The Origins Podcast with Lawrence Krauss - Physics for Everyone, Lecture 2: The Gestalt of Physics, Tools for Seeing

Episode Date: January 22, 2026

Any sufficiently advanced technology is indistinguishable from magic, as Arthur C. Clarke put it. In that spirit, the way we get closest to “magic” in physics is not by memorizing more facts or equations, but by learning a few mental tools that help us see through the illusion of complexity by extracting the wheat from the chaff. They are all simple at heart, but nevertheless quite powerful, and they form the core of what I call the Gestalt of Physics—the worldview that governs how physicists approach nature. And some of them can actually seem like magic to the uninitiated! I’m also pleased to share a quick PSA. We’re organizing our next Origins travel adventure: a sailing expedition through the Greek archipelago (July 24 to 31) with bestselling author and Biblical and ancient civilization scholar Bart Ehrman and me, with a possible Cyprus add-on (July 18 to 23). If you’re interested, it’s worth raising your hand early. These trips tend to fill quickly. Express interest at http://originsproject.org/greece-2026In Lecture 1, I used powers of ten as an intellectual zoom lens, a way to escape the trap of human scale. Lecture 2 steps back and asks a more fundamental question: how do physicists consistently make progress when the world looks hopelessly complicated?This lecture focuses on the fundamental toolkit for seeing. We will use these tools throughout the series, because they are the difference between being dazzled by nature and being able to interrogate it, and ultimately understand it.First, order of magnitude thinking, the art of using powers of ten and rough estimates. It is how you keep your intuition tethered to reality, and how you avoid being bullied by big numbers dressed up with false precision.Second, approximation, which is where I introduce my super cow. It is not only a spherical cow. It’s better. My super cow has exactly the features we need for the question at hand, no more, no less, and it politely agrees to ignore everything irrelevant. I introduce it with a joke, but it is also the core of how we turn messy reality into something we can actually calculate without lying to ourselves.Third, dimensional analysis, one of the great bargains in science. The fact that there are essentially only 3 fundamental ‘dimensional’ quantities describing nature—Length, Time, and Mass—means that all physical quantities can be related to other physical quantities through a small set of relations. Keeping track of dimensions allows us to often guess what the relations are, without knowing any details of specific physical situations. It seems like magic. By keeping track of the dimensions underlying quantities, you can often infer the form of an answer and you can catch nonsense instantly. Sometimes the most important result is realizing something cannot be right, because that is where new physics likes to hide.Along the way I adopt some Fermi style challenges—named after the remarkable physicist Enrico Fermi—to show how these ideas work in real time, and why they are not parlor tricks. They provide a training in scientific judgment. I also end with a preview of what comes next, symmetry, a concept that quietly runs far more of the universe than most people realize.Enjoy, and feel free to share.LawrenceAs always, an ad-free video version of this podcast is also available to paid Critical Mass subscribers. Your subscriptions support the non-profit Origins Project Foundation, which produces the podcast. The audio version is available free on the Critical Mass site and on all podcast sites, and the video version will also be available on the Origins Project YouTube. Get full access to Critical Mass at lawrencekrauss.substack.com/subscribe

Hi, it's Lawrence Krause here, and welcome to the Orgence Podcast. Before we begin our regular program, I wanted to let you know right now about an exciting possibility that's currently being organized, our next Origins Travel Adventure. It should be finalized within the coming week or two, but I can tell you right now, it will involve, at the very least, a seven-day sailing expeditions through the Greek archipelical visiting a variety of islands that are rarely visited, and with guest speaker, Bart Ehrman as well as myself. In addition, we hope to have a pre-trip, an add-on,

or for some of you, you can just do the Cyprus trip, a Cyprus trip between the 18th and 23rd of July. You can do both, obviously, and we'll have travel between Cyprus and the boat in Greece arranged as part of the trip. It should be an amazing adventure. In Cyprus, we'll have a variety of incredible excursions, including a special evening movie presentation on the beach with a special guest.

So consider joining us and what you should do is go to the Origins web page www.orgensproject.org and sign up to express your interest because these things sell out quickly and we'll be reaching those who have expressed interest first as well as our past travelers before we move to open it up to everyone. again July 24th to 31st through the Greek Islands and July 18th to the 23rd in Cyprus. Hope to see many of you there. Thanks. Hi, welcome to the Origins podcast. I'm your host Lawrence Krause. And this is the second in our series of presentations on physics and reality.

The nature of the universe as we now understand it, from the simplest scales to the most complicated scales. and I hope I'll have fun in each lesson. And last presentation, I hope I gave you a comprehensible tour of the universe as we now understand it, everything we understand empirically about the universe and theoretically on a comprehensible scale. And one of the key aspects of that

was the use of powers of 10, of thinking of the scale of objects. Now, what I want to talk about today is probably the most important thing about physics. you saw in that last presentation, the vastly different scales of the universe, the way the universe behaves on the largest scales compared to the small scales, very, very different. How can we try and make sense of a universe that complicated? And the answer is we've used a set of tools, simple tools that allow us to separate

the wheat from the chaff. And that's what I want to spend time today talking about, the gestalt of physics. And I want to go back to Powers of Ten. Because, of course, in the last episode, I showed you how useful they were for allowing you to present vastly different scales on a single image, from the smallest to the largest scales of the universe, but powers of 10 are much more important to that because they make working with numbers easy. And they make, in fact, multiplication and division, they turn it into adding and subtracting, which for most of us is much easier.

And that's very important because in physics, when we try and understand the details of processes, we can make estimates of things that we wouldn't be able to make estimates of otherwise if we didn't use powers of 10. So let me show you what I mean. Well, the basic mathematics of powers of 10 is quite simple. If I take 10 to the power A and I multiply by 10 to the power B, I get 10 to the power A plus B. 10 times 100 is 1, 10 to the 2 is 10 to the 1 plus 2, which is 10 to the 3, which is 1.

And so you see what I've done here, what we do is we turn multiplication into addition, because the powers of 10 just get added when you multiply things. Similarly, when I divide things, 10 to the A divided by 10 to the B, we get 10 to the A minus B. So division becomes subtraction and multiplication becomes addition for the powers of 10. Now, this makes, it's very important because any number we can write down, any physically measurable quantity, we write down as some number A between 1 and 10, like 3.7, times 10 to the power B. Any number, all right, like 376 is 3.76 times 10 squared. 3,720 is 3.720 times 10 to the third.

And this is interesting because it means if I try and think of how to multiply things, most of us are relatively confident multiplying numbers between one and ten with each other. Most of us had that times table drilled into us when we were littler. And that means that multiplying numbers becomes very easy when you think of presenting them as powers of 10, whereas it's more complicated and you need calculators, most of us, if we don't do that. And I want to give you an example. I'd like you at home to try and try. this. Consider, now, I want you to take, I'm going to give you two examples. Take a piece of paper

and try and do this multiplication and division, and you have 15 seconds. Multiply 3,276 by 2438 and divide 376,524 divided by 2,382. Okay, you have 15 seconds. I'll take 15 seconds. I'll let you do this now. Oh, time it myself. I wish we had music to play during this. Okay, times up. Put your papers down if you did this. And normally, I suspect if you try to do this as most people would do it, well, without a calculator, you'll come up with vastly different answers. And some of them might be right. Some of them be off by orders of magnitude. But this is the important point. If I think of these numbers by powers of 10, then it's much simpler. Okay, well, actually, let me go back and just do it for you. Let's do it in 15 seconds. Take these two things. So this is 3.276, or about 3.3 times 10 cubed,

times 2.4 or so times 10 cubed. 10 cubed times 10 cubed is 10 of the 6. And 3.3 times 2.4 is about 7 or 8. So it's something like 8 times 10 to the 6th. Okay? I did that in about five seconds. Now, the right answer may be 7.76 million, but the point is I got approximately the right answer, whereas if you use normal multiplication, many of you probably got answers between a thousand and a billion. Similarly here, okay, this is 3.76 times 10 to the 5th, divided by 2.4.4.5. times 10 to the 3. I'll call this 3.8. 3.8 times 10 to the 5th.

10 to the 5th divided by 10 to the 3 is 10 squared. And 3.8 divided by 2.3 is about 1.5. So this is about 1.5 times 10 squared, or about 150. So you see, I can get close to the answer very quickly, just using powers of 10. Getting close to the answer turns out to be one of the most important things in science. If we can make predictions and get the correct to basically the nearest orders of magnitude, we know at some level that what we're doing is at least not vastly wrong. And this allows you to do lightning fast estimates of things that you wouldn't be able to do otherwise.

And you can have fun with friends trying to estimate things that they wouldn't be able to do. In fact, let's do some estimates. Because the exponents tell the scale of things, most of the information of physics comes from that exponent, comes from that power of 10. Are you talking about a million miles or 100 miles? The physics of what happens on a million miles may be very different than 100 miles, very different, very different than 10 to the minus 7 miles. And if you can get to the nearest power of 10, you can understand things.

And it allows an intuitive understanding of things that are otherwise impossible. It also allows you to answer questions that should are otherwise impossible. I mentioned the last episode in Rico Fermi, one of my physics heroes, one of the last great scientists, a nuclear physicist who was an equally adapted experiment in theory, a Nobel Prize winning physicist. During the building the Manhattan Project, when he was building the first nuclear reactor under the football stadium at the University of Chicago, during breaks, he used to give what it called Fermi problems.

He basically said a physicist should be able to answer any question you ask them. Now, you might not get the right answer, but provide an answer. algorithm to go about to say how you would get the right answer. And when it comes to physics problems, you should be able to make estimates relatively quickly. And I want to give you an example of a few of those Fermi problems that might seem otherwise impossible answer. For example, how many toilets are flushed in New York City each second? If you ask someone that to say, how the heck do I know, of course? And maybe it's not that important to them. But no idea of how to to get any answer that might be within a factor of a million or the right answer. But if you just

break it down into simple thinking of powers of 10, you can do that. Say there are 10 million people in New York and the average person might go to the bathroom five times a day. You can pick three times a day, two times a day, seven times a day. The point is we're trying to get a rough estimate. Well, 10 million people times five visits a day is five times 10 of the seventh. 10 million is 10 of the seventh, five, five times 10 of the seventh toilet visits per day. Now, how many seconds in a day? Well, there's 60 seconds per minute, times 60 minutes per hour, times 24 hours in a day. So that's six times 10 to the first, times six times 10 of the first, times 2.4 times 10 to the first. Multiplying the exponents, one, two, three. Six times six is 36. This is roughly two and a half.

36 times 2.5. Well, this is 3.6 times 10 to the 4th. So we now have 3.6 times 10 to 2.5. That's roughly 9. So they're roughly 9 times 10 to the 4th seconds per day or close to 100,000 seconds per day. And all I have to do is say 5 times 10 of the 7th divided by 9 times 10 to the 4th gives me, well, 5 divided by 9 is roughly a half. 10 to the 7th divided by 10 to the 4th. 7th. 7. minus four is three is 10 to the three. A half times 10 of the three is 500. So roughly 500 toilets are being flushed in New York each second. Now, the actual answer may be 200. It may be a thousand, but it's not going to be a million and it's not going to be five. And so just this kind of estimate tells us that the answer to within, certainly within order of magnitude. Now I want to

take, this may seem kind of a silly question, but at least it gives you a simple example of what I want to try with another Fermi problem, which is one of his famous, famous questions he asked, how many piano tuners are in Chicago? And I often, when I've been teaching physics students, ask them this question. And again, until you take a physics class and learn about powers of 10 estimates, it seems like an impossible question. It's, interesting that no matter what estimates we've used over the 40 years I taught, basically, different estimates in each class, we always came out to roughly the same answer. So I did, this time, I said, maybe there are about 7 million people living in Chicago. I don't know the

exact estimates. Probably not 10 million. It's more than 5, I think. So I said 7 million people. How many people in a household? Let's say about 3 on average. So about 2 million households. How many households have a piano? Now, this is an interesting question. question. It used to be, I would figure maybe thinking about my friends growing up and how many households I used to visit, maybe one in ten houses would have a piano. Now that you can have pianos that are electronic, it may be more. But let's take physical pianos because we're talking about piano tuners, and piano tuners don't have to tune those electronic pianos. So again, one in ten households on piano, you might estimate one in five. You might estimate one in 50.

but I'm going to take 1 in 10. That means if there are 2 million households and 1 in 10 has a piano, then there are 200,000 pianos or 2 times 10 of the 5 pianos. Okay, that's how many pianos roughly might be in Chicago. Now, the next question we have is, how often does your piano get tuned? Well, if you have a piano, you can get the right answer or the answer that works for you.

We've had a piano in my house, and we get a tune roughly every two years. Some people will get a tune much more frequently, especially if you're an avid pianist. Some people might not tune it once a decade. But let's take it every two years as an average. Well, if one piano is tuned every two years, then basically there are 100,000 pianos tuned a year.

Each year, you basically tune half a piano, and so divide by two, and you get 10 of the fifth pianos tuned to. a year. So the 100,000 pianos are tuned in a year. Now you ask, well, how many pianos did a piano tune or have to tune to make a living? 10 a week, maybe? You know, thinking about how much you charge for tuning your piano, two a day, one in the morning, one in the afternoon, five days a week. Not an unreasonable estimate, I would think. Well, that means each pianist can, if there are 50 weeks in a year, each piano tuner, if they're doing 10 pianos a week, then they're tuning 500

pianos a year. There are 50 weeks in the year, 10 times that is 500. So each piano tuner, to make a reasonable living, is tuning 500 pianos per year. We have 10 to the fifth pianos divided by 500 pianos per year. You get 500 piano tuners. And as I say, when I've done this with students, we've varied the estimates, and you could have done it independently yourself, the answer is always going to come up to be between 100 and 1,000. I've never had it vary more than that. And one year, I assigned one of my students who lived in Chicago to go at that time, look at the yellow pages for Chicago to find out how many piano tuners he could find,

and he found 200. That was 30 years ago. But you can see to within a factor two, this very simple analysis gives us the right, order of magnitude. Again, a kind of question which, if I ask someone on the street, how many piano tuners are in Chicago, it would seem like an impossible kind of question to answer. But this kind of algorithmic approach, using powers of 10, takes complicated questions and makes them answerable. It's also the kind of tool that should help us in everyday life. I don't know how many times I've gone to the grocery store when tellers are using their keyboards, their cash

registers, and they're totaling things up. And, you know, you have six items that comes to $500. You say, hold on a second. These six items are $2 a piece. You know, this is $12, not $1,200. Oh, there's a mistake. I hit an extra in fact. It doesn't. But if you're doing this kind of thing, you can automatically, when you're doing your shopping, estimate what the total should be so that you can see if there's an error in that. That kind of thinking can help you in everyday life as well as in trying to assess the frontiers of physics. So powers of 10 are useful because they make handling numbers much easier and they allow you to answer questions that roughly and correctly that you would never be able to answer otherwise. So that's the first bit of the Gestalt of physics.

The second is the idea of approximation more generally. Powers of 10, of course, are approximating numbers to within the nearest order as magnitude, and we can use that idea in a generic sense as well to think of approximating not just numbers, but objects. And one of my books, Fear of Physics, I begin with a joke. It's a physics joke, it's not a very funny one, which, but nevertheless, I'll tell it here. It involves a dairy farm that is trying to increase the production. of milk and they invite three experts in to consult. They give them a week to study the farm.

And they have a physicist, an engineer, and a psychologist. And they asked a psychologist to report after a week. And he said, well, you know, I think you should paint the barn green. It's more mellow for the cows. They'll be happier. They'll produce more milk. Fine. The engineer says, well, I've looked at the milking tubes and you probably should increase their diameter by a factor of two because the flow of milk is not very efficient in these small tubes they're using, et cetera, et cetera. And the physicist has a blackboard, of course, comes up and says, assume the cow is a sphere. Okay? Because that's the kind of thinking that physicists try and do, which is to extract out every extraneous detail and approximate a system with the minimum amount

of information needed to be able to get real results. So it's probably the hardest thing to learn students because we teach physics problems where we give problems and you get exact answers. And even in graduate school, you tend to do that. But in the real world, it's hard to get the exact answer of really complicated systems. You have to learn if you're going to take a complex system and try to understand it, how to simplify it. And simplify it means throwing out information. And throwing out information is just not natural to human beings. Throwing out extraneous information is difficult. It's hard to learn. It takes time and practice. And we're programmed not to. And how do you know what's extraneous? Well, try it and see. If you throw out too much information and you get nonsense,

you know you're throwing out something important. And again, let me give you an example of how powerful they can be using the scaling factor I used in fear of physics. It allows us to do biology without knowing anything about biology, at least certain aspects of biology anyway. Here's the cow as a sphere. Coming back to my joke, I'm going to say the cow has a radius R, because the thing about a sphere that makes it simple

is as described by one number, the radius. Once you give the radius of a sphere, that tells you everything you need to know about a sphere. That's why a sphere is simple. So that's a cow, and let's imagine a super cow that's twice as big, with a radius twice as size, two. are. Why are, why do when we breed cows, do we only breathe them a certain size? Why don't we make

super cows that are 10 times bigger than cows? Well, this kind of analysis, this kind of approximation allows us to think about the answer. Because for example, the mass of a cow is proportional to the volume of the cow. And if, if the size of a cow is R, its volume is roughly the cube of R. How do we know that? Because we talk about volume and cube. cubic yards, cubic meters. If you have a distance scale of R, the volume associated with that scale, if it's R by R by R, is R cubed. So mass, and of course, mass is proportional to volume in this case.

And so the mass of a cow, in this case of scale R is R cubed. But if I double the size of a cow, I make its volume eight times larger because it's, it's The volume goes like R cubed. So 2R cubed is 2 times 2 times R cubed, which is 8 times R cubed. So the volume is 8 times bigger. But the surface area of the cow, which is square meter, square centimeter, square something, is the square of that scale. So the surface area is R squared and the masses are cubed.

And what you see is that the mass increases as one extra power of R compared to the surface area. So as you increase the size of cows, their mass per unit surface area increases like R. Now, if you're trying to hold together a cow, the skin of a cow, you've got all that mass, the cow's skin has a certain strength. Eventually, if this mass per unit area of the cow increases by a large enough amount, the cow's going to fall apart. So you can't breed super cows beyond a certain size simply because, they'll fall apart. You don't have to know anything about cows. You just have to know the mass of a cow

increases as a cube of its size, whereas the surface area increases as the square of its size. And it doesn't matter about the shape of the cow. So we don't have to worry about the complicated shape of a cow. We can approximate the cow as a sphere. And it gives us this simple scaling law in physics. Now you might say that's a very simplistic picture of a cow. Well, let's make a more complicated picture of a cow. A cow as a head and a body, two spheres connected by a rod. And let's consider a super cow, two spheres connected by a rod. Now, the mass of each sphere increases as the cube of this size. So the mass of the head of a super cow is again eight times bigger than the mass of the head of a cow. What about the rod? The strength of a rod depends upon its surface area.

for a given material. And therefore, the rod strength goes as a square of the size of the cow. And you can now see the mass of the head compared to the strength of the rod, again, increases by R. That tells us that you can't expect to have heads that increase in proportion of the size, the same as the rest of the animal increases in size. It tells us that for land animals, the head has to be progressively smaller compared to the size of the body as the animal gets larger.

And if you look at the large dinosaurs for the most part, except for the ones that are extremely strong necks like Tyrannosaurus rex, they have very small heads compared to very big bodies. However, that's, of course, on land. But in water, which is buoyant, which holds you up, you can have animals that have very large heads compared to the size of their body, which is why. the animals that have the largest brains are things like whales and dolphins because they're buoyant. They couldn't keep their heads upright if they were in air, but in water you can do that. Okay, so we can do an incredible amount of biology, understanding the scaling of animals, maybe even why the dinosaurs went extinct because they couldn't have heads that, or brains that were in comparison of their bodies the right size.

Maybe that's not the reason they went extinct. But it tells us a lot about biology without knowing any of the details of biology or even the the kind of details of what the animals look like. Just simple scaling arguments where we can replace all the complexity of an animal by its overall scale size can tell us a lot. Now, the next slide is the universe itself. If we just imagine the universe as a sphere, not just the cow as a sphere, we can do a lot. And I want to show you how much you can get out of imagining just a region of universe as a sphere. So if we look at these are galaxies randomly drawn and may not look like galaxies, but the way I draw galaxies. And we'll consider, the point is we'll consider a scale that's large

enough so that there are many galaxies, like the pictures I showed in the last lesson, large enough so that there's a lot of galaxies in those regions, but small compared to the size of the universe. And what we find is that the universe is more or less the same everywhere. The density of the universe is more or less there are in homogenades. We saw those filaments. but if you consider on average it within a filament anywhere in the universe, the density is about the same. And so the overall scale size we have is R here. We can imagine a sphere of size R.

Now, the one thing that's really interesting is if the universe is the same everywhere, if no region of the universe looks substantially statistically different than any other region on average, then if we think about the dynamics of the universe, what happens to any one galaxy will be the same as what happens to all galaxies. So if we can try and predict the future of the universe, we just have to think about what will happen to a single galaxy, say a galaxy at the edge of a region of size R. Now, to figure what's going to happen,

we rely on an important discovery that was made some time ago by Abram Hubble, which is that the universe is expanding. What Evan Hubble discovered, well, let's see what everyone in hell of discovered. Here's a way to think about what he discovered. And some of you may have seen this before, because I use this slide often to understand the expanding universe. When we try and understand the expanding universe, it's difficult to picture because we're stuck where we are. So one way is to get outside the box, literally, is to stand outside the universe. And I've created two universes here, one at time T1 and one at time T2.

This region of the universe is bigger than this region. This region has expanded into this region. I put galaxies at regular intervals, and this is at time T1, that's a time T2. Now, to see what you'd see at any given galaxy, say this one, I put this one on top of itself to see what you'd see from that galaxy, and what you'd see if you were standing on that galaxy, like our own Milky Way galaxy, is to see all the other galaxies looking like they're moving away from you. And the ones that had gone twice as far would have moved twice as fast, the distance in a given

time, the ones that are three times as far would have increased their size by factor of three and so on and so on, four times as far away, four times further. And it looks like in this picture, you are the center of the universe. And as I like to show, it doesn't matter what galaxy you pick. Every galaxy sees the same thing. Everything appears to be moving away from it. And there's an empirical result that comes out, which is exactly the result that, that, that, that, that, that, when Hubble discovered, when he looked at, for the first time, galaxies outside of our own, and looked at the relative velocity away from us, he discovered that galaxies that were an average twice as far away from us were moving twice as fast and three times as far away, three times as

fast, and he got what we call Hubble's law, that the velocity of the galaxy away from us depends on its distance. And the number doesn't really matter. We call that number is called Hubble's constant. Now it's one of the most important numbers in the universe. It gives us an overall scale size of the universe and how it's expanding. But you can understand, and it took a Hubble a long time to appreciate what he was seeing, is that this apparent fact that we seem like we are the center of the universe is just an accident of our circumstances. What it really tells us is that the universe is expanding. Okay, so let's now use that and go back and understand what's going to happen to the future of the universe. So let's think of a galaxy

at the edge of a region of a lot of galaxies, and compared to the center of that region, like, say, the Milky Way, this galaxy is moving in a way that Mr. Hubble told us. Okay? Its velocity will be the Hubble constant times its distance from the center of that region. Now, the only little bit of physics

that I want to throw in here is that to determine the future of the universe, it's the same as we determine the future of throwing a coin up in the air. If I take a coin and I throw it up, it'll come back down. If I throw it up faster, it'll go up higher and come back down. If I throw it up really fast, it won't come down at all. And the way we can determine that is by the total energy of an object. It comes in two parts, and this may remind some of you from your physics classes. A positive piece, which depends upon the speed of the object, the faster I throw it up, the more energy of motion it has,

we call it kinetic. But the earth is pulling it back down. And it, it, it, it, it, is it, potential energy is negative. This is the kinetic energy, energy of motion. This is the potential energy. And so the closer you are to the earth, the more negative is the potential energy. And in order for an object to escape from the earth, you just look at these two pieces. And if this piece, if the positive piece beats the negative piece, then the object will escape from the earth. If the negative piece beats the positive piece, the object will fall back down to Earth. So that just becomes a counting. Well, if that's on the earth, let's think about the galaxy.

Will this galaxy moving at this speed escape from this region of the universe? And if it escapes, that means our whole galaxy, if that galaxy escapes, then it means our whole universe will continue expanding forever. So if we're interested in predicting the future of the universe, to some extent we can just look in this region and apply this simple picture of a spherical region to see what happens.

So as I say, if this is bigger than zero, this region, if we now apply to a galaxy and not a coin, if the galaxy has positive total energy of motion, then it will move away from ever, forever and the universe will expand forever. If the galaxy has less than zero, if the negative term beats the positive term,

the galaxy will stop and come back down if matters all there is. And so we have to just do that simple, calculation. So I'm going to do something that looks more complicated, but I'm putting in the factors of pie and everything else for a reason. So let's just write this down. I have a half times mass, times the velocity, but velocity is given by HR, so this is HR squared. And then G times the mass of the object. This is the mass of the region from which the object is expanding away from. What's the mass of an object? It's its density times its volume.

This is density. It's a Greek letter, row representing density. And then the volume of a region, of a spherical region, I told you it's like R cubed, but you may remember from high school that it's actually 4 pi over 3 R cubed. The volume of the sphere is 4 thirds pi R cubed.

So this is the total mass pulling that object in, divided by R, and this is its speed. And now you have these two quantities, quantities, and the question is, is this positive or negative? It's a little bit of algebra, but it's not too complicated. I can say, well, if this is the total energy, let me multiply it by two and divide by mass to get

rid of this mass. So the total energy per unit mass of a given galaxy, we're only interested in whether this is positive or negative, so we can multiply or divide by positive numbers. If we do that, we get rid of the factors of a half and m, and we get 8 squared R squared minus this quantity here. And now we realize we have R squareds here. R squared is a positive number. So let me divide by R squared.

And then I get this quantity equals 8 squared minus 8 pi g. This is the strength of gravity, the average density of galaxies in the universe divided by three. Why did I do this? Well, I can call this quantity, this 2E over M, I can give it a constant. I can just call minus k. And I'm going to get minus k over R squared equals this. The reason I did this is what we have just derived is Einstein's equations for an expanding universe.

If K is negative, that means this quantity is positive. The total energy is positive. And if the universe is dominated by matter, this positive energy means. the universe will expand forever. So we're able to understand the dynamics of the universe just by treating a small spherical region in it. If K is positive, the total energy is negative and the universe will collapse.

What's really amazing is this is the equation that Einstein derived in very different ways for a curved universe, and K is the curvature of the universe. And we now learn that if K is negative, that's called an open universe, the universe will expand forever if it's made of matter. If K is positive, the total energy is negative, the universe will fall into itself.

That's called a closed universe. And it turns out positive curvature means that the universe closes in on itself. Literally, if I look out far enough in that direction, I'll see the back of my head. And in such a universe, if it's made of matter, eventually it's doomed to collapse again. If K is negative, so we have an open universe that will expand forever. So the future of the universe can be determined. This was a little bit of complex algebra for some people, but you can spend time working through it.

The point is, just by considering a spherical region and what happens in a very simple way to a coin and applying it to the universe itself, we come up with an equation which determines the future of the universe. And that's really the fundamental equation of cosmology. For the last 60 years, people tried to determine the rate of expansion of the universe and the average density of the universe

to see what the future of the universe should be. That's why I became a cosmologist. I thought if I could determine the average density of the universe by thinking about things like dark matter, I'd be maybe one of the first people to determine the future of the universe. Turns out other complications arose that changed that picture. But it's basically, this was the basis of modern observational cosmology over much of the last 60 years,

just from this simple picture of thinking of the universe as a sphere, or a region of the universe as a sphere. The last thing I want to talk about that takes us again from the simplest bits of physics to the forefront of modern physics, just as I did in scaling before, is a thing called dimensions. The universe is simple. Physics is simple because there are only four fundamental quantities that have characteristics that we call dimensions. Time, space, matter, and charge. All physical quantities can be expressed in terms of basically time, space, matter, and charge. velocity is distance over time space over time every physical quantity depends upon only these four factors and that means there's only a finite number of independent physical quantities which why which is the reason we can create equations that relate very disparate quantities but by thinking of dimensions

you can often come up with the answer to questions that once again you might not have gotten otherwise let me give you an example so i call time t length space l mass M and charge Q. All physical quantities can be described in terms of these four quantities and allows us to make simple relations. There's a finite number of independent physical quantities. All other physical quantities are interrelated, meaning their physics is simple. It's the ultimate free lunch.

Thinking about dimensions appears to give you so much power that you wouldn't have otherwise that it's hard to fathom. and I want to give you a few examples at the forefront of modern physics. But first I want to show you how you can solve simple questions just using what's called dimensional analysis. So the examples I want to talk about are spherical cows, getting the answer without knowing anything and uncovering the unknown, literally uncovering the unknown universe today. So let's go back to the beginning.

Thinking of scaling its volume. Okay, here's a real cow and there's a super cow. Again, I'm going to show it as a cow, but once again, we now know that the volume of a cow is given in terms of dimensions length cubed, and the volume of a super cow is eight length cubed. So you can see that volume, the fact that volume is cubes of distance determines relationships between different objects. Okay, that's trivial.

But let's get a little less trivial. Let's ask the following question. There are two quantities that describe air at a given temperature. It's pressure and its density. Fine. Big deal. Now, if I were to ask you as a physics student, what is the speed of sound and air? Based on that, you might throw up your hands because Newton was one of the first people to calculate this,

but you have to use a lot of physics to think about it. But dimensional analysis allows us to get roughly, as you'll see, the answer right, without knowing anything, except the dimensions of these quantities. What are the dimensions of pressure? Pressure is the force per unit area on an object. We think of it as pounds per square inch, say the air pressure, 15 pounds per square inch here at sea level, which is where I'm at right now. So pounds per square inch.

So this is force per unit area. But what are the units, what are the dimensions of force and the dimensions of area? What are the dimensions of force? To think of the dimensions of force, all you have to remember is probably the only equation you can remember from high school physics, F equals M.A. Right?

F equals M.A. So the dimensions of force are the dimensions of mass times acceleration. And the units of area are length squared. What are the units of acceleration? Well, velocity is length over time. acceleration is the rate of change of velocity. So it's L over T divided by T.

It's L over T squared. So acceleration is V over T and the units of V are L over T. So the units of force are ML over T divided by T because this is mass times the units, the dimensions of velocity, divided by the units of time. And then divided by length squared, well, that's just a little bit of algebra. And that gives you mass divided by length times times squared. Those are the units. Those are the dimensions of pressure.

If you ask what are the dimensions of pressure? It's the dimensions of mass divided by dimensions of length and times square. What are the dimensions of density? It's mass per unit volume. Pounds per, actually it should be pounds per cubic inch. forgive me. Mass per unit, volume. Well, volume is length squared, so the units, the dimensions of density are mass over length cubed. We have the dimensions of pressure and the dimensions of density. What are the dimensions of speed? It's length over time. Well, if these are the only two

quantities, then speed has to be some relationship between, has to be determined by some relationship between density and, density and pressure. Well, let me look at, let me, given this, let me just look at, say, pressure over density. What are the dimensions of this quantity? It's this divided by this. And that happens to be length squared over time squared. You can do it yourself. And length squared over time squared is a dimensions of velocity squared. So if I were going to guess, I would say that the velocity, speed of sound and air is, it goes as a square root of pressure over density. That's what Newton derived, and that's exactly right.

Now, there may be some unit here, there may be some number, usually between 1 and 10. It happens to be a number that's close to 1. So this analysis did not give us the number in front, but it told us how the speed of sound and air depends upon pressure and density without doing any physics. Just knowing the dimensions of these things, the speed of sound must depend as a, must change as a square root of pressure over density. So you get a lot out of very little. Okay, well, this is a basic physics question.

But physicists continue to use this idea of dimensional analysis, the last in the triad of gestalt problems, I told you about orders of magnitude, approximation, and dimensions. Dimensions is sort of the crowning glory of all of this, Gestalta physics, because it allows us, it takes us basically to winning a Nobel Prize. I want to show you how recently a Nobel Prize was won in some sense by dimensional analysis. Now, in order to understand this, you have to take dimensional analysis to its extreme. The people, the physicists who take dimensional analysis to the extreme are particle physicists.

Because particle physicists in some sense are the laziest physicists, because we don't want to have to remember three dimensions, length, mass, and time. Wouldn't it be nice if there's only one fundamental dimensional quantity in physics? Well, in a way there is, because there are two fundamental universal constants in nature that relate length, mass, and time. One is the speed of light. We know the speed of light is a universal constant, three times 10 to the eighth meters per second. And therefore, you can describe all distances by equivalent times. For example, the length of your arm is one meter, is also can be described as three times 10 minus nine seconds. I could say the length of my arm is three times 10 to minus nine seconds.

What do I mean to that? Well, that's how long it takes light to go from one end of my arm to the other. Because lights, the speed of light is universal. The same for all observers. It has absolute meaning. If I say my length is three times 10 and minus nine seconds, anyone can then plug in the speed of light and find out that's one. meter. So suddenly two independent quantities, length and time, are not independent anymore. I can

describe every length in terms of an equivalent time or every time in terms of an equivalent length. That's one of the two fundamental constants in nature. The other is a constant called Planx constant, the fundamental quantity in quantum physics. And Plank's constant comes about by saying the energy of a given wave, a quantum mechanical wave, is Planck's constant times its frequency, the frequency of oscillation, and Planck's constant is a number. Now, we know what the units of energy are,

because we can remember E equals MC squared. Now, C, remember, is length over time. But in my world where I say every equivalent length is equivalent to an inverse time, or time, length, sorry, every equivalent length is equal to an equivalent time, then when I consider length over time, I can consider that to be dimensionless, right? Because I've equated the dimensions of length and time. So C, I can set equal to one. I can just, I can find a set of units where I set the speed of light to be one. And in that case, I can say the dimensions of energy are

effectively just the dimensions of mass. Because remember, I've equated the dimensions of length and time by the speed of light. Whenever I need to put things in, I can always plug the speed of light in after the fact. But if I make it one, and remember that, you know, every length can be described by an equivalent time, then effectively in this weird world, the dimensions of energy is mass. What are the dimensions of frequency? Cycles per second. One over time. So the dimensions of frequency are one over time, if the dimensions of energy or mass and the dimensions of frequency or one over time, then you can see that the dimensions of Planck's constant are mass times time.

So, Plank's constant is a universal constant, and it means for every mass, I can get an equivalent time by just equating, by using Planx constant. So I can describe every unit of mass, every object that has a certain number of kilograms, in terms of a time, and just remember that they're connected by Planck's constant. And that's a universal constant of nature. So with these two constants, basically I've reduced length, mass, and time to one quantity, because every length can be described by a time, and every mass can be described, in this case, by an inverse time, by just remembering that I can get the final correct answer with the right

units by plugging in certain enough powers of the speed of light and Planes constant. So in that sense, any time all objects in the universe, all physical observations can be expressed as your length, time, or mass. And I can go between those dimensions by using the two constants of nature. Okay. I realize that's probably the most complicated idea I'm talking about in this lecture. So I'm going to maybe stop this for a second and think about it. But what it really means is any time I give you a quantity, I can think of it in terms of the equivalent lengths or the equivalent times or the equivalent mass

just by multiplying by powers of the speed of light and Planck's constant, which are universal constants of nature. And because of universal constants of nature, every quantity given in terms of one set of dimensions can be described by an equivalent quantity in terms of another set of dimensions by using those constants of nature. In 1974, a new particle was discovered, called the J-Sy-Particle. And it was a mass about equal to the proton mass,

and its lifetime was 7 times 10 in the minus 21 seconds. Okay, big deal. Why should we be surprised about that? What so makes that discovery so important? Well, remember in particle physics I told you, for every mass, can consider an equivalent time. And if I use that, it turns out dimensional analysis says the equivalent time for that mass, if I plug in all the powers of H and C, a mass of a proton mass, the equivalent time is 10 to minus 25 seconds. So an object that's unstable, remember the proton

is stable, at least as far as we can tell, but an object that's unstable that has a mass of the proton, all other things being equal without any very large or small numbers should live about 10 and minus 25 seconds. And it turns out that if you look at the other elementary particles that we create in accelerators that are around the proton mass, they have a lifetime about this, because there are no large or small numbers. So if there are no larger small numbers in front of the decay time, then the number is the average expected lifetime of such a particle is 10 minus 25 seconds. This particle was discovered to be 10 to the minus 21 seconds, a thousand times longer, more than a thousand times longer. The experimentalists who measured this said it would be like

going into a jungle than describing a new tribe of humans discovering a new tribe of humans who lived 10,000 years instead of 100 years. There's something anomalous, there's some new science. There's some new biology in this case, but in this case, some new physics. Why would a part particle, having a massive proton lived 10,000 times longer or 3,000 times longer or so than you would expect. Something must be happening. It was such a big deal that the Nobel Prize was given two years later for this discovery. And ultimately, this discovery of the physics of why this particle lived so long helped us understand one of the four forces in nature, the strong force, a very weird property of the strong force that allowed you to get a very large or small number

out of the strong force that it turned out the strong force for objects to get quarks they get closer and closer together the force actually gets weaker this quantity called asymptotic freedom which later on another 30 years later produced another Nobel Prize but the bottom line is if you didn't use dimensional analysis then measuring the life of this particle would have just been, okay, wouldn't have meant anything. But dimensional analysis immediately told us there was something strange going on. And that's what's wonderful of dimensional analysis. If you do a calculation, if you make an estimation in physics or in everyday life, and you say, okay, well, this should last 100 seconds, and it lasts 10,000 seconds, it tells you there's something

you're missing in the calculation. And if there's something you're missing in the calculation, there's new physics or new biology or new chemistry. So it's one way to determine if there's something new. And if there isn't, it gives you a way to determine very quickly the approximate answer to any question, just by using dimensional analysis. As I did, the approximate answer to the question of what's the speed of sound in a gas of pressure P and density row gave us a quick answer, assuming there were no large or small numbers, the velocity should go as a square root of p over O.

In this case, the fact that things didn't work, the fact that naive dimensional analysis gave an answer, which is vastly different than the observed answer, told us there was new physics and pointed us in a direction to look for what the source of that new physics was, and ultimately, the experiment garnered a Nobel Prize, the theory would garnered a Nobel Prize,

because it helped us understand the fundamental nature of one of the four forces of nature, the strong force. Ultimately, if you wish, motivated the discovery by dimensional analysis. So you can't get more modern than that, but here is this very basic quantity dimensional analysis, a very basic tool that physicists use to describe a complicated world leading to some new discovery. So to summarize then, these three features of the Gestalt of physics, the overall, Powers of 10, which allow us to calculate, estimate things incredibly fast, and get reasonable estimates accurate to within a factor of 10. The idea of approximation, of throwing out information

and picturing a system as simply as possible, in the cases I showed as a sphere, nothing more simple than that, described by one quantity of R, you then do estimates based on that simplification, and if you come up with reasonable answers, it tells you that what you've thrown out is probably irrelevant. If the answers you get are unreasonable, it tells you you've thrown out the baby with the bathwater, that your approximation was too severe. And then finally, dimensional analysis,

which is really the reason physics is simple, the fact that there are only four dimensions in nature, four dimensional quantities in nature, allow us to make relationships with different quantities, allow us to determine what's sensible and what isn't, and allows us to make really, miraculous estimates of how the universe should behave and what really gets to be interesting in physics is when your estimates are wrong. Because when your estimates are wrong, it tells you you're missing

something and there's something to learn. And that's how science is all about, is learning what's new in the world. So those three tools allow us to take a complicated universe and try and make sense of it. Simplifying nature by powers of 10, approximation, and then ultimately, dimensional analysis, allowing you to do what seem like miraculous estimates of the universe and finding out most interestingly when they're wrong. And when they're wrong like this, you find out there's new physics. There's something new to learn. And that's what science is all about. The most exciting part of being a scientist, ultimately in some sense, is being wrong, because it means there's something new to learn. And this Nobel Prize for the experiment was then

later on Nobel Prize was given for solving that mystery and because that's the heart of science is in this case discovering that nature are one of the four fundamental forces of nature by more or less using dimensional analysis. So that's where we're at. We take a complicated universe and ultimately using those three tools

leading finally to dimensional analysis to take this remarkably complex universe and make it comprehensible. That's the gestalt on which all of modern physics is based. I recommend you go back and think about this and maybe try and do some lightning fast calculations to amaze your friends and neighbors. But these tools will be ones we'll use over and over again

as we try and understand the details of the physical world. And next time, I want to talk about a feature of the physical world that we've discovered using these tools and others, which involves symmetry, a concept which has a very different, meaning in physics than perhaps in the rest of life, but it turns out symmetries guide our understanding of the modern world. Thanks. We'll see you next time.

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