Even professional physicists disagree
on what we expect to happen there. But there is a powerful tool in physics that can give us real intuition into the true nature of the event horizon. Its time you learned it. Black holes, objects with densities so high that there's this region, the event horizon, where the escape velocity reaches the speed of light. Nothing that falls below the event horizon can never escape and is lost to the universe forever while we see falling objects freeze as time stands still at the horizon. And anything that happens below the event horizon stays below the event horizon. That's the official sanitized public version. It's not entirely inaccurate, but the reality is, of course, a good deal more complex and interesting. Even ignoring the complications of Hawking radiation or black hole rotational growth, the simplest black hole of Einstein's general theory of relativity-- purely gravitational, static, and eternal-- is a subtle and misunderstood beast. But we can come to a powerful and intuitive understanding of the beast.
Today I want to teach
you how to use the same tool that physicists use. It's a tool that will let us easily and so the most common questions about black holes. For example, are objects falling through the event horizon really physically frozen there from the point of view of the outside universe? Would you see the entire future history of the universe playing fast forward at the instant that you crossed the event horizon? And do you see anything at all once you're inside the black hole? The tool that will answer these questions is called the Penrose diagram, sometimes also the Carter Penrose diagram. It's a special type of space-time diagram designed to clarify the nature of horizons. But first, a quick refresher on basic space-time diagrams. By graphing time versus just one dimension in space, we can look at the limits of our access to the universe due to its absolute speed limit, the speed of light. With the right choice of space and time units, the speed of light becomes a diagonal line on the space-time diagram. The area encompassed by the so-called light-like paths defines all future events or space-time locations that we could potentially travel to or influence constrained by the cosmic speed limit. That's our forward light cone. Our past light cone defines the region of th
e past universe that could
potentially have influenced us. Let's drop a black hole onto our space-time diagram. It lives at x=0 on the space axis, but exists through all the times on the graph. It has a point of infinite density, the singularity, and an event horizon a bit further out. The mass of the black hole stretches space and time so that light rays appear to crawl out of the vicinity of the event horizon before escaping to flat space-time, no longer following 45 degree paths. Now let's throw a monkey into the black hole. As it approaches the event horizon, its future light cone bends towards the black hole as fewer and fewer of its possible trajectories lead away. Below the event horizon, all possible trajectories lead towards the singularity. The problem with the regular space-time diagram is that the path of light and the shape of the light cone changes as space-time becomes warped. That makes it difficult to figure out what parts of the past and future universe the monkey can witness or escape to. And this is where the Penrose diagram comes in. It looks like this. It transforms the regular space-time diagram to give it two powerful features. It crunches together, or compactifies, the grid lines to fit infinite space-time on one graph-- very useful for black h
oles. It also curves the
lines of constant time and constant space in what we call a conformal transformation so that light always follows a 45 degree path. That means light cones always have the same orientation everywhere. Super handy for understanding monkey trajectories. This is the Penrose diagram for flat space-time with no black holes. Same as with a regular space-time diagram, blue verticalish lines represent fixed locations in one dimension of space and red horizontalish lines are fixed moments in time. Now, those lines get closer and closer together towards the edge of the plot to encompass more and more space-time. They're extremely finely separated at the edges so that any tiny stretch on the graph represents vast distances and/or times. The lines also converge together towards the corners so that light travels a 45 degree path everywhere on the diagram. So a light ray starting from really, really far away and coming towards us hugs the edge of the diagram and crosses an enormous number of time and space steps, only reaching us in our very distant future. OK. Let's drop a black hole into this space-time. Nice and safely far off to the left. And because we only have one dimension of space, and any motion to the left brings us closer to the black hole. Its event horizon becomes the end of the line in that direction. The future cosmic horizon on th
e Penrose diagram is
replaced with a plunge into a black hole. The compactified grid lines there now represent the stretched space-time near the event horizon. An entirely new Penrose region represents the interior of the black hole. Weirdly, the lines of constant position and constant time switch. Space flows at greater than the speed of light inwards, towards the central singularity. It becomes uni-directional, flowing inexorably downwards, just as time flowed inexorably forward in the outside universe. All paths lead to the inevitable singularity. Once you're beneath the horizon, your future light cone still represents all possible paths that you could take. All of them end up at that singularity. The only way to escape back to the outside universe would be to widen your light cone by traveling faster than light. So you're out of luck. Now that we've nailed the Penrose diagram, we can use it to do some serious black hole monkey physics. Our space-faring simian begins its journey and emits a regular light signal that we observe from a safe distance. As it approaches the black hole, these light rays have further and further to travel through increasingly curved space-time and so the interval between receiving signals also increases. The progress of the monkey appears to slow to a halt very close to the event horizon, and the final signal at the moment
of crossing never reaches
us. It's trying to travel at the speed of light against light speed cascade of space-time. With this picture, we can start to answer some very serious questions. First, what would happen if the monkey remembered to fire its jet pack at the last instant before reaching the event horizon? Well, it could still escape. Its future light cone still includes a tiny sliver of the outside universe. It had better be a good jet pack because it's going need to follow a very long near light speed path away. It will, nonetheless, have experienced far less time than us when it emerges into flat space-time in our far future. Assuming no jet packs, the monkey is probably doomed to a graceful reverse swan dive through the event horizon, watching the entire future history of the universe play out above it at that last instant. Yet actually, no. It doesn't see that at all. The monkey's last view of the outside universe is defined by its past light cone that encompasses all of the light that will catch up to it and that light is stuck following these diagonal lines because it has to contend with the same stretched space-time as the monkey. There's no future universe spoiler promo. If it could instead hover above the event horizon, then it would see the universe in fast forward, although that view would be compressed into a small circle directly overhead. Watching the monkey frozen on the event horizon is going to make us feel a bit guilty, after a while. Could we change our minds and launch a daring monkey rescue mission? Sadly, no. Even if we do travel at the speed of light, after a certain point there's no catching the monkey. We would see it suspended above the horizon as we're racing to meet it, but it will always appear to be just a little further ahead no matter how close to that horizon we dare to go. Remember, the monkey isn't actually above the horizon for infinite time, it only appears that way to us because as long as we're outside the event horizon, no times that we can witness correspond to the monkey crossing that horizon. In order to see that crossing, we would have to cross the event horizon ourselves. Once inside the black hole, we could potentially see the monkey below us. All space-time within the black hole is flowing toward the singularity faster than the speed of light. The two neighboring radial layers aren't traveling faster than light relative to each other. That means that the monkey's signal can still reach us, although it might be more accurate to say that we catch up to the monkey's outgoing signal. But even that so-called outgoing light is still moving downwards, doomed to hit the singularity along with the monkey and our rescue mission. All of this describes a non-rotating, uncharged black hole, a Schwarzschild black hole. Even this simple case is a good deal more complicated than I let on. For example, I only showed you half of the Penrose diagram. The complete mathematical solution for a Schwarzschild black hole has two additional regions, one corresponding to a parallel universe on the other side of untraversable wormhole, the Einstein-Rosen Bridge. And down here we have what we call a white hole. These are strange mathematical entities and probably aren't real, but we'll certainly come back to them. We'll also come back to what happens if we set the black hole spinning or add some electric charge. Then our Penrose diagram blooms outwards to include potentially infinite parallel regions of space-time. Thanks to The Great Courses Plus for sponsoring this episode. The Great Courses Plus is a digital learning service that allows you to learn about a range of topics from Ivy League professors and other educators from around the world. Go to thegreatcoursesp lus.com/spacetime and get access to a library of different video lectures about science, math, history, literature, or even how to cook, play chess, or become a photographer. New subjects, lectures, and professors are added every month. For an excellent overview of basically all physics, I really liked Richard Wolfson's "Physics and the Universe" course. It even takes you through the beginnings of Einstein's general relativity. With The Great Courses Plus, you can watch as many different leches as you want anytime, anywhere, without tests or exams. Help support the series and start your one-month trial by clicking the link in the description or going to thegreatcoursesp lus.com/spacetime. Hey, guys. A couple of quick shout-outs. First, a huge thank you to our first Patreon supporter at the Big Bang level. Antonio Park, you rock and I'm really looking forward to our hangouts. Your support is going to make a big difference, as is the support at all levels. So thanks again to everyone who's contributed on Patreon. And a quick announcement of a couple of events I'll be at early next year. First, South by Southwest, March 10 to 19 in Austin, Texas. A few months ago I asked you guys to head to the South by Southwest panel picker to vote for myself, Astrophysicist Katie Mack, and It's OK to be Smart's Joe Hanson to be picked for a panel titled, "We are All Scientists." Well, thanks to you guys, we're scheduled. We're going to talk about the value of and the threat to critical thinking and scientific reasoning and why these skills need to be seen as accessible and important to everyone and how we can act on this idea. Also, for those of you thinking of attending grad school in physics, I'll be talking about studying physics at a professional level at the City University of New York's Student Research Day, April 7 at the CUNY Graduate Center in New York City. More details closer to then, but for now I'll put a link in the description and feel free to reach out to me if you're interested. OK. Onto the comments from last week's episode on De Broglie-Bohm theory. Wow. This is the closest I've seen a YouTube comment section come to looking like a Q&A session after a professional physics seminar. Now, a lot of you wondered why I never mentioned the EM drive when talking about pilot wave theory. The answer is simple, there was nothing useful to say on that connection. In the recent paper out of Eagle Works Labs, Herald White and collaborators present some results on the thrust produced by their EM drive and then go on to talk about how pilot wave theory might explain the apparent conservation of momentum-breaking results. I might get into the details in an upcoming episode, but for the sake of explaining pilot wave theory this paper isn't relevant. The connection is extremely speculative, and honestly I wondered whether pilot wave theory was chosen partly because the internet happens to love it at the moment. DinosaurFromtheFuture asks how it can be that pilot wave theory predicts different particle trajectories, given that the particles supposedly all start at exactly the same point. Well, the simple answer is that the particles don't start at exactly the same points. We just can't know exactly their location at the beginning. See, pilot wave theory states that the particle riding the wave does have a definite position at all times and that position defines its future directory. So if you know the position perfectly and you know the wave function, you can perfectly predict future locations. However, you can't perfectly measure a particle position without changing it slightly in ways that themselves aren't perfectly predictable. As a result, you never know exactly where a particle is. This uncertainty leads to the range of potential future trajectories, including trajectories through one slit or the other. More generally, it allows pilot wave theory to agree with Heisenberg's uncertainty principle. In the Copenhagen interpretation, the uncertainty principle describes the intrinsic randomness of the quantum world. De Broglie-Bohm pilot wave theory states that this uncertainty just arises from our imperfect knowledge and that the universe itself knows exactly where all these particles are. Vacuum Diagrams correctly points out that to know the future trajectory of a particle, you only need position, not velocity, as I had stated. That velocity information is in the guiding wave. Thanks for the correction and thanks also for pointing out those extremely interesting papers that detail certain failings of the pilot wave interpretation. I'll link those and a couple of others that take different sides in the description of this video, as well as in the pilot wave episode. In fact, there was some really heated and fascinating discussion both for and against the pilot wave interpretation and some of it was from people who know a good deal more than I do, like Vacuum Diagrams. Something I took from this is that Bohmian mechanics is, on its own, very unlikely to be the full picture, even ignoring the whole relativity issue. That doesn't necessarily mean, though, that it's not useful. I'll get back to why. But first, as was pointed out to me in a nice email by physicist and science writer Adam Becker, I wasn't entirely accurate when I said that De Broglie, the founder of pilot wave theory, remained convinced by Niels Bohr and his Copenhagen camp, even after Bohm's work. More accurately, De Broglie remained convinced of the objections raised against his idea, even after some of them were addressed in Bohmian mechanics. To quote De Broglie from his 1956 book, he, Bohm, assumes that the [INAUDIBLE] wave is a physical reality, even the [INAUDIBLE] wave in configuration space. I have already stated why such a hypothesis appeared absolutely untenable to me. In fact, De Broglie was never a huge fan even of his own simplistic particle carried by a wave idea. That formulation was a simplified version of what was to be a much more intricate double solution theory in which the so-called particle was actually a matter wave itself embedded in and carried by the sine wave, represented by the wave function. He was unable to pull the math together in time for the fated Solvay Conference, and so derived the simpler description in which the particle is point-like. De Broglie never completed his full double solution theory, but did work on it intermittently throughout his life and was inspired to return to it by Bohm's publication, even if he didn't buy Bohmian mechanics. The fact is we just don't know whether the reality that drives the strange results of quantum experiments is actually deterministic in the way that we understand determinism. But De Broglie-Bohm pilot wave theory is a great example of how a deterministic theory can at least go some way towards predicting the results of quantum experiments. Personally, I'm agnostic towards the relative truth behind the Copenhagen, many-worlds, pilot wave, or the other interpretations of quantum mechanics. I like the idea of a deterministic theory, but the universe has often demonstrated that it couldn't care less about our pet theories.
