The Science of Everything Podcast - Episode 145: Relativity and Black Holes

Episode Date: June 30, 2024

Continuing our series on General Relativity, we discuss the derivation of the Schwarzschild metric as a vacuum solution to Einstein's Field Equations, and analyse the physical meaning of this solution, including the properties of the singularity, event horizon, and effects of time dilation and length compression. We then consider how solutions like the Schwarzschild metric yield testable predictions such as gravitational lensing and graviational redshift, which serve as important evidence in support of General Relativity. We conclude with a discussion about some of the more exotic aspects of black holes, including Hawking radiation, the no hair theorem, and the black hole information loss paradox. Recommended pre-listening is Episode 136: Introduction to General Relativity. If you enjoyed the podcast please consider supporting the show by making a PayPal donation or becoming a Patreon supporter. https://www.patreon.com/jamesfodor https://www.paypal.me/ScienceofEverything

you're listening to the Science of Everything podcast episode 145, Relativity and Black Holes. I'm your host, James Fodor. This episode is a continuation of the discussion of general relativity, which we began in episode 136, which is the prerequisite for this episode. In that episode, we talked about general relativity and explained the notion of space time and how we describe velocity, distance, and curvature of space time using mathematical formalisms and how we combine these formalisms together to yield Einstein's field equations, which loosely say that the curvature of space time is proportional to the energy and matter content of space time.

And I explained how Einstein's field equations are a series of 10 coupled nonlinear partial differential equations, which means that they're very complex and difficult to solve for any realistic cases. However, I did say that there are some closed form, meaning sort of simply mathematically describable solutions known to Einstein's field equations, and I'd talk about them in a future episode. Well, now is that future episode, or at least one of those future episodes, where we'll talk about solutions to Einstein's field equations. And in particular, in this episode, we're going to focus on the Schwartzschild metric and how it's able to describe or predict and describe the existence of black holes. So we'll talk about deriving the Schwarzschild metric, how to

interpret the resulting metric, and then we'll see how the resulting metric yields predictions, which have been experimentally verified and thereby serving as experimental evidence in favor of general relativity. We'll then talk in more detail about Schwarzschild black holes, some of the phenomena there like the event horizon, singularity, and so. forth and we'll conclude by discussing some of the unsolved problems or outstanding issues with black holes including the phenomena of hawking radiation, the no hair theorem and the black hole information paradox. So this will be a pretty dense episode. So hope you're ready and let's then jump into it. But bear in mind though, I will be assuming that you've listened to the prerequisite

on general relativity because that introduces some of the key ideas that I'll be referring to here. So we'll start with exactly where we picked up last time, which is Einstein's field equations, a series of 10 coupled, nonlinear, partial differential equations relating the curvature of space and time to the energy and matter content of space and time. What we're going to do is try to find a solution to these equations. There are a number of closed form solutions known. We're going to focus on one of them today. And essentially what this amounts to is solving for the equations to find the metric, which remember, is a mathematical description of the overall shape of space and time, a metric that satisfies the equations. So the term on the left of the equations is the Einstein tensor, it's denoted as

capital G, and it more or less describes the curvature of the metric, which represents the structure of space and time. On the right hand side is the stress energy tensile, which describes the energy content of space and time. So there's sort of two ways to solve these couple of equations. One is to postulate a stress energy tensor, so to stipulate what the energy content of space is, and then solve for the metric that satisfies the equations given that energy content. You can also do it the other way. You can start with the metric, so specify what the shape of space and time is, and then solve for the stress energy tensor that will give you that metric. So in this particular case, the way we're going to do it is we're going to stipulate what the stress energy tensor is, as well as making a few other assumptions, and then we're going to see what metric that gives us, what metric satisfies the equations when we stipulate what the energy content of the universe is.

So we're going to stipulate at the outset that the energy and matter content of the universe, or at least the region of the universe that we're considering, is going to be zero. So we're looking for a vacuum solution, it's often called, an empty. space solution of the equation. So obviously that simplifies things dramatically because the right-hand side of the equation is just zero. And the equation simplified down to the Rishi-Tensor is equal to zero. The Rishi-tensor, remember, basically describes the curvature of space-time in that region. So the next step, now that we've already simplified things quite a lot, is to make further assumptions to help simplify things. The next thing that we're going to assume is that the Rishi-tensor, all of the components of the Rishi-Tensor are independent of time. So

they're static, they're constant over time, they're not changing. So this is just representing the type of solution that we're looking for, which is a static solution. Again, this is often done for simplicity. Now it turns out when you make this assumption, this dramatically simplifies things further because it means that any terms that interact with the time coordinate, remember there's four coordinates of space and time, there's one time coordinate and three spatial coordinates. So any of them that interact with the time coordinate have to go to zero because otherwise there will be changes over time. And so making this assumption of a static field means that instead of having 16 components of the Rishi tensor, now they're only going to be four components. And they're just the diagonal components.

So those along the diagonal of the matrix, a four by four matrix. It's only the ones along the diagonal that will be non-zero. Everything else goes to zero when we make this assumption of a static field. Things are now very simple because instead of having these 10 coupled partial different nonlinear partial differential equations, now we've got, we've reduced it down to four much simpler equations. all of which are just equal to zero. Four equations for each, essentially one for each of the coordinates, a one time equation and one for each of the three spatial coordinates. Now to make even more simplifications, we introduce a third assumption,

which is that we're going to look for a spherically symmetrical solution. So we're kind of interested in solutions that look the same when you rotate them. That simplifies things a lot further because now we only have to worry essentially about two coordinates. One is the radius, which is the distance from the centre, and then one angle coordinate. Even with all of these simplifications, the equations that we have to solve are still somewhat complicated, but the algebra is at least now solvable, and obviously I can't go through all the details here, but at this point, what we need to do is take the highly simplified form of the Rishi tensor that we've derived

by making these assumptions, and then just substitute in for the actual form of the Rishi Tensor. Remember, the Rishi Tensor is defined in. terms of mathematical objects called Christophil symbols. Christophil symbols basically describe the way in which our path changes owing to the curvature of space as we move. Remember in the last episode we talked about the idea of someone holding a spear out in front of them and starting at the North Pole and walking down towards the equator, their spear will change direction as they walk along even if it locally doesn't look like they're tilting their spear at all. They're holding it in front of them, but it will actually change direction globally, simply

because the earth is curving under them. And so if you're to describe how the direction of that spear is changing with the motion of the person walking with it, you need to consider not just whether the person is rotating or tilting their spear, but also the change in direction of the spear because of the curvature of the earth itself, the path that they're travelling on. Christophel symbols help us to do that. They're mathematical formalism that helps us to describe the change in direction of a path over a curved geometry. And so the Rishi tensor is defined in terms of these Christophel symbols.

Christophal symbols in turn are defined in terms of various derivatives of the metric. The metric is the 4x4 array that describes the overall shape of space and time. What we're going to do at this point, we've got the general form with the Rishi tensor in order to solve for the actual final equation here. All we need to do is plug in the definition of the different components of the Rishi tensor in terms of Christophil symbols and thence in terms of the metric, and then equate that all to zero. Because remember each of these components is equal to zero. There's four components of the Rishi tensor, the diagonal components, one for each of the space-time coordinates.

They're all equal to zero because of the simplifying assumption that we made of a vacuum solution. So we're just going to substitute in the correct forms for each of these components and equate them all to zero. And we've already simplified the equations a lot because of the additional assumptions of the static field and spherical symmetry that we've assumed. So we substitute in the form of the Rishi tensor, do the TDS algebra, rearrange, and combine some things together, and then once we're done, we end up with a solution for the metric. And this metric is called the Schwarzschild metric

after the scientist who derived it originally. So the Schwartzschild metric is, it turns out, not just a solution to Einstein's field equations, It's actually that there's a theorem called Berkov's theorem. We won't go into the details of that, but there's a theorem which states that the Schwarzschild metric is the only spherically symmetric vacuum solution of Einstein's field equations.

So there's only one solution to Einstein's field equations that is spherically symmetric and also describing a vacuum, so no matter or energy. So that's quite an important result, and it turns out that it's not that difficult to derive. There are other closed-form solutions of Einstein's field equations, but this is really the simplest one of interest. And so there's been a lot of, well, over the decades,

there's been a lot of study into this metric and what it tells us. And it turns out that it actually describes a lot of very important objects, including the gravitational effects of many stellar bodies like planets and stars, but also it turns out that it describes a special type of stellar object called black holes,

which we'll come back to in a moment. Now, I won't try to describe the exact mathematical form of the Schweachield metric, because that's not really suitable for this podcast. But what we'll do is we'll sort of talk about the key features of the Schwarzschild metric and what it tells us about space time in this spherically symmetric vacuum solution. Remember I said that there are four components to this metric, well, four non-zero components. There are 16 in the 4x4 matrix, but most of them are zero.

And the four components each describe essentially how the shape of space and time is affected by each of the four coordinates, like one of time and then three of space. So two of the spatial ones aren't really very interesting because of the spherical symmetry. The two coordinates of most interest are the time coordinate and the radial coordinate. So the radius, the radio coordinate describes how far away we are from the central mass. The Swat Shield metric being spherically symmetric, what it fundamentally describes, as I said just before, is the effect of a central spherically symmetrical mass on the space and time around it. So this is applicable to planets and the space around them or stars and the space around them,

even galaxies. Galaxies aren't perfectly symmetric. I mean, neither are stars or planets, but sort of close enough, right? It's an approximation to the effect of these central massive bodies on the space around them. Now, something interesting if you look at the form of the equation for these two coordinates. And that both of them, involve apparent singularities. In this case, it's sort of fairly easy to see because the equation looks like one minus some number over R, where R is the radius, the radial coordinate, how far away we are from the central mass that defines the metric. So when you have an equation like this, one minus some number over R, if you think about what this means, because R is on the denominator,

as R gets bigger, that second term gets smaller. And because there's a minus sign in front of it, it's one minus, then this term that gets smaller, this whole term goes to one as R goes to infinity. Now what this means is that as we move further and further away from the central mass, the metric becomes more and more like the metric just describing flat space. One minus some number over R as R gets big, one over R becomes small, and so we're subtracting off a smaller and smaller number. What we have left with is closer and closer to one, which is just like the metric for a flat number.

universe. It's just basically ones along a diagonal, you know, not very interesting, right? So, but this kind of makes sense. Essentially, as we get very far away from the central mass, the effects of its gravity become smaller and smaller. Space and time become more and more like just empty flat space. So that makes sense. That's a good result. That's not a problem. But what happens with this mathematical structure if R gets really, really small? Instead of getting big, we're now making really, really small. Well, when R gets really, really small, 1 over R actually approaches infinity. It gets really, really big, and it blows up to, theoretically, infinity in the limit of R approaching zero. So this means that the metric is actually not defined

when R equals zero. And that's what we call the singularity. It's a point, well, in this case, it's just sort of in our metric in the description of space and time, where the metric isn't defined. The way you can think about this is at R equals zero, the metric is not continuous. It sort of pinches together at a point. Just as if you were kind of making a funnel out of Play-Doh, but at the end you didn't kind of curve it around, but you just sort of pinched it off. If you imagine sort of rolling up one end, you can't just sort of roll around the tip because it's not continuous. Like it ends at a sharp point. You have to sort of flip your hand all the way around and then go up the other side. That's what's meant by a discontinuity. There's not a

continuous sort of curve motion that you can proceed in from one side to the other. you have to sort of sharply stop and then kind of rotate around. And that's thought to be non-physical, that sort of singularity there, because space-time is thought to be continuous. So singularities like this, sort of these sharp discontinuities in space-time, are not thought to be, well, many people think that they're not real. What they indicate is that there's a breakdown of our theory. The theory is not describing accurately what's happening at that very small spatial scale. Now, this isn't really surprising for our approach as zero. because when R gets small enough, what this means is that our distance scale is getting small enough,

and when it becomes sufficiently small, the effects of quantum mechanics will become relevant. And we don't have a theory of quantum gravity. I'll talk about that in the future episode. General relativity is a description of relatively large distance and time scales, not very small ones. And so it's not really surprising that the theory breaks down at very small scales, because we kind of know that it's not going to be suitable at that scale. So these are two interesting things about the Schwarzschild metric is that when you go very far away from the central mass, the metric just looks like that a flat space. And when you go right to the center, or as you approach the center, there is a singularity. So the metric becomes undefined, telling us that our theory won't be appropriate to describe whatever's happening at that very small scale.

There's another interesting feature of the Schwarzschild metric, which is that when you solve the equations, you find that there's this sort of special length scale. where there's apparently another singularity at this length scale. In other words, we know there's a singularity as R approach is zero, but there's another place where it looks like there's a singularity in the equations, where R approaches this special length scale. This special length scale has been given a name. It's called the Schwarzschild radius. It turns out that the Schwarzschild radius,

it's quite small for most massive objects. So most objects in the universe are much larger than their Schwarzschild radius. The Schwarzschild radius of any object is determined entirely by its mass. So, for example, the Schwartzschild radius of Earth is about one centimeter. The Schwarzschild radius of the sun is about three kilometers. So for most real physical objects, the Swat Shield radius doesn't really matter. Because remember, the Schwarzschild metric only applies in the vacuum. So it's a vacuum solution.

So the Schwarzschild metric will describe the space and time outside of Earth because of the gravitational effects of Earth, but it won't actually describe the space and time inside the Earth itself or inside the sun itself, right? Because obviously that's not a vacuum, right? matter there. So the fact that there's sort of weird stuff happening at the Schwarzschild radius and below that down to the singularity is irrelevant for most stellar objects because most stellar objects are much larger than the Schwarzschild radius and so the Schwarzschild radius sort of never comes into it because, again, the Schwarzschild radius will only exist in space when the object is smaller than the Shreight Shield radius and the Shrachield radius is exposed

and therefore the vacuum actually exists where the Shrachield radius is exposed and therefore the vacuum actually exists where the Schwarzschild radius is. You only see the Shrardt shield radius manifested when the vacuum solution is relevant at the Schwarzschild radius, if you see what I'm saying, because the Shrardt Shard metric is only relevant for a vacuum solution. If you've got matter there, then the Shwight shield radius isn't relevant because the Shwight shield metric isn't relevant. Now, there are special types of objects for which the Shwight Shield radius is relevant because the Shwight shield radius is, so to speak, exposed to space, and therefore it does describe space and time in that area. And these are called black holes, right? We'll come to these in a moment. But the Schwarzschild metric doesn't just describe

the space and time around black holes. It describes the space and time around any spherical symmetrically symmetric stellar object. It's just that only in the case of black holes do you see the Shwashield radius and other sort of bizarre phenomena become manifested. So we'll talk about those a little bit later. It turns out, however, that the apparent singularity that happens at the Shatield radius is not real. It's just because of a poor choice of coordinates. system. There's different coordinate systems that we can use to describe the same metric. So the Schweachield metric, there's only one Schwarzschild metric, but there's many coordinate systems you can use to describe the same metric. Just as I can describe a location

on Earth's surface by using some kind of X, Y coordinate system, or I can use distance from the center of the earth and angle, well, I guess I need two angles, right, but I could use the distance from the center of the Earth, and then I could use two angles to describe the position on the surface of the Earth. Essentially, that would be latitude and longitude. But if I wanted to, I could use a simple Cartesian X, Y, Z coordinate system. Both of those could be used to describe position on the surface of the earth. In practice, probably the radial coordinate plus the latitude and longitude is going to be easier, but either of them describes the same set of positions, right? And so it's the same thing with the Schwachield metric. There's only one Schwartschild metric,

but there are many ways to represent it using different coordinates. And it turns out that the apparent singularity at the Schwartzschild radius isn't real. There's no actual discontinuity of spacetime there, it just appears to be a discontinuity because of poor choice of coordinates. In different coordinate systems, you don't see any singularity there. There's still something important happening at the Schweachield radius, as we'll describe later, but it's not actually a singularity. So there's an important difference between the apparent singularity at the Schwarzschild radius, which is due to our coordinate system.

If we choose a different coordinate system to represent the same Schwarzschild metric, then that singularity goes away. And it's just a regular point in spacetime like any other one. still have important properties, as we'll talk about in a moment, but it's not singular. Space time is still continuous there. Whereas the singularity at R-equal-0 is a real geometric singularity. It exists in any coordinate system for describing the Schwarzschild metric, so you can't transform it away by using different coordinates. And it seems like that according to general relativity, space-time is pinched there and is not continuous. And so

that's an indication that our theory breaks down there. Now, there's some other interesting aspects of interpreting what the Schwarzschild metric is telling us about space and time, apart from the singularity in the Schwarzschild radius. And I think the easiest way to understand what's happening here is that the mass at the center that's generating the distortion that's described by the Schwarzschild metric, that central mass distorts space and time. And it distorts them in a particular way. Specifically, it stretches, elongates essentially, radial distances, or distances to and from the central mass, and it also elongates temporal distances, so time durations. So these effects are specifically described in terms of proper time, so that time

durations measured by an observer, proper time durations nearer to the event horizon are shorter than proper time durations measured further away from the event horizon. You can understand this in terms of imagining a calendar, right? So imagine we have an annual calendar for someone, long way away from the Schwarzschild radius and then we have one for someone very close to the Schwarzschild radius. Let's imagine that we position the person close to the Schwarzschild radius such that their proper times are half as long as those for someone who's further away. So what this looks like is that the calendar of the closer observer is stretched out so that

one year or 12 months for someone far away from the Schwarzschild radius is only equivalent to six months for someone who is close to the Schwarzschild radius. So a 12-month duration for the close observer is stretched out to cover two years for someone further away. So someone who's located further away from the Schwarzschild radius will say that the clock of the closer observer is ticking slower or they're measuring less proper time relative to the faraway observer. And so, for instance, if we imagine that they're both observing signals coming from a distant star or solar system or something, imagine there's one signal sent every month, to the observer who's far away from the Schwarzschild radius, they will observe one of those

signals every month, and so those signals will kind of tick at a rate of one per month. Whereas for the observer who's closer to the Schwarzschild radius, they will see those external signals arriving once every two months, according to their clock. And again, when we say clocks, that means any type of clocks, mechanical clocks, biological clocks, physical clocks of like physical processes. So it's all processes that occur in time. It's not just like a mechanical or digital clock. So clocks for the observer close to the Shwa Chowield radius tick at half the rate, in this example we're positioning it, a position such that they tick at half the rate of the clocks of someone far away. And the closer you go to the Shwachield radius,

the slower and slower your clock ticks so the more and more stretched out your calendar becomes relative to the calendar or time durations of someone further away. So your calendar will stretch out so that 12 months for the distant observer will be three months for you or two months, one month, and so it keeps stretching. You can imagine those sort of timelines, one's stretching out relative to the other. So that's kind of what the effect of the gravity of the massive object is. It stretches out the time durations, the time dimension of the surrounding space. It also stretches out the radial dimension of the surrounding space, such that when objects fall into or get closer to the stradioritial radius, the radial dimension elongates

relative to their perpendicular direction. So if you started with an object that was square and it moved closer to the eventorizing, it would get stretched out. So it would become like rectangular and become elongated in the radial direction. And you can visualize what this effect looks like with a shape called flams paraboloid. If you're interested in Googling that, that's FLA-D-W-M paraboloid, that visualized. That visualized. the effect of radial distances being elongated, stretched out as you move closer to the Schwarzschild radius of a massive object. So these are the main effects that we can interpret from the form of the Schwarzschild metric that we've derived. We've identified that there is a

real geometric singularity at the center where the mass is located. We've identified that there is an apparent singularity at a special distance called the Schwarzschild It's not a real singularity, but there is something weird or interesting that happens at that location. And in particular, it's a radial distance around the singularity. And in particular, we've also identified that the central mass stretches space and time surrounding it in such a way that time passes more slowly, closer to the short-chial radius relative to further away. And also radial distances are stretched, so they're elongated closer to the strutcheel radius relative to further away. So this gives rise, these phenomena give rise to experimentally verifiable predictions.

And so in this next section, we're just going to talk briefly about some of these experimental predictions made by general relativity, and in particular the Schwartschild metric, although some of them are more general, and have these experimental evidences have been used to validate the success of general relativity as a theory of gravity. So let's start with something called the procession of the perihelia. In Newtonian physics, an object in a two-body system, so it's such as a planet orbiting a star, traces out an ellipse, which is kind of like an elegant circle, like it's sort of an overly shape. It's not exactly an oval shape, but that's what sort of an ellipse is a stretched out circle. The planet traces out an ellipse around the star with the center of mass of the system, which usually is the star, located at the focus, one of the focuses of the ellipse. So you have an ellipse. It's got two focuses. At one of those focuses is the star, and then the planet orbits along the ellipse around the star. The closest point of approach of a planet as it orbits its star is called the perihelion.

So because it's an ellipse, it's elongated, so the distance between the planet and the star is not always exactly the same. It changes as it orbits around, and the closest point, the distance between the planet and the star at the closest point is called the perihelion. That's the point of closest approach on the orbit. of the planet. Now, procession of the perihelion refers to the fact that this closest point can move. It can change relative to the background stars. I mean, the planet's orbiting the star along the ellipse, but it's kind of like when there's procession of the perihelion, it's kind of like the orbit itself is kind of rotating. It's a bit hard to explain, but if you imagine an ellipse, as again, it's kind of a stretched out circle, you could imagine that the longer axis,

of the ellipse is drawn so that it's going vertically up and down the page, and the thinner axis is across the page. So then imagine that we put a dot on the ellipse, which represents our planet, so we can then move the dot around the ellipse, and the dots moving around and orbiting around the sun, right? So that's just orbital motion. The point at which the dot gets closest to the star, which is at one focus of the ellipse, that's the perihelion, right? What procession of the perihelon looks like is that the whole ellipse, the line that we've drawn, rotates on the page. So instead of the long axis going directly up and down, it will now go slightly to one side, and then it's slightly down and slightly further down until eventually the orbit will have done an entire 90-degree turn,

where instead of the long axis going up and down the page, is now going crosswise along the page. It's not that the planet has moved, it's actually the orbit itself has kind of rotated relative to, say, there's stars in the background. So that's what procession. of the perihelia is. There are a number of reasons why procession of the periolia happens. Some of them are predicted by Newtonian physics. So one of the reasons is because the solar system does not just consist of one planet and the sun. It consists of many planets in the sun. And the gravitational influence of each planet on all the other planets has an effect on their orbits. So these cause perturbations in the orbit that mean that orbits deviate from true ellipses. So this

caused as procession of the perihilia. So that's known. That was a known effect under Newtonian physics, but that's just due to kind of complex perturbations between planets. However, since the mid-19th century, it had been recognized that Mercury's orbit, Mercury being the planet closest to the sun, and so this effect was most evident, although it's relevant for all the planets, but it was observed first that Mercury's orbit processed about 7% faster than expected by Newtonian physics. So it was processing as expected, but it was a little fast, about 7% faster than was expected. Many attempts to try to explain why Mercury's orbit was processing faster than was predicted had failed. People thought that there might be another planet between the Sun and Mercury,

and that was one thing that was considered and eventually dismissed, and many other potential explanations were considered and rejected. One of the very early successes of general relativity is that general relativity was able to vary out, accurately predict this discrepancy of procession. The reason for this is because in general relativity orbits do not form exact closed ellipses, even in a simple two-body system. And this is what's interesting because in Newtonian physics, yes, you'll have procession of the peritone, but that's just because of complex perturbations with many body systems, like all of the other planets in the solar system causing perturbations. But if you just had a two-body system with just a planet and

one star, in Newtonian physics, the orbits would be perfect ellipses. However, in general relativity, even in a perfect two-body system with just one planet and one star, orbits still don't actually form closed ellipses in general relativity, which is very interesting. The orbits kind of spiral around, not spiral inwards, but kind of spiral around the central star, which, well, it amounts to procession of the perihelion. So that's one difference between Newtonian mechanics and general relativity. is a bit complicated to sort of show that. So I won't explain exactly why, but it's because there's a kind of a change in the potential energy of a planet as it orbits around a star due to the bending of space and time by the central mass. And one manifestation of that is this procession

of the perihelia, even in a two-body system. So because Mercury is so close to the sun, the additional procession due to this general relativistic effect is greater than the other planets, and therefore it was the first to be measured, and that couldn't be explained by Newtonian physics. So that's one experimental vindication of general relativity. Another piece of experimental evidence in favor of general relativity is called gravitational redshift. So this is another novel prediction of general relativity that light should be redshifted as it escapes the gravitational potential of a massive body. You can interpret this as a consequence of the equivalence principle,

the equivalence principle we talked about in the previous relativity episode. it's that the effects of gravity and the effects of acceleration are equivalent to each other, you know, with the right magnitudes. And so we know that when there's a body that's accelerating, that causes redshift. So if a body's coming towards you, light is blue shifted, it's compressed together, that reduces the wavelength, and so it moves to the blue end of the spectrum, whereas if a body's moving away from us, that stretches out the wavelengths and shifts it to the red end of the spectrum, so that's red shift.

So that occurs due to motion and also acceleration. the equivalence principle says that well if we see a phenomena like that due to acceleration we should see a similar phenomena due to gravity and therefore the prediction is that if you're escaping a gravitational well so if light is being emitted by a star for example it's coming from a place of low gravitational potential to moving up to higher gravitational potential just like climbing up a hill is moving from a low pit to a higher gravitational potential it's the same as you're escaping from a star right that means that the light effectively is being, it's like it's being decelerated as it's moving away from the gravitational mass and so it'll be stretched out. Because longer wavelength light has lower energy than higher

wavelength light, this all kind of makes sense, right? Because essentially the photons start off with high energy at the surface of the star, but as they propagate away, they have to use up energy to climb out of the gravitational well. The energy that they gain in escaping the gravitational well of the central mass has to come from somewhere, it comes from the wavelength itself of the light. It stretches out and therefore loses energy as it's stretching, as it's escaping from the gravitational well. Gravitational redshift was relatively hard to measure because the effect is quite small from light emitted by the sun. It wasn't detected, therefore, until the 1950s when we had increasingly precise measurements of the redshift from the sun and also more distant stellar

objects. And so we could measure the effect of the gravitational redshift of light emitted and propagating away from massive bodies like the sun. So that's another vindication of general relativity. A third piece of evidence is something called gravitational lensing. So a gravitational lens is any type of matter, but usually like stars or galaxies even, that bends light from a distant source as it travels towards the observer. So in Newtonian physics, the universe is conceived of as flat, right?

And so light in a Newtonian flat universe will just travel in a straight line. However, in general relativity, space and time are bent, right? They're curved. And in curved space, the paths of objects can be bent. This is just exactly the same as if you go onto Google Earth or something like that, and you choose the shortest path between two points on the surface of the Earth, a so-called Great Circle, and you plot that on a flat map. It will look like it's curved, right? It looks like aircraft if you plot their routes on a flat map that's that's large enough it looks like they take these sort of wild detour curves which look very strange like why is it curving all the way here and then coming back uh wasn't why doesn't it just take a straight line between these points well actually it is taking a straight line it's just it's it's taking a straight line in curved space or on the curve over the curved surface of the earth and so when you project it to flat space it looks like it's it's it's curved but it's actually flat and it's essentially the same when it comes to the um when it comes to the

bending of light, light travels on a straight path but through curved space. And so the way we process visual stimuli or reconstruct them using computers is by making assumptions sort of in our brains or with the computer that when we see an object that's located a long way away, that it is located directly behind where we've observed it, right? So we sort of trace back in a straight line. And we don't factor in the bending of space that occurs in the intervening distance. So to visualize this, let's imagine that we're standing on Earth with a big telescope and we're pointing it at a massive object. Let's say it's a cluster of stars. And if we're just looking at the cluster of stars themselves, we'll see the light that comes from those stars and that's fine.

But that cluster of stars curves the space around them, what can actually happen is that light from objects that are behind that cluster of stars. In fact, like behind relative to us, in fact, very far away behind them. can be bent around so that now we can see them, even though the cluster of stars is between us and the object, if the bending of space is enough and if the relative positions are just so, you can actually see objects that are situated behind the cluster of stars, but because the light is bent around, if you imagine sort of the, I talked in the last episode about placing marbles

and billiard balls on a stretchy sheet, and they make a depression in the sheet, you can sort of roll marbles around the billiard balls and the marbles will bend as they pass by the billiubles because of the depression and curving of the stretchy sort of membrane that they're located on it's the same principle here but those marbles are now photons the photons as they're propagating away from the distant object are bent technically they're traveling a straight line but through bent space they're bent through the depression and the bending to space in time that's made by the massive object, and then they reach our eyes or our telescopes on

Earth, even though they're really located in a place where they wouldn't be able to reach us if they were just traveling through flat space. So this results in a distortion of the position of distant objects or distortion of shape of extended objects as well. So sometimes this can lead to duplication of images so that we can see multiple images of the same thing. Or if the alignment's just right, you can actually give rise to what's called an Einstein ring, where you have a kind of, there's a central object, which is the massive object that does the bending, and then all around it is kind of like this donut of distorted images of the thing that's behind it, because we're kind of seeing versions of the image stretched out all around the central mass that's doing the distortion.

It's probably easier just to look up what that, Google, what that looks like, Einstein Ring, to visualize what I'm talking about. And that's a severe kind of gravitational lensing that can really only be explained by General Reilly. activity. The final experimental evidence that I wanted to mention, I've already kind of touched on, but it's gravitational time dilation. So I talked about that when we were talking about interpreting the Schwarzschild metric, which is the fact that it stretches out, that the central mass in the Shwechield metric stretches out radial distances and as well as time, so that clocks closer to the Shwaichield radius measure shorter temporal intervals than those further away. Or in other

was they run slower than those further away. And this is called gravitational time dilation, and it is a very robust phenomena that's been measured in many different contexts. I think I've talked about this in the special relativity episode, but they've done tests with atomic clocks flown at different altitudes above the earth, and then compared that to atomic clocks that were kept at sea level, and they find that they've indeed found to a very precise accuracy, that the clocks that were on aircraft, and therefore at a higher potential in Earth's gravitational well, recorded slightly longer times than those that were kept at sea level, which is consistent with the gravitational time dilation that clocks lower in a gravitational field

record fewer ticks, or in other words, time passes more slowly. So that's another piece of evidence in favor of general relativity. So I now want to transition to talking about one of the really bizarre implications of the Schweachield metric. And these are a phenomenon called black holes. and you're probably all eagerly awaiting this part of the episode, because after all, that's part of the title. But it takes a little while to get there because you first need to understand what the Schwarzschild metric is

and what it tells us about the nature of space and time surrounding that central massive object. And as I emphasized before, the Schwarzschild metric doesn't just apply to black holes. It actually applies to any central, spherically symmetrical, massive object, and so it's very useful for describing the gravitational effect of stars and their surrounding planets. but there are certain very interesting cases where particularly the

the Schweatield radius and the singularity at the center become more relevant and so these are called black hole so specifically a black hole is a region of space time where gravity is so strong that nothing including light or any other types of radiation is capable of escaping from that region so the black hole is the region that nothing can escape from not even light and that's why it's called black, because obviously you can't see it, no light can escape from it. The region of space within which it's impossible to escape from is called the event horizon. So the event horizon is the boundary, more or less spherical boundary, at least in the simplest form of black holes,

that surrounds the area of space outside of which you can't escape if you enter that space. So the event horizon kind of forms the, if you like the boundary of the black hole, I mean, there's nothing there physically. It's just empty space. but it's a special type of space in which you can't escape from that region, the Vent Horizon being the boundary of that space. And at the very center of the black hole is the singularity that everything's sort of drawn towards.

Now, you might wonder why it's impossible to escape from the inside of the event horizon. And the reason is a bit technical. Now, if you do a fairly sort of simplistic calculation of the escape velocity from a massive object, the escape velocity is the speed that you have to reach in order to, well, escape from the gravitational well, escape from the gravitational well of a massive body, and the escape velocity increases with the mass of the object. So if you do that calculation for an object located at the Schwarzschild radius of a massive object, then you find that the escape velocity is equal to the speed of light,

which is consistent with the fact that you can't escape from the object. that massive object, obviously because you can't go faster than the speed of light. But that's a simple Newtonian calculation, and it doesn't actually capture the full reason as to why black holes are impossible to escape from, because escape velocity is only relevant for a non-accelerating observer. So it's a velocity you travel out that you're gradually decelerated by the gravity of the object. But from a sort of Newtonian black hole, one that had an escape velocity equal to the speed of light, it would still be possible to escape from that, because you just have to keep excessive.

accelerating and eventually you'd be able to get out of the sort of the gravitational well. So you can't account for this inability to escape by just using Newtonian gravity. So black holes really only can be accounted for under general relativity. And the reason why it's impossible to escape from inside the event horizon is because of what happens to space and time as you get closer and closer to the event horizon. It's the actual bending of space and time. One way to think about this is to consider the light cone of an observer located far away from the black hole. The light cone represents the region of space time that it's possible to get to.

So if we think about it in sort of two dimensions, it becomes a light triangle on an axis where the vertical axis is time and the horizontal is space. And so we can travel different distances away from the starting point in space time, depending on how fast we'd travel. So if you stay at the same location, that'll just be traveling vertically upward along the y-axis, right? That's your trajectory at the same location, but over time. Alternatively, if you travel to the right at a slow speed, then that will be like having a straight line with a very steep slope. So you're moving very slightly to the right over time. If you travel at the speed of light to the right, let's say, then that's represented by a line with a 45 degree angle between the vertical and the horizontal axis. That's as fast as you can go.

That's as fast as you can move to the right in time. If you try to move faster than that, then you're traveling faster than the speed of light. So this triangular region formed by a 45 degree line to the left and a 45 degree line to the right, forming an upper triangle, represents the cone, or in this case, triangle of space that you can reach in a certain amount of time. You can go vertically upward if you just stay still or you can move any distance to the right or to the left out to this 45 degree line. But you can't move outside that 45 degree line because doing so would require faster than light travel, right? So the light triangle here represents the region of space time that you can access. In flat space, the triangle is always pointed, if you like, vertically.

So if we imagine lining up the vertical axis of the triangle with respect to the, so that it's parallel with the event horizon of the black hole, What that means is that I can move any distance away from the black hole or any distance towards the black hole in a given amount of time as long as doing so doesn't exceed the speed of light right. So I can move towards it, away from it at whatever speed I like. And there's just as many, this is a key point, there's just as many trajectories towards the black hole as away from the black hole. That's true if you're very far distant from the black hole in like flat space. Now what happens is as you move closer to the black hole, this light cone or light triangle that we have here actually rotates. It tilts progressively more and more towards the event horizon. So instead of the time axis being parallel to the event horizon, the time axis, the vertical axis of my light cone actually angles slightly towards the event horizon and then more and more and more towards the event horizon as I get closer and closer to it.

what this means is that there are now, instead of there being the same number of trajectories towards the black hole as away from the black hole, now as the time axis tilts towards the event horizon, there are actually more trajectories that go towards the black hole than that go away from it. This might seem sort of counterintuitive, like because can't I go in whatever direction in space I like, and there are sort of just as many ways to go backwards as there are to go forwards, right? So can't I just go away from the black hole? What way you can, but effectively it requires higher and higher velocities. Because of the bending of space time, there are more ways to go towards the black hole than away from it. You can still go away from

it as you get closer to the event horizon, but you need faster and faster velocities. Eventually what happens is that this tilting of the light cone becomes greater and greater as you move closer and closer to the event horizon because space time gets more and more bent. Until what happens at the event horizon is that the time axis is rotated a full 45 degrees, relative to where it is if it's very far away from the event horizon. A full 45 degree rotation means that now, even if you were to travel at the speed of light away from the event horizon, you still can't quite escape.

So the very fastest trajectory, even the very fastest trajectory straight away from the singularity, even that doesn't actually get you any further away from the event horizon. So in other words, it means all trajectories now point towards the singularity. And that's why you can't escape. It's not actually because, the escape velocity is too high as such, because that's purely from a non-accelerating trajectory. It's actually that there's literally no paths through space time that you can take,

without exceeding the speed of light, that allow you to get away from the event horizon. And the reason there aren't any is just because it's been bent in such a way that all of those trajectories that would generally allow you to travel in the other direction, away from the event horizon, they now all point towards the event horizon. and actually what happens is that the light cone actually keeps rotating beyond the 45 degree line as you move closer to the singularity until it actually becomes it does a full 90 degree rotation when you reach the singularity which in some loose sense means that time and space have like swapped roles which which is very weird but i won't try to explain what that means exactly but um the point is that

space and time are greatly distorted as you move closer and closer to the event horizon then at the event horizon and inwards, they're so distorted that is actually impossible to move through space away from the event horizon. All of the trajectories that you can take, even accelerating trajectories point towards the event horizon and towards the singularity at the center. So that's why it's impossible to escape a black hole. It's also important to understand that there's a popular misconception about black holes as if they suck things into them, as if they're like some sort of cosmic vacuum cleaner. Now this is not correct. for the most part. If you're a long way away from a black hole, you can't even tell the difference

between whether there's a black hole at the center or whether there's a star or a planet or whatever. Like you can calculate what the mass is based on how much it curves space time, right? But you don't actually know what that object is if you're far away from it. So the point is the Schwartzschild metric just as well describes the gravitational effects of a star and a black hole of the same mass. and just like our son doesn't suck things into it that are far enough away, likewise a black hole doesn't suck things into it that are far enough away from it. The only kind of way in which black holes kind of pull things in is if they get close enough,

in particular, if they are at the event horizon or closer, then yes, it is impossible to escape from black hole. And so at the event horizon or further inwards, it is impossible to get away. And all trajectories inside the event horizon do point towards the black hole. So in that sense, it's still not quite right to think of it as a sucking. It's more just like there's no paths that go any other way other than towards the singularity. But at that point, once you're inside the event horizon, there's no escaping anyway, right? So anything that's outside the event horizon isn't sucked in by a black hole. The only things that are sucked in are those things that are at the event horizon or further inwards.

There are reasons to want to stay away from the event horizon because, for example, the tidal forces close to the event horizon become very strong, which would tend to rip objects at unless the black hole is very, very large. But for stellar-sized black holes, the tidal forces would be very large. There also tends to be a lot of matter that accumulates near the event horizon that heats up because of friction forces. And that results in the emission of a lot of energy and very high intensity radiation, which you don't want to be around. So there's sort of other reasons to want to avoid black holes or getting too close to them. But being sucked into them isn't really one of them. That's kind of a loose explanation of what a black hole

and how it works in the role of the event arise. And I'll just talk a little bit about a few other aspects here. So I've already mentioned that the singularity is located at the center of a black hole. Or at least this is what is described by the Schweitzer metric, right? This is a region, technically, it's a point really, where space-time curvature becomes infinite, and space-time kind of pinches off, and our theories sort of fail to adequately apply to that region. In reality, what we think is that at that scale, quantum effects will become relevant. But we currently don't have a quantum theory of gravity, so we don't actually know whether singularities are real or what real properties they would have, what actual properties they would have.

And so currently, we just, there's not a lot we can say about them. I suspect, I think, along with many physicists that singularities aren't real. There isn't actually a singularity at the center of a black hole. The existence of a singularity is telling us that our theory isn't adequate to cover that that small region there. So whatever does exist there is something currently beyond our understanding. But hopefully we'll one day be able to explain it when we have a quantum theory of gravity. Now, there's an interesting phenomena surrounding objects that are falling into a black hole. We've talked about gravitational time dilation, which means that clocks nearer a black hole,

nearer to the event horizon, tick more slowly. Or in other words, they record fewer ticks, like fewer events happening relative to one clock that's located further away. So due to this, an object that's falling into a black hole that's getting closer and closer to the event horizon appears to gradually slow down as it approaches the event horizon. Theoretically, it would take an infinite amount of time to actually reach the event horizon. So an external observer would see it slowing down or slowing down and eventually kind of freezing to a standstill or kind of a near standstill as it gets very close to the black hole. An external observer therefore would never actually see the object cross the event horizon

because it would kind of freeze or just sort of slow down so much that you would cease to see any motion before it actually crosses the event horizon. Also due to gravitational redshift, the light from the object falling into the black hole would become more and more redshifted and more dim. And therefore eventually you wouldn't actually be able to see it because the wavelength would become too long to detect through any means and the energy would become too low. So eventually the object would sort of slow down and, and redshift and fade away until you couldn't see it anymore. So technically an external observer will never actually see an object fall into a black hole.

And that's kind of interesting because it raises the question as to how a black hole forms in the first place, right? But if nothing can ever actually cross the event horizon, then how does matter accumulate into the black hole? And therefore, how does it form in the first place? So there's a little bit of a puzzle there, but it's important to realize that, remember, the Schwarzschild radius only describes the effects of gravity in empty space. Once there's actually matter there, then you need to use a different metric. The Schwarzschild metric doesn't apply. Now I'll talk more about the detailed processes of black hole formation when we cover

the kind of stellar life cycle and the astronomical aspects of black holes. Here we're focused more on the general relativistic aspects. So I won't talk about the particular life cycle and of the... and how they collapse into black holes. But the basic idea is that many black holes, so-called stellar mass black holes, form by gravitational collapse of large stars when they reach the end of their life. They collapse and become highly compressed, and all the material is pulled down below the Schwarzschild radius, forming a black hole. Now, as that massive object is collapsing, as long as the matter is outside of the event horizon, the metric in that region will not be described by the Schwarzschild metric.

Why? Well, because there's matter in that region, right? So it's not a vacuum, therefore the strutcheel metric doesn't apply. The structural metric only applies in a spherly symmetrical vacuum. Only when the black hole has already formed, when the material has collapsed below the event horizon, only then does the structural metric actually apply, and therefore only then will material not be able to actually cross the strachial metric, right? So the original formation of the black hole is not a problem in this sense, because there is no exposed event. event horizon, well, there is no event horizon before the black hole is formed. So the initial gravitational collapse is not precluded because there's no problem

about things passing the event horizon because the event horizon doesn't exist yet because the black hole isn't formed, right? And therefore the schvartchial metric doesn't apply. So it's sort of a bit weird, but hopefully that's not too confusing. The shrashchial metric will only apply after the black hole is already formed. Now, you can still ask the question, okay, but after it's formed, how does it grow? Because black holes can grow by accumulating matter from, say, a surrounding disk of dust or other debris that they accumulate. Well, there's another factor to bear in mind here, which is that although it's true that when you consider a fixed amount of mass, let's say 10 solar masses, 10 times the mass of our star,

when you consider a fixed mass, a given mass has a specific schwarzschwaite radius. So the schvartier radius depends on the mass of the object, the bigger the mass, the bigger the sphytural radius. As I mentioned before, the shavatural radius of Earth is far, far smaller than the structural radius of the sun, because the sun is more massive, right? The more mass the larger the spartial radius is. So as a black hole accumulates mass, its mass obviously goes up, and therefore a shavatio radius increases. So one way to think about it is as matter gets closer and closer to the event horizon, eventually it gets close enough such that it's sort of, for all intent and purposes indistinguishable from being part of the black hole. Remember, an external observer will not be

able to tell the difference because it will become too close to the black hole and too redshifted to distinguish from it. So it will kind of just look like there's a bunch of frozen lumps around a central mass, which is sort of observational indistinguishable from the black hole. And furthermore, once that matter has become part of the black hole, the black hole will actually increase in mass, and therefore its schvalchal radius will increase. So you can kind of think of it as if the event horizon extends outwards to encompass the new matter that's been falling into it. It's hard to offer a full account of this precisely because remember, the Schwarzschild metric is technically only applicable in a vacuum, in the complete absence of matter and energy.

And if you have a little bit of matter, even a small amount, then technically the Schwarzschild metric will need to be adjusted. So the precise details will never be exactly described by the shavartial metric because it's an idealization. It doesn't really exist in the real world because there's always some matter present. If there's matter falling into the black hole, then you'll need to add perturbations to that metric. So initially it seems a bit paradoxical as to how a black hole could form because if nothing can cross the event horizon, then how does the black hole exist in the first place? But when you factor in that the initial collapse is not of the massive object to form the black hole is not described by the schvartier metric at all because there's matter present, right,

that is collapsing that's before the black hole exists. So that's not an issue. And then when you also consider the fact that even after the formation of the black hole, the structural metric still won't be an exact solution because there is some matter. If there's matter falling into the black hole, then there's still some matter outside that will need to be results in some adjustments to the structural geometric. And then you also consider the fact that as the black hole accumulates mass, the thrushal radius actually expands, it increases. And so it's like a black hole grows outwards to encompass any new material that falls into it. So it's sort of a combination of these effects that allows the black hole to actually grow. Hopefully that kind of clarify some of

the confusions there about how black holes grow. Another thing to note is that the growth of black holes occurs very quickly if new matter falls into it. So for stellar mass black holes, it will take less than a millisecond for new matter to fall across, for new matter that's just outside the event horizon to fall across and then and then reach the singularity at the center. So it's extremely quick. And so from, from an external observer that they'll never be able to distinguish exactly this process anyway. So in other words, whether there's matter that is kind of just frozen at the event horizon versus actually past the event horizon and reaching the singularity, is not something that anyone can differentiate externally because the whole process is actually

very quick. So one last point that I wanted to mention is that for many years, although black holes were by the accepted to exist, there was no way to detect them directly. I mean, obviously they don't emit any radiation. There's nothing to see of a black hole. It's black. So it's sort of hard to know how you would go about detecting them. However, there are ways to observe black holes by the effects they have on their surroundings, though these are kind of subtle and required a long time to develop the tools to do this. Only in 2019 was the first image of a black hole released by astronomers. So this was constructed by, this image was constructed by observing the emission of energy from the gases surrounding the event horizon, because essentially there's a lot of gases that are circling inwards, because they have angular momentum, they circle inwards towards, as they are spiraling inwards towards the event horizon. And the friction between molecules and particles in that gas releases energy, and that energy can be observed if you, you know, detect. the right wavelengths and are able to extract the effects of dust and things that are obscuring it.

So that's very difficult to do, but in theory that is detectable. And so the first image constructed through those methods was released in 2019. You can see that if you Google black hole image. It doesn't look like very much. It looks kind of like a sort of a blobby, glowing donut, really. But the important thing is the hole in the center, because, I mean, there's lots of glowing clouds of gas in the universe. But what's really interesting is a glowing cloud of gas with a hole in the middle, not because there's something obscuring the centre, but because there is no gas at the centre,

because it's fallen over the event horizon, and no light can escape there. Light can escape from outside the event horizon, but not from within it. And so it's that image that clearly shows the presence of the event horizon as that black region at the center, which is what's really exciting about these photos. So the images aren't direct images of the black hole. that's not really possible, but they're images of the hot gas that's falling into a black hole

that clearly show the presence of an event horizon there. So that's pretty cool and I encourage you to check those out if you haven't seen those images before. Okay, so to conclude this episode, I just want to talk briefly about a few what I call unsolved problems of black holes, because black holes are still quite mysterious and there are still a number of issues that have to be addressed. The first that I'll address is hawking radiation. So hawking radiation is radiation that is emitted by black holes over very, very long periods of time, eventually resulting in them black holes losing mass and eventually evaporating completely. Now, it might be wondering, well, hang on, I thought the whole point of a black hole was

it doesn't emit any light or radiation, and therefore you can't detect it in any way, and there's no way for energy or anything else to be, to escape from a black hole. So how can now it be emitting radiation? Well, this is sort of part of what is surprising about the existence of hawking radiation. It's called Hawking Radiation because it was first proposed by Stephen Hawking in the mid-70s. It's quite a tricky phenomenon to get a grasp on. One important thing about Hawking radiation is that it's not like typical radiation where it's emitted by some object. Hawking radiation is emitted by the black hole itself.

For example, I was just talking about how there's hot gases that circle around most black holes and emitting radiation as they heat up and we can detect that, right? Hawking radiation isn't like that at all. It's not emitted by some hot gas or. or a star emitting radiation or something like that. Hawking radiation is emitted by the black hole itself as a gravitational object. It's not emitted by anything falling into the black hole or anything surrounding the black hole or anything like that.

It's a special type of radiation effectively. Another thing about Hawking radiation is, as far as I know, it's only emitted by black holes, or objects with an event horizon technically, but not any other type of object. So it's a special phenomenon that only exists as a result of the effects of general relativity and quantum mechanics. We don't have a full theory, a quantum theory of gravity, so there are still open questions about exactly how Hawking radiation works and how it fits into other things. It turns out that we know enough to be able to predict this specific phenomenon. And that's one of the reasons why explaining Hawking radiation is difficult, because what we have are a number of different derivations or proofs that hawking radiation must exist. given certain assumptions and the assumptions are all very reasonable so everyone is pretty

convinced at this point that Hawking radiation is real. But the problem is that we don't have a precise, at least as far as I've been able to tell, and I did a bit of reading on this in different sources, we don't really have a clear physical description of the actual process by which Hawking radiation is produced and emitted. Different derivations sort of emphasize different things. So Hawking's original derivation focused on a sort of a ray tracing method of of scalar field. if you know what that is, otherwise don't worry about it. But basically he showed that in the presence of an event horizon, there will be a finite amplitude of radiation of scalar field out to infinity,

that is out to a distant observer, which would therefore be observed by an external observer as energy being emitted by the black hole out from the event horizon. And in Hawking's original paper, he calculates the scattering of these particles that are being emitted as a result of Hawking radiation and shows that when you have an event horizon, you get scattering out to infinity and therefore radiation that can actually be detected by external observers. You don't get this when you don't have an event horizon. So this is why not all massive objects emit Hawking radiation. You know, planets and stars don't emit

Hawking radiation. There has to be an event horizon in order for the phenomena to exist. So that's crucial. So he showed all of that. And he gave an analogy, Hawking gave an analogy in this paper for what's happening here. He described it as if outside the event horizon, there are virtual particles, positive and negative energy particles, kind of like matter and antimatter, which pop into existence and usually cancel out and eliminate each other and then disappear in a burst of radiation. But the idea is that sometimes the negative energy particle falls into the black hole, and the positive energy, therefore, the positive energy particle is emitted and sort of turns into a real particle and can be observed by a distant observer as radiation, which,

which is Hawking radiation. You see this heuristic picture of Hawking radiation is repeated in many popular accounts. However, in his original paper, Hawking goes on to say, it should be emphasized that these pictures of the mechanism responsible for the thermal emission and area decrease, that's the decrease in area of the black hole, are heuristic only and should not be taken too literally. So he and others have pointed out that this is not really the mechanism. And it's interesting. As far as I know, Hawking doesn't really explain the mechanism in any very clear way. And I've seen other papers present different derivation. And they seem to give kind of different pictures about exactly what's happening. But what is very clear is that the picture that

Hawking gives is not really a very realistic one. And it's often not presented in a very accurate way. So this idea that there are virtual particles that spontaneously come into existence and then annihilate each other outside of the black hole. That itself is already kind of suspect because I'll discuss this in a future episode, but virtual particles aren't, they don't really exist. They're an abstraction from the calculational process of perturbation theory, which is used to calculate experimental predictions in quantum field theory. But virtual particles, they're not actually real. They're just sort of a fiction that is useful in calculations. So I don't think it's correct to say that virtual particles come into existence and then cancel out each other.

there are kind of fluctuations in the vacuum, but that's, I think it's not true to say that there are actually virtual particles there. Also, virtual particles can't really become real particles, and the idea that one of the virtual particles can fall into the black hole and reduce its mass because the positron falls in and then the electron is emitted, that that doesn't really make sense because obviously if that did happen half of the time, the positron would fall in and half the time the electron would fall in. So on balance, the black hole wouldn't lose. or gain mass, right? And that's not how it works anyway because both electrons and positrons have positive energy, and so this whole notion of a negative energy particle falling in doesn't quite

make sense either. Anyway, this is all very confusing, but the point that I want to make is that this analogy of the positron and the electron popping into existence and then one falling in, and because it's negative energy, it reduces the mass, that's just not really true. It's a very loose metaphor that I think is generally unhelpful. The best description that I've been able to find of the mechanism for hawking radiation is as a quantum tunneling process. So there have been a number of papers that have derived hawking radiation using quite a different method to what hawking used. So they get the same answer but using a different method. And the essential idea here is that the energy comes from the gravitational field of a black hole itself.

So the black hole has a whole bunch of energy stored in the gravitational field. There is no way for any object to travel through space and time. and get out of the event horizon. That's why black holes don't emit radiation, right? However, there is a way for the energy from the gravitational field itself to tunnel out of the potential well of the black hole and be emitted as radiation. So this is kind of how we get around this conundrum of radiation being emitted by black holes, even though no radiation can escape a black hole. It's because the radiation is not emitted by an object,

and it doesn't travel through space time. It's produced by the black hole, and in a sense, the energy tunnels out of a potential well and then comes into existence, or it's radiated out, outside of the event horizon. So I think the right way to think about it is the radiation doesn't travel through space to get out of the event horizon.

That's impossible. It is produced by the black hole with energy coming from the gravitational field of the black hole itself, and some of that energy is converted into radiation that is emitted and like begins to exist and that is emitted outside of the event horizon. So that, that radiation in that form was never inside the event horizon. It only exists outside, but the energy used to be part of the gravitational field of the black hole. It's gradually,

and it's a very slow process, gradually emitted as hawking radiation. How does the energy escape? Well, it's a quantum tunneling process. So I've talked about this in previous episodes about quantum mechanics, if you want to consult some of those. But essentially, because quantum mechanics describes the movement of particles as waves, so waves behave like particles, particles behave like waves in quantum mechanics. That means that it's actually possible for particles and energy to escape barriers that would be impossible classically. So they can kind of jump over potential walls that would normally be impossible for them to jump across. The barrier in question here is that produced by the gravitational field of the black hole.

As I understand it, there's a peak of the potential at the event horizon. So when In other words, if you had an observer inside the event horizon and they tried to escape, they would find that they were sort of climbing up a potential well as they moved towards the event horizon, like from the inside towards the event horizon. I mean, you can't actually do that. But if they tried to, right, that they would find that they kept coming up against this potential that was like trying to sail up a very steep wave, right? They would be sort of trying to go vertically up the wave and it would be impossible.

So they'd sort of fall down again and be pulled inwards towards. the singularity, right? So that's what is meant by the sort of potential well. Now, ordinarily you can't go up that very steep potential well and sort of get outside the event horizon, but a quantum tunneling process can do that. So you can't classically do that, but a quantum process can do that, at least sometimes, like with some probability. And furthermore, once a particle has sort of jumped over that initial potential well, it kind of goes down on the other side of the event horizon. Once you've escaped the event horizon. And the reason is, because remember, as we just talked about

earlier, imagine a piece of matter, I mean, it doesn't quite like that, but a piece of matter or a piece of energy, imagine it somehow jumped out of the event horizon. Once that happens, the mass of the black hole is less. It has been reduced. And so its event horizon shrinks a little bit. So now, if the particle initially was at the old event horizon, now the event horizon's moved in. Now that particle is actually outside of the event horizon. And so that's kind of a lower, a lower energy state than being at the event horizon. horizon. So this is the idea that the energy is actually able to tunnel across the potential barrier that's actually formed by the event horizon and the jumping particle itself. Lusely, you can

imagine as if the particle is able to jump, it's able to jump kind of right to the event horizon, but then it's kind of just out of the black hole, the event horizon moves inwards, because the black hole is shrunk a little bit. Now it's outside of the vent horizon, it can escape away. That's not exactly it, but it's loosely like that, right? So we should think of hawking radiation, I think, as a quantum tunneling process of energy that was previously stored in the gravitational field, gradually able to be converted into electromagnetic radiation and emitted because of the way that it tunnels outside of the gravitational potential, and then the event horizon shrinks, and it can then escape and propagate away. it will be measured by external observers. So hopefully that's a little bit clearer than other

accounts you may have read about Hawking radiation. Now one other thing I should mention here is that Hawking radiation will very slowly cause the energy of a black hole to be converted into radiation and then slowly reducing the mass of the black hole and eventually it will evaporate. But this takes a very, very long time, many, many times longer than the age of the universe. So it's an extremely slow process. And because it's such a slow process, it means that the actual amount of energy emitted as hawking radiation by stellar mass black holes is incredibly tiny. So a black hole of one solar mass will have a temperature due to hawking radiation. So you can kind of convert a hawking radiation to an equivalent temperature. It will have a temperature of only 60 nanofelvens.

So that's compared to the cosmic background radiation of 2.7 Kelvin. So if you're trying to measure 60 nanokalvans compared to about 3 Kelvin, we don't have the sensitivity to measure that kind of small difference in the background radiation. So there's little hope in any time in the near future of actually detecting hawking radiation. So at the moment it remains purely theoretical. No one's actually observed it and it probably won't be observable for a long time. But it is widely accepted because, as I've said, you can derive the existence of walking radiation in quite different ways with relatively simple and fairly widely accepted assumptions. Hawking radiation is very important because one of the key predictions of Hawking, Stephen Hawking and others studying Hawking

radiation is that black holes should emit Hawking radiation as perfect black bodies. So a black body is any type of body really which absorbs all of the radiation that's incident on it and then reemits radiation at a certain defined temperature. Each temperature has a particular distribution of rate of frequencies that it emits. So the sun, for example, is very close to a perfect black body, and its surface temperature is about 6,000 Kelvin. The important point here is that the distribution of frequencies emitted by any black body is exactly the same, as long as those two black bodies have the same temperature. So you specify the temperature of black body, and then you fully specify precisely what its distribution of emitted wavelengths will look like. This is a very

useful phenomena, and I've discussed it in previous physics episodes, the prediction that was derived from Hawking's derivation and from other derivations is that hawking radiation should be exactly thermal radiation. So in other words, a black body emits hawking radiation at, let's say, 60 nanokalvens, then the distribution of wavelengths of that radiation should look exactly like that of a theoretical black body at 60 Kelvin. So that was a very clear prediction of Hawking's derivation. That actually has very interesting implications, which I'll get to in just a second. So another important phenomena of black holes is something called the no hair conjecture. It's sometimes called the no hair theorem, though it's not actually a theorem, it's just a conjecture.

So I don't know why it's called that. So this is a postulate, which says that once a black hole has formed and sort of reached a stable condition, then there are only three independent physical properties of that black hole. that is mass, electric charge and angular momentum. Otherwise, all black holes are exactly the same. They only have those three different properties. Now, in this episode, we've focused on what are called Schwarzschwarzschild black holes, which are fully defined only by their mass.

So these are electrically neutral, non-rotating black holes that only have mass. There are other types of black holes as well that have electric charge and also that have angular momentum that rotate. They have different metrics that describe them, because you need to add on the effects of angular momentum and electrical. charge, but many of the features are pretty similar, and so we'll just hear focus on the Schwarzschild metric for simplicity. But the point is the no-hair theorem says that at most the black hole is completely described by just three properties. The mass, which is what we focused on,

that defines the Schwarzschild radius, for example, and then charge if it has one, and then angular momentum, if it has angular momentum. Otherwise, all black holes are completely featureless. They have no hair, this idea, right? Now, it's not known if this conjecture is true, but if it is true, if it is true, it gives rise to what's called the black hole information paradox, and this is the last thing that I want to talk about, because this is a very important unsolved problem in physics. The black hole information paradox arises from the no hair theorem and also is related to the thermal radiation, the thermal nature of hawking radiation. The problem arises because of this. Currently, all known laws of physics preserve the information

about physical states that produce them. So, for example, if we look at quantum mechanics, it's also true in general relativity, you compute the equations for predicting the outcome of a certain physical process. You can just as well run the equation sort of backwards and work out what would happen if everything ran in reverse, right? So you normally would predict what happens next, but you could as equally well start with the end state and work out what happened previously. The equations run in kind of both directions. And this is called unitarity. Essentially, it means that information is preserved about the physical state of a system as it changes over time. By the information we mean the detailed physical description of the micro-state of the system.

We might not have access to that information. If you burn a piece of paper, it's not like you still can find out what was written on the paper, but theoretically, if you could track down all of the burnt pieces and then measure all of the detailed properties down to the atomic level and everything, or even subatomic level, of fragments of that piece of paper and put it all together with all of the equations of motion and so forth, technically all the information about what was written on the paper would still be preserved. It wouldn't be accessible in any macroscopic way, right? We don't mean information in that sense, but in a microscopic kind of physical sense,

the information is still there. It didn't disappear. It just became inaccessible, right? So all known laws of physics currently obey this unitarity principle. However, if the no-hair theorem or no-hair conjecture is true, and also if hawking radiation is truly thermal, that means that two black holes that formed through different methods, but still have the same mass, would have nothing to distinguish them. There's no properties that you could distinguish between them that if they have the same mass and, let's say, the same charge and angular momentum, their hawking radiation would also be the same because it depends only on the mass, and there's no other information in hawking radiation other than just the temperature of the body, which is a 10 bytes mass.

And so therefore, there's no way for information to be stored anywhere about the initial material or processes that led to the formation of those two different black holes. So there would be information loss, right? Different initial states, but the same final state and no way to distinguish between what gave rise to that final state. So if Hawking radiation is truly thermal and the no-hear conjecture is correct, then there seems to be information loss. But many physicists regard this as problematic because no-no-no. physical mechanism can produce that. All known physical processes are unitary. So what's going on here? Where does the information go? Or what's happening? There seems to be a paradox, a puzzle here. Now, there are some physicists who've kind of countered this, and it sort of seems to me that there's

a potential issue here as well, because wave function collapse in quantum mechanics is also a non-unitary process. It destroys information because once the wave function collapses, it sort of selects the state that the system ends up in, then there's no way to work out what happened before that. The information is lost about the superposition that existed prior to collapse. Although it's true that not every interpretation of quantum mechanics regards collapse as a real thing. So if you don't think wave function collapse is a real phenomenon that actually occurs in reality, if that's just part of our description of reality, then you might not think that

this is a way function collapse is a violation of unitarity. So that seems to be a point of dispute. As far as I know these days, most physicists think that the information is preserved by black holes. So this would mean that the no hair conjecture is false and that hawking radiation is not true thermal radiation. So one possibility here is that hawking radiation contains information about the physical state that gave rise to the black hole. And in order for that to be true, it would have to be the case that hawking radiation isn't true thermal radiation, but actually has extra like deviations and perturbations and things

that contain the information. But no one seems to really know exactly how that would work, or where in Hawking's derivation, where he proved that the radiation would be thermal, where exactly he went wrong. It's one thing to say that, well, there must be deviations in there, but no one's been out actually proved to my knowledge

of where they would come from. So that's still an open question. Now, another possibility, or I think they could go together, would be that is that the information about the formation of a black hole and physical states that led up to it would be encoded on the surface of the event horizon of the black hole in some form. Now this relates to a result called the called ADS-CFT correspondence. And I won't really try to explain this here because it gets well beyond what I can talk about here.

But basically this theorem is a way of explaining how you can relate the description of general relativity to descriptions of quantum field theory, which ordinarily are hard to conundering. connect to each other in a way whereby the information about a volume is described or encoded on the surface of the region of that volume. So in the case of a black hole, it will be on the surface of the event horizon surrounding the black hole. This principle indicates that it may be possible to encode the information about the system that formed the black hole on the event horizon of the black hole.

That being said, this theory doesn't, the ADS CFT correspondence, doesn't directly prove this. It's more an illustrative result about how it might kind of work, but it's still, it doesn't directly apply into black holes. It's more of a proof of principle, proof of concept, and not an actual demonstration that it actually applies here. So from what I understand, people like Stephen Hawking have been convinced that ADS-CFTA correspondence was strong evidence that there is some kind of way of encoding the information about the formation of a black hole on its surface, which would then be omitted.

in the Hawking radiation as perturbations in the thermal spectrum there, thereby preserving the information about the formation of the black hole and resolving the black hole information paradox. But at the same time, these are all kind of heuristic arguments. It's like, well, we kind of think that it could work this way, but there's no actual proofs here, and there's no actual experimental evidence, certainly either way. So I think this is still an open question, and there's still a lot that we don't understand about exactly how

Hawking radiation works, exactly how information might be stored in or kind of on the surface of a black hole. And what it comes down to is exactly what the relationship is between a sort of a quantum field theory description of a black hole and the general relativistic description of the black hole because ultimately we don't have a full quantum theory of gravity. And I think perhaps that might be required to really resolve these questions. Though we will see. But anyway, that brings me to a conclusion. I hope you found this episode interesting. It's certainly a deep topic. thanks very much for listening if you want to support the show you can do so in a number of means you can make a one-off donation via PayPal to my email address fos12 at gml.com that's fodesds 1.2

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