Science & Technology Q&A for Kids (and others) [Part 140]

Hello, everyone. Welcome to another episode of Science and Technology. Q and A for kids and others. I'm still not back in my natural habitat, so not as quite as well set up as usual. But see what we can do here. Let's see. I see a bunch of questions that came in. There's one from Anonymous. Can you explain dark matter in layman's terms? Okay, let's see what we can do with that. So one of the things that. So if you look at a galaxy, for example, what holds a galaxy together? Well, it's gravity. There are gravitational attraction between the stars and the galaxy. And like, our galaxy's been around. Oh, I don't know. How long has it been around? Probably, I'm not sure, actually, it's known. But certainly five, six billion years at least, maybe longer than that. And the galaxy, our galaxy is a spiral galaxy. It's a fairly big galaxy, and it's rotating around, and it's rotating at a certain rate. Takes a few hundred million years, I think, to do a rotation. And the thing is not flying apart because it's held together by the force of gravity. And it's like the stars on the outside of our galaxy are rotating around the center, and it's like they're orbiting around the center of the galaxy and they're being held in their orbits by gravity, just like the Earth is being held in its orbit around the sun by gravity. So that's kind of how the thing is set up. And you might say, well, gosh, we know how gravity works. We have this formula, this inverse square law of gravity, that the strength of gravity is proportional to 1 over the square of the distance between things and so on. We, we know how all of that works. So we should be able to work out how fast the galaxy can be rotating and still be held together with gravity and so on. We should be able to just work all that stuff out. Okay, so people starting 100 years ago tried to work those kinds of things out, and it didn't work out. Right. They said there's this amount of mass in the galaxy, and they can say, well, there's this amount of gas because, look, we can see all the stars in the galaxy. We know what the typical masses of stars are. The sun is a fairly massive star. On the scale of masses of stars, there are a lot of much smaller, fainter stars in the galaxy. They're also much bigger stars than the sun. But the sun is kind of on the upper end. But we know There are about 100 billion stars in our galaxy, and we know roughly what the kind of number of stars is in different regions of the galaxy. And we should be able to work out kind of how the amount of gravity produced by those stars and what that means for the possible rotation of the galaxy. And in particular, what's called the rotation curve, which is how fast are things rotating as a function of how far out from the center of the galaxy they are. Now, one thing that has been known in more recent times is that at the center of a galaxy like ours, there's essentially a galaxy giant garbage dump, a very big black hole, maybe 100,000 times the mass of the sun. That's gradually sort of eating up stars and so on. And if we were near the center of the galaxy, we're not. We're on the outer edge of the galaxy. It's kind of probably a good thing we're not near the center of the galaxy. Because this big garbage dump is coming through and eating up stars near the center of the galaxy. Another feature of being near the center of the galaxy Is the density of stars is much higher. So in our night sky, we see a certain number of stars. We see a certain number of stars that are bright because they're fairly near us. If we were at the center of the galaxy, our night sky would be much more lit up, so to speak, with stars. There'd be just a lot more stars nearby. But in any case, from all of this, one could work out sort of what's the rotation that you would expect in the galaxy. Relative to the distance from the center of the galaxy. And by golly, it didn't fit. Well, why not? Well, the idea is, well, there must be something else in the galaxy that we're not seeing. Some kind of dark matter. The matter that is in stars is sort of the luminous matter, the matter that's producing light. We can tell it's there because it's stars and they're shining. But there must be something else, some kind of dark matter that is not visible in the galaxy. So that was kind of the first tip off. Then there are other places where one can kind of see that there's something we don't see that still has a gravitational effect. Another one. There are two galaxies that collided with each other. And they kind of went through each other. But you can tell from the motion of the things that there's this sort of blob in the middle. Where there doesn't seem to be anything there. Where it's still having gravitational effect. Part of the way you can tell that is by this Phenomenon of gravitational lensing. So when one of the features of gravity is gravity attracts everything, everything that is affected by the force of gravity. The way one thinks about gravity in general relativity, the kind of standard theory of gravity Is that gravity causes a distortion in the structure of space time. Gravity causes. You can think of it this way, that without gravity, the shortest path between two points is a straight line. And if you shine a laser beam or something, the laser beam will always follow the path of the shortest path in the direction it's going. So it will just be a straight line. But if you introduce gravity, gravity produces a curvature in space time. And that curvature, just like if you were to say, I've got this curved surface, I've got this dip in the surface, and I say, what's the shortest path from this place to that place, including the dip? Well, that path will not be straight anymore. That path will deflect as it goes through the dip. That will correspond to the shortest path. And so it is when there's gravity that the shortest path is not the straight line path. And in fact, we can think of gravity as being our way of describing the fact that the shortest path is deflected. So we can describe that as the force of gravity. We can also say, oh, space time is curved and so. Or space is curved. And so that's why, even though these things really are following shortest paths, the shortest path is no longer a straight line. Well, that phenomenon of the way that gravity works affects not only, you know, spacecraft in orbit around the earth, or planets or stars in orbit around this or that. It also affects light. And when one goes. When one has a massive object, maybe a planet or the sun or a galaxy, Light is deflected by the force of gravity associated with that object. And so that means, for example, light traveling around the sun is bent by the presence of gravity associated with the sun. And that was observed first observed in 1919 when there was a total solar eclipse. No doubt we'll be able to see it again in the April 8th of this year's total solar eclipse. If you look at a star that's really close to the sun, that is sort of obviously behind the sun because the star is far away, and the disk of the sun is covered by the moon. And so you get to see the stars because the sky is this burst of light from the sun. You get to see those stars. And so you look at a star that's very close to the edge of the sun as you see it in the sky. And now with the moon blocking the sun. You kind of look at that star and you ask the question, did the presence of the sun move the effective position of that star? And what was observed in 1919, probably not as dramatically as people claimed at the time, but doesn't really matter, was that, yes, the. The effective position of that star has changed because the light coming from that star is bent as it went around the Sun. And so that phenomenon of light is sort of pulled in as it goes around a massive object. That is the phenomenon of gravitational lensing. And when light goes through a converging lens, like a convex lens, a lens that bulges out, what happens is that light from that's coming in, lots of parallel rays of light are coming in. And that lens will make those rays of light converge and, for example, come to a focus. And that's how, for example, when we look at a distance and we are trying to look at things that are far away, the way that those things get focused onto the retina and our eye is that the lens in our eye is a converging lens. And it does lensing, so to speak, to make light be bent and converge together. Well, the same thing happens with light going around a galaxy that there is a lensing effect. That sort of stars from or galaxies that you see behind that galaxy, the light from them is sort of pulled in. Now, the really dramatic thing is that you can end up with something, and these are really cool pictures. You can end up with something where there are multiple images of one galaxy. There's a galaxy behind some galaxy here. And the light is bent. And it ends up that where we are, the light is bent going one way around that galaxy in the middle and going the other way around the galaxy in the middle. And you end up with two different images of the galaxy behind. So that's a way. And you can kind of measure the amount of gravity in that galaxy in the middle by seeing how much of those images, where those images end up, how much gravitational lensing is there. So it's another way of kind of measuring mass out there in the universe. Another thing that has where you can kind of see the effects of kind of mass is from the cosmic microwave background. Actually, we don't need to talk about that right now. Just say there are places where there's stuff that produces gravity, but it isn't. We don't see it as luminous stars. That's what dark matter is. What do we know about dark matter, what it might be? Well, it's kind of the very fact that it's called dark Matter means that we think there's something there that is some kind of matter, like, thing. It's not completely obvious. That's correct. In fact, I suspect it isn't correct. I suspect that dark matter isn't matter at all. It's some feature of the structure of space that we don't yet understand. Now, if you think it's matter, you say, well, what kind of matter? What might it be? Might it made little tiny black holes? Might it be little particles that are incredibly light relative to the electron or something? Might it be. What might it be? And there are some constraints that people have found over the years. It can't be particles that are too light. They may have to be, you know, it can't be particles that are, you know, a billion trillion times lighter than an electron, for example, because if it was, it would have different effects in the early universe. It can't be things that are too massive in this way or another way. Another thing is it can't be anything that interacts with ordinary matter because we don't see something where there's actual sort of processes of interaction. We don't see, for example, particles that, I don't know, electrons or something like that going through these regions of dark matter and scattering from them and so on. So the assumption is that if it's. That it must be some kind of matter that interacts incredibly weakly. And there are all sorts of experiments that involve trying to find particles. Different kinds. Axions are one popular possibility. There are a variety of others that are particles that occasionally interact. If they never interacted at all except through gravity, it'd be really hard to detect their presence. But the idea is, if you have a big enough tank of water or something, or you're looking at some. Some region underneath the ice in Antarctica or something where everything is very still, then just occasionally, maybe one of these dark matter particles would hit a particle of ordinary matter and interact with it. And that particle of ordinary matter will be kicked by the dark matter particle. And it would be zooming off in some direction, and you'd be able to detect that. And a typical way you detect it is with a thing called Cherenkov radiation, which is something where if a particle is kicked so that it is going faster than light goes in that medium. So in a vacuum, light goes at the standard speed of light. But in something like a piece of glass, light goes 1 1/2 times slower than it does in the vacuum. But you can have a particle that proton, electron, whatever else that was sort of pushed hard enough that it's going faster. It's going close to the actual speed of light in the vacuum and faster than the speed of light as it exists in a piece of glass or water, which will be 1.33 times slower than the speed of light in a vacuum. And when you have a particle that's going faster than the local speed of light, it has this effect of producing essentially a light shock front, kind of like a sonic boom for light. Like when you have a plane going faster than the speed of sound or all the kind of air, the sound waves kind of pile up on the plane and you get this kind of sonic boom. You get the same kind of thing with light. You get this kind of cone of light that gets produced. That's called Cherenkov radiation, typically bluish light. And you can detect that potentially, you know, if you've got this big, big area of water deep under Antarctica or something. And there's. There's nothing. Well, actually, that, the Antarctic case is an actual tank of stuff that was put there. But if you. If you have in the ocean, various places in the deep ocean. I think there was one near Hawaii. I'm not sure if that's still being experiment that's still being done, where usually there's nothing that produces light, but you end up with something that suddenly produces this little flash of light. So in any case, people have been looking for these things. There are a large number of experiments that have been done. Nobody's ever found anything. So my own guess is that one's looking in the wrong place. And that dark matter isn't in fact, matter at all. It's not a sort of unseen particles. It's rather some feature of the structure of space. Not quite sure what feature of the structure of space. I have some ideas. Let me give an analogy, though, which I think is perhaps useful in. One of the things that really mystified people in the 1800s was what heat is. You have a piece of material, a piece of metal or something like this. You heat it up. We know that heat can flow from one piece of material to another. You can have a hot piece of metal next to a cold piece of metal. The heat will flow from the hot piece of metal to the cold piece of metal. Well, what is the heat? So people said it must be a fluid. It must be something like water that somehow gets into the pores inside the metal. And when you put the hot metal next to the cold metal, that, that fluid that represents heat, they called it caloric fluid. When that fluid is, there's more caloric fluid in the hot metal than there is in the cold metal. So it will somehow flow from the hot metal to the cold metal. And there were equations, mathematical equations that described how that worked. And those equations actually look very much like the equations of fluid flow. So people said, yep, it must be a fluid. Well, nobody ever found caloric fluid. And turned out it wasn't a fluid after all. What is heat? Heat is the microscopic motion of molecules. Heat is the. When you heat something up, what's happening is that the individual molecules are bouncing around more quickly. That's what temperature, for example, is the average rate at which those molecules are bouncing around. So heat is really a feature of the microscopic structure of the material, not some kind of add on thing of, oh, we've got this caloric fluid. Okay, so what about dark matter? Well, is there an analogy there? Well, it depends what you think space is. Kind of the view of space that people have tended to have is that space is just sort of a background. You just put things in different places in space. Space isn't a thing. And you might have said that about water or something. It's just this background that can flow from here to there. But actually, we know space is made of stuff. It's made of molecules. It's made of lots of little molecules bouncing around, hitting each other. The large scale effect of all those molecules is to have something which behaves like water, but there's an underneath, there's a bunch of molecules there. Well, I strongly suspect that the same is true of space. And that space is kind of made of some discrete elements that we can think of as being sort of related to each other by some kind of big network. And that that network is progressively changing. It's not like a mechanical collision between sort of atoms of space. It's more a more abstract kind of process that involves the relations between atoms of space gradually being updated according to some rule that we can think of as kind of a computational rule. And so we built this whole model of physics that seems to work remarkably well that is based on kind of this idea of a discrete space and sort of these rules being applied to this network that represents discrete space. And if you ask, well, you know what emerges, it's kind of like when you have a bunch of molecules bouncing around underneath. What emerges is something of fluid like water. What emerges as the structure of space? And the answer is gravity and the things we know about space. So that's very encouraging. But now the question is, well, are there features of space that we're not kind of taking into account when we look at this kind of large scale structure of space that corresponds to things like gravity. Well, what else is there about space? One of the things I kind of suspect is that there is what we might call space time heat, that is that the microscopic structure of space associated with exactly how is this network set up, exactly what kinds of updates happen exactly where in the network. Those things don't have an effect on the sort of overall sort of structure of space, but they probably do have some effect. They have an effect that probably changes the amount of gravity that's there, the sort of base amount of gravity that's there, and possibly does so in a kind of way that's a bit like dark matter. So it would be kind of an ironic thing if it turns out that dark matter is actually a feature of the microscopic structure of space, just like heat turned out to be a feature of the microscopic structure of materials. And my somewhat guess is that that's what dark matter is. It has nothing to do with matter of an ordinary kind. It's rather microscopic structure of space. And well, what is matter? In this view, a matter something like an electron or a photon or a quark or something like this. In this view of the universe, where everything is this kind of giant network of atoms of space, a particle is some kind of big sort of persistent object formed from pieces of this network. Much like in a fluid, you can make a little eddy that's a bunch of fluid going around that's formed from lots of underlying molecules. But there's kind of this large scale kind of piece of the fluid that has a certain persistence. The eddie moves through the fluid without change. So similarly, that's what we suspect particles are in our models of physics. So if dark matter was particles, it would be a bunch of sort of persistent large scale, large scale relative to the sort of atoms of space, large scale kind of excitations, motions inside the structure of space as opposed to space time heat, which would be something much more microscopic, much less coherent, not formed into big blobby particles, so to speak. So anyway, that's my guess about how it works. And we'll see maybe this year we'll get some sort of experimental implications of that and we'll be able to actually tell people use this telescope, do this and you'll see this weird, weird phenomenon. And that's what dark matter is. I will say that another related thing, a bit related to spacetime heat, is the idea of dimension fluctuations. We believe we live in a three dimensional universe with three dimensions of space, you go forward, sideways, up, down. But in fact, in our models, this network that represents space doesn't have to be exactly 3 dimensional. It could be 3.01 dimensional. What does that mean? It means if you say, for example, if in three dimensions you say you start from a point, you say, how much stuff is there? If I go out a distance, a certain distance from that point? Well, you know, if you know the formula for the volume of a sphere, it's 4/3 PI r cubed and where R is the radius. Well, if space isn't 3 dimensional, but it's 3.01 dimensional, the corresponding formula is the amount of stuff is R to the power 3.01. And we kind of suspect that formulating kind of. I talked about gravity as being associated with the curvature of space, that actually you can as well think about gravity as a feature of small changes in effective dimension of space. And that that's sort of an equivalent, an alternate view of what's going on there. And that difference between sort of the view of it in terms of curvature, the view of it in terms of dimension change might have something to do with sort of the changes to sort of the, the way gravity works. That would be, that would account for things like dark matter, we don't know yet. Still, still up, up in the air. Just trying to. Something we'd like to figure out. Let's see. Okay. IP asks, could dark matter just not be interacting with light and photons? Yes, that's. I mean, many of the models of dark matter just say it's particle. Well, I think it can not be interacting both with light and with other kinds of things because there's lots of gas in the interstellar medium, in galaxies and things like that. And you would expect to see if there was sort of interactions between that and dark matter, you would expect to see some kind of effect on the gas. And that isn't seen. Let's see. Kathy asks, if our universe is accelerating faster than the speed of light, does that mean we cannot see anything past the speed of light? Well, so this is a complicated mess. This question about sort of just how fast is our universe expanding and how does that all work? One of the things that there are sort of equations that describe the effective size of the universe, but it's not like when you hit the edge of the universe, you fall off. It's more complicated than that because the universe is expanding. But let's say we try to reach the edge. Well, at most we can go at the speed of Light. And there's sort of this question about, do we reach the edge? And are we reaching the edge at the point where the universe has sort of expanded past that edge? In the end, it's a sort of thing that you can best describe with the kind of math associated with kind of the sort of the. The structure of space. It's a weird thing because you're describing all of space and that's the only place you can go. So it's not like you'd say, well, there's an edge and you reach that edge and there's nothing there. That's not how things work. It's more like if you're on a sphere, for example, and you say, let's just keep going. Oh, do I reach the edge of the sphere if I'm a little creature on the surface of the sphere? No, I don't. At worst, I go back around and come back around the sphere, but I never reached the edge of the sphere. That's just not a concept. That is a thing in the case of the surface of a sphere. So in the expansion of the universe, one of the questions is. So at the beginning of the universe, there was a. We think of it as a Big Bang. The universe just started going outwards, started expanding at a certain rate. Well, one feature is the universe has stuff in it, and that stuff has mass, which produces gravity. So what you might expect is that the universe would expand out for a while and then the force of gravity would like, pull it, pull it, pull it back. And eventually it would get. The expansion of the universe would reverse and the universe would start coming back. And eventually there'd be a Big crunch to go along with the Big Bang. And a long time from now, so to speak. That's one possibility. Another possibility is there's kind of just enough sort of outward motion associated with the Big Bang that just keeps going. And eventually it's just sort of coasting out at a certain rate. That is basically the speed of light. Another possibility is that there's something. Gravity is a force that just always pulls things together. Imagine there was anti gravity. Imagine there was a force that could push things apart. Well, if that was happening in the universe, the universe could be perhaps accelerating in how fast it was expanding. Well, there's this weird idea, dark energy. Dark energy is basically matter with negative mass. And matter with negative mass has the feature that instead of producing an attractive force of gravity, it produces a repulsive force of gravity. Now, we don't know that matter with negative mass exists. We don't know that anything with negative mass exists, it's probably not matter. People say, oh, that's not really matter because matter is something with positive mass, but something with negative mass. Now it gets very tricky because when you say there's a thing there that has negative mass, well, okay, when you say there's nothing there, turns out in theories of physics for the last hundred years, it's not been possible to have a situation where there's truly nothing there, where there's truly a quotes vacuum with absolutely nothing there. And that's because there are always quantum fluctuations in quantum mechanics. One of the features of it is to say you never know exactly what's going on. There's always multiple possibilities and there's always certain probabilities of different things happening. And so it is with knowing. You say there are no particles here. Well, quantum mechanics says you'll never be able to say that. You'll always have some uncertainty in the number of particles. And so that means that you've always got kind of in what would normally be the vacuum. You've got this sort of bubbling of particles. They're virtual particles that are coming into existence. Particles and antiparticles. They're coming into existence for a short time, then getting destroyed. And this is happening all over the universe all the time. I have to say that in our models of physics it's a little bit cleaner what's happening. But that's the traditional view. In our models of physics, the structure of space is this network. And in a sense, what holds together that network is the fact that every piece of the network is continually being updated. So if you say I've got two atoms of space and are they continuing to be related to each other? Are they continuing to even be connected in space? The answer is what makes them continue to be connected is that they're constantly being rewritten, that the sort of network around them is constantly being rewritten. Those very atoms of space are constantly being rewritten. But you can always say, can I get from this place to that place? Well, yes, because there are these rewrites that are happening that are kind of making this change, this change, this change all the way along some line so that you can always get from one place to another. And that's kind of what keeps the universe together. It knits together the structure of space. So in our models of physics, most of the activity of the universe is about knitting together the structure of space. That's most of what's happening in the universe. Everything that we care about is a tiny little Piece of fluff on top of all this activity, knitting together the structure of space. So, for example, it might be, let's say, one in 10 to the 120 or something. So one in one part in a trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion of what's happening in the universe is all the stuff we care about, and most of it is just knitting together the structure of space. So, in sort of traditional view of physics, something rather similar happens, except it's a bit messier and a bit harder to understand and doesn't really quite work. It's this virtual particle thing, these vacuum fluctuations. One of the things that's always been embarrassing about those vacuum fluctuations is that the vacuum fluctuations are associated with a certain amount of mass. If there was that much mass in the universe. When you sort of separate out the theory of gravity from the theory of quantum mechanics, you get the implication that all this quantum mechanical stuff that's making this effective mass at every place in space would also produce gravity, and that gravity would be so big that it would curl the universe up into a tiny ball. And, well, we're here. The universe isn't curled on that tiny ball. So something went wrong with that picture. And in our models, nothing like that happens, because all those little effects which in quantum mechanics will be in standard quantum mechanics, will be interpreted as vacuum fluctuations. Those are all part of this phenomenon of how space is knitted together. And so the fact that. And gravity is part of that, too. So it's not that there are two separate things where one has a bizarre effect on the other. These are two things that all go together. That's all associated with knitting together the structure of space. So, okay, so we've got space, and we've got this vacuum fluctuations. We've got this knitting together of the structure of space, and we've got. So now the question is, well, what happens if there's a piece of space where there are fewer vacuum fluctuations or a piece of space where there's somehow less knitting? Well, then that would correspond to a piece of space that effectively has less mass and relative to the rest of space. If we say the most of space, we say, oh, it has zero mass because it's just what we call vacuum, then the places where there's kind of less activity will appear to have negative mass. So that's kind of a picture of how you would get something like dark energy. Now, you could have other phenomena that lead to that. There's kind of a big hack in current physics to do with the Higgs particle And the origin of mass in current physics. Oh heck, I can explain that. I suppose so. Okay, this is a bit of a. This is the sideshow of how does current physics account for the fact that particles like electrons and so on have non zero mass. Photons have zero mass. And this is going to be related to, you'll see how it's related to this whole question about negative mass things and so on. Okay, so here's how it works. When you make a theory of physics in the most elegant mathematical approach is a thing called the local gauge theory. And essentially what happens there is you say, you're kind of saying at different places in the universe, you say that things are set up in. You're kind of describing what's happening in a way that may be different in different places. So for example, if you say, what's a good example here? If you say at every place in space, I'm going to say there's a quantity that is somewhere on a circle. So it's at some angle on the circle. And we say it's 30 degrees at this place in space. It's 60 degrees at that place in space. It's 20 degrees at that place in space. Well, at any local place in space you just say you don't know what angle it's at because you have no way to measure it. All you can tell is it's at a different angle in this place in space than it is in that place in space. And so this question about how do the angles relate between what it's like in this place in space and what it's like in that place of space, the relation between those angles and is associated with a thing called the connection. And the connection is kind of what tells you how does the angle in this place connect to the angle in that place. So we took all of space and we just said we're putting down these different angles in different places. Now, this is not quite as abstract as I'm making it sound because when we think about electromagnetism, you can say electromagnetism is associated with essentially an arbitrary choice of what amounts to an angle. How does this work? Okay, a little bit of kind of in electromagnetism there's this idea of a voltage. What is the, we say in standard electricity and so on, we say this place is at such and such a voltage, that place is at such and such another voltage. When you have a 9 volt battery, what it means is that the difference in voltage, the difference in electric potential between the two poles of the battery is 9 v. Volts is a measure of. You can think of it as a measure of electric potential. And it's saying there's a 9 volt difference between the positive pole and the negative pole. And that 9 volt difference, that voltage difference is something which, for example, pushes electrons in one way or another. So it pushes electrons through wires, makes an electric current. That's what the battery does. Okay, but the choice of what the, the choice of what, what we say, we say, okay, one side of the Battery, it's at 712 volts. Well, the only thing we know is the other side of the battery is 9 volts different from, from that. We don't know that the absolute voltage doesn't really mean anything. The only thing that means something is the voltage difference. And so that idea kind of translates itself into saying, well, at every point in space you could say there's a voltage. But all that matters is the difference of voltages between these different places in space. And the question is sort of how does one place in space kind of, if you, you know, how, how do you relate the sort of the voltages effectively in different places in space? And one, one way to think about that is let's imagine that you have this voltage arranged in space and suddenly you change. Yeah, I should say when there's a, when there's a difference in voltage, you can represent that difference by an electric field and you can kind of say when there's a. Yeah, roughly what's going to happen is we've got one place where there's lots of voltage. Another place where there's low voltage, there's an electric field that kind of is produced between those two different, Two different places. Well, let's say we make a change in the voltage in one of these places, and that's got to change the electric field. Because when we change the voltage here, it means the difference between the voltage here is different. And now the electric field will change. Okay, so we make that change somewhere. And now the question is, so the electric field's got to change. How fast does the electric field change? We make a change here and somehow 10 miles away, eventually the fact we made this change here has to somehow propagate out to have an effect 10 miles away. Well, it doesn't do it instantaneously. How does it do it? Well, it does it at the speed of light. It does it. There's a kind of wave of change that propagates out. That wave of change is essentially an electromagnetic wave. And the fact that there has to be this sort of wave of change is the reason that photons exist in this picture. It's the reason you have to have something that's sort of a carrier. You can think of it as a carrier of change where it's sort of telling the rest of the universe that oh, there was a change in this voltage. And so it is a necessary feature of this idea of oh, there's a voltage and it's sort of arbitrary. That's sometimes called the gauge. And you can make these different choices of gauge. The difference between gauge in different places has a measurable effect. It's something like an electric field because there's this difference in voltage. But then the question is, as you, as you make a change in that gauge, how does that change engage? Kind of how does the rest of the universe get to know about that change? And the answer is you have to have this is the so called gauge fields, like photons that are kind of the carriers of that change. Boy, that's a pretty abstract set of things. And I actually think I did okay at explaining that because that's a, you know, for those who might be interested, there's this whole notion of a covariant derivative. When you say there's a change that's happening, you can say there's a change in the field and well, maybe it's not worth saying this. There's a change in this electric potential. And well, okay, you have to kind of include this change in conventions and that's associated with this connection thing. But anyway, bottom line is a connection field connection. This sort of carrier of change is something like a photon. One feature of those things that carry change is they have to have zero mass. That's just a feature of the way the math works out is that that sort of wave of change, it always goes at the speed of light and it always effectively has zero mass like a photon. However, in that kind of setup, one would start concluding, well, everything in the universe has to have zero mass, but whoops, it doesn't. Electrons don't have zero mass, quarks don't have zero mass, et cetera. And so how do you get mass into the picture? Well, it turns out that the mathematical theory of zero mass gauge fields is very elegant. It has all kinds of nice features. There are all sorts of infinities that often show up. They don't show up quite the same way. There you can get rid of them. So called renormalizable theory. It's all good. But then there's this ugly thing that Actually, in the universe, there's mass, okay? So there was this chap called Peter Higgs, along with a couple of other people who in the early 60s, had this idea, kind of a hack, actually. And the idea was imagine that you have something which would normally be zero mass, but that thing interacts with something else. And it interacts with what we'll call it the Higgs field. It interacts with this kind of background thing, and it keeps on, you know, it's just going happily along, but it actually is always being kicked by the fact that there's a sort of a background field. Okay, now, normally, if you just have, let's say, electrons and photons, they interact with each other. Just aren't very many electrons around. There aren't many, very many photons around. So most of the time, it's like, oh, yeah, electron hits a photon or whatever, and the electron is deflected, and it's all good. And it just sort of happens once. And you only have to take account of it very rarely, so to speak. But let's imagine instead in this Higgs field, it's a particle as well. But let's imagine that the whole universe is full of Higgs field. That in some way, there's a kind of. It's like there's a condensate. Like, instead of there being Higgs particles in a gas that just bouncing around very, very sparsely, the. Let's imagine that. It's like a liquid that it condensed, that there's just Higgs particles everywhere. They're just, you know, the whole universe is full of densely filled with Higgs particles. Okay? If that's the case and your little electron, which was massless, keeps interacting with these Higgs particles, it will keep on kind of. It will keep on kind of getting slowed down by, oh, I had to interact with this Higgs particle. Oh, it kind of bounced me this way or it bounced me that way. That effect of that process will end up being giving the electron an effective mass. It gives the. It's. It's the. The fact that there's this Higgs condensate, this kind of everywhere will make the electron have a mass. So you went from this thing where the electron really has zero mass, but because there's this Higgs condensate everywhere, which it keeps on interacting with, it's slowed down by that, and it effectively has a non zero mass. Okay, so then that's all good. But the question is. And that allows one to sort of say, well, where does mass come from? That's where mass comes From. Okay, well, how did there come to be this Higgs condensate thing? Well, there. You have to assume that there's a self interaction between Higgs particles, that the Higgs particles, like the fact that if you have particles in a. In, I don't know, a water, for example, the. The molecules of water interact with each other. They're kind of attracting each other. And that's how you get liquid water, as opposed to just water molecules floating around and just not really paying attention to each other. The fact that they form this condensate, they form this condensed water where everybody's gotten together, that's a consequence of the fact that they have self interactions. And I think that the thing that. So there's this assumption that the Higgs field has the self interaction and that things work out just so that there's a certain condensate, certain density of Higgs particles in the universe. I always thought this was just a total hack. But as soon as you know that there are Higgs particles and they form this condensate and so on, that means you should be able to get a situation where there's just one isolated Higgs particle somewhere. And eventually, in the Large Hadron Collider, Higgs particles were discovered as isolated Higgs particles. And they have a definite mass. 120 times the mass of the proton, I think, roughly. And right. I think so. 125, I think. I used to. When I was a kid, I was into the masses of all the elementary particles. I could quote you all those masses that existed at that time, but I haven't paid attention as well since then. And so I'm a little less up to my precise quoting of the numbers for these things. But any case, Higgs particles were discovered. So some part of this picture is like, we know there's something there. The fact that this condensate forms is just a hack. It's just, let's assume these particles interact with each other. It will form this condensate. Okay. One feature of that is that means there's a thing throughout the universe. The universe is full of Higgs particles. Higgs field, at least. And when I say particles, a particle is more like an isolated sort of excitation, sort of. It's like the eddy in the fluid, whereas the sort of the underlying field is like the whole fluid itself. So in any case, the thing that happens is a little bit like liquid water. If you heat it up, eventually there isn't a condensate anymore. Eventually it turns to steam. And the same exact thing Happens with the Higgs condensate. If you heat it up enough, eventually it no longer condenses like that, and there isn't a high density of Higgs field. So in the early universe, when things were hotter, there was. The Higgs field no longer formed the condensate. If you go back early enough, less than what is it, maybe a microsecond after the beginning of the universe, there won't be. One wouldn't expect this condensate to exist. Well, what effect does that have? Well, this condensate has a bunch of mass associated with it. It's just a whole bunch of particles. So it's a bunch of mass. So that means that the sort of underlying the effective mass everywhere in the universe is much lower earlier on than it is later on. And in fact, one of the things I noticed with a friend of mine back in, when was it, 1978, was if that happens in the early universe, there's this period of time when there isn't a big density of mass which produces gravity, which prevents the universe from expanding so quickly. Instead, the presence of the. If you get rid of that, the universe expands incredibly quickly. In fact, it expands exponentially quickly. And so at that point where it's sort of an uncancelled expansion like that, you get this very rapid expansion of the early universe. And that turned into later on, the inflationary scenario for the early universe, which I now think is kind of nonsense, but that was. It was one of the sort of pieces of personal scientific history. We had noticed this phenomenon, but for various reasons, I just didn't think it could be a significant enough phenomenon in the history of the universe, because in order for it to be significant, it would have to be the case that it would be as if one took steam and made it colder, colder, colder, and made it incredibly, incredibly cold. And it still hadn't turned into water. And it would have to have be cooled by a factor of a billion and not turn into water. And for various reasons that didn't seem plausible because there didn't seem like in order for that to happen, you have to not have any sort of density changes in the material and so on. Didn't think that would happen for the universe. But later on, people said, look, it's such a cool effect that you can get this exponential expansion, that it has to be the case that something must work out okay with respect to this super cooling phenomenon. And that led to a lot of complicated scenarios that I think are very patched and so on. I think the actual story there about the early universe and its expansion has to do with more with the change of dimension of the universe, that the universe started infinite dimensional and then gradually kind of cooled down to be three dimensional. And that accounts for lots of phenomena that have required all sorts of weird patches in the existing theories. Okay. Which is to all to say that if you want to get negative mass stuff, one way to get it, this Higgs condensate thing is one way to get it, that normally there's Higgs condensate and it leads to the masses of electrons and things like that. But for example, if there's a high temperature, then that condensate gets sort of evaporated away and you have lower mass density everywhere in the universe, which shows up as. Relative to the usual baseline of the universe, shows up as negative mass. So that's another example of how that kind of thing works out. Okay. What happens if there's negative mass in the universe? Well, the expansion of the universe accelerates. And that means that relative to us at some particular place in the universe, there are parts of the universe that are effectively receding from us faster than the speed of light. So we'll never know what happened to those parts of the universe. It's as if the universe sort of tears itself apart and there are pieces where we just don't know what's going to happen to them. Because you might be exchanging radio signals and the thing that's exchanging radio signals, the radio signals will never get to you. Very similar to what happens in a black hole. It's in fact, a very similar kind of event horizon phenomenon where inside a black hole, that radio signal was trying to get out, it's being pulled back by gravity, and that radio signal will never get out. And it's the same type of thing here. It's because of this thing called principle of equivalence, equivalence between gravity and acceleration. There's a very close correspondence between these things, and you just never get the signal, so to speak. So there are parts of the universe we never hear from. Well, anyway, I think that the. Yeah, there's some question here from Lolly about is the viscosity of the Higgs field. What gives mass to particles? It's not quite viscosity, unfortunately. Let me explain it a little bit. Okay. So in light, in a vacuum, goes at a certain speed, about 300 million liters per second. In a material like water, go slower. Water, it's 1.33 times slower. Glass, 1.5 times slower. I mentioned that earlier in quite different context. Why does that happen? Well, what's happening is in the material, like water, for example, the light goes into the water. The photon of light keeps on hitting water molecules. Every time it hits a water molecule, there's the water molecule that says, okay, I just got, I got kicked by this photon. I'm going into an excited state. I've got energy sort of inside the atom. And after a little while that energy, the atom says, oh no, I want to get rid of that energy again. I emit another photon. So photon comes in, excites the atom. Then the atom takes a little while to decide it's going to re. Emit the photon. Onward. And the reason that light is going slower in these materials is, is because the photons are continually getting absorbed and re emitted. Absorbed and re emitted. And the amount that they're getting absorbed, the amount of slowdown is not very great. In the case of those materials in diamond, for example, it's a factor of 2.7. In the sun, for example, the density of material is high enough that a photon from the center of the sun will take, I think it's 10 million years to get to the surface. So it's continually being absorbed, re emitted, absorbed, re emitted. And it just keeps on, keeps that keeps on happening. And the density material is high enough that it really slows it down a lot. So what's happening with, with particles like electrons in the Higgs field is more or less a phenomenon like that. That what's happening is that the, the with relative to this Higgs field, the electrons coming along and it keeps on interacting with the Higgs field. And it's not quite an absorption re emission story. The, the sort of official version of this. Gosh, let me see if I can reduce the math to something more like this. I mean the, the official version. Okay. Just to say the words, there's this thing called the Lagrangian. The Lagrangian is. It's both a representation of energy and it's also the thing that, is the thing that appears in the formula that tells you how things progress with time in quantum mechanics. The path integral. Oops. The. And so, so let's see how does, how does this work? The. This is, this is going to get mathy and technical. When you represent a particle in this Lagrangian, you have a formula. It's the derivative of the field associated with the particle all squared and plus the mass squared times the. Mass squared times the particle field squared. Why am I telling you all of this? Well, because the term that's associated with mass is mass squared. Times field squared. And what happens is the interaction between the Higgs field and a particle field, like let's say the electron field is called psi. There's a. Let's see. This is a little tricky. Okay, let's. Roughly, there's a mass squared psi squared term. And when you have interaction with the Higgs field, it becomes a phi for the Higgs field squared psi squared term. And so what happens is the. It's a little tricky because. Anyway, never mind footnotes to this whole story. But roughly what happens is when you say there's a condensate of Higgs field, that phi squared, which would normally be the Higgs field, that's wiggling around, doing all kinds of things, changing a position with the position in space and so on. That thing is just a constant associated with the density of Higgs field. And so that's the thing that masquerades as a mass because it's just that thing that appears in that formula. It plays the role of a mass, even though it really is not a mass. It's really just the density of that field. Roughly. That's roughly how it works. So let's see. Alexander. Alexander asks how might the topological stability and quantization of Skymions initially proposes models for nucleons relate to our understanding of gravity, considering their presence in solid state physics? Okay, so this is going, this is jumping to a higher technical level of, of elaboration. But a skyrmion is something a little bit like an eddy in a fluid. It's something where once you've got that eddy going, you can't kind of get rid of the eddy without. You can kind of slow down the eddy, but you can't just say, oh, I'm going to sort of undo the eddy, because there's pieces of fluid that are going one way and going the other way around that core, and they don't. You can't get them to just sort of cancel out. It's kind of like you, you have a piece of string and you've wound it around a pencil. You can't, you know, you pull on the ends of the piece of string, it's still going to be wound around the pencil. You have to make a sort of discrete change to unwind it around the pencil. And that's what's called topological stability. It's topology because it relates to kind of the, not the detailed geometrical position. It doesn't really matter whether the string is aligned this way or that way. It's just the overall structure. Like topology is about something like you can have a sphere or you can have a donut with a hole in it. It always confused me because when I was growing up in England, donuts have jam injected into the middle of them. It's only American donuts that have holes in the middle of them intrinsically made so that the donut is like a torus. And when I read things when I was a kid that talked about donuts and that had holes in them, it's like, I don't understand what's going on, because the doughnuts that I get in donut places just have jam injected into this thing that is basically a sphere, okay? But independent of that, an American donut has. Is a torus with a. With a hole in the middle, so to speak. And the point is that you can't continuously go from the. You have to kind of rip. If you've got donut material, if you've got a blob of whatever dough the doughnut is presumably made of, and you want to turn it into a doughnut with a hole in it, at some point, you have to rip the dough. You can't just continuously deform it to go from the no hole to the whole situation. And so that's kind of the concept of topology is this. You can wiggle it around. That's sort of the geometry changing, but you can't change this sort of overall, how many holes does it have? Type, structure, and so on. And that whole idea shows up in all sorts of places. For example, in our models of physics, we believe that there's a similar kind of topological phenomenon, something where sort of the overall structure, the overall kind of arrangement of things is such that you get something that represents a particle. So skyrmions are a particular version of that for a particular model of. Of how. How a field fluid is like a fe. So fluid. There are lots of kinds of things that we describe as fields in a fluid. You're looking at this fluid and it's got. It's got all these pieces. But the thing that's important is it has a certain velocity, has a certain speed and a certain direction at every place in the fluid. And you say that that's the velocity field of the fluid. And what happens in an eddy, for example, is that the velocity field has this kind of topological stability to it. It can't get rid of that circulation that it has. Skermions are more elaborate version of a field. And that's a case where. And they were a model at one point for things like protons that model didn't pan out, but this general idea that things that are stable are stable because they are topologically stable, because there's something you just can't get rid of. Some. Some kind of donut hole that you can't get rid of. That's some. That, that's. That's kind of the thing. All right, I should wrap up soon here, but let me see. Healy asks, can the probability of a random walk returning to the origin be another way of defining local dimension on a graph? All right, again, quite technical. Let me just unpack that just for fun. Okay, what is a random walk? So let's say you're on a line and you can take a step up, you could take a step down, and let's say that you flip a coin at every moment to say, I'm going to take my next step up or down, up, down, up, down, etc. And you're always flipping a coin. You say, well, where do I get to? Well, you get all these wiggly curves and they'll keep going up and then they'll come down and et cetera, et cetera, et cetera, and they wiggle around. And you can say, what's the average distance that this curve goes away from? How far are you typically away from zero? If zero is in the middle and you're randomly going up or going down? And the answer is your average distance is about the square root of the number of steps you've taken, the average. The overall distribution of how far you got is so called Gaussian distribution, normal distribution. It's a distribution. It's kind of a bell curve. It's something that shows up all over the place. And as. So the random walk has this feature that that's the average distance you've gone. Now, the question is, if you go up and you're wiggling up and you're wiggling up, you keep on doing, hitting heads, heads, heads, heads, heads. Then you get tails, tails, tails, tails, tails. One feature of random walks is that anything that comes up goes down again. Every random walk eventually gets back to where every random walk in one dimension, just going up and down, eventually gets back to zero again. And that's a phenomenon that people talk about in the following way. Let's say you're. You're betting based on coins and you say, I'm ahead, I'm ahead, I'm winning, I'm winning. I've got a zillion of run of heads and I'm just going to win. I'm going to keep betting. I'M going to keep betting. Well, it turns out it's the phenomenon. It's called gambler's ruin. Eventually you're going to lose because eventually that random walk is going to come right back to zero again. Eventually you're going to have enough tails that you get right back to zero again. In one dimension, it's guaranteed that a random walk, if you keep going long enough, you'll always hit zero again. So it's never a good thing to. You can. That's kind of like even though you thought you were ahead, you're going to be tailed down again, so to speak, and reach zero. That's true in one dimension. In two dimensions, a random walk on a square grid, for example, has a certain probability to return to the origin. In three dimensions, a random walk, chances are it'll never return to the origin. Chances are it'll just keep wandering around and it'll just keep wiggling around and going where it goes and it'll never get back to the origin again. And there's a critical dimension which I should know and I don't for when. Well, I think it depends on the kind of grid you're on, actually. And that is something you can't really define outside of an integer number of dimensions. The. Let's see. Well, I'm not quite sure there's a notion of. I mean, fancy words, upper critical dimension. There's a thing called the renormalization group, which is a way of analyzing kind of things like these kinds of random processes. Anyway, a bunch of fancy stuff which I can't immediately unravel for myself. So I don't know the answer to whether there's a. If you say one dimension, two dimensions, three dimensions. There are definite things that happen. If you say, if you could make something 2.5 dimensional, what would it be like? Now, there are many in these graphs that we have that represent kind of the structure of space in the universe. It's sort of the first time where we have a robust way of talking about fractional dimensions. These things called fractals, these nested patterns, they're a rather specific way of talking about fractional dimensions. And it doesn't quite work generically there, but. So there may be something which I. There may very well be something. Yes, yes, yes, there will be. Yes, in fact, there is. Okay, so yes, it is right, that you could think about sort of take a random walk on this graph. The graph has a certain effective dimension which you can associate with. Well, you start at a point. How. How many points do you get to by going a certain distance out. And that is closely related to the question of, well, what's the probability that you come back on the random walk? And that in turn is related to if you had heat. You have the equation for heat is the diffusion equation. That's actually the diffusion equation. The way to think about it is instead of one random walk just wiggling around here, you say, start a very, very large number of random walks. What is the density of where all those different random walks will reach? That's the thing that gives you the Gaussian distribution, the Bell curve. And it does that in any number of dimensions. There's a kind of an analog of that in any number of dimensions. And so there is even in fractional dimensions. And I think that's closely related to this random walk thing, I think to the return time of random walks, which gives you the. The kind of things. The probability of coming back to the origin. But there's a bit of random walkology there that I can't in real time decode and would have to think more carefully about it. All right, let's see. 1. Oh, gosh. All right, I'll try one more. And then we should wrap up for today. From tude saying I'm quite confused with heat and temperature and the general introduction of thermodynamics in this term in school. Well, not sure how totally much I can help with that, but I'll try. What is heat? What is temperature? So when you have a material, it's made of molecules, and the molecules are, well, they're typically wiggling around. They're bouncing around. Maybe it's a gas. They bounce until they hit another molecule. Maybe it's a solid. They're always wiggling around close to their usual position, but they're always wiggling around a bit. Okay, when they wiggle around, they're moving. They have a certain kinetic energy. They have a certain energy of motion. So in the standard kind of way things are set up, if they have a velocity V, their kinetic energy is a half mv squared, where M is their mass. So anyway, there's an energy associated with that motion and that energy. There's little microscopic motion of all of these molecules, whether it's in a gas where they move maybe a millionth of a meter before they hit another molecule, whether maybe it's a solid where they move only a billionth of a meter before they get stopped and they have to go back the other way type thing. But in any case, they're all wiggling around. Okay, so that amount of wiggling around. There's an average amount of wiggling around that these molecules do. That average amount of wiggling around is determined by the temperature. In fact, there's a formula. It says that if you take the average kinetic energy associated with these molecules, it's a constant called Boltzmann's constant multiplied by the temperature, and then there's a three halves that we won't talk about for a second. But basically the. Well, actually, I can say it in every direction. The molecule might be wiggling. So, like, if there's three dimensions of space, every dimension, there's three. Three directions. It might be wiggling every wiggling direction. The amount of energy is a half times kt, where K is Boltzmann's constant and T is the temperature. Okay, so temperature is defined in terms of the average kinetic energy of the molecules wiggling around. Now, okay, how do you measure temperature there? Well, it's important that if you reduce the temperature enough, eventually you get to the point in principle where the molecules aren't wiggling anymore. There is no kinetic energy. The kinetic energy is zero. What temperature is the kinetic energy of molecules zero? Well, the answer is it's that the absolute zero of temperature and it's minus 273.16 degrees Centigrade is absolute zero at that temperature. Then all the kinetic energy is gone. There's no kinetic energy that's making the molecules wiggle around. Okay, footnote. Because of these quantum fluctuations that I actually mentioned earlier, there is still kinetic energy, even at absolute zero, associated with quantum mechanics. And that means that. Well, in the particular example of helium, helium 4. Helium 4 is still a liquid at absolute zero because of quantum effects. But most of the time, most materials are just solids because all of their atoms are just locked in place, not moving. So in a first approximation, in sort of the simplest way of thinking about the theory, at absolute zero of temperature, there is no kinetic energy. Everybody is just locked in place, not moving. Okay, so then you. You. You add. So when you add. Okay, so can average kinetic energy of molecules is proportional to temperature. Okay, what's heat? Heat is the total energy of all those molecules. So you've got a big block of metal, and every individual molecule is wiggling around a certain amount. It has a certain average kinetic energy. But when you take all of those energies put together, you add them all up, then you get the total heat energy associated with that block of metal. And those things are related by a thing called the specific heat that tells you the amount of heat per degree of temperature. So to speak. So as you increase the average kinetic energy of those individual molecules, you're increasing that. And there's a certain number of molecules in the block of metal or whatever it is. And then that with a certain, there's a certain mass of the metal depending, et cetera, that constant proportionality is the so called specific heat. So as you put heat into the material. So now here's where things get tricky. How do you put heat into a material? Well, you know, one thing you might do is just rub the material back and forth. You might have friction or something, or you might, you know, does it make, does it put heat in? If you say, let me take this big block of metal and let me throw it at high speed, the big block of metal has a certain kinetic energy. And you might say, well, I put energy into this block of material, but doesn't heat the block of material up. So just throw it, doesn't heat it up. The heating it up would be you've got microscopic motion of molecules in the block of metal. But that isn't the macroscopic. The large scale motion of the block of metal isn't the same kind of thing. So the relationship between the large scale motion and the small scale wiggling around of molecules is a very subtle thing. And that's the thing called the second law of thermodynamics. Maybe I should explain the. Well, there are sort of three or four common laws of thermodynamics. The first law basically says that energy is conserved. You can have energy that's in the overall motion of things. You can have energy that's in the microscopic motion of molecules. But if you add up the energies of those things, it'll always be a constant. You can't. Without sort of adding stuff from outside the system, you can't change the energy of the system. That's the first law of thermodynamics. The second law of thermodynamics says what was originally energy associated with large scale motion gets ground down into energy associated with small scale microscopic motions. So let's say you've got friction, you're rubbing two things against each other. That's a large scale motion. But the effect of that is they get heated up microscopically. What's happening is the molecules are getting pulled around and each individual molecule is doing its little thing. And in the end, what was kind of large scale, very organized motion of just rubbing the two blocks of metal turns into small scale disordered motion. And that's the phenomenon that large scale motion tends to turn into small scale Motion that sort of. And that's an irreversible thing. Once it's heat, it doesn't suddenly. The molecules don't suddenly arrange themselves so that the block of material, so that all the molecules go the same way at the same time. And the block of material sort of takes off running in one direction. It's just stuck in this sort of random motion. So that phenomenon of the fact that you get from sort of organized motion, it's sort of ground down into random motion. That's the second law of thermodynamics. The tendency towards sort of randomness of motion. That was the originally formulated version. There's a much more general version of that that's really about the fact that simply defined initial conditions tend to become more random. It's something I've studied for many years, and I just wrote this book actually about the second law of thermodynamics. That is, I think, finally an explanation of how this all works that people have been really confused about actually for 150 years. So it's kind of cool to be able to unravel that and see how it really works. But that's a different thing. For these purposes, the second law just says large scale motion grinds down into small scale microscopic motion. That's heat. By the way, just for fun, I can say the third law of thermodynamics is the statement that it's really hard to get to absolute zero, that you really. There's enough randomness in all that motion of molecules that if you want to take heat out and you want to just get those molecules to all be sort of stationary, that you kind of have to poke every molecule separately. It's in a finite number of steps, it's not possible to get to absolute zero. And, and that has the practical effect that there are all kinds of schemes for refrigeration that get you to microkelvins. Even, I think, nanokelvins now within a millionth of a degree centigrade from absolute zero or even a billionth of a degree centigrade. And those things are sort of fighting the third law of thermodynamics. There's also, by the way, a 0th law of thermodynamics, which is the idea that there is a concept of temperature and the idea that if you have a block of metal and that there's a single temperature, that all the molecules are wiggling around with about the same kinetic energies in different parts of that piece of metal after enough time that eventually the piece of metal comes to Equilibrium. That's a very rough introduction to thermodynamics. I mean, I will say that there are all kinds of fun things like there's the notion of negative temperature. Normally. Oh, I should say that when there's like I say, the average kinetic energy is this or that. There's actually a distribution of kinetic energies. They're not all the same. There's a distribution, it's always E to the minus, the kinetic energy divided by KT, the temperature. There's an exponential distribution. And so normally things of higher kinetic energy are less likely. But there's this notion of inversion that happens in lasers, for example, that effectively corresponds to negative temperature because you'll have more things in that higher energy state than in the lower energy state. And that's a sort of thing not really covered in thermodynamics. But that's a sort of related phenomenon anyway, that's. I don't know whether that was helpful in kind of intro to thermodynamics. But what happens in thermodynamics there is, you know, that's kind of the difference in heat and temperature, macroscopic the whole energy versus the individual molecules energies. In traditional thermodynamics there are a lot of relations between heat, internal energy work that's mechanical motion type things, pressure, volume, temperature, et cetera, et cetera, et cetera. And it's kind of all very useful if you're doing chemical thermodynamics or engineering thermodynamics or H Vac systems or something like this. It is a little bit confusing with respect to how it all works microscopically because it's really very. It's a sort of a simple mathematical theory that sits on top of kind of a a. The foundations of it are kind of complicated. But once you have decided that things just work according to, let's say, the second law of thermodynamics, once you've bought into the second law, then you can deduce all these mathematical relations. All right, we should probably wrap up here. Just to mention it's always fun for me to give these explanations. I have to say that many of the things I talk about here. I've known about these things most of my life and I don't think I've ever explained them to people. So it's really fun for me to have a chance to try and explain them and see how well I can manage to sort of get to the end result without having to sort of yak about fancy math and things like this. Just one thing I might mention, I'VE been this explaining things as people may know, there's a total solar eclipse. I happened to mention that earlier in different contexts that's coming up on April 8th of this year, visible across the US and in sort of recognition of this, I had done some work back in 2017 when there was another total solar eclipse, kind of tracing the history of how one predicted when an eclipse would occur. And we put up a website that predicted to the second when the eclipse would arrive at a particular place. We're going to do that again in April of this year. And I have a little book that I put together about how one predicts when eclipses will happen. And that book, I think, is coming out this coming week ahead of the eclipse. You can't dilly dally because the sun and the moon, they move as they move and the eclipse will happen on April 8th. And there's none of this, oh, whoops, we're late. It's, you know, let's postpone the eclipse. It's going to happen when it happens. So we kind of have to get our book out in time. And it's coming out next week. So I just wanted to mention that maybe in one of these live streams I can talk a little bit about the theory of eclipses and how one predicts them and so on, and maybe a bit about the history of how that was figured out. All right. Well, thanks for joining me and thanks for asking lots of interesting questions and see you another time. Bye for now.