Article Derived From Transcript of YouTube Video: Something Strange Happens When You Follow Einstein's Math

Transcript of YouTube Video: Something Strange Happens When You Follow Einstein's Math

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The video delves into the peculiarities of black holes as predicted by Einstein's general theory of relativity, illustrating the slowing of time for an object nearing a black hole's event horizon. It traces the evolution of gravity theory from Newton to Einstein, highlighting Einstein's field equations and the Schwarzschild metric, which describe spacetime curvature around mass. The concept of white holes and wormholes is introduced, along with the theoretical framework that allows for parallel universes. The video also covers the significance of the Chandrasekhar limit in star collapse and the formation of neutron stars. It presents the Penrose diagram for visualizing cosmic time and space, and discusses the properties of rotating black holes, including the ergosphere and inner horizon. Finally, it addresses the theoretical possibility of traversable wormholes, the potential existence of an anti-verse, and the current scientific skepticism surrounding these phenomena due to the necessity of exotic matter with negative energy density.

Detailed Transcript of YouTube Videos

Introduction to Black Holes and Einstein's Math

You can never see anything enter a black hole. 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 out of 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 and Its Origins

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.

Einstein's Field Equations

Can you write down the Einstein field equation? This was 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.

Understanding Spacetime with Light Cones

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.

Schwarzschild's Solution and the Concept of Singularity

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.

The Existence of Black Holes and the Role of Astrophysics

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?

The Life Cycle of Stars and the Formation of Black Holes

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.

The Chandrasekhar Limit and the Formation of Neutron Stars

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 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.

The Existence of Neutron Stars and the Possibility of Black Holes

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.

The Theory of Black Holes and Oppenheimer's Solution

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.

The Schwarzschild Solution and the Concept of Wormholes

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.

The Coordinate System and the Perception of Singularity

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 mis 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.

The Concept of White Holes and Parallel Universes

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? 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.

The Visualization of Space Flowing into a Black Hole

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.

The Oppenheimer's Theory and the Reality of Black Holes

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.

The Sponsor's Introduction: Incogni

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The Explanation of Incogni's Service

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The Benefits of Using Incogni

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How to Get Started with Incogni

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The Kruskal-Szekers Diagram and the Concept of Time

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.

The Penrose Diagram and Its Implications

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.

The Exploration of Black Holes and the Possibility of White Holes

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.

The Hypothetical Nature of Wormholes and Parallel Universes

What color is a white hole? I mean, 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.

The Discussion of the Existence of Other Universes

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. 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. 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?

The Mathematical Representation of Universes and Wormholes

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 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.

The Existence of Wormholes and the Einstein Rosen Bridge

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. 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.

The Hypothetical Use of Black Holes for Inter-universal Travel

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. 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.

The Kerr Solution and the Properties of Rotating Black Holes

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. Physicists tried, but 10 years after Schwarzschild solution, they still hadn't solved it. 10 years turned into 20, which turned into 40 and then in 1963, Roy Kerr found the solution to Einstein's equations for a spinning black hole, which is far more complicated than Schwarzschild solution and this comes with a few dramatic changes.

The Structure of a Rotating Black Hole and Its Implications

The first is that the structure is completely different. The black hole now consists of several layers. It's also not spherically symmetric anymore. This happens because the rotation causes it to bulge around the equator. So it's only symmetric about its axis of spin. Alessandro from science click simulated what happens around this spinning black hole. Space gets dragged around with the black hole taking you and the particles along with it. When you get closer, space gets dragged around faster and faster until it goes around faster than the speed of light, you've now entered into the first new region, the ergosphere. No matter how hard you fire your rockets here, it's impossible to stay still relative to distance stars, but because space doesn't flow directly inward, you can still escape the black hole.

The Inner Layers of a Rotating Black Hole

When you travel in further, you go through the next layer, the outer horizon, the point of no return. Here you can only go inwards, but as you get dragged in deeper and deeper, something crazy happens, you enter another region, one where you can move around freely again, so you're not doomed to the singularity. You're now inside the inner event horizon. Here you can actually see the singularity - In a normal black hole, it's a point, but in a rotating black hole, it actually expands out to be a ring and there are weird things happened with spacetime inside the center of a black hole, a rotating black hole, but it's thought that you can actually fly through the singularity.

The Penrose Diagram for a Rotating Black Hole

We need a Penrose diagram of a spinning black hole, where before the singularity was a horizontal line at the top here, the singularity lifts up and moves to the sides, revealing this new region inside the inner horizon. Here we can move around freely and avoid the singularity, but these edges aren't at infinity or a singularity, so there must be something beyond them. Well, when you venture further, you could find yourself in a white hole, which would push you out into a whole nother universe.

The Concept of an Anti-verse and the Possibilities It Entails

You can have these pictures whereby you're in one universe, you fall into a rotating black hole, you fly through the singularity, and you pop out into a new universe from a white hole, and then you can just continue playing this game. Extending this diagram infinitely far. but there is still one thing we haven't done, brave the singularity. So you aim straight towards the center of the ring and head off towards it, but rather than time ending, you now find yourself in universe, a strange universe, one where gravity pushes instead of pulls. This is known as an anti-verse.

The Speculations and Limitations of Wormholes

If that's too weird, you can always jump back across 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 at, you know, essentially at face value and add on a little bit, which is what Penrose does here, he says this, "oh look, these shapes are very similar, I can just stick these together." Then this is the conclusion that you get. Now we have effectively an infinite number of universes all connected with black hole, white holes all the way through and you, of you go to explore, but it'll be a very brave person who's the first one who's gonna leap into a rotating black hole to find out if this is correct?

The Realities and Limitations of Wormholes and Black Holes

Yeah, I would not sign up for that. So could these maximally extended Schwarzschild and Kerr solutions actually exist in nature? Well, there are some issues. Both the extended Schwarzschild and Kerr solutions are solutions of eternal black holes in an empty universe. As you say, it's an eternal solution. So it stretches infinitely far into the past and infinitely far into the future and so there's no formation mechanism in there, it's just a static solution and I think that is part of the, part of the reason why black holes are realized in our universe and white holes aren't- Or might not be. Or might not be, or I'm reasonably I, personally, I'm reasonably confident that they don't exist, right?

The Current Understanding of Wormholes and the Future of Astrophysics

For the maximally extended Kerr solution, there's also another problem. If you're an immortal astronaut inside the universe, you can send light into the black hole, but because there's infinite time compressed in this top corner, you can pile up light along this edge, which creates an infinite flux of energy along the inner horizon. This concentration of energy then creates its own singularity, sealing off the ring singularity 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 go through these second copies. So all the white holes, wormholes, other universes and anti universes disappear.

The Conclusion on the Existence of Wormholes and Parallel Universes

Does that mean that real wormholes are impossible? In 1987, Michael Morris and Kip Thorne looked at wormholes that an advanced civilization could use for interstellar travel, ones that have no horizons, so you can travel back and forth, are stable in time, and have some other properties like being able to construct them. They found several geometries that are allowed by Einstein's general relativity. In theory, these could connect different parts of the universe, making a sort of interstellar highway. They might even be able to connect to different universes. The only problem is that all these geometries require an exotic kind of matter with a negative energy density to prevent the wormhole from collapsing.

The Final Thoughts on the Existence of Wormholes

This exotic kind of matter, is really against the loss of physics, so it's, I have the prejudice that it will not exist. I'm bothered by the fact that we say that the science fiction wormholes are mathematically possible. It's true, it's mathematically possible in the sense that there's some geometry that can exist, but Einstein's theory is not just geometries, it's geometries plus field equations. If you use the kinds of properties of matter that matter actually has, then they're not possible. So I feel that the reason they're not possible is very strong.

The Future of Astrophysics and the Possibility of Surprises

So according to our current best understanding, 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. So maybe we'll be surprised again. I mean, we have one universe, right? Good, why can't we have two.


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