# Black holes and why time slows down when you are near one

## Transcript

BRIAN GREENE: Hey, everyone. Welcome to this next episode of Your Daily Equation, or maybe it's going to be your every-other-day daily equation, your semi-daily equation, whatever it is, your bi-daily equation. I never know what the right usage of those words actually is. But in any event, I'm going to focus today on the question, the issue, the subject, of black holes. Black holes.

And black holes are an amazingly rich arena for theorists to try out ideas, to explore our understanding of the force of gravity, to explore its interaction with quantum mechanics. And as I mentioned, black holes are now also an arena which is rich in fertile for observational astronomy. We've gone beyond the era in which black holes were merely theoretical ideas to now the recognition that black holes are real. They are really out there.

I'll also note at the end that there are great many puzzles to do with black holes that have yet to be resolved. And maybe if I have time, I'll mention a few of those. But I'd like, for the most part, to focus here, in this episode, on the traditional, more straightforward, widely-- well, not completely but more widely accepted historical version of the trajectory that led us to recognize the possibility of black holes and some of the properties that emerge from the basic mathematics of Einstein's equations.

So, to get us going, let me just give a little bit of historical background. The story of black holes begins with this fella right here, Karl Schwarzschild. He was a German meteorologist, mathematician, really smart guy, astronomer, who was actually stationed on the Russian front during World War I. And as he's there, and he's charged with actually calculating trajectories of bombs. You hear them going off and so on.

And somehow, in the trenches, he gets hold of Einstein's paper in the general theory of relativity, does some calculations on it. And he realizes that if you have a spherical mass and you crush it to a very small size-- the bombs are still going off all around him-- it will create such a warp in the fabric of space that anything that gets too close will not be able to pull away. And that's really what we mean by a black hole.

It is a region of space in which enough matter has been crushed to a sufficiently small size that the warpage is so significant that anything that gets too close, closer than, as we will see, what is known as the event horizon of the black hole, cannot escape, cannot run away. So the sort of image that you can have in mind is if we have a little animation here of the moon going around the Earth. This is the usual story of warped environment around the vicinity of a spherical body like the Earth.

But if you crushed the Earth down to a sufficiently small size, the idea is that the indentation will be far greater than what we saw for the Earth. The indentation would be so significant that at least, metaphorically speaking, if you're hanging out near the edge of a black hole and you were to turn on a flashlight, if you're within the event horizon, the light from that flashlight would not go off into deep space. Instead, it would go into the black hole itself. This image is a little bit off, I should say.

But it kind of gives you at least a mental toehold for the idea of why it is that light can't get away from a black hole. When you turn on a flashlight, if you're within the event horizon of a black hole, the light shines inward not outward. Now, another way of thinking about this idea-- and look, I know this is quite familiar territory. Black holes are in the culture, you know the phrase falling into a black hole. Or he did something, and it created a black hole. We use that kind of language all the time. So all these ideas are familiar.

But it's good to have mental imagery to go along with the words. And the mental imagery that I'm about to give you, I find particularly interesting and useful. Because there is a mathematical version of the story that I'm going to show you visually right now. I'm not going to describe that mathematical story right now. But just know that there is a version of the so-called waterfall analogy that really can be fully articulated in a mathematical way that makes it rigorous. So here's the idea.

If you are near a waterfall and you are, say, paddling your kayak-- is that the right word? Yeah. Paddling your kayak. If you can paddle faster than the rate at which the water is flowing toward the waterfall you can get away. But if you can't paddle faster than the water is flowing, then you can't get away. And you're doomed to fall down the waterfall. And here's the idea. The analogy is space itself falls over the edge of a black hole. It's sort of like a waterfall of space.

And the speed at which space travels over the edge of a black hole is equal to the speed of light. Nothing can go faster than the speed of light. So near a black hole, you're doomed. So you might as well just paddle right toward the black hole and go on a joyride down the throat of the black hole itself. So that's another way of thinking about it. Edge of a black hole event horizon, space, in some sense, is flowing over the edge. It's flowing over the edge at a speed equal to the speed of light.

Since nothing can go faster than the speed of light, you can't paddle upstream. And if you can't paddle upstream, you can't get away from the black hole. You're doomed, and you will fall into the black hole. Now, that all is highly schematic and metaphorical. I hope that it is useful for thinking about black holes. But for a long time, we knew what black holes should look like if we were to ever see them. We wouldn't literally see the black hole itself.

But in the environment around a black hole, as material is falling over the event horizon of a black hole, it heats up. The material rubs against the other material. That's all falling inward. It gets so hot that the frictional forces heat up the material, and they generate x-rays. And those x-rays go out into space. And those x-rays are things that we can see.

So let me now just show you, therefore, the expected view of a black hole would be something like this. Around the edge of the black hole, you see the swirling maelstrom of material giving off these high energy x-rays. I've put them in the visible, so we can see them. And within that maelstrom of activity is a central region from which no light itself is being released. No light is being emitted.

And that would be the black hole itself. Now, Schwarzschild is doing his work, as I said, it was World War I. So, we're back in 1917 or so. And so, he puts forward this idea of this solution. I show you the mathematical form of that solution as we go forward. But there's a real curious feature of-- well, there are many curious features of the solution. But one in particular is for an object to become a black hole, you've got to squeeze it down.

But how far do you have to squeeze it down? Well, the calculations show that you'd have to squeeze the sun down to about three kilometers across or so to be a black hole. The Earth, you'd have to squeeze it down to a radius of about centimeter or so to be a black hole. I mean, think about the Earth down to a centimeter. It doesn't seem like there'd be any physical process that would ever allow material to be compressed to that degree.

So, the question is are these objects just mathematical implications of the general theory of relativity? Or are they real? And a step in the direction of showing that they're real was taken some decades later when scientists realized that there is a process that could actually lead to matter collapsing in on itself and thereby crushing it down to the small size as required for the black hole solution to be realized, physically.

What are those processes? Well, here is the canonical one. Imagine we were looking at a large star, like a red giant. That star supports its own hefty mass through nuclear processes in the core. But those nuclear processes, which give up the heat, the light, the pressure, ultimately, they will use up the nuclear fuel. And when the fuel is used up, the star will now begin to implode in on itself, getting hotter and denser toward the core, until ultimately, it'll heat up to such a degree that an explosion will take place.

That explosion will ripple through layer upon layer of the star until the explosion ripples right to the surface blows off the surface of the star supernova explosion. And what remains is a core that doesn't have any nuclear reaction to support it. So that core will collapse all the way down into a black hole. A black hole in space taking the form that I showed you a moment ago, a region from which no light is escaping.

In this image here, you see the black hole's gravity is bending the starlight around it creating this interesting lensing effect. But that's at least a process in principle that could lead to the formation of a black hole. Now, what about actual observational data that supports these ideas? All of this is highly theoretical at the moment. And look, there's been data accumulated for a long time.

Observations of the center of our Milky Way galaxy show that stars were whipping around the center at such fantastically high velocities. And the entity responsible for creating the gravitational pull that was whipping them around was so incredibly tiny, that for a tiny region to give rise to the gravity necessary to explain the whipping motion of the orbiting stars, scientists concluded that the only thing capable of doing that would be a black hole.

So that was interesting indirect evidence for the existence of black holes. Perhaps, the most convincing evidence from a few years ago was the detection of gravitational waves. So you may recall that if you have two orbiting objects-- I'll do this at some point in some episode-- as they orbit, they ripple the fabric of space. And as they ripple the fabric of space, they send out these wave train of distortions in the space-time fabric that, in principle, we can detect.

And in fact, we did detect it first time back in 2015. And when the scientists did the analysis as to what was responsible for the squeezing and stretching. Not of this degree as we see in this animation of planet Earth but a fraction of atomic diameter, the arms of the LIGO detector stretched and contracted in schematic manner shown by this Earth that's being distorted. When they worked out the source of the gravitational waves, the answer came out to be two black holes that were orbiting each other rapidly and collided.

So that was nice evidence in support of black holes. But of course, the most convincing evidence of all is to see a black hole. And indeed, that's what, in some sense, the Event Horizon Telescope did. So a consortium of radio telescopes around the world were able to focus on the center of a distant galaxy. It may be seven, I believe.

And they combined data that they were able to amass from those observations gave rise to this famous photograph. Photograph in quotes. It's not actually of cameras. It's radio telescopes. But this famous photograph where you see the telltale ingredients. You see the glowing gas around a dark region, black hole. Wow. Amazing, right? Imagine that chain of events.

Einstein writes down the general theory of relativity, 1915. It's published in 1916. Some months later, Schwarzschild gets a hold of the manuscript, works out the solution to the equations for a spherical body. He beats Einstein to the punch. I probably should have emphasized that early on. Einstein wrote down Einstein's equations of course. But he was not the first person to solve those equations, to solve them exactly.

Einstein wrote down approximate solutions that are really good in situations that are not too extreme, like the bending of starlight near the sun, the motion of mercury in its orbit. These are situations in which the gravity is not strong. So an approximate solution to his equations is all they actually need to work out the trajectory of starlight or the trajectory of mercury. But Schwarzschild writes down the first exact solution to Einstein's equations of the general theory of relativity. Wonderful achievement.

And embedded in that solution to those equations is the possibility of black holes. And then, in whatever it is, 2017? What was-- 2018? When was the Event Horizon Telescope deployed? Time goes so fast. Whenever it was-- 2018? '19? I don't know. Somewhere in there. So roughly speaking, 100-- roughly speaking, 100 years later, we actually have the closest you can imagine to a photograph of a black hole.

So that's a beautiful scientific story, a beautiful scientific achievement. What I want to do now in the remaining time is just quickly show you some of the math behind all of this. So let me actually switch over to my iPad here. Why is it not coming up? Oh, please, don't mess me up here. OK. Yes. I think we're good.

Let me just write and see if it's coming up. Yes. Good. All right. So, we are talking about black holes. And let me just write down some of the essential equations. And then, I want to at least show you in the math how you can get to some of the iconic features of black holes that you may know a lot about or at least you may have heard of. If you haven't, they're kind of mind boggling in their own right. So what's the starting point?

The starting point, as always, in this subject is Einstein's equations for gravity in the general theory of relativity. So you've seen these before, but let me write it down. R mu nu minus 1/2 g mu nu R equals 8 pi Newton's constant G speed of light fourth times the energy momentum tensor T mu nu. So this first guy over here, this is the so-called Ricci tensor, scalar curvature, energy-momentum tensor, metric on space-time.

And again remember, we are describing curvature in terms of a distortion to the distance relationships between points in a space. A good example-- if I can just switch back over half a second here. I showed you this earlier, but here's the Mona Lisa painted on a flat canvas. But if we curved the Canvas, if we warp it, if we distort it, look what happens. The distance relationships between points on her face, for example, are being changed. So curvature is reflected in this way of thinking about things.

As a distortion in those distance relationships, the metric-- oh, let me go back. Good. The metric over here is what allows us to measure distance relationships. It defines the distance relationships on a geometrical space. And that's why it comes into the story. So what we want to do now is to take these equations and try to solve them in a certain circumstance. What's that circumstance? Imagine you have some central mass M.

Imagine let's say, at the origin of the coordinate system. And imagine that it's spherical and that everything else is spherically symmetric. And that gives us a simplification on the metric because a general metric will have distance relationships that can vary in a non-symmetric manner. But if we're looking at a physical circumstance in which we have a spherically symmetric mass, then the metric will inherit that symmetry.

It will be spherically symmetric. And that allows us to simplify the analysis because the metric now has a particularly special form. So our goal then is to do the following. Outside of this mass-- let me just use a different color here-- and say any of the regions-- oh, come on, please. Any of these regions out here, outside of the mass itself, there is no energy-momentum at all. So that will be T mu nu equals 0.

And the only place in which the mass is going to come into the story is when we solve the differential equations, the boundary conditions at infinity. We'll need to reflect the fact that the space does have a body within it. But the equations that we are going to solve are the equations that are relevant external to that body. And outside of that body, there is no additional mass or energy. We're not going to imagine that there's any swirling gas or any of the things that I showed you in the animation.

And we'll keep it real simple, so we're going to solve the Einstein field equations in a-- sorry-- static spherically symmetric circumstance in which the energy-momentum tensor outside of the central mass is equal to zero, it vanishes. So now, let's do that. Now, I'm not going to actually take you through the detailed analysis of finding the solution, not particularly illuminating. And I think you'd find it a little bit boring for me to write down all the terms.

But what I will do is I just want to give you a feel for how complicated the Einstein field equations, in general, are. So now, what I'm going to do is very quickly just write down those equations in a more specific form. So, here we go. So I'm going to write down here the Riemann tensor quite quickly. Riemann tensor in terms of the Christoffel connection that gives us parallel transport. I will then write down the Ricci tensor and the scalar curvature which has come from contracting the Riemann tensor along various indices.

I then write down the connection in terms of the metric and its derivatives. And this is the metric compatible connection that ensures that underpowered translation, the length of vectors do not change. And therefore, we have the chain of events that we start with a metric that gives us the connection in terms of that metric, that gives us the curvature, Riemann curvature, in terms of the connection, in terms of that metric. And then, we contract it in the various places I've shown you. And that gives us the left hand side of Einstein's equation.

It's a complicated nonlinear differentiable function of the metric. So we have a differential equation that we need to solve. And what happened is-- now, get to what Schwarzschild did. He took that complicated mass that I just rapidly showed you, and he found an exact solution to the equations. Some of you write down the solution that he found.

So, as is conventional, I'll write down the metric as g equals g alpha beta dx alpha dx beta. Repeated indices are summed over. I don't always say that. I don't always write it. But just recognize we're using the Einstein summation convention. So alpha and beta are repeated which means they run from 1 to 4. Sometimes people say 0 to 3.

They're running over T, x, y and z, whatever numbers you care to assign to those particular variables. So that is the metric. So what I need to write down now is the particular coefficients g alpha beta that Schwarzschild was able to find inside of those equations in the circumstance that we were just looking at. And here is the solution that he finds in the trenches when it should have been calculating artillery trajectories during World War I.

So he finds that the metric g is equal to-- let's write it in this form. 1 minus 2GM over c squared r times-- well, times c squared. I should write down here. If I'm going to keep c's in, I should at least be consistent. c squared dt squared minus-- well, where should I write that? I write over here.

Minus 1 minus 2GM over c squared r to the minus 1 times dr squared plus the angular part of the metric, which I'll just write down is r squared s omega. So I'm not going to talk about the angular part at all. I'm just really interested in the radial part and the temporal part. The angular part is symmetric, so there's nothing particularly interesting happening there.

So there it is. There is the solution that Schwarzschild writes down. Now, when you look at the solution, there are a number of interesting things. Let me just give myself a little bit of space. I wrote too big, but I'll try to squeeze it in over here. So first of all, you might say to yourself, the situation of having a massive object m-- I mean not to do it there-- the situation of having a massive object.

Well, far away from that massive object, yeah, it should sort of look like Newton, you would think. All right. And does it look like Newton? Is there any hint of Isaac Newton in the solution that Schwarzschild found to this complicated nonlinear partial differential equations from Einstein's field equations? And indeed, there is. Let me set c equal to 1 to make it easier for us to recognize what we're driving at.

Just use the units where c is equal to 1, 1 light year per year, whatever units you want to use. And then, you will note that this term over here has within it the combination GM over r. GM over R. Ring a bell? Right. That's the Newtonian gravitational potential for a mass m, say, sitting at the origin of the coordinates. So you see there is a remnant of Newton in that equation.

In fact, truth be told, the way you solve this equation is by making contact with Newtonian gravity far away from the origin. So the solution itself builds it in, from the get go, is part of the way to find the solution. But be that as it may, it's beautiful to see that you can extract the Newtonian gravitational potential from the Schwarzschild solution of the Einstein field equations. OK. That's point number one which is kind of nice.

Point number two that I want to make is that there are some special values. Special values of r. Well, let me just-- I'm still like I'm lecturing in front of a class, but let me just write this now. So point number one, we see Newtonian gravitational potential in the solution. That's cool. Point number two is that there are some special values, special values of r.

What do I mean by that? When we look at this solution, you notice in particular that if r equals 0, then some funny stuff happens because you divide them by 0 in those coefficients of the metric. What does that mean? Well, it turns out that that's a big deal. That's the singularity. The black hole singularity you see right there, the infinity that crops up as r goes to 0 and the coefficient of the metric.

But now, you might say, well, wait. What about also the value of r equals 2GM or to 2GM over c squared. But c is equal to one in these units. That's a value for which this term goes to 0. And if it goes to 0, then this term is going to infinity. So another version of infinity cropping up is that a singularity. And people thought that that was a singularity. So r equals to 0 is right here.

But r equal to what's known as rs, the Schwarzschild value. And let me call this rs 2GM over r. People thought-- and of course, it's a whole sphere that I'm only drawing part of it. Early days, people thought that might be a singularity, but it turns out it's not actually a singularity. It's what's known as a coordinate breakdown, or some people say coordinate singularity. It's where the coordinates don't work well. You're familiar with this from polar coordinates, right?

In polar coordinates, when using r and theta-- r theta, well, that's a perfectly good way of talking about a point such as that away from the origin. But if you're actually at the origin, and I say to you, OK, r is equal to 0 but what's theta? Theta could be 0.2, 0.6 pi, pi, it doesn't matter. Every angle at the origin is the same point. So, the coordinates are not good at that location.

Similarly, the coordinates rT and then the angular part, theta and phi are not good all along r equals rs. So people have understood this now for a while. But r equal to rs, even though it's not a singularity, it's a special location because look at it. When you are, say, heading in from infinity, and you get to r equal to rs. And then, say, you cross over r equals to rs, look what happens here.

This term and this term, they change their signs, right? When r is bigger than rs, then this quantity over here is smaller than 1. And therefore, 1 minus it is a positive number. But when r is smaller than rs, this term is now bigger than 1. Therefore, 1 minus it is negative. And therefore, this picks up a negative sign as does this. Now, the only difference between a T and an r, as far as this metric is concerned, is the sign.

So if there's signs flip, then in some sense, space and time flip. Wow. Space and time flip. So as you go across the edge, what you thought was time becomes space and what you thought was space becomes time-- again, because the only difference between space and time as far as the metric is concerned is this minus sign over here. Oh, and I wrote down things funny here. That was confusing. This should be a minus sign also if I'm putting the minus in front of my space. Sorry about that. So go all the way back and imagine that.

But the point is, again, focusing just upon the radial and the temporal part. The only thing that distinguishes the radial from the temporal, as far as the metric is concerned, is the sign, a plus or a minus. And when you cross over r equal to rs, the plus and the minus interchange, space and time interchange. And that actually gives us one way of thinking about why you can't escape from a black hole. When you cross over r to rs, the spatial direction is now better thought of as a time direction.

And just as you are unable to go back in time, once you cross over the event horizon, you can't go back in the r direction because the radial direction is like a time direction. So just as you are ineluctably driven forward in time, second after second after second, once you cross over the edge of a black hole, you are ineluctably driven to smaller and smaller values of r because it is if you are being pulled forward in time.

So that's another way of understanding this. So in particular, the following is the black hole summary that I want to give. For a physical body-- so, I mentioned this before. If you're talking about the mass of the sun and you work out the Schwarzschild radius, just stick into this formula 2GM or to 2GM over c squared, you'll get that number that I mentioned before. I think it's-- I'm working from memory here. I think it's about 3 kilometers.

Now, that means that for a body like the sun-- let me make it nice and orange. For a body like the sun-- here's the sun-- the Schwarzschild radius is deeply embedded within the sun. And you will recall that the solution that we derived is only valid outside the spherical body. I set T mu nu on the right hand side of Einstein's equations equal to 0.

So the solution for the sun, say, the Schwarzschild solution, is really only valid outside the sun itself, which means that you will never get to the Schwarzschild radius because it's not part of the solution. It's not that you can't solve the Einstein equations inside the body. You can. But the point is everything that we're talking about is only relevant outside the physical boundary of the object itself.

And for a body like the sun or any typical star, the Schwarzschild radius is so small that it's well within the object, well beyond the reach of the solution that we are talking about. Similarly, if you look at the Earth, as I mentioned before, if you plug that in, Schwarzschild radius 2GM Earth, this is massive sun, Earth over c squared, you get something on the order of centimeters.

And again, a centimeter is so small compared to the size of the Earth that that's Schwarzschild radius is deeply embedded within the core of the Earth. But what is a black hole then? A black hole is an object whose physical size is smaller than its own Schwarzschild radius. So if you take any mass at all and you squeeze that mass down to a size rs equals 2GM over c squared, just calculate that. If you can take that mass and squeeze it down to a smaller than rs size, so squeeze it down so that r is less than rs.

A lot of squeezing but whatever. Imagine that it happens. Now the Schwarzschild radius is outside the physical boundary of the object itself. Now the Schwarzschild radius really matters. It is part of the domain within which the solution holds. And therefore, you have the possibility of crossing over the edge of the Schwarzschild radius as we were talking about over here. And then, space, and time interchange, you can't get out. All that good stuff follows from there.

That's really what a black hole is. Final point that I want to make. You may have heard this idea that when you get closer and closer to a massive body-- I'm going to stick with black holes just because it's more dramatic. But it's really for any massive body at all. As you get closer and closer to the edge of a black hole-- so imagine we have a black hole. Again, the singularity at the center, what does that mean?

It means we don't know what's going on there. The metric blows up, our understanding breaks down. Now I'm not going to try to explain that any further here, basically because I don't have anything to say. I don't know what happens there. But if this, say, is the event horizon that I just drew over there. You may have heard that as you head in from infinity and you get closer and closer and closer to the event horizon of the black hole, you find that time elapses slower and slower and slower.

Clocks tick ever slower compared to the rate at which they tick, say, way out here at infinity. So if you have a clock out here and you bring in a clock over here, the idea is that it tick slower and slower. Let me actually show you that. I have a nice little visual on that. So here you have clocks that are ticking next to each other far away, say, from a body like the sun. Bring one clock closer and closer to the surface of the sun. It actually is ticking slower.

It's just, to the effect, is so small for a regular, ordinary object like a star, like a sun that the effect is too small to see. But now, if you squeeze the sun down into a black hole, now, you're allowed to bring the clock closer and closer. The sun doesn't get in the way. The clock can get closer and closer to the event horizon. And look at how that clock is ticking, ever more slowly. Good. Now, going back over here. Can we see that effect in the equations?

And indeed, you can. My equations have become so incredibly messy as I draw all these little things that maybe I can clean up. Oh, that's pretty. In fact, I can get rid of all these things and the fact that I can change this little guy over here from a plus to a minus, everybody's looking real cool here. What's my point though? My point is I want to focus my attention-- here I go again-- on this term over here.

So let me just rewrite that term without the mess around it. So that first term just looked like-- it's not what I want. All right. The first term I choose a different color. Something-- that's good. So, I had 1 minus 2GM over r, putting the c equal to 1, times dt squared. That's what the metric looks like. Now, this dt part over here, think about that as the time interval, ticking of a clock.

Delta t is the time between the clock being in one location and say, a second later. Now when r goes to infinity, this term over here goes to 0. So you can think about dt or dt squared as measuring how a clock ticks far away, infinitely far away from A black hole where this coefficient goes to 1 because the 2GM over r goes to 0 at infinity.

But now, as you go on your journey toward the edge of a black hole-- this is the journey we are going on-- this r now is getting smaller and smaller. This quantity over here is getting bigger and bigger, still less than 1 outside the Schwarzschild radius, which means that this combined guys is getting smaller and smaller. What does that mean? Well, what that means is we have a number in front times dt squared.

This number is getting small as r approaches the Schwarzschild radius. And it goes to 0 there. That small number is multiplying the time interval delta t squared or dt squared. And that's giving you the physical time that it takes for a clock to tick at a given radius. And because that number is getting smaller and smaller, time is ticking slower and slower. So there it is.

It's the fact that this term over here is getting smaller and smaller as you get closer and closer, as approaching 0, as r goes to rs, it's that coefficient getting smaller and smaller that is giving the slower and slower rate at which clocks tick as they go on this journey toward the edge of a black hole. So, there it is. That is the slowing of time near the edge of any mass. But it didn't have to be a black hole.

Black hole again, as we saw in the animation just allows you to get closer and closer to the Schwarzschild radius where that coefficient gets closer and closer to 0 making the effect more and more manifest. All right. Look. There are a lot, a lot of puzzles of black holes. I've just scratched the surface here. We're only talking about black holes that have mass. They don't have charge. That's another black hole solution. You also can have black holes with angular momentum, which in the real world they typically will have those solutions have and written down, too.

Exactly, what happens at the deep interior point of a black hole, the singularity there are still things that people struggle with. And in fact, when you put quantum mechanics into the story - this is just classical general activity, no quantum mechanics - when you put quantum mechanics into the story, even what happens at the edge, the event horizon of a black hole is now open for discussion. Oh, sorry. There's something right here. Even that is open for discussion and has been vigorously discussed in recent years. And there's still questions that people argue about even there.

But this gives you at least the classical story. The basic underpinnings of the history of how we came to this possibility of black holes. The observational story which establishes that this stuff is not just in the mind but is actually real. And then, you see some of the mathematical manipulations responsible for some of the essential conclusions about how big an object needs to be squeezed down to for it to be a black hole, and the fact that time itself elapses slower and slower.

Even that shape the usual funnel shape, you can see from the mathematics as well-- I should probably stop, but I'm getting carried away as I often do. Look at this term over here. So much as this term showed us that time is elapsing ever slower toward the edge of a black hole. The fact that you've got this guy over here with a minus 1 there, means that in some sense, distances are being stretched out as you get closer and closer to the edge of a black hole. How do you stretch out those distances?

Well, one way to graphically represent that is you take that plane and you stretch it out. And you get that big indentation. That big indentation is representing this term that we have over here because it's getting ever bigger as you get ever closer to the edge of a black hole. Ever bigger means ever bigger stretch. Anyway, it's kind of fun to see the pictures come to life through the mathematics. And that was really the point that I want to get across here today.

With this first exact solution of the Einstein field equations coming from Karl Schwarzschild, the Schwarzschild solution, which again works not just for black holes but for any spherically symmetric massive body, like the Earth and the sun. But black holes, it's a particularly dramatic solution as we can get right down to the event horizon and probe gravity in unusual domains that Newton would not have been able to understand or reveal to us based on his own equations.

Of course, if Newton were around today, he would totally understand what's going on. He would be leading the charge. OK. That's really all I want to talk about here today. I'll pick this up again shortly, not exactly sure if it will be everyday as I mentioned before. But until next time, this has been Your Daily Equation. Take care.

And black holes are an amazingly rich arena for theorists to try out ideas, to explore our understanding of the force of gravity, to explore its interaction with quantum mechanics. And as I mentioned, black holes are now also an arena which is rich in fertile for observational astronomy. We've gone beyond the era in which black holes were merely theoretical ideas to now the recognition that black holes are real. They are really out there.

I'll also note at the end that there are great many puzzles to do with black holes that have yet to be resolved. And maybe if I have time, I'll mention a few of those. But I'd like, for the most part, to focus here, in this episode, on the traditional, more straightforward, widely-- well, not completely but more widely accepted historical version of the trajectory that led us to recognize the possibility of black holes and some of the properties that emerge from the basic mathematics of Einstein's equations.

So, to get us going, let me just give a little bit of historical background. The story of black holes begins with this fella right here, Karl Schwarzschild. He was a German meteorologist, mathematician, really smart guy, astronomer, who was actually stationed on the Russian front during World War I. And as he's there, and he's charged with actually calculating trajectories of bombs. You hear them going off and so on.

And somehow, in the trenches, he gets hold of Einstein's paper in the general theory of relativity, does some calculations on it. And he realizes that if you have a spherical mass and you crush it to a very small size-- the bombs are still going off all around him-- it will create such a warp in the fabric of space that anything that gets too close will not be able to pull away. And that's really what we mean by a black hole.

It is a region of space in which enough matter has been crushed to a sufficiently small size that the warpage is so significant that anything that gets too close, closer than, as we will see, what is known as the event horizon of the black hole, cannot escape, cannot run away. So the sort of image that you can have in mind is if we have a little animation here of the moon going around the Earth. This is the usual story of warped environment around the vicinity of a spherical body like the Earth.

But if you crushed the Earth down to a sufficiently small size, the idea is that the indentation will be far greater than what we saw for the Earth. The indentation would be so significant that at least, metaphorically speaking, if you're hanging out near the edge of a black hole and you were to turn on a flashlight, if you're within the event horizon, the light from that flashlight would not go off into deep space. Instead, it would go into the black hole itself. This image is a little bit off, I should say.

But it kind of gives you at least a mental toehold for the idea of why it is that light can't get away from a black hole. When you turn on a flashlight, if you're within the event horizon of a black hole, the light shines inward not outward. Now, another way of thinking about this idea-- and look, I know this is quite familiar territory. Black holes are in the culture, you know the phrase falling into a black hole. Or he did something, and it created a black hole. We use that kind of language all the time. So all these ideas are familiar.

But it's good to have mental imagery to go along with the words. And the mental imagery that I'm about to give you, I find particularly interesting and useful. Because there is a mathematical version of the story that I'm going to show you visually right now. I'm not going to describe that mathematical story right now. But just know that there is a version of the so-called waterfall analogy that really can be fully articulated in a mathematical way that makes it rigorous. So here's the idea.

If you are near a waterfall and you are, say, paddling your kayak-- is that the right word? Yeah. Paddling your kayak. If you can paddle faster than the rate at which the water is flowing toward the waterfall you can get away. But if you can't paddle faster than the water is flowing, then you can't get away. And you're doomed to fall down the waterfall. And here's the idea. The analogy is space itself falls over the edge of a black hole. It's sort of like a waterfall of space.

And the speed at which space travels over the edge of a black hole is equal to the speed of light. Nothing can go faster than the speed of light. So near a black hole, you're doomed. So you might as well just paddle right toward the black hole and go on a joyride down the throat of the black hole itself. So that's another way of thinking about it. Edge of a black hole event horizon, space, in some sense, is flowing over the edge. It's flowing over the edge at a speed equal to the speed of light.

Since nothing can go faster than the speed of light, you can't paddle upstream. And if you can't paddle upstream, you can't get away from the black hole. You're doomed, and you will fall into the black hole. Now, that all is highly schematic and metaphorical. I hope that it is useful for thinking about black holes. But for a long time, we knew what black holes should look like if we were to ever see them. We wouldn't literally see the black hole itself.

But in the environment around a black hole, as material is falling over the event horizon of a black hole, it heats up. The material rubs against the other material. That's all falling inward. It gets so hot that the frictional forces heat up the material, and they generate x-rays. And those x-rays go out into space. And those x-rays are things that we can see.

So let me now just show you, therefore, the expected view of a black hole would be something like this. Around the edge of the black hole, you see the swirling maelstrom of material giving off these high energy x-rays. I've put them in the visible, so we can see them. And within that maelstrom of activity is a central region from which no light itself is being released. No light is being emitted.

And that would be the black hole itself. Now, Schwarzschild is doing his work, as I said, it was World War I. So, we're back in 1917 or so. And so, he puts forward this idea of this solution. I show you the mathematical form of that solution as we go forward. But there's a real curious feature of-- well, there are many curious features of the solution. But one in particular is for an object to become a black hole, you've got to squeeze it down.

But how far do you have to squeeze it down? Well, the calculations show that you'd have to squeeze the sun down to about three kilometers across or so to be a black hole. The Earth, you'd have to squeeze it down to a radius of about centimeter or so to be a black hole. I mean, think about the Earth down to a centimeter. It doesn't seem like there'd be any physical process that would ever allow material to be compressed to that degree.

So, the question is are these objects just mathematical implications of the general theory of relativity? Or are they real? And a step in the direction of showing that they're real was taken some decades later when scientists realized that there is a process that could actually lead to matter collapsing in on itself and thereby crushing it down to the small size as required for the black hole solution to be realized, physically.

What are those processes? Well, here is the canonical one. Imagine we were looking at a large star, like a red giant. That star supports its own hefty mass through nuclear processes in the core. But those nuclear processes, which give up the heat, the light, the pressure, ultimately, they will use up the nuclear fuel. And when the fuel is used up, the star will now begin to implode in on itself, getting hotter and denser toward the core, until ultimately, it'll heat up to such a degree that an explosion will take place.

That explosion will ripple through layer upon layer of the star until the explosion ripples right to the surface blows off the surface of the star supernova explosion. And what remains is a core that doesn't have any nuclear reaction to support it. So that core will collapse all the way down into a black hole. A black hole in space taking the form that I showed you a moment ago, a region from which no light is escaping.

In this image here, you see the black hole's gravity is bending the starlight around it creating this interesting lensing effect. But that's at least a process in principle that could lead to the formation of a black hole. Now, what about actual observational data that supports these ideas? All of this is highly theoretical at the moment. And look, there's been data accumulated for a long time.

Observations of the center of our Milky Way galaxy show that stars were whipping around the center at such fantastically high velocities. And the entity responsible for creating the gravitational pull that was whipping them around was so incredibly tiny, that for a tiny region to give rise to the gravity necessary to explain the whipping motion of the orbiting stars, scientists concluded that the only thing capable of doing that would be a black hole.

So that was interesting indirect evidence for the existence of black holes. Perhaps, the most convincing evidence from a few years ago was the detection of gravitational waves. So you may recall that if you have two orbiting objects-- I'll do this at some point in some episode-- as they orbit, they ripple the fabric of space. And as they ripple the fabric of space, they send out these wave train of distortions in the space-time fabric that, in principle, we can detect.

And in fact, we did detect it first time back in 2015. And when the scientists did the analysis as to what was responsible for the squeezing and stretching. Not of this degree as we see in this animation of planet Earth but a fraction of atomic diameter, the arms of the LIGO detector stretched and contracted in schematic manner shown by this Earth that's being distorted. When they worked out the source of the gravitational waves, the answer came out to be two black holes that were orbiting each other rapidly and collided.

So that was nice evidence in support of black holes. But of course, the most convincing evidence of all is to see a black hole. And indeed, that's what, in some sense, the Event Horizon Telescope did. So a consortium of radio telescopes around the world were able to focus on the center of a distant galaxy. It may be seven, I believe.

And they combined data that they were able to amass from those observations gave rise to this famous photograph. Photograph in quotes. It's not actually of cameras. It's radio telescopes. But this famous photograph where you see the telltale ingredients. You see the glowing gas around a dark region, black hole. Wow. Amazing, right? Imagine that chain of events.

Einstein writes down the general theory of relativity, 1915. It's published in 1916. Some months later, Schwarzschild gets a hold of the manuscript, works out the solution to the equations for a spherical body. He beats Einstein to the punch. I probably should have emphasized that early on. Einstein wrote down Einstein's equations of course. But he was not the first person to solve those equations, to solve them exactly.

Einstein wrote down approximate solutions that are really good in situations that are not too extreme, like the bending of starlight near the sun, the motion of mercury in its orbit. These are situations in which the gravity is not strong. So an approximate solution to his equations is all they actually need to work out the trajectory of starlight or the trajectory of mercury. But Schwarzschild writes down the first exact solution to Einstein's equations of the general theory of relativity. Wonderful achievement.

And embedded in that solution to those equations is the possibility of black holes. And then, in whatever it is, 2017? What was-- 2018? When was the Event Horizon Telescope deployed? Time goes so fast. Whenever it was-- 2018? '19? I don't know. Somewhere in there. So roughly speaking, 100-- roughly speaking, 100 years later, we actually have the closest you can imagine to a photograph of a black hole.

So that's a beautiful scientific story, a beautiful scientific achievement. What I want to do now in the remaining time is just quickly show you some of the math behind all of this. So let me actually switch over to my iPad here. Why is it not coming up? Oh, please, don't mess me up here. OK. Yes. I think we're good.

Let me just write and see if it's coming up. Yes. Good. All right. So, we are talking about black holes. And let me just write down some of the essential equations. And then, I want to at least show you in the math how you can get to some of the iconic features of black holes that you may know a lot about or at least you may have heard of. If you haven't, they're kind of mind boggling in their own right. So what's the starting point?

The starting point, as always, in this subject is Einstein's equations for gravity in the general theory of relativity. So you've seen these before, but let me write it down. R mu nu minus 1/2 g mu nu R equals 8 pi Newton's constant G speed of light fourth times the energy momentum tensor T mu nu. So this first guy over here, this is the so-called Ricci tensor, scalar curvature, energy-momentum tensor, metric on space-time.

And again remember, we are describing curvature in terms of a distortion to the distance relationships between points in a space. A good example-- if I can just switch back over half a second here. I showed you this earlier, but here's the Mona Lisa painted on a flat canvas. But if we curved the Canvas, if we warp it, if we distort it, look what happens. The distance relationships between points on her face, for example, are being changed. So curvature is reflected in this way of thinking about things.

As a distortion in those distance relationships, the metric-- oh, let me go back. Good. The metric over here is what allows us to measure distance relationships. It defines the distance relationships on a geometrical space. And that's why it comes into the story. So what we want to do now is to take these equations and try to solve them in a certain circumstance. What's that circumstance? Imagine you have some central mass M.

Imagine let's say, at the origin of the coordinate system. And imagine that it's spherical and that everything else is spherically symmetric. And that gives us a simplification on the metric because a general metric will have distance relationships that can vary in a non-symmetric manner. But if we're looking at a physical circumstance in which we have a spherically symmetric mass, then the metric will inherit that symmetry.

It will be spherically symmetric. And that allows us to simplify the analysis because the metric now has a particularly special form. So our goal then is to do the following. Outside of this mass-- let me just use a different color here-- and say any of the regions-- oh, come on, please. Any of these regions out here, outside of the mass itself, there is no energy-momentum at all. So that will be T mu nu equals 0.

And the only place in which the mass is going to come into the story is when we solve the differential equations, the boundary conditions at infinity. We'll need to reflect the fact that the space does have a body within it. But the equations that we are going to solve are the equations that are relevant external to that body. And outside of that body, there is no additional mass or energy. We're not going to imagine that there's any swirling gas or any of the things that I showed you in the animation.

And we'll keep it real simple, so we're going to solve the Einstein field equations in a-- sorry-- static spherically symmetric circumstance in which the energy-momentum tensor outside of the central mass is equal to zero, it vanishes. So now, let's do that. Now, I'm not going to actually take you through the detailed analysis of finding the solution, not particularly illuminating. And I think you'd find it a little bit boring for me to write down all the terms.

But what I will do is I just want to give you a feel for how complicated the Einstein field equations, in general, are. So now, what I'm going to do is very quickly just write down those equations in a more specific form. So, here we go. So I'm going to write down here the Riemann tensor quite quickly. Riemann tensor in terms of the Christoffel connection that gives us parallel transport. I will then write down the Ricci tensor and the scalar curvature which has come from contracting the Riemann tensor along various indices.

I then write down the connection in terms of the metric and its derivatives. And this is the metric compatible connection that ensures that underpowered translation, the length of vectors do not change. And therefore, we have the chain of events that we start with a metric that gives us the connection in terms of that metric, that gives us the curvature, Riemann curvature, in terms of the connection, in terms of that metric. And then, we contract it in the various places I've shown you. And that gives us the left hand side of Einstein's equation.

It's a complicated nonlinear differentiable function of the metric. So we have a differential equation that we need to solve. And what happened is-- now, get to what Schwarzschild did. He took that complicated mass that I just rapidly showed you, and he found an exact solution to the equations. Some of you write down the solution that he found.

So, as is conventional, I'll write down the metric as g equals g alpha beta dx alpha dx beta. Repeated indices are summed over. I don't always say that. I don't always write it. But just recognize we're using the Einstein summation convention. So alpha and beta are repeated which means they run from 1 to 4. Sometimes people say 0 to 3.

They're running over T, x, y and z, whatever numbers you care to assign to those particular variables. So that is the metric. So what I need to write down now is the particular coefficients g alpha beta that Schwarzschild was able to find inside of those equations in the circumstance that we were just looking at. And here is the solution that he finds in the trenches when it should have been calculating artillery trajectories during World War I.

So he finds that the metric g is equal to-- let's write it in this form. 1 minus 2GM over c squared r times-- well, times c squared. I should write down here. If I'm going to keep c's in, I should at least be consistent. c squared dt squared minus-- well, where should I write that? I write over here.

Minus 1 minus 2GM over c squared r to the minus 1 times dr squared plus the angular part of the metric, which I'll just write down is r squared s omega. So I'm not going to talk about the angular part at all. I'm just really interested in the radial part and the temporal part. The angular part is symmetric, so there's nothing particularly interesting happening there.

So there it is. There is the solution that Schwarzschild writes down. Now, when you look at the solution, there are a number of interesting things. Let me just give myself a little bit of space. I wrote too big, but I'll try to squeeze it in over here. So first of all, you might say to yourself, the situation of having a massive object m-- I mean not to do it there-- the situation of having a massive object.

Well, far away from that massive object, yeah, it should sort of look like Newton, you would think. All right. And does it look like Newton? Is there any hint of Isaac Newton in the solution that Schwarzschild found to this complicated nonlinear partial differential equations from Einstein's field equations? And indeed, there is. Let me set c equal to 1 to make it easier for us to recognize what we're driving at.

Just use the units where c is equal to 1, 1 light year per year, whatever units you want to use. And then, you will note that this term over here has within it the combination GM over r. GM over R. Ring a bell? Right. That's the Newtonian gravitational potential for a mass m, say, sitting at the origin of the coordinates. So you see there is a remnant of Newton in that equation.

In fact, truth be told, the way you solve this equation is by making contact with Newtonian gravity far away from the origin. So the solution itself builds it in, from the get go, is part of the way to find the solution. But be that as it may, it's beautiful to see that you can extract the Newtonian gravitational potential from the Schwarzschild solution of the Einstein field equations. OK. That's point number one which is kind of nice.

Point number two that I want to make is that there are some special values. Special values of r. Well, let me just-- I'm still like I'm lecturing in front of a class, but let me just write this now. So point number one, we see Newtonian gravitational potential in the solution. That's cool. Point number two is that there are some special values, special values of r.

What do I mean by that? When we look at this solution, you notice in particular that if r equals 0, then some funny stuff happens because you divide them by 0 in those coefficients of the metric. What does that mean? Well, it turns out that that's a big deal. That's the singularity. The black hole singularity you see right there, the infinity that crops up as r goes to 0 and the coefficient of the metric.

But now, you might say, well, wait. What about also the value of r equals 2GM or to 2GM over c squared. But c is equal to one in these units. That's a value for which this term goes to 0. And if it goes to 0, then this term is going to infinity. So another version of infinity cropping up is that a singularity. And people thought that that was a singularity. So r equals to 0 is right here.

But r equal to what's known as rs, the Schwarzschild value. And let me call this rs 2GM over r. People thought-- and of course, it's a whole sphere that I'm only drawing part of it. Early days, people thought that might be a singularity, but it turns out it's not actually a singularity. It's what's known as a coordinate breakdown, or some people say coordinate singularity. It's where the coordinates don't work well. You're familiar with this from polar coordinates, right?

In polar coordinates, when using r and theta-- r theta, well, that's a perfectly good way of talking about a point such as that away from the origin. But if you're actually at the origin, and I say to you, OK, r is equal to 0 but what's theta? Theta could be 0.2, 0.6 pi, pi, it doesn't matter. Every angle at the origin is the same point. So, the coordinates are not good at that location.

Similarly, the coordinates rT and then the angular part, theta and phi are not good all along r equals rs. So people have understood this now for a while. But r equal to rs, even though it's not a singularity, it's a special location because look at it. When you are, say, heading in from infinity, and you get to r equal to rs. And then, say, you cross over r equals to rs, look what happens here.

This term and this term, they change their signs, right? When r is bigger than rs, then this quantity over here is smaller than 1. And therefore, 1 minus it is a positive number. But when r is smaller than rs, this term is now bigger than 1. Therefore, 1 minus it is negative. And therefore, this picks up a negative sign as does this. Now, the only difference between a T and an r, as far as this metric is concerned, is the sign.

So if there's signs flip, then in some sense, space and time flip. Wow. Space and time flip. So as you go across the edge, what you thought was time becomes space and what you thought was space becomes time-- again, because the only difference between space and time as far as the metric is concerned is this minus sign over here. Oh, and I wrote down things funny here. That was confusing. This should be a minus sign also if I'm putting the minus in front of my space. Sorry about that. So go all the way back and imagine that.

But the point is, again, focusing just upon the radial and the temporal part. The only thing that distinguishes the radial from the temporal, as far as the metric is concerned, is the sign, a plus or a minus. And when you cross over r equal to rs, the plus and the minus interchange, space and time interchange. And that actually gives us one way of thinking about why you can't escape from a black hole. When you cross over r to rs, the spatial direction is now better thought of as a time direction.

And just as you are unable to go back in time, once you cross over the event horizon, you can't go back in the r direction because the radial direction is like a time direction. So just as you are ineluctably driven forward in time, second after second after second, once you cross over the edge of a black hole, you are ineluctably driven to smaller and smaller values of r because it is if you are being pulled forward in time.

So that's another way of understanding this. So in particular, the following is the black hole summary that I want to give. For a physical body-- so, I mentioned this before. If you're talking about the mass of the sun and you work out the Schwarzschild radius, just stick into this formula 2GM or to 2GM over c squared, you'll get that number that I mentioned before. I think it's-- I'm working from memory here. I think it's about 3 kilometers.

Now, that means that for a body like the sun-- let me make it nice and orange. For a body like the sun-- here's the sun-- the Schwarzschild radius is deeply embedded within the sun. And you will recall that the solution that we derived is only valid outside the spherical body. I set T mu nu on the right hand side of Einstein's equations equal to 0.

So the solution for the sun, say, the Schwarzschild solution, is really only valid outside the sun itself, which means that you will never get to the Schwarzschild radius because it's not part of the solution. It's not that you can't solve the Einstein equations inside the body. You can. But the point is everything that we're talking about is only relevant outside the physical boundary of the object itself.

And for a body like the sun or any typical star, the Schwarzschild radius is so small that it's well within the object, well beyond the reach of the solution that we are talking about. Similarly, if you look at the Earth, as I mentioned before, if you plug that in, Schwarzschild radius 2GM Earth, this is massive sun, Earth over c squared, you get something on the order of centimeters.

And again, a centimeter is so small compared to the size of the Earth that that's Schwarzschild radius is deeply embedded within the core of the Earth. But what is a black hole then? A black hole is an object whose physical size is smaller than its own Schwarzschild radius. So if you take any mass at all and you squeeze that mass down to a size rs equals 2GM over c squared, just calculate that. If you can take that mass and squeeze it down to a smaller than rs size, so squeeze it down so that r is less than rs.

A lot of squeezing but whatever. Imagine that it happens. Now the Schwarzschild radius is outside the physical boundary of the object itself. Now the Schwarzschild radius really matters. It is part of the domain within which the solution holds. And therefore, you have the possibility of crossing over the edge of the Schwarzschild radius as we were talking about over here. And then, space, and time interchange, you can't get out. All that good stuff follows from there.

That's really what a black hole is. Final point that I want to make. You may have heard this idea that when you get closer and closer to a massive body-- I'm going to stick with black holes just because it's more dramatic. But it's really for any massive body at all. As you get closer and closer to the edge of a black hole-- so imagine we have a black hole. Again, the singularity at the center, what does that mean?

It means we don't know what's going on there. The metric blows up, our understanding breaks down. Now I'm not going to try to explain that any further here, basically because I don't have anything to say. I don't know what happens there. But if this, say, is the event horizon that I just drew over there. You may have heard that as you head in from infinity and you get closer and closer and closer to the event horizon of the black hole, you find that time elapses slower and slower and slower.

Clocks tick ever slower compared to the rate at which they tick, say, way out here at infinity. So if you have a clock out here and you bring in a clock over here, the idea is that it tick slower and slower. Let me actually show you that. I have a nice little visual on that. So here you have clocks that are ticking next to each other far away, say, from a body like the sun. Bring one clock closer and closer to the surface of the sun. It actually is ticking slower.

It's just, to the effect, is so small for a regular, ordinary object like a star, like a sun that the effect is too small to see. But now, if you squeeze the sun down into a black hole, now, you're allowed to bring the clock closer and closer. The sun doesn't get in the way. The clock can get closer and closer to the event horizon. And look at how that clock is ticking, ever more slowly. Good. Now, going back over here. Can we see that effect in the equations?

And indeed, you can. My equations have become so incredibly messy as I draw all these little things that maybe I can clean up. Oh, that's pretty. In fact, I can get rid of all these things and the fact that I can change this little guy over here from a plus to a minus, everybody's looking real cool here. What's my point though? My point is I want to focus my attention-- here I go again-- on this term over here.

So let me just rewrite that term without the mess around it. So that first term just looked like-- it's not what I want. All right. The first term I choose a different color. Something-- that's good. So, I had 1 minus 2GM over r, putting the c equal to 1, times dt squared. That's what the metric looks like. Now, this dt part over here, think about that as the time interval, ticking of a clock.

Delta t is the time between the clock being in one location and say, a second later. Now when r goes to infinity, this term over here goes to 0. So you can think about dt or dt squared as measuring how a clock ticks far away, infinitely far away from A black hole where this coefficient goes to 1 because the 2GM over r goes to 0 at infinity.

But now, as you go on your journey toward the edge of a black hole-- this is the journey we are going on-- this r now is getting smaller and smaller. This quantity over here is getting bigger and bigger, still less than 1 outside the Schwarzschild radius, which means that this combined guys is getting smaller and smaller. What does that mean? Well, what that means is we have a number in front times dt squared.

This number is getting small as r approaches the Schwarzschild radius. And it goes to 0 there. That small number is multiplying the time interval delta t squared or dt squared. And that's giving you the physical time that it takes for a clock to tick at a given radius. And because that number is getting smaller and smaller, time is ticking slower and slower. So there it is.

It's the fact that this term over here is getting smaller and smaller as you get closer and closer, as approaching 0, as r goes to rs, it's that coefficient getting smaller and smaller that is giving the slower and slower rate at which clocks tick as they go on this journey toward the edge of a black hole. So, there it is. That is the slowing of time near the edge of any mass. But it didn't have to be a black hole.

Black hole again, as we saw in the animation just allows you to get closer and closer to the Schwarzschild radius where that coefficient gets closer and closer to 0 making the effect more and more manifest. All right. Look. There are a lot, a lot of puzzles of black holes. I've just scratched the surface here. We're only talking about black holes that have mass. They don't have charge. That's another black hole solution. You also can have black holes with angular momentum, which in the real world they typically will have those solutions have and written down, too.

Exactly, what happens at the deep interior point of a black hole, the singularity there are still things that people struggle with. And in fact, when you put quantum mechanics into the story - this is just classical general activity, no quantum mechanics - when you put quantum mechanics into the story, even what happens at the edge, the event horizon of a black hole is now open for discussion. Oh, sorry. There's something right here. Even that is open for discussion and has been vigorously discussed in recent years. And there's still questions that people argue about even there.

But this gives you at least the classical story. The basic underpinnings of the history of how we came to this possibility of black holes. The observational story which establishes that this stuff is not just in the mind but is actually real. And then, you see some of the mathematical manipulations responsible for some of the essential conclusions about how big an object needs to be squeezed down to for it to be a black hole, and the fact that time itself elapses slower and slower.

Even that shape the usual funnel shape, you can see from the mathematics as well-- I should probably stop, but I'm getting carried away as I often do. Look at this term over here. So much as this term showed us that time is elapsing ever slower toward the edge of a black hole. The fact that you've got this guy over here with a minus 1 there, means that in some sense, distances are being stretched out as you get closer and closer to the edge of a black hole. How do you stretch out those distances?

Well, one way to graphically represent that is you take that plane and you stretch it out. And you get that big indentation. That big indentation is representing this term that we have over here because it's getting ever bigger as you get ever closer to the edge of a black hole. Ever bigger means ever bigger stretch. Anyway, it's kind of fun to see the pictures come to life through the mathematics. And that was really the point that I want to get across here today.

With this first exact solution of the Einstein field equations coming from Karl Schwarzschild, the Schwarzschild solution, which again works not just for black holes but for any spherically symmetric massive body, like the Earth and the sun. But black holes, it's a particularly dramatic solution as we can get right down to the event horizon and probe gravity in unusual domains that Newton would not have been able to understand or reveal to us based on his own equations.

Of course, if Newton were around today, he would totally understand what's going on. He would be leading the charge. OK. That's really all I want to talk about here today. I'll pick this up again shortly, not exactly sure if it will be everyday as I mentioned before. But until next time, this has been Your Daily Equation. Take care.