes the balance of forces within a star and the role of electron degeneracy pressure in preventing a star from collapsing into a black hole. The narrative follows the contributions of Ralph Fowler, Pauli's exclusion principle, Heisenberg's uncertainty principle, and the concept of electron degeneracy pressure. The paragraph also highlights the work of Subrahmanyan Chandrasekhar, who identified a limit to the mass a star can have before it collapses beyond a white dwarf, known as the Chandrasekhar limit. The discussion then moves to the discovery of neutron stars and the role of neutron degeneracy pressure in supporting stars heavier than the Chandrasekhar limit.

15:03

đŸ–„ïž Coordinate Systems and the Illusion of Singularities

This paragraph discusses the concept of coordinate systems in the context of black holes and how they can influence our understanding of singularities. It explains that the appearance of a singularity at the event horizon is an artifact of the chosen coordinate system and not a physical reality. The text introduces the idea of space flowing into a black hole, akin to a waterfall, and how photons struggle to escape this flow. It also touches on the concept of a white hole, which is the time-reversed counterpart of a black hole, where matter is ejected instead of absorbed. The paragraph concludes with a humorous analogy to spam calls and a brief mention of a sponsor, Incogni, which helps combat spam calls.

20:04

🌟 Penrose Diagrams and Cosmic Topology

The fifth paragraph introduces Penrose diagrams as a way to visualize the entire universe's spacetime, including black holes, white holes, and parallel universes. It describes how the Penrose diagram represents the infinite past, present, and future as well as infinite distance, and how it can illustrate the concept of a white hole and its relationship to a black hole. The discussion continues with the idea that black holes may lead to the existence of white holes and parallel universes, and how these concepts can be mathematically represented. The paragraph also explores the notion of an Einstein-Rosen Bridge, a theoretical passage through a black hole to another universe, and the limitations of this idea in the context of real-world physics.

25:05

🌀 Rotating Black Holes and the Kerr Solution

The sixth paragraph shifts the focus to rotating black holes and the Kerr solution, which is a more complex solution to Einstein's equations that describes a spinning black hole. It explains how the rotation of a black hole changes its structure, leading to an ergosphere where spacetime is dragged along with the black hole's rotation. The narrative describes the layers of a rotating black hole, including the outer and inner horizons, and the possibility of traversing the singularity in a different manner than with a non-rotating black hole. The paragraph also discusses the theoretical implications of a rotating black hole, including the potential for white holes and the existence of multiple universes, while acknowledging the speculative nature of these ideas.

30:07

🚀 Wormholes, Exotic Matter, and the Future of Relativity

The final paragraph addresses the possibility of traversable wormholes and the challenges associated with their existence. It discusses the work of Michael Morris and Kip Thorne, who explored wormholes that could be used for interstellar travel and the requirements for such structures to remain open. The text highlights the necessity for exotic matter with negative energy density to stabilize wormholes, which is considered unlikely to exist according to current physics. The paragraph concludes by reflecting on the history of scientific discovery and the potential for future surprises, suggesting that while current understanding indicates that white holes and traversable wormholes may not exist, there is always the possibility of new discoveries that could challenge this view.

Mindmap

Keywords

💡Black Hole

A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing, not even particles or electromagnetic radiation such as light, can escape from it. The video discusses the behavior of objects, including light, as they approach a black hole, and how they appear to an outside observer to slow down and eventually freeze at the event horizon. The concept is central to the video's theme of exploring the extreme predictions of general relativity.

💡Event Horizon

The event horizon is the boundary around a black hole beyond which events cannot affect an outside observer. It is the point of no return, where the gravitational pull becomes so strong that not even light can escape. The video uses the event horizon to illustrate the concept of time dilation and the redshifting of light as it approaches this boundary.

💡General Theory of Relativity

The general theory of relativity, proposed by Albert Einstein, is the current description of gravitation. It describes gravity not as a force, but as a curvature of spacetime caused by mass and energy. The video explains how this theory predicts the existence of black holes, white holes, and wormholes, which are all explored in the narrative.

💡White Hole

A white hole is a hypothetical region of spacetime that cannot be entered from the outside but can only be exited. It is the time-reverse of a black hole. The video mentions white holes in the context of exploring the theoretical possibilities beyond the event horizon and the concept of an object being expelled rather than attracted.

💡Wormhole

A wormhole is a hypothetical topological feature of spacetime that would fundamentally be a 'shortcut' through space and time, connecting two distant points in the universe. The video discusses the concept of wormholes in relation to black holes and the theoretical possibility of traveling between universes.

💡Singularity

In the context of black holes, a singularity is a point at the core of the black hole where the laws of physics as we know them break down, and the gravitational field becomes infinite. The video explains that at the singularity, spacetime curvature becomes infinite, and it is the point where all matter within a black hole is predicted to be crushed.

💡Spacetime

Spacetime is any four-dimensional continuum that combines the three dimensions of space and the one dimension of time into a single entity. The video uses the concept of spacetime to explain how gravity affects the curvature of spacetime around massive objects like stars and how this curvature influences the motion of objects within it.

💡Einstein's Field Equations

Einstein's field equations are a set of ten interrelated equations that describe the fundamental interaction of gravitation as a result of the distribution of mass and energy. The video highlights these equations as the mathematical representation of the general theory of relativity, which, when solved, can predict phenomena like black holes.

💡Schwarzschild Radius

The Schwarzschild radius is the critical radius or distance from the center of a mass at which the escape velocity equals the speed of light. The video discusses this concept to explain the formation of a black hole, where anything within this radius, including light, cannot escape the gravitational pull.

💡Redshift

Redshift, in the context of the video, refers to the phenomenon where light or other electromagnetic radiation from an object moving away is increased in wavelength, or shifted to the red end of the spectrum. It is used to describe the observation of light from an object falling into a black hole, which gets redshifted to the point of invisibility as it approaches the event horizon.

💡Rotating Black Hole

A rotating black hole, as described by the Kerr solution, is a black hole that possesses angular momentum, causing an effect known as frame-dragging, where spacetime is 'dragged' around the black hole. The video explains the unique structure of a rotating black hole, which includes an ergosphere and an inner event horizon, and how it differs from a non-rotating black hole.

💡Exotic Matter

Exotic matter, as mentioned in the video, is a hypothetical form of matter that would have negative energy density and negative pressure. It is often invoked in the context of wormholes to keep them open against the natural tendency to collapse. The video discusses the theoretical need for such matter to stabilize wormholes but also expresses skepticism about its existence based on current physical laws.

Highlights

Objects entering a black hole appear to slow down and freeze at the event horizon from an outside observer's perspective.

Light from an object near a black hole gets redshifted to the point of invisibility due to the intense gravitational pull.

Einstein's General Theory of Relativity predicts not only black holes but also white holes and the possibility of parallel universes.

Newton's gravity theory had a fundamental flaw, which Einstein resolved by describing gravity as the curvature of spacetime caused by mass.

Einstein's field equations describe how the distribution of matter and energy curves spacetime.

The Schwarzschild metric is the first non-trivial solution to Einstein's equations, describing the spacetime curvature outside a spherically symmetric mass.

The concept of a singularity arises from the breakdown of equations at points of infinite density, such as at the center of a black hole.

The idea that a star's collapse could be halted by electron degeneracy pressure was challenged by Chandrasekhar's limit, which defines a maximum mass for white dwarfs.

The existence of black holes was doubted until Oppenheimer and Snyder's work showed that stars could collapse indefinitely, forming black holes.

The spacetime diagram of a black hole shows how light cones change as they approach the event horizon, illustrating the black hole's one-way membrane effect.

The Kruskal-Szekers diagram and Penrose diagram provide visualizations of black holes, including the concept of a white hole and a final moment in time for anything entering a black hole.

The theoretical existence of white holes, which expel matter, is a time-reversed concept compared to black holes.

The Schwarzschild and Kerr solutions suggest the existence of wormholes that could connect different parts of the universe or even different universes.

The stability and existence of wormholes may require exotic matter with negative energy density, which is currently speculative and not proven to exist.

Rotating black holes, as described by the Kerr solution, have a complex structure with multiple layers, including an ergosphere where space is dragged along with the black hole's rotation.

The singularity in a rotating black hole is not a point but a ring, and there are theories that one might be able to fly through it.

The Penrose diagram for a spinning black hole suggests that there are regions inside the inner horizon where one could move freely and potentially avoid the singularity.

While the theoretical models suggest the possibility of multiple universes and exotic phenomena, practical observations and physical laws may limit the actual existence of such entities.

Transcripts

00:00

- You can never see anything enter a black hole.

00:03

(bell dings)

00:04

Imagine you trap your nemesis in a rocket ship

00:07

and blast him off towards a black hole.

00:10

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

00:14

As he zooms in, gravity gets stronger,

00:17

so you would expect him to speed up,

00:19

but that is not what you see.

00:21

Instead, the rocket ship appears to be slowing down.

00:25

Not only that, he also appears

00:27

to be shaking his fist slower and slower.

00:30

That's because from your perspective,

00:32

his time is slowing down

00:35

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

00:38

the point beyond which not even light can escape,

00:41

he and his rocket ship do not disappear,

00:45

instead, they seem to stop frozen in time.

00:51

The light from the spaceship gets dimmer and redder

00:54

until it completely fades from view.

00:57

This is how any object would look

00:59

crossing the event horizon.

01:01

Light is still coming from the point where he crossed,

01:04

it's just too redshifted to see,

01:08

but if you could see that light,

01:10

then in theory you would see everything

01:12

that has ever fallen into the black hole

01:14

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

01:19

but in practice, photons are emitted at discreet intervals,

01:22

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

01:26

and therefore these images will fade after some time.

01:29

- This is just one of the strange results

01:32

that comes outta the general theory of relativity,

01:34

our current best theory of gravity.

01:36

The first solution of Einstein's equations

01:38

predicted not only black holes,

01:40

but also their opposite, white holes.

01:43

It also implied the existence of parallel universes

01:46

and even possibly a way to travel between them.

01:50

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

01:53

white holes, and wormholes.

01:56

- The general theory of relativity

01:58

arose at least in part due to a fundamental flaw

02:00

in Newtonian gravity.

02:02

In the 1600s Isaac Newton

02:04

contemplated how an apple falls to the ground,

02:06

how the moon orbits the earth and earth orbits the sun

02:09

and he concluded that every object with mass

02:12

must attract every other,

02:14

but Newton was troubled by his own theory.

02:17

How could masses separated by such vast distances

02:20

apply a force on each other?

02:22

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

02:26

through a vacuum without the mediation of anything else

02:29

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

02:33

who has a competent faculty of thinking

02:34

could ever fall into it."

02:38

One man who definitely had a competent faculty of thinking,

02:42

was Albert Einstein and over 200 years later,

02:45

he figured out how gravity is mediated.

02:48

Bodies do not exert forces on each other directly.

02:52

Instead, a mass like the sun curves the spacetime

02:55

in its immediate vicinity.

02:58

This, then curves the spacetime around it

03:00

and so on all the way to the earth.

03:03

So the earth orbits the sun, because the spacetime

03:06

earth is passing through is curved.

03:09

Masses are affected by the local curvature

03:12

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

03:16

Mathematically, this is described

03:18

by Einstein's field equations.

03:20

Can you write down the Einstein field equation?

03:23

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

03:26

after special relativity

03:28

and essentially what we've got in the field equations

03:30

on one side it says,

03:32

tell me about the distribution of matter and energy.

03:34

The other side tells you what the resultant curvature

03:37

of spacetime is from that distribution

03:40

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

03:43

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

03:46

But it's not really one equation.

03:47

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

03:51

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

03:54

and they are differential equations,

03:56

so it means that there are integrals

03:58

that have to be done, da, da da.

04:00

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

04:02

to solve the field equations.

04:04

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

04:07

we need a tool to understand spacetime.

04:11

So imagine your floating around in empty space.

04:14

A flash of light goes off above your head

04:16

and spreads out in all directions.

04:19

Now your entire future, anything that can

04:22

and will ever happen to you will occur within this bubble

04:27

because the only way to get out of it

04:29

would be to travel faster than light.

04:31

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

04:35

If we allow time to run up the screen

04:37

and take snapshots at regular intervals,

04:39

then this light bubble traces out a cone,

04:41

your future light cone.

04:43

By convention, the axes are scaled so that light rays

04:46

always travel at 45 degrees.

04:48

This cone reveals the only region of spacetime

04:51

that you can ever hope to explore and influence.

04:55

Now imagine that instead of a flash of light

04:57

above your head, those photons were actually traveling in

05:00

from all corners of the universe

05:02

and they met at that instant

05:03

and then continued traveling on

05:05

in their separate directions.

05:08

Well, in that case then into the past,

05:10

these photons also reveal a light cone,

05:13

your past light cone.

05:15

Only events that happened inside this cone

05:17

could have affected you up to the present moment.

05:21

We can simplify this diagram even further

05:23

by plotting just one spatial and one time dimension.

05:26

This is the spacetime diagram of empty space.

05:29

If you want to measure how far apart

05:31

two events are in spacetime, you use something called

05:34

the spacetime interval.

05:36

The interval squared is equal to minus dt squared,

05:39

plus dx squared, since spacetime is flat,

05:43

the geometry is the same everywhere

05:45

and so this formula holds throughout the entire diagram,

05:48

which makes it really easy to measure the separation

05:50

between any two events, but around a mass,

05:54

spacetime is curved and therefore you need to modify

05:57

the equation to take into account the geometry.

06:00

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

06:04

They tell you how spacetime curves

06:06

and how to measure the separation between two events

06:09

in that curved geometry.

06:12

Einstein published his equations in 1915

06:15

during the First World War,

06:16

but he couldn't find an exact solution.

06:19

Luckily, a copy of his paper made its way

06:22

to the eastern front where Germany was fighting Russia,

06:25

stationed there was one of the best astrophysicists

06:27

of the time, Karl Schwarzschild.

06:30

Despite being 41 years old, he had volunteered

06:33

to calculate artillery trajectories for the German army.

06:36

At least until a greater challenge caught his attention,

06:40

how to solve Einstein's field equations.

06:45

Schwarzschild did the standard physicist thing

06:47

and imagined the simplest possible scenario,

06:49

an eternal static universe with nothing in it

06:52

except a single spherically symmetric point mass.

06:55

This mass was electrically neutral and not rotating.

06:59

Since this was the only feature of his universe,

07:01

he measured everything using spherical coordinates

07:04

relative to this center of this mass.

07:06

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

07:10

For his time coordinate, he chose time as being measured

07:13

by someone far away from the mass,

07:15

where spacetime is essentially flat.

07:18

Using this approach, Schwarzschild found the first

07:20

non-trivial solution to Einstein's equations,

07:23

which nowadays we write like this.

07:26

This Schwarzschild metric describes how spacetime curves

07:30

outside of the mass.

07:32

It's pretty simple and makes intuitive sense,

07:34

far away from the mass spacetime is nearly flat,

07:37

but as you get closer and closer to it,

07:39

spacetime becomes more and more curved,

07:41

it attracts objects in and time runs slower.

07:45

(gunshots firing)

07:47

Schwarzschild sent his solution to Einstein,

07:49

concluding with, "The war treated me kindly enough

07:52

in spite of the heavy gunfire

07:54

to allow me to get away from it all

07:55

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

08:00

Einstein replied, "I have read your paper

08:02

with the utmost interest, I had not expected

08:04

that one could formulate the exact solution to the problem

08:06

in such a simple way."

08:12

But what seemed at first quite simple,

08:15

soon became more complicated.

08:17

Shortly after Schwarzschild solution was published,

08:19

people noticed two problem spots.

08:22

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

08:25

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

08:30

and therefore this equation breaks down

08:32

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

08:35

This is what's called a singularity.

08:38

Maybe that point could be excused,

08:40

because it's in the middle of the mass,

08:42

but there's another problem spot outside of it

08:45

at a special distance from the center

08:47

known as the Schwarzschild radius, this term blows up.

08:50

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

08:57

Well, at the Schwarzschild radius,

09:00

the spacetime curvature becomes so steep

09:02

that the escape velocity, the speed that anything would need

09:06

to leave there is the speed of light

09:10

and that would mean that inside the Schwarzschild radius,

09:13

nothing, not even light would be able to escape.

09:17

So you'd have this dark object

09:18

that swallows up matter and light,

09:22

a black hole, if you will,

09:26

but most scientists doubted that such an object could exist,

09:29

because it would require a lot of mass

09:31

to collapse down into a tiny space.

09:35

How could that possibly ever happen?

09:39

(thrilling music)

09:40

Astronomers at the time were studying

09:42

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

09:44

During its lifetime the inward force of gravity is balanced

09:47

by the outward radiation pressure

09:49

created by the energy released through nuclear fusion,

09:52

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

09:55

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

10:01

Most astronomers believed some physical process

10:04

would hold it up and in 1926,

10:07

Ralph Fowler came up with a possible mechanism.

10:10

Pauli's exclusion principles states that,

10:11

"Fermions like electrons cannot occupy the same state,

10:15

so as matter gets pushed closer and closer together,

10:18

the electrons each occupy their own tiny volumes,"

10:21

but Heisenberg's uncertainty principle says that,

10:23

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

10:26

with absolute certainty, so as the particles become

10:29

more and more constrained in space,

10:32

the uncertainty in their momentum,

10:34

and hence their velocity must go up."

10:37

So the more a star is compressed,

10:39

the faster electrons will wiggle around

10:41

and that creates an outward pressure.

10:44

This electron degeneracy pressure would prevent the star

10:47

from collapsing completely.

10:49

Instead, it would form a white dwarf

10:51

with the density much higher than a normal star

10:54

and remarkably enough astronomers had observed stars

10:57

that fit this description.

10:58

One of them was Sirius B.

11:04

But the relief from this discovery was short-lived.

11:06

Four years later, 19-year-old Subrahmanyan Chandrasekhar

11:09

traveled by boat to England to study with Fowler

11:12

and Arthur Eddington, one of the most revered scientists

11:15

of the time.

11:17

During his voyage, Chandrasekhar realized

11:19

that electron degeneracy pressure has its limits.

11:22

Electrons can wiggle faster and faster,

11:24

but only up to the speed of light.

11:27

That means this effect can only support stars

11:30

up to a certain mass, the Chandrasekhar limit.

11:33

Beyond this, Chandrasekhar believed,

11:35

not even electron de degeneracy pressure

11:37

could prevent a star from collapsing,

11:40

but Eddington was not impressed.

11:42

He publicly blasted Chandrasekhar saying,

11:45

"There should be a law of nature

11:47

to prevent a star from behaving in this absurd way"

11:51

and indeed scientists did discover a way

11:53

that stars heavier than the Chandrasekhar limit

11:55

could support themselves.

11:58

When a star collapses beyond a white dwarf,

12:00

electrons and protons fuse together

12:02

to form neutrinos and neutrons.

12:05

These neutrons are also fermions,

12:07

but with nearly 2000 times the mass an electron,

12:10

their degeneracy pressure is even stronger.

12:13

So this is what holds up neutron stars.

12:16

There was this conviction among scientists

12:19

that even if we didn't know the mechanism,

12:21

something would prevent a star from collapsing

12:23

into a single point and forming a black hole,

12:28

because black holes were just too preposterous to be real.

12:34

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

12:38

when Jay Robert Oppenheimer and George Volkoff

12:40

found that neutron stars also have a maximum mass.

12:44

Shortly after Oppenheimer and Hartland Snyder

12:47

showed that for the heaviest stars,

12:49

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

12:53

they wrote, "This contraction will continue indefinitely,"

12:58

but Einstein still couldn't believe it.

13:00

Oppenheimer was saying that stars can collapse indefinitely,

13:03

but when Einstein looked at the math,

13:05

he found that time freezes on the horizon.

13:08

So it seemed like nothing could ever enter,

13:11

which suggested that either

13:12

there's something we don't understand

13:14

or that black holes can't exist,

13:17

(star explodes)

13:21

but Oppenheimer offered a solution to the problem.

13:23

He said to an outside observer,

13:26

you could never see anything go in,

13:27

but if you were traveling across the event horizon,

13:31

you wouldn't notice anything unusual

13:33

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

13:37

So how is this possible?

13:39

We need a spacetime diagram of a black hole.

13:43

On the left is the singularity at r equals zero.

13:46

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

13:49

Since the black hole doesn't move,

13:51

these lines go straight up in time.

13:55

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

13:58

in this curved geometry.

14:01

When you're really far away,

14:02

the future light cones are at the usual 45 degrees,

14:06

but as you get closer to the horizon,

14:07

the light cones get narrower and narrower,

14:11

until right at the event horizon,

14:13

they're so narrow that they point straight up

14:16

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

14:22

but something strange happens with ingoing light rays.

14:26

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

14:29

they actually asymptote to that value

14:32

as time goes to infinity,

14:34

but they don't end at infinity, right?

14:36

Mathematically they are connected and come back in

14:41

and they're traveling in this direction

14:44

and this bothered a lot of people,

14:46

this bothered people like Einstein,

14:48

because he looked at these equations and went,

14:50

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

14:55

then how could there be black holes?

14:57

How could black holes even form?"

15:00

- So what is going on here?

15:02

Well, what's important to recognize

15:04

is that this diagram is a projection.

15:06

It's basically a 2D map

15:08

of four dimensional curved spacetime.

15:12

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

15:15

When you do that, you always get distortions.

15:18

There is no perfectly accurate way

15:20

to map the earth onto a 2D surface,

15:22

but different maps can be useful for different purposes.

15:25

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

15:28

like if you're sailing across the ocean

15:30

and you need to find your bearings,

15:31

you can use the Mercator projection,

15:33

that's the one Google Maps uses.

15:35

A downside is that it misrepresent sizes.

15:39

Africa and Greenland look about the same size,

15:42

but Africa is actually around 14 times larger.

15:46

The Gall-Peters projection keeps relative sizes accurate,

15:49

but as a result, angles and shapes are distorted.

15:53

In a similar way, we can make different projections

15:56

of 4D spacetime to study different properties of it.

16:00

Physical reality doesn't change,

16:01

but the way the map describes it does.

16:05

- He had chosen to put a particular coordinate system

16:07

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

16:11

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

16:14

- [Derek] People realize that if you choose

16:15

a different coordinate system

16:17

by doing a coordinate substitution, then the singularity

16:20

at the event horizon disappears.

16:23

- It goes away.

16:24

That problem goes away and things can actually cross

16:27

into the black hole.

16:30

- What this tells us is that there is

16:32

no real physical singularity at the event horizon.

16:36

It just resulted from a poor choice of coordinate system.

16:41

Another way to visualize what's going on

16:44

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

16:48

like a waterfall.

16:49

As you get closer, space starts flowing in

16:52

faster and faster.

16:54

Photons emitted by the spaceship have to swim

16:56

against this flow, and this becomes harder and harder

17:00

the closer you get.

17:02

Photons emitted just outside the horizon

17:04

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

17:09

At the horizon, space falls in

17:11

as fast as the photons are swimming.

17:13

So if the horizon had a finite width,

17:16

then photons would get stuck here,

17:18

photons from everything that ever fell in,

17:21

but the horizon is infinitely thin.

17:23

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

17:29

Inside the horizon, space falls faster

17:31

than the speed of light,

17:33

and so everything falls into the singularity.

17:36

So Oppenheimer was right.

17:38

Someone outside a black hole can never see anything enter

17:42

because the last photons they can see

17:44

will always be from just outside the horizon,

17:48

but if you yourself go,

17:50

you will fall right across the event horizon

17:52

and into the singularity.

17:55

Now you can extend the waterfall model

17:57

to cover all three spatial dimensions,

17:59

and that gives you this, a real simulation

18:02

of space flowing into a static black hole

18:05

made by my friend Alessandro from ScienceClic.

18:08

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

18:10

falling into a rotating black hole.

18:16

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18:18

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18:20

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18:22

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18:25

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20:19

and now back to spacetime maps.

20:22

If you take this map and transform it

20:25

so that incoming and outgoing light ray

20:27

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

20:30

then something fascinating happens.

20:33

The black hole singularity on the left

20:35

transforms into a curved line at the top

20:40

and since the future always points up in this map,

20:44

it tells us that the singularity is not actually

20:47

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

20:52

the very last moment in time for anything

20:55

that enters a black hole.

20:58

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

21:02

but this only represents a portion of the universe,

21:04

the part inside the black holes event horizon

21:07

and the part of the universe closest to it,

21:10

but what we can do is contract the whole universe,

21:13

the infinite past, infinite distance, and infinite future,

21:17

and morph it into a single map.

21:20

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

21:24

That gives us this penrose diagram.

21:28

Again, light rays still always go at 45 degrees.

21:31

So the future always points up.

21:33

The infinite past is in the bottom of the diagram.

21:36

The infinite future at the top

21:39

and the sides on the right are infinitely far away.

21:42

The black hole singularity is now a straight line

21:44

at the top, a final moment in time.

21:49

These lines are all at the same distance

21:51

from the black hole.

21:52

So the singularity is at r equals zero,

21:55

the horizon is at r equals 2M,

21:57

this line is at r equals 4M,

21:59

and this is infinitely far away.

22:02

All of these lines are at the same time.

22:05

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

22:09

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

22:12

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

22:15

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

22:19

and you can see and receive information from this area,

22:23

but if you go beyond the horizon,

22:25

your only possible future is to meet the singularity.

22:29

You can still, however, see and receive information

22:32

from the universe.

22:33

You just can't send any back out.

22:36

Now think about being at this point in the map.

22:39

This is at the event horizon,

22:41

and now your entire future is within the black hole,

22:45

but what is the past of this moment?

22:48

Well, you can draw the past light cone

22:51

and it reveals this new region.

22:54

If you're inside this region,

22:56

you can send signals to the universe,

22:58

but no matter where you are in the universe,

23:00

nothing can ever enter this region

23:02

because it will never be inside your light comb.

23:06

So things can come out, but never go in.

23:09

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

23:15

What color is a white hole?

23:18

(Geraint exhales)

23:19

(Derek laughs)

23:20

- I mean, it's gonna be the,

23:22

it's not gonna have a color, right?

23:24

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

23:27

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

23:31

that's what you are going to see.

23:33

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

23:35

it's all gonna be ejected.

23:36

So the white hole kind of picture

23:39

is the time reverse picture of a black hole,

23:42

instead of things falling in, things get expelled outwards

23:46

and so whilst a black hole has a membrane,

23:51

the Schwarzschild horizon, which once you cross,

23:53

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

23:56

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

23:59

so it kicks you out kind of thing, right?

24:01

Relativity doesn't tell you which way time flows.

24:04

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

24:08

and that is the past.

24:10

When you are doing your mathematics

24:12

and you're working out the behavior of objects,

24:15

you make a choice about which direction is the future,

24:19

but mathematically, you could have chosen

24:21

the other way, right?

24:22

You could have had time point in the opposite direction.

24:25

Any solution that you find in relativity,

24:28

mathematically, you can just flip it

24:30

and get a time reverse solution

24:32

and that's also a solution to the equations.

24:36

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

24:37

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

24:40

be ejected to the left.

24:42

So what's over there?

24:44

This line is not at infinity,

24:46

so there should be something beyond it.

24:49

If we eject things in this direction,

24:51

you find that they enter a whole new universe,

24:55

one parallel to our own.

25:01

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

25:03

and somebody in this universe here

25:05

could fall into this black hole in their universe,

25:08

and we would find ourselves in the same black hole.

25:11

(Derek chuckles)

25:12

- The only downside is that

25:14

we'd both soon end up in the singularity.

25:18

I guess I'm just trying to understand

25:20

where that universe appears

25:22

in the mathematical part of the solution.

25:24

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

25:27

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

25:30

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

25:32

Like- - Yeah,

25:33

well, it's coordinates, right?

25:35

Imagine somebody, right, came up with a coordinate system

25:40

for the earth, but only the northern hemisphere

25:43

and you looked at that coordinate system, right?

25:45

And you looked at it and you said,

25:47

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

25:50

but mathematically latitudes can be negative, right?

25:55

You've only got positive latitudes in your solution.

25:58

What about the negative ones?"

25:59

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

26:02

No southern hemisphere, right?"

26:05

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

26:08

you can have negative latitudes.

26:09

Maybe we should go and look over the equator

26:12

to see if there is something down there"

26:13

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

26:16

because we know we live on a globe,

26:17

but we don't know the full geometry

26:20

of what's going on here in the sense that

26:22

Schwarzschild laid down coordinates

26:24

over part of the solution.

26:26

It was like him only laying down coordinates

26:28

on the northern hemisphere

26:30

and other people have come along and said,

26:32

"Hey, there's a southern hemisphere"

26:34

and more than that, there's two earths.

26:36

That's why it's called maximal extension.

26:39

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

26:43

then what is the extent of the coordinates

26:47

that I can consider?

26:49

And with the Schwarzschild black hole,

26:51

you get a second universe

26:52

that has its own independent set of coordinates

26:56

from our universe.

26:57

I want to emphasize right, this is the simplest solution

27:00

to the Einstein field equations,

27:02

and it already contains a black hole,

27:03

white hole and two universes.

27:05

- [Derek] That's what you get

27:07

when you push this map to its limits

27:09

so that every edge ends at a singularity or infinity.

27:13

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

27:16

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

27:19

that is an Einstein Rosen Bridge.

27:24

- To see it, we need to change coordinates.

27:27

Now this line is at constant crustal time

27:30

and it connects the space of both universes.

27:33

You can see what the spacetime is like

27:35

by following this line from right to left.

27:38

Far away from the event horizon,

27:39

spacetime is basically flat,

27:41

but as you get closer to the event horizon,

27:43

spacetime starts to curve more and more.

27:46

At this cross, you are at the event horizon,

27:49

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

27:53

that gives you a wormhole that looks like this.

27:59

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

28:04

to travel from one universe to another.

28:06

- Hypothetically, because these wormholes

28:08

aren't actually stable in time.

28:11

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

28:14

and then becomes shorter and then becomes long again

28:17

and if you try to traverse this bridge,

28:19

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

28:21

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

28:23

But as you start crossing the bridge and start running,

28:25

your speed is finite, right?

28:27

The speed of light roughly and then the bridge starts,

28:30

becoming stretching and you never come out the other side.

28:35

- [Derek] This pinching off always happens too fast

28:38

for anything to travel through.

28:40

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

28:43

because when you're inside one universe,

28:45

there isn't a light cone that can take you

28:47

to the other universe.

28:49

The only way to do that

28:50

would be to travel faster than light,

28:54

but there might be another way.

28:57

Schwarzschild solution describes a black hole

28:59

that doesn't rotate.

29:00

Yet, every star does rotate

29:02

and since angular momentum must be conserved,

29:04

every black hole must also be rotating.

29:08

While Schwarzschild found his solution within weeks

29:10

after Einstein published his equations,

29:12

solving them for a spinning mass

29:14

turned out to be much harder.

29:15

Physicists tried, but 10 years after Schwarzschild solution,

29:19

they still hadn't solved it.

29:21

10 years turned into 20, which turned into 40

29:24

and then in 1963, Roy Kerr found the solution

29:28

to Einstein's equations for a spinning black hole,

29:32

which is far more complicated than Schwarzschild solution

29:35

and this comes with a few dramatic changes.

29:40

The first is that the structure is completely different.

29:43

The black hole now consists of several layers.

29:47

It's also not spherically symmetric anymore.

29:50

This happens because the rotation

29:52

causes it to bulge around the equator.

29:54

So it's only symmetric about its axis of spin.

29:59

Alessandro from science click simulated what happens

30:02

around this spinning black hole.

30:07

Space gets dragged around with the black hole

30:10

taking you and the particles along with it.

30:13

When you get closer, space gets dragged around

30:15

faster and faster until it goes around faster

30:19

than the speed of light,

30:20

you've now entered into the first new region,

30:24

the ergosphere.

30:26

No matter how hard you fire your rockets here,

30:29

it's impossible to stay still relative to distance stars,

30:33

but because space doesn't flow directly inward,

30:36

you can still escape the black hole.

30:39

When you travel in further, you go through the next layer,

30:42

the outer horizon, the point of no return.

30:46

Here you can only go inwards,

30:49

but as you get dragged in deeper and deeper,

30:52

something crazy happens, you enter another region,

30:57

one where you can move around freely again,

31:00

so you're not doomed to the singularity.

31:03

You're now inside the inner event horizon.

31:07

Here you can actually see the singularity

31:12

- In a normal black hole, it's a point,

31:13

but it in a rotating black hole,

31:14

it actually expands out to be a ring

31:17

and there are weird things happened

31:19

with spacetime inside the center of a black hole,

31:22

a rotating black hole,

31:23

but it's thought that you can actually

31:24

fly through the singularity.

31:29

- [Derek] We need a Penrose diagram

31:31

of a spinning black hole, where before the singularity

31:35

was a horizontal line at the top

31:37

here, the singularity lifts up and moves to the sides,

31:40

revealing this new region inside the inner horizon.

31:45

Here we can move around freely and avoid the singularity,

31:49

but these edges aren't at infinity or a singularity,

31:53

so there must be something beyond them.

31:55

Well, when you venture further,

31:57

you could find yourself in a white hole,

32:00

which would push you out into a whole nother universe.

32:05

- You can have these pictures whereby

32:08

you're in one universe, you fall into a rotating black hole,

32:12

you fly through the singularity,

32:14

and you pop out into a new universe from a white hole,

32:18

and then you can just continue playing this game.

32:21

- Extending this diagram infinitely far.

32:25

but there is still one thing we haven't done,

32:27

brave the singularity.

32:30

So you aim straight towards the center of the ring

32:33

and head off towards it, but rather than time ending,

32:37

you now find yourself in universe, a strange universe,

32:40

one where gravity pushes instead of pulls.

32:44

This is known as an anti-verse.

32:48

If that's too weird, you can always jump back

32:50

across the singularity and return to a universe

32:53

with normal gravity.

32:55

- And I know this is basically science fiction, right?

32:57

But if you take the solutions of relativity at,

33:02

you know, essentially at face value and add on a little bit,

33:05

which is what Penrose does here, he says this,

33:07

"oh look, these shapes are very similar,

33:10

I can just stick these together."

33:12

Then this is the conclusion that you get.

33:14

Now we have effectively an infinite number

33:17

of universes all connected with black hole, white holes

33:20

all the way through and you, of you go to explore,

33:25

but it'll be a very brave person who's the first one

33:28

who's gonna leap into a rotating black hole

33:30

to find out if this is correct?

33:31

(Derek chuckles)

33:33

- Yeah, I would not sign up for that.

33:35

So could these maximally extended Schwarzschild

33:38

and Kerr solutions actually exist in nature?

33:41

Well, there are some issues.

33:43

Both the extended Schwarzschild and Kerr solutions

33:46

are solutions of eternal black holes in an empty universe.

33:50

- As you say, it's an eternal solution.

33:52

So it stretches infinitely far into the past

33:55

and infinitely far into the future

33:57

and so there's no formation mechanism in there,

33:59

it's just a static solution

34:02

and I think that is part of the,

34:07

part of the reason why black holes

34:10

are realized in our universe and white holes aren't-

34:15

- Or might not be.

34:16

- Or might not be,

34:16

or I'm reasonably I,

34:18

personally, I'm reasonably confident

34:20

that they don't exist, right?

34:22

- [Derek] For the maximally extended Kerr solution,

34:24

there's also another problem.

34:26

If you're an immortal astronaut inside the universe,

34:28

you can send light into the black hole,

34:31

but because there's infinite time compressed

34:34

in this top corner, you can pile up light along this edge,

34:37

which creates an infinite flux of energy

34:40

along the inner horizon.

34:42

This concentration of energy

34:43

then creates its own singularity,

34:46

sealing off the ring singularity and beyond.

34:50

- My suspicion and the suspicion

34:51

of some other people in the field is that

34:55

this inner horizon will become singular

34:56

and you will not be able to go through these second copies.

34:59

- So all the white holes, wormholes, other universes

35:03

and anti universes disappear.

35:06

Does that mean that real wormholes are impossible?

35:10

In 1987, Michael Morris and Kip Thorne looked at wormholes

35:13

that an advanced civilization could use

35:15

for interstellar travel, ones that have no horizons,

35:18

so you can travel back and forth, are stable in time,

35:20

and have some other properties like

35:22

being able to construct them.

35:24

They found several geometries that are allowed

35:26

by Einstein's general relativity.

35:28

In theory, these could connect different parts

35:31

of the universe, making a sort of interstellar highway.

35:34

They might even be able to connect to different universes.

35:39

The only problem is that all these geometries

35:42

require an exotic kind of matter

35:44

with a negative energy density

35:45

to prevent the wormhole from collapsing.

35:48

- This exotic kind of matter,

35:50

is really against the loss of physics, so it's,

35:54

I have the prejudice that it will not exist.

35:56

I'm bothered by the fact that we say that

35:58

the science fiction wormholes are mathematically possible.

36:01

It's true, it's mathematically possible

36:03

in the sense that there's some geometry that can exist,

36:05

but Einstein's theory is not just geometries,

36:09

it's geometries plus field equations.

36:12

If you use the kinds of properties of matter

36:14

that matter actually has, then they're not possible.

36:17

So I feel that the reason they're not possible

36:20

is very strong.

36:22

- So according to our current best understanding,

36:25

it seems likely that white holes, traversable wormholes,

36:28

and these parallel universes don't exist,

36:32

but we also used to think that black holes didn't exist.

36:35

So maybe we'll be surprised again.

36:38

- I mean, we have one universe, right?

36:41

Good, why can't we have two.

36:46

(whimsical music)

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Verwandte Tags
Black HolesWhite HolesWormholesGeneral RelativitySpacetimeEinsteinSingularityAstronomyCosmologySciFi
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