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0:00
Black holes are not just the strangest objects in the universe. They're the sharpest tests we have of how reality actually works. They form when mass is compressed beyond a critical limit. But their importance goes far beyond how they're made. Black holes are the most extreme laboratories in the universe, forcing general relativity and quantum mechanics into direct confrontation. In this longplay episode, we're looking back through our 10-year
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history and giving you 90 minutes to explore how black holes form, evolve, and ultimately help us test and possibly reconcile our deepest theories of reality. Black holes are one of the strangest objects in our universe. To make one, we need both general relativity and quantum mechanics. Today, I'm going to show you how. In a previous episode, we discussed the true nature of black holes. We talked about them as general relativistic
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entities, as space-time regions whose boundary curvature effectively removes the interior from our observable universe. Now, it would be a great idea to watch this video first if you haven't already. Now these are some abstract ideas and really black holes were at first just a strange construction of general relativity and just because something exists in the mathematics does not mean it has to exist in reality. So are black holes real? The answer is yes. Black holes
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are astrophysical realities that we have ample evidence for. Yet to actually form a black hole, Einstein's descriptions of mass, energy, and spaceime are not enough. We need quantum mechanics. If you're up for it, let's build a black hole. First step, find a very massive star and wait. Let it cook. Not for long, because these guys have very short lives. Just wait a few million years for the supernova. If you get impatient, you can turn
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up the core temperature by bombarding it with gravitational waves. It'll be done quicker. The details of the deaths of massive stars are pretty awesome, but they can be found in lots of places, so we'll just gloss over them here. In the last throws of a very massive stars life, increasingly frantic fusion in the interior produces one periodic table element after another in Russian doll shells of increasingly heavy nuclei that finally surround an iron core. The formation of
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that core represents the end of exothermic fusion. Fusing two iron nuclei absorbs energy. It doesn't release it. So, starved of an energy source, the stellar core collapses on itself. Electrons are slammed into protons in the iron nuclei, forging a neutron star. The collapsing outer shells ricochet off this impossibly dense nugget in a supernova explosion, enriching the galaxy with juicy new
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elements. The leftover core, the neutron star, is a very weird beast. A ball of neutrons the size of a city with the mass of at least 1.4 suns and the density of an atomic nucleus. We see them when we see them as pulsars. Now beneath a thin atmosphere of iron plasma, a neutron star is a quantum mechanical entity and it's a quantum phenomenon that saves it for the moment from final collapse. It's also a different quantum phenomenon that will let us push it over the edge creating a
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black hole. To understand how space works for a quantum object like this, we need to think not in regular 3D space or even 4D spaceime, but rather in sixdimensional quantum phase space. For a neutron star, this is the space of both 3D position and 3D momentum. And it defines the volume that can be occupied by the strange matter in a neutron star. Now the exact way that the matter of a neutron star fills this 6D quantum phase space depends on two important principles
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of quantum theory. The pi exclusion principle and the Heisenberg uncertainty principle. These govern the delicate balance between stability and collapse. The ply exclusion principle basically just says that two things can't occupy the same place at the same time. And by thing I mean firm the particle type comprising all regular matter for example electrons, protons, neutrons. Now by place I mean location in quantum phase space. So two firmians can occupy the same physical location
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just fine as long as their momenta or any other quantum property is different. Now this rule is what keeps electrons in their separate stable orbits and in turn is part of what allows solid matter to have its structure. In the case of a neutron star, position momentum phase space is completely full of neutrons. Every spatial location and every momentum location connected to those spatial locations contains a neutron. Okay, jargon alert. This weird state of matter
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where phase space is completely full, we call it degenerate matter. And the degeneracy pressure resulting from particles not having anywhere else to collapse into is incredibly strong. strong enough to initially resist the insane gravitational crush of a neutron star. As far as we know, there's no way to overcome par exclusion, at least not directly. See, it's not a matter of force. Two firmians just can't ever occupy the same quantum state, and that's that. So,
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the neutron star is safe. But come on, we want to build a black hole. Fortunately, there's another quantum phenomenon that lets us get around the pi exclusion principle. The Heisenberg uncertainty principle tells us that the properties of a quantum entity are fundamentally uncertain. The details may be a topic for another episode, but in short, quantum mechanics describes matter as
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a distribution of possibilities. Certain numerical properties that you can assign to a particle exist in a wave of varying degrees of maybe. Location is one such property. A neutron, for instance, is not in any one place, but exists as a cloud of possible locations that might be tightly constrained or maybe very spread out. Location remains a possibility cloud until the neutron interacts with another particle, at which point its location is resolved. This is the weirdest,
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coolest aspect of quantum mechanics, and we'll try to get back to it in another episode. But for now, we have a black hole to make. The Heisenberg uncertainty principle tells us that particular pairs of quantities position and momentum or time and energy must when taken together contain a minimum degree of uncertainty. If one is tightly constrained, then the other must be uncertain and span a wide range of potential values. So a neutron star is comprised of the densest matter
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in the universe. Its constituent neutrons are about as constrained in position as you can get. Therefore, the Heisenberg uncertainty principle tells us that they must have highly undefined momenta. Very very large neutron velocities become part of the possibility space. To put it another way, the neutrons are packed so close together in position space that their momentum space becomes gigantic. Phase space expands. And here's the thing, the denser the neutron star becomes,
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the more momentum space you get. So, Heisenberg lets us circumvent that pesky degeneracy pressure. If we can somehow add more matter to a neutron star, throw another star at it, maybe it won't get spatially larger. The extra matter certainly needs somewhere to go. The star must expand. But it doesn't expand in position space. The star expands in momentum space. In position space, it actually
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gets smaller. The more mass of the neutron star, the smaller its radius. This is a quantum effect even though it's happening on the scale of a star. Until now, the neutron star has hovered above a critical size. The space-time curvature at the neutron star surface is pretty extreme. Clocks run noticeably slower, and the densities inside the star produce some very strange states of matter.
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However, despite this, the star is still very much a thing in this universe. And yet below the stars surface there lurks the potential event horizon. The surface of infinite time dilation. Now the event horizon doesn't actually exist as long as the neutron star stays larger than the wouldbe horizon. However, if we can increase the mass of the neutron star, the actual star shrinks and the
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event horizon expands. You can see where I'm going with this. There's a mass where the radius of the neutron star and the event horizon overlap. It's three times the mass of the sun. At this point, the event horizon actually comes into being and the neutron star submerges beneath it. We finally created our black hole. But what happens to the star when it slips below its event horizon? Everything inside is lost from this universe. Spacetime is radically altered inside the star
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with all geodics space-time paths turning inward towards the center. When the black hole first forms, the material inside must resemble the stuff of the original neutron star. But there's no stopping ultimate collapse. All paths lead to the central point of infinite curvature, the singularity. From the point of view of the star itself, the inward cascade happens. All position space collapses towards the singularity while momentum space expands accordingly with the
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corresponding enormous velocities all inward pointing. Neutrons are certainly shredded into component quarks and gluons. But what happens to these as the star approaches an infinite decimal point, the plank scale? Physics cannot yet tell us. From the point of view of an outside observer, so us, this never happens. The black hole forms. The stellar core goes dark. But on our timeline, nothing ever happens beyond the event horizon again. We can't meaningfully think about what's
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happening now beneath the event horizon. There is no corresponding now. The material of the star and all events that happen to it are no longer a part of the timeline of the external universe. On our clock, the singularity forms infinitely far in the future. To us there is only the event horizon. So this is how a real astrophysical black hole is made. The mass of the stellar core becomes the apparent mass of the black hole and very few other properties of the
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collapsed material are remembered. The black hole retains mass, electric charge and spin and these continue to influence the outside universe sometimes in very important ways. Of course, a real black hole is not the static creature that we sometimes describe in theory. They grow, they leak, they change. We'll get to what this means for black holes and for the universe in another episode of Spacetime. In the very first instant after the Big Bang, the density of matter was so
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great everywhere that vast numbers of black holes may have formed. These primordial black holes may still be with us. There's no longer any question that black holes exist. LIGO's recent observation of gravitational waves for emerging black holes is a stunning confirmation of this fact. Of course, we already thought they must exist as long as a volume of space contains a high enough density of
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mass or energy. General relativity tells us that a black hole will form. In the modern universe, there's only one natural way to get such insane densities. That's in the core of the most massive stars when they die. The process is awesome and we look at it in a previous video, but that's the modern universe. Once upon a time, the entire universe had the density of a stellar corpse. In
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fact, soon after the big bang, the density of the universe was vastly higher. So why didn't all the matter in the universe become black holes then? Well, actually, some of it may have formed what we call primordial black holes, and they may still be around today. Let's back up a bit. In order to make a black hole, extremely high density isn't enough. You need a density differential. Otherwise, there's no preferred direction for all that gravitational attraction. Also,
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the gravitational pull needs to be strong enough to overcome the expansion of the universe. Now, matter in the early universe was pretty smoothly spread out and the universe was expanding fast. That means most of it avoided collapsing into black holes. And that's a very good thing, by the way. However, it wasn't perfectly smooth. There were lumps. The oldest light we can see is the cosmic microwave background radiation. It reveals tiny differences in the density of matter from one
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point in space to the next. The universe was very slightly lumpy at the moment the CMBB was created about 400,000 years after the Big Bang. These density fluctuations were enough to kickstart the formation of galaxies, but certainly not enough to immediately collapse into black holes. Yet, if we rewind time, those fluctuations must have been much stronger. It's thought that these fluctuations originally formed when the entire observable universe was smaller than a single
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atom. Back then, quantum fluctuations caused a sort of static fuzz across the minuscule cosmos. There are several different stories for the initial size and growth of these fluctuations and cosmic inflation certainly plays a role. But it's well within the possibility of many models that some of these fluctuations were at some point in the early expansion intense enough to resist the local expansion of the universe and form black holes. Some highly speculative big bang physics
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also predicts primordial black holes. For example, the collapse of cosmic string loops and the collision of bubble universes. Awesome. Now, these models can predict a huge range of possible masses for primordial black holes. PBH's as we like to call them in the biz. PBHs could have been formed at a few grams to tens of thousands of times the mass of the sun depending on which formation model you go with, or they might not exist at all. That's a big possibility. If they do exist,
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then there's probably a particular mass range that most of them formed at. Discovering PBH's and learning their masses would tell us a huge amount about the earliest moments of our universe. We need to hunt for these black holes or their influence in the modern universe. First of all, we aren't going to find primordial black holes less than around a billion tons or the mass of a small asteroid. They would have all evaporated away due to Hawking radiation.
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I'll get back to that. Black holes larger than this should still be around, but they'd be very difficult to spot being so black and all. If PBHs are rare, then it may be impossible to confirm or disprove their existence entirely. However, there is a question that we can answer with some certainty. Could primordial black holes be dark matter? This is a slightly terrifying possibility that 80% of the mass in the universe is in the form of countless swarming black holes.
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That's a lot of primordial black holes. And so we expect them to leave their mark on the universe in different ways. For one thing, if these little knots of warped spaceime are everywhere, then they should produce obvious gravitational lensing. We'd expect them to frequently pass in front of other space stuff. Depending on PBH mass, this would cause a twinkling effect. microl lensing in stars in our galaxy, in distant quazars, even in gammaray bursts. And well, we just don't see
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enough of this twinkling, which rules out a lot of possible masses. There's also the fact that swarms of black holes would mess up their surroundings. As the heavier ones buzz around the galaxy, they should pull apart loosely bound binary systems and have an effect on the structure of star clusters. The smallest should fall into neutron stars, causing them to either explode or become black holes themselves, but we see loosely bound binaries and normal star clusters and plenty
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of neutron stars. These arguments let us rule out all but a very narrow set of mass ranges for primordial black holes as an explanation for dark matter. The options we're left with are either lots of PBH's with masses similar to a large asteroid like series, so around 10 ^ of 21 kg, or a much smaller number of really big PBHs, around 20 to 100 times the sun's mass. Now, this last possibility is sketchy. Some scientists think that the voracious feeding of lots of really
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big primordial black holes would have left their mark on the cosmic microwave background. However, others argue that the recent LIGO detection of the merging of two approximately 30 solar mass black holes is evidence in favor of this idea. With new observations from both regular telescopes and LIGO, we're rapidly closing all of these mass windows. Before too long, we'll either spot the signature of primordial black holes at these masses or discover that PBH's
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are actually very rare and that they're certainly not dark matter. This latter is more likely, but we'll see. Of course, primordial black holes that have already evaporated due to Hawking radiation definitely are not dark matter, and that rules out any PBH is lighter than about a billion tons. But that last stage of Hawking evaporation is very fast. In fact, it's explosive. It's possible that
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certain types of very short gammaray bursts are these final flashes from PBH's evaporating in our galaxy. Some highly speculative stuff, but also some highly awesome possibilities. It wouldn't be right to end a discussion on primordial black holes without talking about what would happen if one passed through the solar system. Even a close encounter with a black hole as massive as the sun or higher would be pretty catastrophic. If it passed anywhere near
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the planetary system, the gravitational tug would disrupt the planet's orbits. Even if it passed by the outskirts of the solar system, it could shake up the orc cloud and send a nice rain of comets to pepper the inner solar system. Of course, regular black holes from supernova can and perhaps have done that. Having highmass primordial black holes just makes it more likely. If PBH's are closer to the mass of a large asteroid, then they're too low in mass and
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probably moving too fast to do any gravitational damage. They'd just zip right through the solar system unnoticed. It's a different matter if one hit the Earth traveling at a couple hundred km/s, it would punch straight through the planet, but certainly leave a narrow column of vaporized rock behind it. These sorts of hits would be incredibly rare and may never happen. However, if primordial black holes have approximately the minimum possible mass to not have evaporated
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around a billion tons, these would be much more abundant than asteroid mass PBH's. In fact, they may pass through the planet frequently. A billion ton black hole has an event horizon around the size of a proton. So, it would pass through the planet as though the Earth were made of air. However, it would deposit something like a billion jewels of Hawking radiation on its way through. This should leave detectable traces in crystalline material in Earth's crust. In fact,
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perhaps geologists will be the first to discover primordial black holes. If they're out there, someone will figure it out. I mean, how long can the universe expect to hide vast numbers of holes punched in the fabric of spaceime? The singularity, the point of infinite density at the core of a black hole, but also so much more. In mathematics, singularities come in wild and wonderful varieties. The black hole itself contains more than one. Isaac Newton's universal
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law of gravitation was an incredible insight when he figured it out in the late 1600s. In fact, we still use it to fly spacecraft around the solar system today. However, it has its problems. Let's look at the math. Newton's equation gives you the gravitational force exerted between two masses m1 and m2 that are distance r apart. Straightforward enough, that r squared in the denominator spells trouble. It means the force gets larger the closer the masses are to each other. That
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makes sense. But what about when r gets really close to zero? Then the result of the equation, the force becomes extremely large and is infinite when r becomes equal to zero. That doesn't really make a lot of sense. Infinite force means infinite acceleration, which means well, physics breaks. According to Newton's law, in order to feel that infinite gravitational acceleration, you need to get zero distance from an object's center of mass. That means all of that object's
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mass would need to be concentrated at that center, a single point of zero size, which means infinite density, and that of course would make it a black hole. We often use the word singularity to describe the hypothetically infinitely dense core of a black hole. But in math, the meaning of this word is much more general. You know what? Instead of me trying to explain mathematical singularities, how about we get a real mathematician to do this properly? Guys,
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meet Kelsey Houston Edwards of the new PBS show, Infinite Series. Hey, Kelsey. Hey, Matt. Thanks for having me on. Kelsey, the math for black holes goes to infinity for different properties and in different locations. What does this mathematical weirdness tell us? Well, mathematicians use the word singularity pretty broadly. It's really just any point that causes problems. Commonly, these problematic points are where quantities become bigger and bigger, approaching infinity,
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as they do near a black hole. Some singularities come about from your choice of reference frame or coordinate system. An example of a frame dependent singularity that might be familiar to space-time viewers is the event horizon of the black hole. I'll leave that to you to explain. Here on Earth, the north and south pole are examples of coordinate singularities. It's possible to pass through time zones infinitely quickly, but only because of your choice of spherical coordinates.
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All right, that makes sense. But the gravitational singularity at the center of a black hole is a so-called real singularity, right? I mean, the curvature and the density are infinite from any frame of reference, right? And there's no way to avoid a horrible crushing death just by switching coordinate systems. But the reality of the black hole singularity may give reason to doubt the theory that predicts such a thing. In fact, it's happened many times before. From models of the
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movement of water to human population growth. Mathematics predicts a physical singularity and we've been forced to reject the corresponding theory. So you're saying Einstein is wrong? blasphemy. Actually, Einstein himself agreed on this point. Guys, you should check out Kelty's show, Infinite Series, where she goes into much more depth on the nature of singularities. It's a math show, by the way, so it's sometimes about real stuff. Mathematicians are lucky. Being
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limited by reality is so boring. So, does the fact that it includes a singularity mean there's something fundamentally wrong with Newton's law of gravitation? Well, we already know the law isn't really so universal. When the gravitational field is too strong, say near a star or a black hole, Newton's law gives the wrong answers, and we need Einstein's general theory of relativity, which
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is the far more complete theory of gravity. So, does general relativity rid us of Newton's pesky singularity? Uh, no. In fact, it gives us even more singularities. To understand this, we need to look at something called the Swatshield metric. It's what you get when you solve the delightfully complicated Einstein field equations for the simple case of a spherically symmetric mass in an
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otherwise empty universe. We're going to simplify it to only allow movement directly towards or away from our massive object. In that case, it looks like this. Okay, that sure is some math. Hey, this is spacetime. we can deal. Actually, it's really easy to see the singularities in this equation. But let me first walk you through what it tells us. The swast shield metric allows us to compare two points or events in spaceime around a massive object from the perspective of different
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observers. For example, a short space-time path of some object, so its world line, might move an object a distance deltar over a short time step delta t. That motion is towards or away from the mass of object which is a distance r away. That delta s squ thing is the space-time interval. And it's a strange and interesting quantity. Every inertial so non-acelerating observer will agree on the same space-time interval for every pair of events and for every world line.
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We talk about this in a lot more detail in our relativity playlist. Today, we're going to keep it simple. As long as our object's world line doesn't require faster than light motion, then the square root of the space-time interval is equal to the amount of time that the object itself feels over that interval. We call that the object's proper time. Oh, and r subscript s is a measure of the mass of the mass of object. In fact, it's 2 times the gravitational constant
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times the mass. There would have been some speed of lights through the equation, but we set them equal to one because we're that cool. Now, the first thing to notice is that the singularity is still present in the SWAT shield metric. R the distance to the center of mass remains in the denominator just as it was in Newton's law. When you use the swast shield metric to calculate the curvature at r equals0 that curvature is infinite. This gives us the same infinite
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gravitational pull as Newtonian singularity. And just as with the Newtonian case this gravitational singularity can only exist if infinite densities are possible. But unlike Newton's law of gravity the swast shield metric actually tells us whether or not that infinite density is expected. To see how we need to look at the second singularity in this equation, a singularity that Newton's
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law does not contain. See, when distance to the center of mass is exactly equal to this RS thing, then RS over R is equal to 1, at that point the entire equation starts behaving very badly. It's as much a mathematical singularity as the one in the center of the black hole. If you haven't guessed, this bad behavior corresponds to the event horizon and RS is the SWAT shield radius. Imagine an object sitting at the event horizon but not moving. So it's delta R would be zero. But
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this bracket is zero also because 1 minus one. The entire space-time interval for a non-moving point at the event horizon is zero. But remember for sub lighteed world lines the space-time interval tells us the rate of flow of proper time. So does that mean time doesn't pass for an object hovering at the event horizon? Not quite. Time certainly doesn't pass at the event horizon. No clock ticks
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can ever happen there. But the prohibition against objects experiencing time at the event horizon is actually a prohibition against objects spending time at the event horizon. No temporal thing, nothing that normally experiences the passage of time can have a space-time interval of zero. At the event horizon, the only way to get a nonzero space-time interval is to have a nonzero delta r. An object at the event horizon has to change its distance from the black hole to keep its
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clock ticking. That means falling below the event horizon. And once inside, inward spatial movement continues to be the only way to fuel the ticking of an object's proper time clock. We'll come back to that bit of awesome weirdness in a future episode. There is one thing that can have a space-time interval of zero. Light. Actually, anything capable of traveling at light speed can only have a space-time interval of zero. From its perspective, a photon exists in a single instant,
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and so it can hang out at the event horizon, which also only exists at one infinitely stretched out instant. The act of crossing the event horizon is where this singularity really starts to behave badly. At the moment of crossing, the denominator here in the swar metric is zero. and the whole equation blows up to infinity. But what is actually infinite here? It's nothing physical. It's the fact that even an outgoing light ray takes infinite time to move any distance. So using
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boring old time and distance, delta t and delta r, doesn't let us trace a world line smoothly across the event horizon. That horizon is a coordinate singularity, just like Kelsey talked about. But that means we can fix it. There are ways to construct our space-time axes. So this singularity just evaporates. For example, Edington Finkelstein taught us coordinates, the compactify with the stretching of spaceime to cancel out the infinities. That's a bit much for right now,
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but Google away, my friends. Anyway, the upshot is that it's really a breeze to drop through the event horizon, both physically and mathematically. Of course, once inside the event horizon, we still have that central singularity to deal with. Unfortunately, that one can't be done away with by a simple change in coordinates. But can that point of infinite density really exist? Actually, Einstein's theory and the swaste solution that is derived from it suggests it must exist. The
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apparent inevitability of this singularity may be evidence that general relativity is incomplete. But to better understand why the central infinity is unavoidable in Einstein's theory, we have to go back to that coordinate shift at the event horizon there. The causal roles of space and time switch places, and the central singularity becomes not so much a location in space, but an inevitable future. Actually, to really get this, we're going to need another entire episode. Stand
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by to explore what happens when you switch the causal roles of time versus space to space time. The special theory of relativity tells us that one person's past may be another's future. When time is relative, paradoxes threaten. Today, we peer deeper into Einstein's theory to find that the immutable ordering of cause and effect emerges when we discover the causal geography of spacetime. Recently, we've been talking about the weirdness of spacetime in the vicinity of a black
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hole's event horizon. Very soon, we'll be dropping below that horizon to peer at the interior of the black hole. There, space and time switch roles. But to truly understand that bizarre statement, we need to think a little bit more about how the flow of time is described in relativity. Today, we're going to look at the amazing geometric structure that time, or more accurately, causality,
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imprints on the fabric of spacetime. First, let's recap a little bit of Einstein's special theory of relativity. There are two previous episodes in particular that will be useful here if you find you need more background. Special relativity tells us that our experience of both distance and time are well relative. If I accelerate my rocket ship to half the speed of light, the distance I need to
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travel to a neighboring star shrinks dramatically from my point of view. An observer I leave behind with an amazing telescope observes me traveling the entire original distance, but will perceive my clock as having slowed. The combination of this length contraction and time dilation allows both moving and stationary observers to agree on how much older everyone looks at the end of the journey. Everyone agrees on the number of ticks that occurred on everyone else's clock.
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They just don't agree on the duration of all of those ticks. Reminder, time measured by a moving observer on their own clock is called proper time. But counting those clock ticks isn't the best way for everyone to agree on space-time relationships. There's this thing called the space-time interval that relates observer dependent perspectives on the length and duration of any journey that
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all observers will agree on. even if they don't agree on the delta x and deltat t of that journey. We've talked about it before, but it's a tricky concept to understand intuitively. But we want that intuition because more than proper time, the space-time interval defines the flow of causality. In relativity, 3D space and 1D time become a single 4D entity called spacetime. To preserve our sanity, we represent this on a space-time diagram, plotting time and only one dimension of
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space. We'll see our causal geometry emerge plain as day. Even in this simplified picture, there is no standing still on a space-time diagram. If I don't move through space, I still travel forward in time at a speed of exactly 1 second per second. According to my proper time clock, motion at a constant velocity appears as a sloped line and the time axis is scaled so that the speed of light is a 45° line. Now, let's say we have a group of space-time travelers. They start at the
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origin where x and t equal zero. They race away to the left and the right for 5 seconds according to their own watches. They all travel at different speeds, some close to the speed of light, but never faster. The path they cut through spaceime is called their world line. My world line is only through time. And the tick marks on the time axis correspond to my own proper time clock ticks. The
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faster a traveler moves, the longer their world line. That's not just because of their speed, though. To me, their clocks tick slow. They time their journey on these slow clocks. So I perceive them traveling for longer. Accounting for this, we find that our space-time travelers are arranged on a curve that looks like this. This shape is a hyperola. Drawing a connecting line at the tick of every traveler's proper time clock gives us a set of nested hyperbole. But these
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aren't just a pretty pattern. These curves are kind of the contours defining the gradient of causality down which time flows and etched into spaceime by the equations of special relativity. But to understand why, we need to see how these proper time contours appear to other space-time travelers. Instead of doing that with equations, we can see it with geometry. First, we need to draw the space-time diagram from the perspective of one of the other travelers.
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To transform the diagram, we need to figure out what they see as their space and time axes. Time is easy. They see themselves as stationary. So, their time axis is just their own constant velocity world line. and their x-axis. Well, from my stationary point of view, I define my x-axis as a long string of space-time events at different distances, but that all occur simultaneously at time t=0. To observe those points, I just wait around until their light has had time to reach me.
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At every future tick of my clock, a signal arrives from the left and the right, and I use that to build up a set of simultaneous events, defining my tals 0 xaxis. Our traveler does the same thing, but from my point of view, their clock is slow, so I see them register signals at a different rate. At the same time, they're moving away from the signals coming from the left and towards the ones originating on the right, affecting which signals are seen at a given instant.
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The traveler infers a set of simultaneous events that to me are not simultaneous, but there is no preferred reference frame. Their sloped xaxis is right for them. Even just doing this graphically, we see that the traveler's x-axis is rotated by the same angle as their time axis. That comes from insisting that we all see the same speed of light, 45° on the spac-time diagram. Moving between these reference frames is now a simple matter of squaring up our traveler's axes. In fact, we
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grid up the diagram with a set of lines parallel to these new axes and square up everything while maintaining our intersection points. My world line is now speeding off to the left while our traveler is motionless. We just performed a Lorent transformation but using geometry rather than math. This transformation allows you to calculate how properties like distance, time, velocity, even mass and energy shift between reference frames. But check out what happens if
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I attach pens to all of the intersections. When I transform between frames, they trace out our hyperbole. Those intersections represent locations of space-time events relative to the origin. They will always land on the same hyperola, no matter the observer's reference frame. I told you that these contours show where clocks moving from the origin reach the same proper time count. But more
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generally, each represents a single value for the space-time interval. The delta x and delta t of the event at the end point of a traveler's world line might change depending on who is watching but the hyperbolic contour that they landed on the space-time interval will not. This is because the space-time interval itself comes directly from the lorren transformation as the only measurement of space-time separation that is unchanging or invariant under that transformation.
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Now we can finally get to why this thing is so important and what it really represents. It may seem counterintuitive that an event very close to the origin in both space and time can be separated from that origin by the same space-time interval as an event that is very distant in both space and time. The hyperbolic shape seems to demand that. But remember, it takes the same amount of proper
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time to travel from the origin to a nearby near future event compared to a distant far future event on the same contour. From the point of view of a particle communicating some causal influence, those points are equivalent. The space-time interval tracks this causal proximity. We can think of these lines as contours on a sort of causal geography. The way I define the space-time interval, it becomes increasingly negative in the forward time direction. So we can represent this
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as a valley dropping away from me here at the origin. I naturally slide through time by the steepest path straight down. I can change that path by expending energy to change my velocity. Although doing so realines the contours, so I always slide down the steepest path. There's no point anywhere downhill that I can't reach as long as I can get close enough to the speed of light. In fact, the nearest downhill contour defines the forward
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light cone for anyone anywhere on the space-time diagram. But uphill is impossible as long as the cosmic speed limit is maintained. Breaking that speed limit and sliding uphill are equivalent. To reverse the direction of your changing space-time interval is to reverse the direction of causality. To travel backwards in time, the space-time diagram we looked at today was for a flat or manowski space in which faster than light travel is the only way to flip your
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spac-time interval. But in the crazy curved space within a black hole, it gets flipped for you. We'll soon see how this requirement of a forward causal evolution leads to some incredible predictions when we try to calculate the subevent horizon interval of spaceime. Today on spacetime, we're going to talk about time space or the strange switching in the roles of space and time that occurs in the mathematics when we drop below the event horizon of a black hole.
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What does this bizarre statement space and time switching roles even mean? Is this space-time dyslexia purely a mathematical quirk or does it correspond to real tiny wimy weirdness? We've been working up to this one, so you might want to hit pause and check out these episodes if you think you need some more background. Okay, let's get started. First, we'll think about what the flow of time looks like without black holes or even space-time curvature. When we talked about the
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geometry of causality, we saw that this quantity that we called the space-time interval governs the flow of cause and effect. The only reliable ordering of events in a relative universe. I'm going to show you the math one more time, and then we'll get back to doing all of this graphically. The spac-time interval is defined like this for boring old flat or Minkovsky space. Different observers may report that two events are separated by different distances delta x and by different
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amounts of time delta t. However, all observers record the same space-time interval. If one event causes a second event, the space-time interval must be zero or negative. That just means that a light speeded causal link may have traveled between them. You could say that an object at a given space-time instant is caused by whatever version of itself existed an instant earlier. So, world lines of objects have decreasing space-time intervals. In fact, forward temporal evolution
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requires a negative space-time interval. In flat spacetime, that negative sign in front of the delta t drives that forward evolution. This makes t the timelike coordinate while x is the space-like coordinate. For causality to be maintained, the timelike coordinate must always increase. Reversing causality means flipping the sign of the space-time interval. In our episode on super luminal time travel, we saw that in flat space, this means traveling faster than light,
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which is of course impossible. But if we introduce a black hole, we now have a second way to flip the sign of the space-time interval. We're going to see how this changes the behavior of time in very strange ways. Add a non-rotating uncharged black hole and the space-time interval becomes this. This comes from Carl Schwartzshield's solution to the Einstein field equations, the very first accurate description of a black hole. I've left out a few terms. This equation
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assumes no orbital motion, only motion towards or away from the center of the black hole, which is a distance r away. That RS is the swast shield radius, the radius of the event horizon. Very far from the event horizon, the swast shield interval becomes the good old Minkovsky interval. and time and space are nicely separated. But if an object gets close to the event horizon, so are just a
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little bit bigger than RS, that stuff in the two brackets describes extreme warping of spaceime. But as long as you're outside the event horizon, time behaves itself mostly, a negative space-time interval still means causal movement. And the only way to break causality is still with faster than light travel. Things change radically below the event horizon. When R gets smaller than RS, then
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both of these brackets become negative. The entire deltar stuff is now negative and the delta t stuff is positive. Below the event horizon, there is only one way to maintain the respectable causal progression expected of a well-mannered temporal entity. That's to fall inwards to have a nonzero delta r. As it happens, you don't have a choice. Space itself is falling inwards faster than the
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speed of light towards the central singularity. It carries you with it and drives your personal clock forward as it does so. In the mathematics, the coordinate r which once represents a distance now grants the negative sign needed to maintain your causal flow. It becomes timelike. It's unidirectional. Meanwhile, the coordinate previously known as time t lost its negative sign and become space-like. So, it can be traversed in any direction or not traversed at all. But what
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does all of this time space switching actually look like? Let's fall into the black hole one more time now graphically instead of mathematically. Back out here in the regular universe, it's pretty obvious where the past and the future are. On our ever popular space-time diagram, we see a sharp division between the two. Our past light cone encompasses all of spaceime that could have influenced us. While our future light cone shows us the parts of the universe that we might ever
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hope to encounter or influence. Which direction is the future? ahead along our time axis and at right angles to all of our space axes. Our future light cone stares fixedly forwards, encompassing all spatial directions equally. This is no longer true if we introduce gravity. Close to a massive object, your future is no longer at right angles to space. It becomes slightly tilted in the direction of that mass. send out a burst of future defining light rays and they won't spread out
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evenly because they bend towards the gravitational field. As you approach the event horizon of a black hole, more and more light rays are turned towards the event horizon. Your future light cone and your time axis begin to blur together with the inward radial axis of the black hole. At this point, it's time we switch diagrams. Close to and within the black hole. The Penrose diagram is much
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more useful. It deals with the extreme stretching of space and time by compactifying lines of constant space or time close to its boundaries. We talked about these diagrams previously, but an important thing to remember is that the lines of constant space and time are curved so that light cones remain upright and light always travels at a 45° angle even inside the black hole. This entire diagonal line represents the event horizon. Watch what happens to our view of the
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universe as we approach it. Our entire future light cone encompasses more and more of the event horizon. That last tiny sliver is a narrowing window directly above that you could escape to at close to the speed of light. Meanwhile, our past light cone now encompasses light that has been struggling to escape from just above the event horizon since the distant past. But we still see nothing from below the horizon. Yet, as soon as we pass the horizon, everything changes. The outside
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universe exits our future light cone, which now just contains the singularity. We also begin to encounter a new set of photons from the past. At the moment of crossing, light rays from the event horizon itself are suddenly visible. In fact, we plummet through a sea of light that is eternally climbing outwards but getting nowhere. After that, we have access to the history of the interior of
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the black hole. As we fall with the faster than light flow of spaceime, we overtake light that is outward pointing. That light isn't actually making headway outwards. It's trying to swim upstream and failing against the faster than light cascade of spaceime. Some of this light might be from the collapsing surface of the star that first formed the black hole, emitted long before we entered
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the event horizon. It appears to come from below us because it's trying to climb upwards. In fact, though, it was emitted at larger radi than wherever we encounter it. Also, in our past light cone are light rays that are pointed inwards. some of them coming from the outside universe. This light overtakes us as we fall. This is light that entered the event horizon after we did and appears to reach us from above. We can try to move towards either source of light, down towards light
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from the black holes past or up towards light from the black hole's future. Those directions, those spatial freedoms are now described by what was once the time coordinate, but it's no longer timelike. You can traverse it in either direction, making it space-like. Doing so isn't actually traveling in time, even though there's a sense of past events in one direction, the collapsing star, and future events in the other, everything that fell into the black hole after us. But remember
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that our future light cone actually just points towards the singularity. If we try to accelerate in either direction, up or down, we just quicken our demise. Best just to fall. It's the last mercy granted by the black hole. It transports us to our doom by the slowest path unless we resist. Below the event horizon, there's still a sense of spatial upness and downness. However, the old radial dimension isn't space-like. It's timelike. Every photon that reaches us was emitted
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at some larger radius than wherever we encounter it. Even if it's old light struggling outwards, the past is radially outwards. and all possible future directions lead radially inwards in the same way that all world lines move towards the future in the outside universe. Time is layered radially and r is timelike unidirectional. The singularity becomes a future time not a central place. In fact, the swast shield metric really gives two separate space-time maps in a
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single equation. One for above and one for below the event horizon. The coordinates R and T play different roles in those regions. There are other coordinate systems in which that switch never happens. But this mysterious dimensional flip does give us some fascinating insight into how time and space blend together in what is perhaps the strangest place in all of spaceime. Lurking in the depths of the mathematics of Einstein's general relativity is an object even
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stranger than the mysterious black hole. In fact, it's the black holes mirror twin, the white hole. Some even think that these could be the origin of our universe. The astrophysical phenomenon of the black hole has captured the imagination of scientists and science enthusiasts alike for many decades. When the idea first emerged from Einstein's general theory of relativity, physicists wondered how seriously to take this mathematical description of an inescapable region
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of spaceime. Astronomers have since demonstrated that black holes are very real with convincing evidence that quazars, x-ray binaries, even the center of our own Milky Way galaxy harbor these gravitational monstrosities. But the mathematics that predicts the existence of the black hole also describes entities that are even stranger but whose relationship with reality is still unclear. One such entity is the white hole. A white hole is the opposite of a black hole in a very literal
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mathematical sense. In fact, it's a time reversed black hole. A black hole is defined as a region of inward flowing spaceime with a one-way boundary called the event horizon from inside of which nothing can ever escape. That makes a white hole a region of outward flowing spaceime. It also has an event horizon, but that horizon prohibits entry, not exit. Nothing outside a white hole can ever
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enter, and everything inside must be ejected. Not even light can leave a black hole, hence the whole black thing. But light can only leave a white hole. So, these might be expected to radiate like crazy, and white would be an understatement. Now, before everyone gets too excited, white holes are probably a figment of mathematical imagination, but they're a fascinating one, and the idea may
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help us understand the origin of the universe. White holes first emerged in the very earliest mathematical description of black holes. Only a few months after Einstein published his general theory of relativity, Carl Schwartzshield solved its equations for a very particular case. a single point of mass in an otherwise empty spaceime. The resulting SWAT shield metric actually describes a black hole, the simplest black hole possible, one without spin, without charge,
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or without change. An eternal black hole that doesn't grow or shrink and has always existed. We've talked quite a bit about the bizarre behavior of space and especially time at and below the event horizon of a black hole. Here's a little playlist if you want a refresher. But here's the lowdown. The time that happens inside a black hole is not part of the past or future history of the outside universe. From the perspective of an outside observer, any events occurring at the
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event horizon, including falling into it, happen infinitely far in the future. Once you fall into the black hole, the swath shield metric tells us that space and time switch their roles. The singularity no longer occupies a central location. It now occupies an inevitable future time. Now, a real black hole forms from the gravitational collapse of a massive stars core. After the collapse, the future singularity comes into being. And in the past, well, there's just a star. But
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what does this idealized eternal black hole look like in the past? If we follow the Swatshield metric back in time, we find something very strange. We find the singularity again lurking infinitely far in the past. From the point of view of the outside universe, the eternal black hole singularity exists both in the infinite future and in the infinite past. That may sound strange, but it gets stranger. To really understand what this eternal black hole looks like, we're going to
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need to use a tool that we've already played with the Pinrose diagram. To refresh your memory, in a Penrose diagram, the X and Y axes are redefined from space and time to merge space and time into new coordinates. They compactify spaceime so that time bunches up towards the edges and the borders correspond to infinite past and future. Also, lines of constant distance and time curve so that light paths always travel on 45° paths. We are hanging out here and now at the center of the
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diagram. If we place an eternal black hole far to the left, then the future left boundary represents the black holes event horizon. Any movement to the left brings you closer to that event horizon. The event horizon itself is a 45° line. In our weird Penrose coordinates, this represents a constant distance from the center of the black hole. Light traveling at that 45 degree angle takes infinite
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time to escape the event horizon. And the region beyond that line represents the interior of the black hole. There the dimensions of space and time switch roles. The once vertical contours of space are now timelike and flow inexorably towards the future singularity. These two regions, our universe and the black hole interior, are just the swast shield metric mapped out using Penrose coordinates. But our map isn't complete. Remember, this is an eternal black hole. So,
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it must exist in the past. Map into the past and we see a time reflected version of our future black hole. Everything about it is time reversed. The singularity is a past event. Space within is timelike. But instead of flowing towards the singularity, it flows away. And the event horizon is now a barrier to entry, not to exit. We can make some sense of the behavior of this strange region by using the Penrose diagram. Imagine that something in
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our past was traveling at the speed of light and trying to reach the past event horizon. There's no way it can get there unless it goes faster than light. Oh, it'll reach an event horizon, but only the event horizon of our future where it plunges into a regular old black hole. Remember that all of this is from our perspective, far from the event horizon. We can never see anything cross the horizon. The light rays from any crossing reach us infinitely far in the future. Even if
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the black hole plunge began far in the past. So the past region of the eternal black hole has an event horizon that's a barrier to entry. But also light rays within that region must move up on the diagram that suggests they must exit into the outside universe. Anything inside the past eternal black hole must be ejected. So far, this region fits perfectly the description of a white hole. The eternal black hole of the past technically is a white hole. However,
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it's not one that we can ever observe for two reasons. One, light rays exiting that past white hole can never reach us. The past singularity and past event horizon are infinitely far in the past from our point of view. light has to traverse infinite time to reach our location. And two, there's no such thing as an eternal black hole. The universe hasn't existed for eternity. And it didn't even begin with black holes in place. Even though this type of white hole isn't observable,
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some physicists have taken the description very seriously. The math describing the white hole is a perfectly good use of the swat geometric. It obeys general relativity. It really is just a black hole but viewed backwards in time. Yet general relativity is time reversal symmetric. Something that can happen forwards in time should also be able to happen in reverse. So can new white holes actually form? Well, theoretically yes. But to make one, you need to reverse entropy.
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See, although it's possible to build a white hole in general relativity, there are other laws of physics that the universe needs to obey. For example, the second law of thermodynamics. It demands that entropy, a measure of disorder, always increase. This law defines the direction of the flow of time. To reverse time, you need to break the law. You need to decrease entropy. Now, this is technically possible because entropy is a statistical phenomenon. Very rare reductions
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in entropy do happen. As long as globally entropy increases on average, it's conceivable that an incredibly rare entropy dip could lead to an effective reversal of time and a white hole could form. However, it would immediately explode in a burst of energy as soon as entropy and time resumed their normal flow upwards and forwards. We actually did talk about a case where a random drop in entropy led to something very much like a white hole in this episode. It's been speculated that
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the Big Bang itself came from such a profoundly improbable entropy dip. And as it happens, the Big Bang looks mathematically at least much like a white hole. It's an expanding outpouring of spaceime containing a vast amount of energy. And the bang itself can never be entered. After all, it's in the past. The difference between the Big Bang and a white hole is that the former possesses
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no singularity. It happened everywhere at the same time. Still, that hasn't stopped physicists from having fun with the idea. It's been proposed that when a black hole forms, a white hole forms on the opposite side. Energy entering the black hole exits the white hole. Physicist Lee Smolen takes it a step further to suggest that the resulting white hole is the big bang of a new baby universe
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and that in fact our universe formed that way. More on that another time. But speaking of other universes, it turns out that we haven't finished building our Penrose diagram yet. The past white hole was revealed when we traced the eternal black hole backwards in time. In fact, what we did was to maximally extend spaceime. We required that all paths be traceable through infinite past and future space, provided they don't hit the singularity. But what about light rays entering or
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leaving our eternal black hole from the opposite side? The mathematics of the swast shield metric describes an entirely independent region of spaceime parallel to our own. It looks like an identical alternate universe on the other side of the black hole accessible through what we call an Einstein Rosen bridge, better known as a wormhole. In the not too distant future, we'll investigate the reality of this mysterious parallel patch of spaceime. He was perhaps the greatest genius of
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our time. Steven Hawking peered behind the curtain of reality and glimpsed the true workings of the universe. He inspired all of us to pursue our curiosity no matter the obstacles. However, his true legacy is his work. He made profound contributions across physics from quantum theory to cosmology. Our tribute is to bring you Steven Hawkings most famous discovery. I'm Matt Odow. This is spaceime and it's time for Hawking radiation. Soon after Einstein revealed his great
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general theory of relativity in 1915, physicists realized that it allowed for the possibility of catastrophic gravitational collapse in places of extreme density like the dead core of a massive star. Space and time could be dragged inwards to create a hole in the universe, a boundary in spaceime called an event horizon that could be entered but from beyond which nothing could return. Once formed, there was nothing in theory or imagination that could bring material consumed
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back to the outside universe. These black holes should exist forever, only growing, never shrinking. Or so we thought, until 1974 when a young physicist named Steven Hawking published a paper in Nature entitled Black Hole Explosions. In this and in a follow-up 1975 paper, he attempted a new union of quantum mechanics and general relativity to show that black holes should not be so black after all. They should leak. They should emit what we now know as Hawking radiation.
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There's a popular description of how Hawking radiation works. It goes something like this. Empty space seas with activity as pairs of virtual particles matter and antimatter spontaneously appear and then annihilate each other briefly borrowing energy from the vacuum itself. But when this happens near a black hole, sometimes one of the pair will be swallowed by the event horizon, leaving the other free to escape and taking its stolen energy with it. That energy can't
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come from nothing, and so the black hole itself pays the debt by slowly leaking away its mass. This is a nice picture, but how accurate is it? In fact, if we follow the narrative of Hawkings original calculation, the story sounds rather different. We've come a long way over the past few months, building up the knowledge we'll need to follow that calculation. Re-watching some of those episodes either now or after this video will be helpful. But if you think you're ready, let's
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take a deep dive into the quantum field theory of curved spaceime to glimpse the true nature of Hawking radiation. Actually, a quick QFT refresher can't hurt. Space is filled with quantum fields. They can oscillate with different frequencies, much like the many possible vibrational modes on a guitar string. A particle is like a note on the string. And just like a real guitar note, real particles tend to be comprised of many vibrational modes. Those underlying vibrational
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modes are still present in the absence of real particles. They fluctuate in energy due to quantum uncertainty. And those fluctuations give us what we think of as virtual particles. Now, don't take the existence of virtual particles too seriously. They're really just a tool for calculating the infinite ways in which a fluctuating quantum field can behave. One way that quantum fields are very different to guitar strings is that they can have both positive and negative frequencies.
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A negative frequency can be thought of as a mode that travels backwards in time and can be interpreted as corresponding to antimatter. Now, that's a whole level of weird all on its own. And we talk about it here. When a quantum field is in a vacuum state, there's a balance between positive and negative frequency modes, which you can crudely think of as a balance between virtual matter and antimatter particles. These all virtually annihilate or cancel out so that no real
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particles exist. This is all fine in flat space, but spatial curvature can mess with the balance of the underlying quantum field modes by introducing horizons. Horizons cut off access to certain modes of the quantum fields, disturbing the balance that defines the vacuum. Steven Hawking knew that black holes with their insane space-time curvature would wreak havoc on quantum fields in their vicinity.
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But what would the effect be? To answer that properly, he would need a full union of general relativity and quantum mechanics, a theory of quantum gravity, a theory of everything. It didn't exist then and it doesn't exist yet. Not to be deterred by the impossible, Hawing came up with an ingenious workaround. The narrative of Hawings mathematics goes something like this. He imagined a single space-time path, a light speeded trajectory called a null geodisic. It
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extends from far in the past to far in the future. This is a perilous path. It passes through the location of a black hole in the instant before it forms. In fact, it is the very last trajectory to do so. It emerges barely ahead of the forming event horizon. Hawking imagined a simple quantum field tracing this path. a field that is in a perfect vacuum state before the formation of the black hole. But he found that the close shave with the
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black hole disturbs the fundamental vibrational modes that define the fluctuations of the vacuum. By the time this trajectory has found its way back out into flat space again, those fluctuations look like real particles. A distant future observer sees radiation coming from the black hole. Hawkings imaginary path from the distant past to the distant future was brilliant. It allowed him to compare the state of the vacuum in two regions of flat space far from the black hole. Regions
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where the nature of vacuums, quantum fields, and particles are perfectly well understood. But to understand the effect of the close encounter with the black hole, he required an uneasy marriage of quantum mechanics and general relativity. In the absence of a theory of quantum gravity, Hawking needed a hack. That hack was the Boliv transformations. Say that three times fast. These can be used to approximate the effect of curved spacetime on quantum fields by smoothly
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connecting regions of flat space. They describe a sort of mixing of the positive and negative frequency vibrational modes that are caused by that curved space. The physical interpretation of this mixing via the Bolivia transformations is tricky. In fact, there isn't just one valid interpretation. Hawings calculation talks about scattering. Certain modes of the quantum field are scattered or deflected by the gravitational field of the forming black hole. They are nudged off
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their narrow escape path and so are lost behind the forming event horizon. Meanwhile, other modes avoid scattering and continue unscathed. With the loss of certain fundamental modes, the vacuum state must be constructed from the remaining modes. That distorted vacuum looks like it's full of particles. The nature of the lost modes tells us what Hawking radiation should look like. Black holes tend to scatter modes with wavelengths similar to their own sizes. The quantum field that
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emerges is distorted in the same wavelength range. And so it produces wave packets. It produces particles that also have wavelengths about as large as the event horizon. So the more massive the black hole, the longer the wavelength of its radiation. Hawking calculated the frequency distribution of this radiation and found something incredible. It should look exactly like thermal radiation. Black holes should have a heat glow with an apparent temperature that depends on
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their mass. More directly, it's proportional to the surface area of the event horizon. Large black holes should appear cold, radiating excruciatingly slowly, but small black holes should appear hot and the smallest should radiate explosively. Okay, so what about the whole picture of particle antiparticle pairs being pulled apart by the event horizon? So, Hawings math describes splitting or mixing of these pure positive and negative frequency modes. It's fair to interpret this
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mixing as the promotion of what were once virtual particles into reality. And for the escaping modes, there exist a corresponding set of modes linked by quantum entanglement that are trapped behind the event horizon. We can interpret those as corresponding to the swallowed antiparticle partner. So the split matter antimatter part of the picture is reasonable. But there are reasons to dismiss aspects of this picture. Firstly, this radiation is not localized. Remember the Hawking
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radiation has wavelengths the size of the event horizon, the size of the entire black hole. Well, these are the deu wavelengths of created particles and they tell us that there is an enormous quantum uncertainty in the location of these particles. Hawking radiation must appear to come from the global black hole, not from specific points on the event horizon. In fact, an observer in freef fall through the horizon sees nothing. To them, space is locally flat. The
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vacuum should look like a vacuum. This radiation is visible only to distant observers. Well, there is one exception. When you turn on your jetpack and hover a fixed distance above the horizon, then you do see particles. you see unrew radiation. We'll look at its relationship to Hawking radiation in the future. By the way, Hawking radiation is mostly going to be photons and other massless particles. To produce particles with mass, the energy of the radiation
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has to be high enough to cover the rest mass of the particle. So, it's okay to interpret the narrative of Hawkings calculation as the splitting of entangled matter and antimatter pairs. Even if it really is just a huristic interpretation, it's the cause of the splitting that's hard to pin down. We can think about positive and negative frequency modes being mixed due to scattering, perhaps by the as yet undiscovered graviton. Other physicists have derived Hawkings result
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with very different seeming narratives. For example, in 2001, Periq and Wilchek got the same thermal spectrum for Hawking radiation by thinking about particles escaping from beneath the event horizon through quantum tunneling. The common thread is quantum uncertainty. For example, uncertainty in positional momentum can lead to particle pairs that were once in the same location or modes that were once on the same world line becoming separated by the event horizon.
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Alternatively, uncertainty in energy can lead to particle creation. Whichever way you interpret it, it's hard to avoid the conclusion that black holes emit particles. The fact that different derivations lead to exactly the same result or that the radiation looks thermal can't be by chance. It's hard to make Hawking radiation go away in the math. And believe me, Steven Hawking himself tried. Ultimately, however, these calculations are all hacks, albeit utterly
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brilliant ones. Without a full quantum theory of gravity, the origin of Hawking radiation will remain mysterious. And there are other mysteries that we haven't touched on. For example, what happens to the particles or modes trapped by the black hole? How do they end up reducing the black hole's mass instead of increasing it? And then there's the famous information paradox in which Hawking radiation appears to destroy what should be a conserved quantity, quantum information.
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We'll tackle all of these in future episodes. For now, we must conclude that black holes radiate and in doing so evaporate. The scariest monsters of general relativity are ultimately unraveled by the brilliant mind of Steven Hawking and a mysterious quirk of quantum spacetime.
1:19:03
Black holes seem like they should have no entropy, but in fact, they hold most of the entropy in the universe. Let's figure this out. At first, it seemed that black holes were so simple they should have no entropy. Well, it turns out they contain most of the entropy in the universe. Let's see why. Because this fact may force us to conclude that the universe is a hologram. Black holes are a problem. They are the inevitable result of extreme gravitational collapse. At least
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they are inevitable according to the equations of Einstein's general theory of relativity. That theory is one of the most thoroughly tested in all of physics, which means we should probably believe in black holes. Also, we've seen them in their gravitational effects on their surrounding space and in the gravitational waves caused when they merge. And yet, if black holes exist, which apparently they do, they contradict other theories in physics that are as sacred as general
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relativity. They cause all sorts of problems with quantum theory, which we've talked about before and we'll review in a sec. But they also present an apparent conflict with the notion of entropy and the second law of thermodynamics. It was while pondering that conflict that Jacob Beckenstein realized an incredible connection between black holes and thermodynamics. His insight launched an entire new way of thinking about the universe in terms of information theory and ultimately
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led to the holographic principle, which I promise we're getting to and are almost there. But first, you are going to need to know more about why black holes contain most of the universe's entropy. Okay, I'm getting way ahead of myself. Let's actually rewind back to those episodes where we laid out the black hole information paradox because they're going to be critical to a proper understanding. We're also rewinding to the late '60s, early '7s when physicists realized something
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odd about black holes. What they realized is that it doesn't matter what material goes into one. From the point of view of the outside universe, black holes can only have three properties: mass, spin, and electric charge. This is the so-called no hair theorem. And it suggests that most of the information about anything that falls into a black hole is lost to the outside universe. But
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a fundamental tenant of quantum mechanics is that quantum information can never be destroyed. So if black holes evaporate, as Hawking discovered and we also covered, this evaporation should destroy a black hole's internal quantum information, giving us the black hole information paradox. Eventually, a possible resolution to this paradox was found by Gerard. He described a mechanism by which the information contained by infalling particles could be preserved on the event horizon
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of the black hole. From there, it could be imprinted on the outgoing Hawking radiation, allowing the information to escape back into the universe. Okay, problem solved. But in our previous episodes, we skipped the key insight that started all of this. It all began with Jacob Beckinstein thinking about black hole entropy. Okay, first entropy. Yeah, we talked about that a lot recently. Also, you know, it's almost like all of those episodes are starting to come together.
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Almost like we planned this. Go and watch that background stuff if you're behind. But of course, for now, I'll give you a quick TLDW on entropy. So, we can think of entropy in two ways. One, it's a measure of how evenly energy is spread out. High entropy means thermal equilibrium. So energy is very evenly distributed and can't be extracted in a useful way. And two, entry measures the amount of unknown information that you would need to perfectly describe the systems internal
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state like all the particle positions, velocities, etc. The higher the entropy, the more randomly distributed its particles and the more possible configurations lead to the same macroscopic state. The higher the entropy, the less you can guess about the properties of individual particles based on the global properties like temperature, volume, pressure, etc. Okay, so the second law of thermodynamics states that entropy of an isolated system must
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always increase which means energy tends to spread out evenly and particles tend to randomize reducing our information about their microscopic states. How does this relate to black holes? Let's make a black hole and see what happens to entropy. We start as usual by collapsing the core of a dead star. Now that's a high entropy beast, super hot and full of randomly moving particles. We have almost no information about the individual particles, but that information still exists
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in the universe, like I guess the particles know where they are. At the instant the star collapses far enough to form an event horizon, it becomes a black hole. We go from knowing next to nothing about the object to knowing everything. We can easily measure its mass, spin, and electric charge. And according to the no hair theorem, that's all there is to know. The region of space in which the black hole formed appears to have gone from high entropy to zero entropy in an
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instant, shattering the second law in the process, which to put it mildly is a problem. But if you paid attention to the whole information paradox bit, you might be able to think of a solution. If quantum information is stored on the surface of the black hole, can't we store entropy there also? And then why not radiate the entropy back into the universe as Hawking radiation? Actually, yeah. The
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resolution to the information paradox also saves the second law of thermodynamics. That was easy. I thought physics was supposed to be hard. Okay, hang on. Let's think about this a little bit more. It was this seeming violation of the second law that got Jacob Beckinstein thinking about the connection between black holes and information in the first place. The breakthrough insight was this simple observation. The surface area of a black hole event horizon can never decrease
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at least not according to general relativity. So you know how nothing can escape black holes ignoring corking radiation for the moment. That should mean that black holes can only grow. They can never shrink in mass or radius. Well, that's not quite true. If you merge two black holes, some of their mass gets converted to the energy radiated away in gravitational waves. There's also the Penrose process in which you can extract rotational energy of a spinning black hole. And
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by you, I mean not you, I mean super advanced far future civilizations. Gravitational radiation and the Penrose process reduce black hole mass and radius or the sum of masses and radio merging black holes. But there's one property of black holes that no process other than Hawking radiation can decrease. That's the surface area of the event horizon. Do anything to black holes and their total surface area can only grow or stay constant. Beckenstein saw a close correspondence
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between the always increasing event horizon surface area and the always increasing nature of entropy. He also realized that the equation relating the change in black hole surface area to the change in its mass closely resembles the original definition of thermodynamic entropy. Just replace change in entropy and internal thermal energy with change in black hole surface area and black hole mass respectively. You can also add the work done when you extract energy from the black
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hole and it looks the same as the equation for the work extracted from a thermodynamic system. Beckinstein had just discovered black hole thermodynamics, but that didn't give him the exact definition for black hole entropy. For that, he turned to Ludvig Boltzman'sformational definition for entropy. So entropy can be defined as the information hidden in a systems microscopic configuration times the Boltzman constant. Beckenstein estimated the amount of
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information that would be lost into a black hole as it grew. Essentially, he built a black hole out of idealized elementary particles that each contained a single bit of information. And guess what? The information content of a black hole is proportional not to its mass or radius or volume, is proportional to its surface area. In fact, the information content is very close to that surface
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area divided by the number of plunk areas. It's as though each of these minimum possible quant of area each contain a single bit of information. Now just multiply that information content by the Boltzman constant and you have the entropy of a black hole which is going to be directly proportional to the surface area of the event horizon. Beckenstein's connection between surface area and entropy could have been a coincidence. At least until Steven Hawking came along. In 1974,
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a year after Beckenstein's first paper on black hole thermodynamics, Hawking published his first Hawking radiation paper. He showed that black holes radiate random particles exactly as though they have a heat glow of a particular temperature that depends on their mass. So if black holes have a temperature, then they also have entropy. Good old-fashioned thermodynamic entropy tells us that
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change in entropy is change in internal thermal energy divided by temperature. So Hawking just plugged his Hawking temperature into that equation along with black hole mass for internal energy and figured out the total entropy contained in a black hole. He got an expression almost identical to Beckenstein's but just a slightly different constant of proportionality. So you get the same result for black hole entropy whether you figure it out from the amount of information that gets
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trapped building a black hole or the amount of heat that leaks as it evaporates. And it's proportional to the surface area. How bizarrely consistent. I'd say that means it's right. The second law of thermodynamics is saved because black holes do have entropy. In fact, they have enormous entropies, the maximum possible. So much that black holes are now believed to contain most of the entropy in the universe. But the real importance of this work
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wasn't the solution to some obscure conundrum. It changed our thinking about the informationational content of the universe. Beckinstein's formula was derived for black holes, but it also gives the maximum amount of information that can be fit into any volume of space. In this respect, it's called the Beckenstein bound and it's proportional to the surface area of that space. This is unexpected. Surely, the maximum amount of information you can fit into some patch of space
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depends on the volume of that space, as in one bit per tiny volume element inside that space. But in fact, the rule is one bit per tiny area element on the surface of that space. That also means that the information needed to describe any volume of space, no matter its contents, is proportional to the area bounding that space. I've hinted once or twice that this simple idea led to the holographic principle. The idea that the entire 3D volume of the universe is just a projection of
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information encoded on a 2D surface surrounding the universe. You just need to add a little bit of string theory. It's a hell of a conceptual leap given it started with Jacob Beckenstein noticing a peculiar similarity between some formula. It might also be true and obviously we'll be back before too long to talk about string theory and the holographic nature of spaceime. These black stones are volcanic rock and this is one of the youngest patches of land on planet Earth. But that
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same geological event that built this land has provided another window. It allows us to observe a time when the universe was still cooling from the fire of its own formation. And to see this, all we have to do is travel to a telescope on top of the tallest volcano in the world. So we're driving up to the summit of Monaco on the big island of Hawaii. This is the tallest volcano on the planet that 4200 m. The oxygen up here is 60% sea level, but astronomers deal with it
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because it is the premier astronomical observing site in the northern hemisphere. To Hawaiians, it is a sacred site. And to astronomers, it's where the Earth meets the universe. Wow, it's amazing up here. It's like being on another planet. I can already feel the effect of the thinner atmosphere. My natural impulse, bizarrely, is to hold my breath. Must remember to keep breathing. Here we have 13 of the greatest telescopes in the world operated by 11 different
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countries. We have the Japanese Subaru telescope, the Twin Kemes. Over here we have the Canada, France, Hawaii telescope, and this is Gemini. That's where we're going. We're here to talk about a very special observation. In the spring of 2017, astronomers turned Gemini's great mirror towards the constellation of Buotis, the plowman. They were looking for a faint speck of light that had been noticed in one of our great surveys of the sky. Astronomers
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guessed the speck was a quazar, a vortex of radiant matter falling into a giant black hole.
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Now quazars are the most luminous objects in the universe. What was strange about this one was its distance. Its light was so red that astronomers realized that that light must have been stretched out, redshifted by traveling many billions of years through our expanding universe. The quazar appeared to be more distant than any we had ever seen. But that doesn't mean we can't unravel their mysteries. And Gemini did exactly that. To find out how,
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we're going to need to go inside. You've got to see this. It's incredible. Meet the Gemini telescope. This is what a worldass telescope looks like these days. It is enormous. I still remember the first time I came to a telescope like this. It blew me away. Look at the size of this thing. This is our window to the universe. It's cold in here. They keep the dome at the temperature of the upcoming night so that giant structure
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doesn't warp and twist with the change in temperature. That's a little below freezing right now. And you hear that sound? That's the cryogenics. They keep the sensitive infrared cameras at 15 above absolute zero. Let's actually talk about light for a second. Light is a wave and the wavelength of that wave determines the properties of light. For example, visible light, the wavelength range that our eyes are sensitive to,
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spans only a tiny fraction of the spectrum. That's why we create telescopes. The universe looks very, very different at different wavelengths. For example, viewed in visible light, the Andromeda galaxy shows us newborn stars. Our atmosphere is transparent to visible light. So, a groundbased telescope can see a visible universe as can we. Gemini is built to be sensitive to the infrared.
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The infrared andrometer is a swirl of star forming clouds and gas. Some infrared light also makes it through the atmosphere, though it helps to be up here on a mountaintop. Although the air above the observatory is crystal clear, it still blurs distant light somewhat. Turbulence in the atmosphere causes incoming wave fronts of light to be warped, and it blurs our view. To correct this,
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Gemini uses adaptive optics. It has a deformable mirror that flexes and bends to match and correct the warping of incoming light. To do this in real time, Gemini creates its own artificial guide star by shooting lasers to twinkle off sodium atoms at 90 km height, right off the edge of space.
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This is the instrument used to analyze the most distant quazar. It's the Gemini North infrared spectrograph. Genius. A spectrograph takes incoming light and breaks it into its component wavelengths similar to a prism. And it records how much energy is received at each wavelength. We call that a spectrum. When the light analyzed by this machine left its quazar, it was ultraviolet. But traveling through the expanding universe sapped energy and stretched the wavelength of
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that light so that it was infrared by the time it reached the earth. And this spectrograph the red shift tells us how long that light has been traveling. 13.1 billion years, meaning the quazar lived when the universe was only 5% its current age. There's a broad blank patch in the quazar spectrum. It's a stretch of nothing that tells us a ton. Shortly after the Big Bang, when things had cooled down a bit, the universe was filled with hydrogen gas. It was murky,
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especially for ultraviolet light. Now, that gas collapsed into the very first stars. Then the very first galaxies. Those stars eventually melted away the remaining hydrogen in a process called reionization, leaving a crystal clearar universe. But this quazar shines out from the era of those first stars before they'd finished the job of reionization. Much of the quazar's once ultraviolet light was sucked up before it escaped the early universe. And what about the super
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massive black hole at the center of the quazar? The same signature wavelengths used to measure red shift are also broadened due to the extreme speeds of matter moving near the black hole. That allows us to estimate the mass of the black hole. 800 million suns. If it replaced our sun, it would easily swallow Saturn's orbit. Scientists struggle to figure out how it could grow to that insane size in a tiny fraction of the age of the universe. We are expanding our understanding of
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physics to figure this one out. That tiny speck is both a revelation and a mystery. It literally shines a light on the earliest epochs of our universe, teaching us about our most fundamental origins. But it also opens new questions. And our great telescopes, our portals to the universe, past and present, will tackle those questions, too, and ultimately bring us closer to understanding this mysterious, this magnificent spaceime.
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