# Something Strange Happens When You Follow Einstein's Math

May 14, 2024
- You can never see anything enter a black hole. (bell rings) Imagine you trap your nemesis in a rocket and launch him into a black hole. He looks at you shaking his fist at a steady pace. As it gets closer, gravity becomes stronger, so you would expect it to accelerate, but that's not what you see. Instead, the rocket appears to be slowing down. Not only that, he also seems to be shaking his fist slower and slower. That's because from your perspective, his time is slowing down at the very instant he should cross the event horizon, the point beyond which not even light can escape, he and his rocket don't disappear, instead they appear to stop. frozen. time.
The spaceship's light becomes dimmer and redder until it completely disappears from view. This is what any object crossing the event horizon would look like. The light is still coming from the point where it crossed, it's simply too redshifted to see, but if you could see that light, then in theory you would see everything that ever fell into the frozen black hole on its horizon, including the star that formed it, but in practice photons are emitted at discrete intervals, so there will be a last photon emitted outside the horizon and therefore these images will fade after a while. - This is just one of the

### strange

results that emerge from the general theory of relativity, our best current theory of gravity.

## something strange happens when you follow einstein s math...

The first solution of Einstein's equations predicted not only black holes, but also their opposites, white holes. It also implied the existence of parallel universes and possibly even 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, because of a fundamental flaw in Newtonian gravity. In the 17th century, Isaac Newton watched how an apple falls to the ground, how the Moon orbits the Earth, and the Earth orbits the Sun, and concluded that all objects with mass must attract each other, but Newton was concerned about his own theory. .
How could masses separated by such great distances apply a force to each other? He wrote: "That one body can act on another at a distance through a vacuum without the mediation of anything else is to me so great and absurd that I believe that no man who has a competent faculty of thinking could ever fall into it." One man who definitely had a competent faculty of thought was Albert Einstein and more than 200 years later, he discovered how gravity is mediated. Bodies do not exert forces on each other directly. On the other hand, a mass like the Sun bends space-time in its immediate vicinity.
This then curves spacetime around it and so on until it reaches Earth. So the Earth orbits the Sun, because the space-time that the Earth passes 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 the Einstein field equation? - This was the result of Einstein's decade of hard work after special relativity and essentially what we have in the field equations on one side says, tell me about the distribution of matter and energy. The other side tells you what the resulting curvature of space-time is from that distribution of matter and energy and it is a single line.
It seems like this is a simple equation, right? But it's not really an equation. It's a family of equations, and to make life harder, they're coupled equations, so they depend on each other, and they're differential equations, so it means there are integrals that have to be done, da, da da. So there are a lot of steps you have to

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to solve the field equations. To see what a solution to these equations would look like, we need a tool for understanding spacetime. So imagine that you are floating in empty space. A flash of light shoots over your head and spreads in all directions.
Now, their entire future, anything that can and will ever happen to them, will happen inside this bubble because the only way out of it would be to travel faster than light. In two dimensions, this bubble is just a growing circle. If we give time to scan the screen and take photographs at regular intervals, this bubble of light will trace a cone, its future cone of light. By convention, the axes are scaled so that light rays always travel at 45 degrees. This cone reveals the only region of space-time that you can hope to explore and influence. Now imagine that instead of a flash of light overhead, those photons were actually traveling from all corners of the universe and they met at that instant and then continued traveling in separate directions.
Well, in that case, then into the past, these photons also reveal a light cone, their past light cone. Only the events that happened within this cone could have affected you up to the present moment. We can further simplify this diagram by plotting only one spatial and one temporal dimension. This is the space-time diagram of empty space. If you want to measure how far apart two events are in space-time, you use

## something

called a space-time interval. The squared interval is equal to minus dt squared, plus dx squared, since spacetime is flat, the geometry is the same everywhere and that is why this formula holds throughout the diagram, which makes it really easy to measure the separation between any two events, but around a mass, spacetime is curved and therefore it is necessary to modify the equation to take the geometry into account.
These are the solutions to Einstein's equations. 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 World War I, but he could not find an exact solution. Fortunately, a copy of his article reached the Eastern Front where Germany was fighting Russia, where one of the best astrophysicists of the time, Karl Schwarzschild, was stationed. Despite being 41 years old, he had volunteered to calculate artillery trajectories for the German army. At least until he was drawn to a bigger challenge: how to solve Einstein's field equations.
Schwarzschild did the typical 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 did not rotate. 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 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 we write like this today.
This Schwarzschild metric describes how spacetime curves away from mass. It's pretty simple and makes intuitive sense, away from mass space-time is almost flat, but as you get closer and closer to it, space-time becomes more and more curved, attracts objects and time runs faster. slow. (gunshots) Schwarzschild sent his solution to Einstein, concluding with: "The war treated me kindly enough despite the intense shooting to allow me to get away from it all and take this walk through the land of your ideas." Einstein responded: "I have read his article with great interest. I did not expect that the exact solution to the problem could be formulated in such a simple way." But what seemed simple enough at first soon became more complicated.
Shortly after Schwarzschild's solution was published, people noticed two problem points. At the center of mass, at r equals zero, this term divides by zero, so it explodes to infinity and therefore this equation breaks down and can no longer describe what is happening physically. This is what is called a singularity. Maybe that point could be excused, because it is in the middle of the mass, but there is another problem point outside of it, at a special distance from the center known as the Schwarzschild radius, this term explodes. So there is a second singularity. What's going on here?
Well, at the Schwarzschild radius, the curvature of spacetime becomes so steep that the escape velocity, the speed at which anything would need to get out of there, is the speed of light and that would mean that within the Schwarzschild radius , nothing, not even light, could escape. Then we would have this dark object swallowing 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 into a tiny space. How could that ever happen? (exciting music) Astronomers of 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 stellar material inward, but how far? Most astronomers believed that some physical process would slow it down, and in 1926, Ralph Fowler devised a possible mechanism. The Pauli exclusion principles state that "fermions like electrons cannot occupy the same state, so as matter gets closer and closer, the electrons each occupy their own tiny volume", but the Pauli exclusion principle Heisenberg's uncertainty says that "you cannot know the position and momentum of a particle with absolute certainty, so as particles become increasingly limited in space, the uncertainty in their momentum and therefore in its speed must increase." So the more a star is compressed, the faster the electrons will move and that creates outward pressure.
This electronic degeneracy pressure would prevent the star from collapsing completely. Instead, a white dwarf would form with a density much greater than that of a normal star, and surprisingly, astronomers had observed stars that fit this description. One of them was Sirius B. But the relief of this discovery was short-lived. Four years later, 19-year-old Subrahmanyan Chandrasekhar traveled by ship to England to study with Fowler and Arthur Eddington, one of the most revered scientists of the time. During his journey, Chandrasekhar realized that the pressure of electron degeneration has its limits. Electrons can move 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 that not even the degeneracy pressure of electrons could prevent a star from collapsing, but Eddington was not impressed. He publicly criticized Chandrasekhar by saying, "There should be a law of nature to prevent a star from behaving in this absurd way" and, in fact, scientists discovered a way in which stars heavier than Chandrasekhar's limit could be held up at themselves. When a star collapses past a white dwarf, electrons and protons fuse to form neutrinos and neutrons. These neutrons are also fermions, but at almost 2,000 times the mass of the electron, their degeneracy pressure is even greater.
So this is what sustains neutron stars. There was a 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 too absurd to be real. The big blow to this belief came in the late 1930s,

## when

Jay Robert Oppenheimer and George Volkoff discovered that neutron stars also have a maximum mass. Shortly after Oppenheimer and Hartland Snyder showed that the heaviest stars have nothing left to save them when they run out of fuel, they wrote, "This contraction will continue indefinitely," but Einstein still couldn't believe it.
Oppenheimer said that stars can collapse indefinitely, but when Einstein looked at the

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ematics, he discovered that time freezes on the horizon. So it seemed like nothing could ever get in, suggesting that there is something we don't understand or that black holes can't exist (the star explodes), but Oppenheimer offered a solution to the problem. He told an outside observer that you could never see anything coming in, but if you were traveling through the event horizon, you wouldn't notice anything unusual and would walk right past it without even knowing it. How is this possible? We need a space-time diagram of a black hole.
On the left is the singularity at r equal to zero. The dotted line at r equals 2M is the event horizon. Since the black hole does not move, these lines rise in time. Now let's see how the incoming and outgoing light rays travel in this curved geometry. When you are very far away, the future light cones are at 45usual degrees, but as you get closer to the horizon, the light cones become narrower and narrower, until just at the event horizon, they are so narrow that they point up and into the horizon, the light cones tilt to the left, but something

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#### happens

to the incoming light rays. - They fall, but they do not reach r equal to 2M, in reality they asymptote to that value as time reaches infinity, but they do not end at infinity, right?
Mathematically they are connected and they come back and travel in this direction and this bothered a lot of people, this bothered people like Einstein, because he looked at these equations and said, "well, if nothing can cross this kind of limit, then how could the black holes? How could black holes even form? - So what's 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 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 on a 2D surface, but different maps can be useful for different purposes.
You want to keep the angles and shapes the same, like if you were sailing across the ocean and need to find your orientations, you can use the Mercator projection, which is what Google Maps uses. One downside is that it misrepresents sizes. Africa and Greenland look about the same size, but in reality Africa is about 14 times larger. The Gall-Peters projection keeps relative sizes accurate, but as a result, angles and shapes become distorted. Similarly, we can perform different projections of 4D spacetime to study different properties of it. Physical reality does not change, but the way the map describes it does. - I had chosen to put a particular coordinate system of a space and have a time coordinate, and that's it.
It's the most sensible thing to do, right? - People realize that if you choose a different coordinate system by performing coordinate substitution, the singularity on the event horizon disappears. - It goes. 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 was simply the result of a poor choice of coordinate system. Another way to visualize what is happening is to describe space as flowing towards the black hole, like a waterfall. As you get closer, space begins to flow faster and faster.
The photons emitted by the spacecraft have to swim against this flow, and this becomes increasingly difficult the closer you get. Photons emitted just outside the horizon can barely make it out, but they take longer and longer. At the horizon, space falls as fast as photons swim. So if the horizon had a finite width, then photons would get trapped here, photons from everything that ever fell there, but the horizon is infinitely thin. So in reality, the photons eventually escape or fall. Within 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 will never be able to see anything enter because the last photons they will be able to see will always be the ones just outside the horizon, but if you go yourself, you will cross the event horizon and enter 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 will use this model to see what it is like to fall into a rotating black hole.
Now, I've never been sucked into a black hole, but sometimes I feel that way when I'm stuck on the phone with a spam necklace. Fortunately, today's sponsor, Incogni, can help. You know, I used to get several unwanted calls a day and they frustrated me so much that I wrote a letter to the Do Not Call registry to get my number removed, but that didn't work and that's why I wanted to go offensive. I even considered making a video where I just mess with spam callers to get revenge for all the frustration and wasted time, but I don't have to do that anymore thanks to Incogni.
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So, to try Incogni and fight data brokers, visit incog.com/veritasium. You can click the link in the description or scan the QR code right here and be sure to use the code veritasium to get 60% off. Head to incog.com/veritasium to get started. I want to thank Incogni for sponsoring this part of the video and now let's get back to the space-time maps. If you take this map and transform it so that the incoming and outgoing light rays travel at 45 degrees like we are used to, then something fascinating will happen. The black hole singularity on the left transforms into a curved line at the top and since the future always points up on this map, it tells us that the singularity is not actually a place in space, but a moment in time. time, the same. last moment in time for anything that enters a black hole.
The map we just created is a Kruskal-Szekers diagram, but this only represents a portion of the universe, the part within the event horizon of the black holes and the part of the universe closest to it, but what we can do is Collapse the entire universe, the infinite past, the infinite distance and the infinite future, and transform it into a single map. It's like using the best fisheye lens in the universe. That gives us this Penrose diagram. Once again, the light rays always go at 45 degrees. That's why the future always points up. The infinite past is at the bottom of the diagram.
The infinite future at the top and right sides are infinitely far away. The black hole singularity is now a straight line at the top, a last moment in time. All of these lines are 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 these lines are at the same time. The nice thing about this map is that it's very easy to see where you can still go and what might have affected you.
For example, when you are here, you have a lot of freedom. You can enter the black hole or fly 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 encounter the singularity. However, you can still see and receive information from the universe. You simply can't return any. Now think about being at this point on the map. This is on the event horizon, and now your entire future is inside the black hole, but what is the past of this moment? Well, you can draw the light cone from the past and it reveals this new region.
If you are within this region, you can send signals to the universe, but no matter where in the universe you are, nothing can ever enter this region because it will never be within your light comb. So things can go out, but never come 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 will be him, he will have no color, right? It will be whatever is spit out. It depends on what's inside and what's thrown out, that's what you're going to see.
So if there's light there, if it has mass, everything will be ejected. So the white hole image is the time reverse image of a black hole, instead of things falling in, things are thrown outwards and therefore, while a black hole has a membrane, the horizon of Schwarzschild, that once you cross, you cannot get out again, the white hole has the complete opposite. If you're within the event horizon, you have to be kicked out, so you get kicked out, right? Relativity doesn't tell you which direction time flows. There's nothing there that says that, that's the future and that's the past.
When you do

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and calculate the behavior of objects, you choose which direction the future is, but mathematically, you could have chosen the other path, right? You could have made time point in the opposite direction. Any solution you find in relativity, mathematically, you can just invert it and get an inverse solution in time and that is also a solution to the equations. - Now, we've been showing things being ejected to the right, but they could also be ejected to the left. So what's there? This line is not at infinity, so there should be something beyond it.
If we expel things in this direction, you will find that they enter a completely new universe, parallel to our own. -We could fall into this black hole, and someone in this universe here could fall into this black hole in his universe, and we would find ourselves in the same black hole. (Derek laughs) - The only downside is that we would 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. Can you point to the part of the equation and say, 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, they're coordinates, right? Imagine someone 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 said, "Oh, I can see the coordinate system, it looks good, but mathematically latitudes can be negative, right? You only have positive latitudes in your solution. What about the negative ones?" And they told you, (scoffs) "Negative? There is no southern hemisphere, right?" And you have to say, "Well, the math says you can have negative latitudes. Maybe we should go and look over the equator to see if there's something down there" and I know that's kind of an extreme example, because we know we live in a globe, but we don't know the full geometry of what's happening here in the sense that Schwarzschild set coordinates over part of the solution.
It was like he just set coordinates in the northern hemisphere and other people came and said, "Hey, there's a southern hemisphere," and more than that, there are two Earths. That is why it is called maximum extension. It's like, if I have this mathematical structure, 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 set of coordinates independent of ours. I want to emphasize that this is the simplest solution to Einstein's field equations and already contains a black hole, a white hole and two universes. - That's what you get when you push this map to the limit so that every edge ends in a singularity or infinity. - And actually, there's another little feature here, which is that little point where they intersect, which is an Einstein Rosen bridge. - To see it, we need to change the coordinates.
Now, this line is in constant cortical time and connects the space of both universes. You can see what spacetime is like by

#### follow

ing this line from right to left. Far from the event horizon, spacetime is basically flat, but as you get closer to the event horizon, spacetime begins to curve more and more. At this cross, you are at the event horizon and if you go further, you end up in the parallel universe which gives you a wormhole that looks like this one. - This is how we could hypothetically use a black hole to travel from one universe to another. - Hypothetically, because these wormholes are not actually stable over time. - It's a bit like a bridge, but it's a bridge that is long and then it gets shorter and then it gets longer again and if you try to cross this bridge, at some point, the bridge is very short, right?
And you say, "Oh, well, let me cross this bridge." But when you start crossing the bridge and you start running, your speed is finite, right? The speed of light is abrupt and then the bridge starts, stretches and you never get out.other side. - This pinch always happens too quickly for anything to get through. You can also see this if you look at the Penrose diagram, because when you are inside one universe, there is no cone of light that can take you to the other universe. The only way to do this would be to travel faster than light, but there could be another way.
Schwarzschild's solution describes a non-spinning black hole. However, every star spins and since angular momentum must be conserved, every black hole must also be spinning. While Schwarzschild found his solution a few weeks after Einstein published his equations, solving them for a rotating mass turned out to be much more difficult. Physicists tried, but 10 years after Schwarzschild's solution, they still hadn't solved it. 10 years became 20, which became 40 and then, in 1963, Roy Kerr found the solution to Einstein's equations for a rotating black hole, which is much more complicated than Schwarzschild's solution and comes with some dramatic changes. . The first is that the structure is completely different.
The black hole now consists of several layers. It is also no longer spherically symmetrical. This happens because rotation causes it to bulge around the equator. Then it is only symmetrical with respect to its axis of rotation. Science Click's Alessandro simulated what happens around this spinning black hole. Space is swept away and the black hole takes you and the particles with it. As you get closer, space crawls faster and faster until it is spinning faster than the speed of light. You have now entered the first new region, the ergosphere. No matter how hard you fire your rockets here, it's impossible to stay still relative to distant stars, but since space doesn't flow directly in, you can still escape the black hole.
When you travel further, you pass through the next layer, the outer horizon, the point of no return. Here you can only go inward, but as you get dragged further and further, something crazy happens: you enter another region, one where you can move freely again, so you're not doomed to the singularity. You are now within the internal event horizon. Here you can see the singularity: in a normal black hole, it is a point, but in a rotating black hole, it actually expands into a ring and strange things happen with spacetime inside the center of a black hole . a rotating black hole, but it is believed that you can actually fly through the singularity. - We need a Penrose diagram of a rotating black hole, where before the singularity was a horizontal line at the top, here the singularity rises and moves sideways, revealing this new region within the inner horizon.
Here we can move freely and avoid the singularity, but these edges are not at infinity or a singularity, so there must be something beyond them. Well, when you venture further, you might find yourself in a white hole, which would push you into a completely different universe. - You can have these images where you are in a universe, you fall into a rotating black hole, you fly through the singularity and come out to a new universe from a white hole, and then you can continue playing this game. . - Expand this diagram infinitely. but there is still one thing we have not done: face the singularity.
So you point directly at the center of the ring and head towards it, but instead of time ending, you now find yourself in a universe, a strange universe, one where gravity pushes instead of pulls. This is known as the antiverse. If that seems too strange, you can always jump through the singularity and return to a universe with normal gravity. - And I know this is basically science fiction, right? But if you take the solutions of relativity, you know, essentially at face value and add a little bit to them, which is what Penrose does here, he says this, "oh, look, these shapes are very similar, I can just paste them together." . together." So this is the conclusion you get.
Now we have effectively an infinite number of universes, all connected with black holes, white holes on all sides and you are going to explore, but it will be a very brave person who is the first to jump into a spinning black universe. Hole to know if this is correct? (Derek laughs) - Yeah, I wouldn't sign up for that. So could these maximally extended Schwarzschild and Kerr solutions actually exist in nature? Both the extended Schwarzschild and Kerr solutions are eternal black hole solutions in an empty universe - As you say, it is an eternal solution. So it extends infinitely into the past and infinitely into the future and that's why there is no. formation mechanism there, it's just a static solution and I think that's part of the reason why black holes are created in our universe and White holes are not... - Or maybe they aren't. - Or it might not be, or I'm reasonably, personally, I'm reasonably sure that they don't exist, right? - For the Kerr solution extended to the maximum, there is also another problem.
If you are an immortal astronaut inside the universe, you can send light into the black hole, but since there is infinite time compressed in this upper corner, you can accumulate light along this edge, which creates an infinite flow of energy along the interior. . horizon. This concentration of energy then creates its own singularity, sealing the singularity of the ring and beyond. - My suspicion and the suspicion of some other people in the field is that this inner horizon will become singular and you will not be able to get through these second copies. - Then all white holes, wormholes, other universes and anti-universes disappear.
Does that mean real wormholes are impossible? In 1987, Michael Morris and Kip Thorne observed wormholes that an advanced civilization could use for interstellar travel, that have no horizons so you can travel back and forth, that are stable over time, and that have other properties. , how to build them. They found several geometries allowed by Einstein's general relativity. In theory, these could connect different parts of the universe, forming a kind of interstellar highway. They could even connect to different universes. The only problem is that all of these geometries require an exotic type of matter with a negative energy density to prevent the wormhole from collapsing. - This exotic type of matter is really against the loss of physics, which is why I have the prejudice that it will not exist.
The fact that we say that science fiction wormholes are mathematically possible bothers me. It is true, it is mathematically possible in the sense that there is some geometry that can exist, but Einstein's theory is not just geometries, it is geometries plus field equations. If we use the kinds of properties matter actually has, then they are not possible. So I feel that the reason why they are not possible is very strong. - So to the best of our current knowledge, it seems likely that white holes, traversable wormholes, and these parallel universes don't exist, but we also used to think that black holes didn't exist.
Then maybe we will be surprised again. - I mean, we have a universe, right? Well, why can't we have two? (whimsical music)