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Fundamental Constants Of Nature Part 01

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TITLE: Fundamental Constants of Nature - Part 01 CHANNEL: SWAYAM Prabha IIT Madras Channels DATE: 2026-04-22 ---TRANSCRIPT--- [music]

Swayamprabha, digital India, educated India. [music] All right, so let’s start. Let me start by asking you what we mean by a constant. If I say a physical constant of nature, what do I mean by that? Something that changes, doesn’t have a fixed value, doesn’t change at time, etc. Then there are these so-called fundamental constants of nature. So, let us start with that and then we’ll go on to various other things. The so-called fundamental constants of nature. Can you give me an example of such a constant from your physics program, for instance? Which gravitational constant? Small G or big G? Capital G, Newton’s universal gravitational constant. Very good. So, let’s write that down. Does anyone know its value? Minus 11 what? Very good. So, what is important here? The 10 to the minus 11 or the factor in front of it? 10 to the minus 11 because that gives you the order of magnitude of this quantity, okay? Is this experimentally determined or is it determined to an infinite number of decimal places or So, it’s within some experimental accuracy that this is true. Capital G is an excellent constant of nature, universal gravitational constant, okay? We’ll come back to this. Any other constant? Speed of light changes depends on the medium. Speed of light where? Speed of light in a vacuum, no material at all and yet you have an oscillation of electric and magnetic fields and you have light propagating in it. What do you denote it by? C, all right. Small c, okay? What is the value What is its value? 3 * 10 to the power 8 m/s. So, we’ll agree to use standard international units because otherwise there are lots and lots of systems of units, but these were very arbitrary units and even these units are arbitrary, the meter. For how is the meter defined, for instance? Right, it used to be determined by a the mark between two on two markers on a platinum bars kept in Paris and so on under temperature control, etc. Before that they had units like yards and furlongs and feet and so on which are arbitrary things like the distance between the nose and the tip of the fingers of some king or the other. All that is gone now. We base our constants of nature on physical phenomena like this, for instance, like the wavelengths, the number of wavelengths and so on. Okay. What other constant of nature can you think of? What’s that denoted by? I’ll write it like that with a slash. What does that stand for? Yeah, h slash or h cross stands for h over 2 pi because that occurs more often than h. All the formulas which involve h generally have this 2 pi also added to it, so you might as well give it a separate symbol and call it. So, when I say Planck’s constant, I mean h cross. What’s the value of this quantity? 10 to the minus 34 what? Joule second. Joules per second or joule second? Joule second, okay, very good. That’s awfully small, 10 to the minus 34. Because a joule is like the kind of energy you see in daily life. When I take a small piece of kilogram weight and lift it by a certain amount like a meter or so, it’s about a joule of energy, potential energy. But this 10 to the minus 34 is incredibly small compared to it. It’s telling us something that this is so small and this is telling us something, we’ll come to that in a minute. Anything else? I’m going to draw a bar here and write whatever constant you give me here on this side and I’ll explain why. What’s the constant you mentioned? Gas constant. What does that refer to? What’s meant by the gas constant? Yeah? Uh what do you call that energy when you increase by 1 Kelvin? You take a thermodynamic system, you increase its temperature by 1, then what do you call it? What’s the rate at which it’s increasing? The rate of change of the energy with respect to temperature is called If I supply an amount of heat dQ or delta Q to a system and its temperature increases by delta T, what is delta Q over delta T? It’s the specific heat capacity under whatever conditions you presume. So, R is coming from some specific heat of an ideal gas in some sense. All right, but it refers to an ideal gas. What’s meant by an ideal gas? If that is true, if there’s no interaction between molecules at all, then if I take a piece of material and put a container and put some ideal gas in it, whatever state I start with will remain as it is. It won’t change at all. It won’t reach thermal equilibrium at all. The fact is it’s in thermal equilibrium, right? So, that means there are some interactions. What kind of interactions in an ideal gas? Oh, that’s a very good point. There’s collision with the walls of the gas. That’s certainly true, but there are a lot more collisions with the gas molecules, between the gas molecules themselves. So, even an ideal gas, while the attractive interaction at long distances is negligible or neglected, does undergo elastic collisions. The particles have to undergo elastic collisions. So, there is a contact potential. There is an interaction, but it’s a repulsive interaction. Okay? That’s an ideal What else is an ideal gas? What other condition do you need? When we say PV equal to RT, that’s the equation of state of an ideal gas, what gas are we talking about? We’re talking about point particles undergoing elastic collisions with each other. Okay? And what else is involved? The gas obeys classical statistics, Maxwell-Boltzmann statistics. You’re familiar with the Maxwellian distribution of velocities? Yes, no? No, we’ll come, we’ll talk about that also, okay? These particles are assumed to be classical particles. They don’t obey quantum mechanics. So, that’s the assumption of a classical ideal gas and for that such a gas the equation of state for 1 mole of the gas is PV equal to RT. And all the other laws, Charles’ laws, Boyle’s law, etc., they follow from this single equation, PV equal to RT, which by the way is extremely hard to derive from fundamental principles. It’s not so easy. It’s derivable, but not so trivial at all. Whatever it is, this R refers to a classical ideal gas. It comes from that gas constant. What other constant can you think of? Stefan’s constant. Now, we’re getting a little more very very very technical. Stefan’s constant, what does that say? It essentially says that the internal energy of a gas of photons in a black body cavity in thermal equilibrium is some constant of proportionality multiplied by T to the power 4. That’s the Stefan-Boltzmann law. And that constant of proportionality essentially has the Stefan’s constant sitting in it. So, it refers to a system. It refers to black body radiation. This refers to a classical ideal gas. What other constant can you think of? Hm? Planck length, where did that come from? We’ll come back to Planck length. It’s going to be derivable from these things, so we won’t talk about the Planck length at the moment. What about the mass of an electron? That doesn’t change. Every electron is exactly the same mass. So, let’s write it mE. Stefan-Boltzmann constant was sigma. What other constant can you think of? Uh the charge of an electron, that’s a very good guess. So, all charges of elementary particles come in units of the electronic charge, either with a plus sign or a minus sign. So, that’s a quantized charge, definitely. Okay, that’s a constant of nature. What else? Permittivity of the vacuum. How about the magnetic permeability of the vacuum? Actually, these are units which dependent on units. Because we’re using standard international units, you have this epsilon naught mu naught. You could set them all equal to 1. You could use units such that these are all set equal to 1 and so on. Whatever it is, they refer to properties of the vacuum. Anything else? We’ve got a wild set of very large number of things. You can talk about the mass of a proton, the mass of a neutron, the mass of a uranium nucleus. You can go on and on in this fashion, right? So, there are lots and lots of constants of nature, but there’s something very special about these three constants of nature that is not shared by the rest of these constants. There are literally hundreds of these things, but that’s they don’t they have a property which is not shared by these three. There’s something fundamental about these three. What is that? Take a guess. In some sense, they’re universal constants. They don’t refer to any particular system at all. They refer to deep properties of space-time itself. Now, let me explain what that is in words without saying why it’s so. The existence of Planck’s constant tells you that there are fundamental limitations on what we can say about the universe, physical universe. There are fundamental limitations like the Heisenberg uncertainty principle about the universe. In essence, the existence of Planck’s constant is telling you that the universe is quantum mechanical. Not doesn’t obey the laws of classical physics except as an approximation. So, the quantum nature of the universe is buried in the fact that H cross sits and is not equal to zero. It tells you that there are fundamental limitations on what we can know about nature, what we can measure about systems, etc. For instance, among other things, it says that the position and velocity of a momentum of an electron or an elementary particle cannot be specified to infinite precision simultaneously. It is not a fault of the experiment. It’s not a fault failure of our measuring apparatus or anything like that. It’s an intrinsic limitation in nature which says nature is intrinsically quantum mechanical and all the consequences that follow. So, this is telling you that nature is quantum mechanical. So, it’s telling you a deep deep property of the physical universe as we as we know it. The quantum nature of things. There may be reasons, there may be approximations under which the classical physics holds good and Newton’s laws hold good and so on and so forth. But, deep inside it’s quantum mechanical to the best of our knowledge. What does the existence of C tell you? This is the speed of light in vacuum. But, it also says it’s a fundamental speed in the sense that nothing can go faster than that. So, it’s a limitation in nature. It says nature is relativistic in nature, not non-relativistic. Nature is intrinsically relativistic, yes. It so hap Yes, that’s called the special theory of relativity that tells you that that’s the ultimate speed possible. Because it turns out in relativity, they can show that if you reach the speed C, you need an infinite amount of energy to proper to accelerate anything to that speed, anything material to that speed. Okay? It so happens and is lucky for us that there exists an entity which actually travels with that speed and we call that light. But, with or without light, there is this fundamental limitation in speeds. C is the ultimate one that you can have. Because the momentum of the particle becomes infinite by that time. Yeah. Yes, it also says simultaneously that particles which do not have what’s called a rest mass in invariably can only travel with speed C. The photon has zero rest mass. Okay? There are other particles if you have another particle which has zero rest mass, that will also travel with the speed of light. For a long time, it was believed that neutrinos were massless. In which case, the speed could only be plus or minus C. The the velocity could only be plus or minus C in some direction. But, now we know that the neutrinos, at least the lightest except for the lightest one, the neutrinos have mass in general. So, they cannot travel with the speed of light, such neutrinos. Okay? So, anyway, this is telling you something about the nature of the universe. It says nature is relativistic in its in its character. So, relativistic. You need to use special relativity or general relativity. So, this is telling you that at the tiniest levels, quantum mechanics takes over. This is telling you that the universe has quantum intrinsic limitations in it like the limitation on the speed, maximum C. What is G telling you? Says any two objects will attract each other with this force given by Newton’s law of gravitation, etc. But, deeper than that, Einstein, as you know, showed in general relativity that gravitation was a consequence of the curvature of space-time, so geometrical effect in some sense. Okay? So, capital G is telling you something about the structure of space-time except that this so weak, this capital G in absolute terms compared to other interactions is so weak that you require a lot of mass, a lot of distortion of space-time before you can feel its effects. It is true that if I step across step over this step [snorts] by mistake and fall down, gravity is going to affect me quite severely. That is certainly true. But, it takes the whole earth to produce this gravity. And it’s a weak force, really weak force. Every time you take a bar magnet and pick a pin off the table, you’re outpulling the entire earth with this little bar bar magnet. So, gravitation is a much weaker force than electromagnetism, for instance, or all the other forces. To see strong gravitational effect, you need a lot of mass. You need a huge amount of mass packed in a small volume. The gravitational effect of two the gravitational repulsion between two electrons set meter apart is infinitesimal compared to that between the electro that the force between the electrostatic force between them, the repulsion due to the electrostatic force between them. It’s 10 to the 38 times stronger, the electromagnetic force. Okay? So, gravitation is very weak except that if you have large masses like galaxies, then gravitation comes into its own. Or you have very dense matter like neutron stars or black holes, then gravitation comes into its own. Okay? So, gravitation is telling you something about the large-scale structure of space-time itself. So, this is describing space-time. And together, these are the three fundamental constants, really fundamental constants of nature that we’ve come across. Yeah. It’s It’s very common to say that quantum mechanics applies at the atomic level or the microscopic level, but not at the classical level, not at the macroscopic level. This piece of chalk obeys classical mechanics, etc. This is certainly true, but the consequences of this quantum mechanics appear everywhere. The reason I don’t fall through this floor, why matter is impenetrable, why solids do not penetrate each other, and so on. Why do you think that is so? Because after all, this consists of atoms, they all consist of atoms, and atoms are mostly empty space. How big is a nucleus of an atom typically? 10 to the minus?

  1. Yeah, it’s a Fermi. 10 to the minus 15 m is a Fermi. So, 10 to the minus 15 m is the size of a nucleus, roughly speaking. Now, what’s the size of an atom typically? So, that’s five orders of magnitude bigger. Most The atom is mostly empty space. So, if I put two atoms on each other, collide, they should pass right through each other most of the time. Why is that not happening? That’s very weak compared to what the effect of pushing through is. I can go overcome that and push it through. It’s mostly empty space. It’s true that the electron clouds will repel each other and so on. That’s a negligible interaction. I can force my way through it. Why is matter impenetrable? It’s impenetrable because of the Pauli exclusion principle which says the electrons cannot be in the same space state at the same time. Two electrons cannot be in the same state. So, that acts as a statistical repulsion which makes matter in solid matter impenetrable. That’s quantum mechanics. It’s completely quantum mechanical effect. Classically, you would not have magnetism, you would not have electric you would not have electrical conduction, you cannot have impenetrable matter, etc. classically in pure classical physics. So, while it is true that quantum mechanics operates at the tiniest levels, it’s the its consequences do not. Its consequences are in our world around us, all around us. Uh I’ll come back to that. Remind me in try to give another talk on this about the Pauli exclusion principle. Basically, it’s because of Well, we’ll start right at the beginning. It’s because of the fact that all the physical laws we know happen to be relativistic in nature. They’re invariant under what are called Lorentz transformations. And as a consequence of that, there are certain properties that elementary particles have which arise from this Lorentz invariance. One of them is called mass and the other is called spin. And the spin is quantized. And we turns out that in some units, in units of H cross, etc., the spin quantum number can have integer or half integer values like 0, 1/2, 1, 2, 3, etc. And those which have half integer values fall into a class called fermions because they obey Fermi-Dirac statistics. Those which have integer spin fall into another class of particles called bosons which obey Bose-Einstein statistics by something called the spin-statistics theorem. And for fermions it turns out that they can only have occupation number 0 or 1 for any given state which is the Pauli exclusion principle. So it’s buried deep in quantum field theory in something called the spin-statistics connection. So that’s the ultimate reason. Going back to my question, if you take a few bricks, put them together, then it behaves like the entire system behaves like a bigger brick, right? Then if you stack these all small atoms to create like ourselves, then logically thinking, they should behave like they would at the microscopic level. Why the distinction? It’s a very, very deep and very good question. It turns out, and this is the crux of a lot of modern physics, it turns out that a collection of identical objects can have properties which individual objects cannot do not have. It’s called emergent behavior. For instance, I take a set of water molecules. There’s an interaction between individual two every pair of water molecule. There’s a given interaction which doesn’t change. Attractive at long distances, repulsive at short distances, etc. And yet as you change the temperature, this water goes from a solid state to a liquid state to a vapor state. It’s the same water. It’s the same interaction. But this magic happens because of collective effects called emergent properties. This piece of chalk is white. On the other hand, individual atoms in it or molecules in it have no color at all. So where does this property called color start? How many molecules do we need before we see color? There is no gray There is no definite boundary at all. Exactly as the fact that there is no sharp boundary between classical and quantum mechanics. It’s a gradual onset of quantum behavior, more and more quantum behavior as you have fewer and fewer particles. But the collection of such particles, a large enough collection, can have properties which the individuals do not have. Okay? So that’s responsible for the phases of matter. And by the way, when you study, how many phases of matter are there? Five. Five. But at least we’ve been told five. You’ve been told five. How do you distinguish the What are these five states of matter? Solid, liquid, gas, plasma, and Bose-Einstein condensates. Wow, is that what they’re saying these days?

[laughter] Okay. Do you think phase of matter is the same as state of matter? Do you think there’s a difference between these two? Sometimes you hear phase of matter, sometimes you hear state of matter. What would you say? Sir, one of them refers to it not being able to being converted. I don’t know if I’m putting them in very good words. But one of them refers to I think state refers to the fact that if it’s in gaseous in the gaseous state, then it can’t be converted to liquid, but gaseous phase represents from liquid water as it evaporates. I’m not sure I understand what you’re saying, but let me say it this way. Would you agree that if I take a crystal, there are different kinds of crystalline order that you can have. Iron, for instance, can exist in something called a body-centered cubic phase state phase, can exist in something called face-centered cubic state, and so on. These are phases of different phases of crystalline materials. Okay? They are phases, genuine phases. But then if I say that a solid is something which has shear modulus, and a liquid is something which cannot support shear, just shear under shear it flows, that is one distinction I can make. On the other hand, if I take a lorry load of bricks and pour it down, it flows exactly like a liquid. Or take brick powder and pour it out, it pours exactly like a liquid. And yet it’s in a solid phase. So phase and state are completely different objects altogether. You can have a granular state, you can have a crystalline state, you can have a polycrystalline state, all of the same phase. There are many, many phases that matter has, different kinds of matter have, but states of matter is a smaller number. And that’s presumably what you’re talking about when you say it’s got the solid, liquid, gas, plasma, and this weird thing called Bose-Einstein condensate. There are actually many, many such phases possible, very large number. We’ve not managed to succeed in classifying all of them yet. There are new phases appearing all the time called topological phases. So phase of matter is very different from state of matter. You’ve got to be careful about that distinction. Okay, yeah. For like the Bose-Einstein condensate, Yeah. they say that it reaches to certain temperatures that it is in the equilibrium state. Now, there’s another statement they make that the kinetic energy of the particles are completely removed. How is it possible? Completely? Removed in that process. Is that possible? For it to be in equilibrium. Oh, no, no, no. It’s not that it’s static. No, that’s not true. It means it’s not moving at all. How is that possible? In a Bose-Einstein condensate, you’re saying the particles are not moving at all? No, no, no. Like it’s in equilibrium, but they say that the energy is removed from it or like something like that. How is it possible? Well, Bose-Einstein condensates, as the name suggests, would have applied to boson systems. So it’s a collection of bosons which obey Bose statistics. Roughly speaking, bosons have the behavior opposite to that of fermions. Where fermions like to stay in different states. They don’t like to stay in the same state. They cannot because of the Pauli exclusion principle. Bosons, on the other hand, would like to be in as many as as many of them would like to be in the same state as possible. They’d like to condense, especially as the temperature goes down. And it gets to a stage where there’s something called a Bose-Einstein condensation takes place, below which the number of bosons in the system which get into the same state increases gradually till all the bosons in the system are in the same ground state at absolute zero temperature, theoretically. Okay? So they’re in what’s called the ground state of the system. Not no excited states are occupied at all. Whereas in a fermion system, it’s exactly the opposite happens. Things fill up up to something called a Fermi level, and beyond that there are no states. So they have very different, opposite behaviors. And the Bose-Einstein condensate has many, many weird properties of this kind which require quantum statistics to explain in detail. Okay? Yeah. So you mentioned because of the Pauli’s exclusion state, matter is infinite, and I understand that. But why do fermions have to stay in different states just because of the spin? It’s a consequence of Again, the the deep reason for it, the reason why it happens is buried in quantum field theory. Because Fermi particles obey a certain statistics, quantum statistical relationships which are different from that for bosons. And they make sure that the occupation number of any state is either 0 or 1. Either there’s no fermion in that state or there’s just one, exactly one. Two is not allowed. [clears throat] But for bosons, 1, 2, 3, 0, 1, 2, 3 is allowed. All the all the particles could be in the same state as happens in Bose-Einstein condensate at absolute zero. Okay? Okay, let’s talk about physical dimensions now of these objects. Yeah, is there a question? What exactly is spin and what is matter? What exactly is? Spin. Uh what is spin? That’s a very, very I’ll say it in words. Okay? It’s not an explanation, it’s a statement of fact. Okay? It turns out that due to this principle called Lorentz invariance, namely invariance under Lorentz transformations of the laws of nature, it turns out that elementary particles can have two attributes, at least. One of them is a rest mass which could be zero as in the case of the photon, and the other is called spin which is an intrinsic angular momentum. So it is something like saying it is not related to mechanical motion. That’s the first thing you should get straightened out. It’s not related to saying that the electron has spin half and it’s plus half if it’s rotating clockwise and then minus half if it’s rotating anticlockwise or that assumes that this is a classical object with an axis of rotation and so on. No such thing happens at all. The The phenomenon of spin is an intrinsic angular momentum. Just as the electron has an electric charge, it has a mass, it has an intrinsic angular momentum called it called it spin. If you measure that spin indirectly through the magnetic moment of the electron about any direction, you measure what is this angular momentum about this direction, the answers you get are either plus half h cross or minus half h cross. That’s why you say it’s got spin half. It has nothing to do with spinning clockwise or anticlockwise. It has nothing to do at all with mechanical rotation of any kind. Spin is a historical name that was given to this in the 1920s when spin was first discovered, postulated, and subsequently verified. And it’s an unfortunate name. It should have been called intrinsic angular momentum, but that’s too big a name. So the originators of the name Goudsmit and Uhlenbeck called it spin. And the name stuck. Okay. I have seen in your textbook in ninth standard or tenth standard textbook the statement that uh in the in the chemistry section it says that the electron rotates clockwise it’s plus one and so on. When something is rotating clockwise I have to look at it from underneath it will look anticlockwise. So, it’s complete nonsense. And if you know anybody who’s written such a book, please call our attention to it. We’ll get him shot. NCERT book. NCERT is a law unto itself, so nobody can correct them. There are incorrigible effects. Yeah. That person should be shot and hanged. [laughter] You can measure it, but not directly. You cannot measure that angle How do you measure angular momentum of anything? How do you measure the angular momentum of a spinning top? Take an axis. There’s an axis of rotation. Find out what the angular velocity is. Find out what the moment of inertia is. Multiply the two together and then you have the angular momentum, right? That’s not what you can do for an atom. You can’t do that kind of thing for an electron for instance. But then the electron has a charge and when an object has an intrinsic angular momentum, which is called spin, it also and a charge, it also has what’s called a magnetic dipole moment, intrinsic magnetic dipole moment. So, the electron will act like a tiny bar magnet. And you can now use a magnetic field in order to measure this diamagnetic moment about any direction. And because it’s quantum mechanical, angular momentum gets quantized in quantum mechanics and it can only value have values which are half integral or integral by multiples of H cross about any axis at all. Why do we need to know what charge is or mass is or anything is? So, an intrinsic It it gives a magnetic moment to the electron. The reason why you have magnets in nature is because of that, because of the intrinsic spin angular momentum. Ultimately. Yeah. [snorts] Anytime you measure an axis you you measure a direction it’s with respect to something, isn’t it? It’s always there’s a reference. Uh tell me, when I write F is equal to MA in textbooks Newton’s equation written down in textbooks. Force is mass times acceleration. It says if you apply a force to an object of mass M, then the force is with F, then the acceleration is given by A, where A is F divided by M. This is the inertia of the object M. Right? That’s Newton’s law. Is it always true? Let’s assume that it’s true here and it’s true in Mumbai. Okay? These are vectors. Now, I ask you what is meant by vector? Direction with respect to what? Which which origin? Which coordinate system? Why should I Why should I believe your reference system? I have another reference system. Ah, now they’re getting Now we’re getting to deep waters here. When they write F is equal to MA, does this textbook also tell you what axis to choose? It’s completely arbitrary. Somebody reading it in Timbuktu will have a different set of axes. So, is there any sense in this whole thing at all? No, you have to be precise now. In what sense is somebody in Mumbai going to be find their equation to be as valid as I’m going to find it here? In what sense? That’s a circular argument. Newton’s law is obeyed because it’s frame invariant and frame invariant because of Newton’s law being obeyed. No. This brings us to a deep question which is generally unanswered in textbooks at the school level and even college level, which is what is the meaning of a vector? You say it’s a quantity with magnitude and direction and I immediately ask direction with respect to what and you say a frame of reference. Then you’re telling me it can be chosen arbitrarily. We have to make that statement more precise. Take force in a particular direction. Okay, I’ll draw two coordinate systems. So, here’s one coordinate system and here’s another coordinate system. This is observer O. This is observer O prime. This guy finds F is equal to MA. What The experiment is done by this person here in this observer and he is there say that’s his coordinate system. What is he going to find? He’s going to find a force F which is different from what I’m finding. He’s going to find some F prime. His components are different. His axes are different. So, the component numerical values of the components of the force are going to be different. So, he’s going to find this. So, what’s meant by a vector equation is that the vector is a quantity which has certain transformation properties in going from one frame to another and you’re guaranteed that if I find F is equal to MA the other person is going to find F prime is equal to MA prime, where F prime and A prime are what he measures. So, the form of the equation is not changed. The numerical values are changed, but the form doesn’t change. Which means that these quantities F and A, if they’re vectors, must carry their own transformation properties with them. In other words, if you give me F and you give me A and you tell me how that coordinate system is situated with respect to mine, I should be able to compute F prime and A prime and guarantee that F prime is equal to MA prime. So, if you want the form invariance of equations, you must express the equation in terms of physical quantities whose transformation properties are known to you. They’re called covariant objects. So, the invariance of laws requires the use of covariant quantities, which carry their own dictionary of transformation with them. That’s what a vector really means. In that definition a scalar is an object which doesn’t change under rotations. If I rotate the coordinate system, the value of the scalar doesn’t change. The value of a vector certainly changes, right? So, the next time you hear the statement that a vector is a quantity with magnitude and direction, ask the person direction with respect to what. So, it’s a very deep statement here. More generally you have scalar equations, tensor equations, etc. The moral of the story that you’ve learned is that you must express physical laws in terms of quantities which are whose transformation properties are known to you beforehand. In other words, quantities which carry their own dictionary with them. When you go from one country to another, it changes the language automatically. Okay? What are the physical dimensions of this G? You have F is equal to G M1 M2 over R squared. So, this implies that M L squared M L T to the minus two is equal to the physical dimensions of G times M squared divided by L squared. So, the physical dimensions of G are M inverse L cubed T to the minus two. Okay? What would be those units? We don’t care about the units. We’re worried about the physical dimensions M, L, and T. Okay? What are the physical dimensions of C? What are the physical dimensions of H? Well, you can use any equation which involves Planck’s constant. For instance, E equal to H new. The energy is H times the frequency. You can use that equation for instance. Okay? Then what would you get for H? It’s energy multiplied by time, joule second. So, it’s M L squared T to the minus one. Okay? From these three quantities, G, H, and C, can you find a quantity of dimensions length? How do you go about it? Well, I would assume that that quantity that you calculate which is dimensions of length would be some combination of H L and C H L and G G H C and G. So, I would say length is equal to G to the power A, that means M inverse L cube T to the minus two to the power A multiplied by A to to the power A should be equal to L to the power one multiplied by C, that’s L T inverse to the power of B times M L square T to the minus one to the power C. Let’s call it alpha, beta, and gamma. I don’t want to confuse with C. So, this means that my length is supposed to be G to the power alpha C to the power beta and H cross to the power gamma. Now, I have three equations here. I equate powers of M, powers of L, powers of T to each other and solve for alpha, beta, gamma. And you get non-trivial solutions. You discover that this quantity G H cross over C cube square root of this guy is a length and it’s called the Planck length. That’s the definition of the Planck length in honor of Planck. So, weird combination of constants G H cross and C cube square root of the whole thing. Check this out. So, this means the power alpha is equal to half the power beta is equal to [snorts] minus three halves and the power gamma is equal to half. Those are the solutions. Now, you can find the numerical value of this quantity. We’ll come to what it signifies in a minute. It’s a quantity of dimensions length, so it’ll turn out the units you’ll use it measure it in is in meters. You can put in and find the numerical value because you know the numerical values of the various constants of nature G, H, and C. [snorts] And this quantity, check this out is of the order of 10 to the minus 35 meters. That should immediately give you pause because it’s at the order of 10 to the minus 35 meters. What the hell does that mean? A nucleus is the smallest object we are familiar with and that’s 10 to the minus 15 meters. This is 20 orders of magnitude smaller than that. 20 orders of magnitude. We don’t know of anything that’s got that length. But we’ve got to now find out what’s the significance of this in a minute. Yeah. Yeah, we’ll we’ll come We will we’ll talk about that. Yeah, we we’ll talk about that. Yeah. But whatever it is, it’s much much smaller than any length we know. There’s no object that at that length that we know of. Pardon? No, they’re they’re much bigger. They’re much bigger. Indirectly, they’re much 20 orders of magnitude. So, once you found a length, you can also ask can you find a quantity of dimensions time? Can I find a quantity of dimensions time from this? Well, I have a length Planck length. If I divide by C it gives you a time. L divided by L L and T inverse gives you T. So, immediately you know that there is also a solution which tells you that there is a time called the Planck time which is equal to C H cross H cross G H cross over C to the power five. Because there’s a square root here, I divide by C square. And I can find out what this numerical value is. And this value turns out to be of the order of 10 to the minus 42 seconds. 43 seconds. Check out I’m being careless about numerical values, but check it out. The order of magnitude 42 or 43, I don’t remember. That too is incredibly small. It’s much smaller than any other time scale we can think of. What is the smallest physical time scale that you can think of? Physical phenomenon occurring on that time scale. Well, electromagnetic interactions happen on the time scale of 10 to the minus 15 seconds. Femtoseconds. Okay, we’ve got femtosecond spectroscopy now and lasers and so on. We’ve got attosecond one one three orders of magnitude smaller than that also. But 10 to the minus 43 seconds is unbelievably small. The smallest time scales we can infer directly by experiment or indirectly by experiment are what are called strong interaction decay times and they’re of the order of 10 to the minus 23 seconds. This is 20 orders of magnitude smaller than that. Okay? Now, the physical meaning of this is that this is the time scale on on length scale on which gravity and quantum mechanics simultaneously play a role. Gravitation has to do with the structure of space-time. So, this is telling you something about quantum effects of space-time. It’s telling you something about the fact that in simple terms, it’s trying to tell you that the laws of physics as we know them would probably be not valid at this range. The concept of a continuous length and the continuous time, continuously flowing time and become meaningless and in some sense, space-time looks like a foam which is constantly rearranging itself. So, the granular nature of space-time due to quantum effects is becoming manifest at the Planck length and Planck time scale. C is very much there. Yeah. No, relativity is there. C is sitting there. It’s because of relativity that you have this problem. Well, no. I should say that this at this stage to understand what’s happening in this time scale and length scale, you need a quantum theory of gravity. You need to incorporate quantum mechanics as well as gravity. And that’s a task which you’ve not succeeded in. Yes. We don’t know what happens below this time scale and length scale. We think that quantum that the concept because Planck’s constant has to do with fluctuations, quantum fluctuations, we think space-time itself is fluctuating. [snorts] In a quantum way. So, we cannot say for certain what’s happening. You can ask what happens to the mass. What if I what if I now calculate the Planck mass? So, M Planck turns out to be C H cross over G square root of Yeah. By that time, no, gravity is playing a strong role here. Special relativity is true for non relative for special relativity is true in the absence of a strong gravitational fields. So, you really have to do full general relativity with quantum mechanics at that level. We don’t even think it’s the right theory. So, what about the Planck mass? You can this turns out to be C H cross over G. And this is of the order of 10 to the minus eight kilograms. Does that strike you as funny? Because this is the smallest length scale I can think of. This is the smallest time scale I can think of. Below that, quantum fluctuations take over and I can’t talk about definite lengths and time anymore time period intervals anymore. This, on the other hand, is 10 to the minus eight kilograms. That’s almost macroscopic, 10 to the minus five grams. You can weigh it on a scale. What’s the mass of an electron? 10 to the minus 31, something like that. 30 or something like nine nine times 10 to the power minus whatever, 31. That’s enormously small compared to this guy. What’s the mass of a proton? Minus 27 Minus 27 10 to the minus 27. That is enormously small compared to 10 to the minus 8 kg. So while those are the smallest length scale and time scale you can think of, this in some sense is the largest mass an elementary particle can have. And matter and energy are equivalent in relativity, they’re convertible to each other. So it is that length at which it is the largest possible Schwarzschild radius that you can have. If you pack in as much energy as possible into a small smaller and smaller region of space volume LP cubed for instance, that would be a natural length scale. If you pack in this much energy in the system, beyond this energy becomes a black hole and disappears. So in some that sense, this is the largest mass an elementary particle can have. While the other two are the smallest length and smallest time scale you can think of. Okay? So you already see that there are limitations as to what we can do and what we can’t do. Till we tackle the problem of quantization of gravity properly, till we understand it properly, and people have been trying for the last 100 years, we’re not through yet. We’re not through yet. Many alternatives have been proposed, strings, higher dimensions, blah blah blah, etc. We haven’t gotten very far. But we understand what’s happening and why it’s happening. The problem lies in the fact that we don’t fully understand quantum mechanics. We have a set of rules which are very efficient which calculate and describe everything. We have a standard model of particle physics and so on, but we don’t have what’s called a theory if at all such a thing exists. So I’m saying this particularly because I want you people to understand that problems are open yet. We barely scratched the surface. This whole enterprise called science is less than 400 years old. We have a long way to go. Yeah. Sir, why is there a distinction between [snorts] mass and then length and time? Because for length and time you can immediately tell that it’s the smallest possible scale. But for some reason, mass alone is the largest possible scale. Why is this distinction even there? That’s the way it is. Largest and smallest depend on what you call the number. If you talk about the inverse of that mass, it becomes the smallest. Right? It’s like absolute zero of temperature. That by the way is a mistake, we shouldn’t have done that. We made a lot of historical mistakes, just as we named spin the wrong way and gave it a misleading name to it and so on. Similarly, this business of absolute zero of temperature, you can’t reach it in a finite number of steps and so on and so forth and absolute zero is some limit. Limit, etc. There’s no such thing. What we call temperature should have been called the reciprocal of the temperature. Then this absolute zero would have become infinite temperature in the new terminology and then we all know we can’t reach infinity. That wouldn’t be easy, but the fact that the boundary is set at zero, you ask why can’t I go to the absolute zero? The mystery arises because you called it the wrong name. You should have called it infinite temperature. Yeah. Is it possible to get a relation between the mass, the time, and the length? Seeing that this it’s clear that mass is the inverse here. Is there some relation that you know involves these three? I’ve got these three numbers. I’ve got these three numbers for which I’ve given some physical interpretation. I’ve said capital G occurs because of the large scale structure of space time is controlled by gravitation. C occurs because the universe is relativistic in nature and H cross occurs because the universe has got to obey the rules of quantum mechanics. That’s my input, right? And I’ve got these three constants. What else can I do with them? I found mass, length, and time because we are more familiar with those quantities than capital G. We’re not so familiar with Newton per meter squared per something, etc. We chose M, L, and T as our fundamental quantities. Okay? The interesting question that arises is can I find a dimensionless combination of G, L, and H? And H cross, G, C, and H cross. Can I find a dimensionless combination? What would that mean? It would be an absolute number. So can I have this? Can I Can I have G to the power alpha, C to the power beta, H cross to the power gamma is equal to one on the right hand side? No dimensions. M to equal to M to the zero L to the zero, E to the zero. Can I find alpha, beta, and gamma such that I have completely dimensionless pure number over there? Do you think that’s possible? Try it out. But if you did find such a number, remember that that number wouldn’t be independent of units. If you had such a number, it would be completely independent of what units you chose because it’s a pure number. The number three for instance doesn’t have any units. So this means there would be a special number in nature. If it’s a non-trivial number, if you got a solution, then this would be a special number. 42 maybe, with apologies to Douglas Adams. 42 is a special number. 6 and 1/2 is a special number or something. Do you think this is possible? What do you think on general grounds? What’s your educated guess? Do you think there’s a special number in nature? What’s that number? This is not the golden ratio, it’s not the golden It would be a special nature, right? Here’s the fundamental constants of nature, they’ve combined conspired to give you a dimensionless absolute number. You don’t think it’s possible. It’s not possible because you solved these equations M to the alpha, I mean G to the alpha, beta, L to the C to the beta, etc. And you put that equal to either length, mass, or time. So you put a one or a zero zero or a zero one zero or a zero zero one on the right hand side and you got solutions. Non-trivial solutions. That was a set of inhomogeneous simultaneous equations and you got non-trivial solutions. The homogeneous set of equations cannot have a solution in that case. Must only be the trivial solution. Isn’t that true? If you have two simultaneous equations AX + BY = C, CX and DY DX + EY = F, either the homogeneous set has a solution an infinite set of solutions and the inhomogeneous one doesn’t have any solution, or the inhomogeneous equation has a unique solution and the homogeneous set has only the trivial solution. It’s called Cramer’s alternative. Cramer’s rule. By that rule, it’s clear that since we found an LP and TP and MP, it’s clear that there cannot be a dimensionless combination at all. It’ll turn out that the solutions are zero, zero, and zero. But M to the zero, T to the zero, L to the zero, C to the zero, whatever, is equal to one. So if you have a special number, it’s one. And one is indeed a special number because the base of the number system is either zero or one. You must have one non-trivial non-zero number in order to have a number system. So you got nowhere. You proved one equal to one. Which is important. Maybe you want one better be equal to one or zero equal to zero. So you cannot find a dimensionless combination from G, H, and C. Okay? We’ve been doing some very esoteric things, but I’m going to stop here now and resume it in the next talk. But I’m going to leave you with a question. [snorts] Now that we are in the technological institution, we should also take a bow and you know, give a nod to applied physics. So let’s look at the most mundane thing of all which concerns all Chennai citizens, namely the water supply. Okay? So I’ve got water flowing in a pipe, non-turbulent flow in a pipe, horizontal pipe. I have a system from which I have pressure head. Let’s say the length of this segment is L and the pressure here is P one and the pressure at the output here in this pipe is P two. And this surface area this cross-sectional area is A. Radius of the pipe is constant, it’s A, and water is flowing out of it. I ask for this quantity Q equal to the rate of discharge of water, namely for a given pressure head, for a given P one and P two, we assume the level of the system doesn’t come down too much. There’s water flowing in a jet out of this water unhindered flowing out of this pipe. The radius of this pipe is R. And this is the volume of water coming out per unit time. Q. What proportionality What is the What power of R does it depend on? In this steady flow. Question. What power of our will it depend on? I argue in the following way. I say this is steady flow and not turbulent, so the velocity of the water is the same. It’s a uniform velocity. Right? Therefore, Q must be equal to the surface area pi r squared. That’s the area of the outlet. The hole here. Multiplied by V. So, it’s pi r squared times the velocity of the water, which is supposed to be steady. Do you agree with this? Well, what else could it be? All the water that’s within length V of the outlet pipe is going to flow out in unit time. And the cross-section area is pi r squared. So, pi r squared V is our discharge rate. So, I conclude that the Q is proportional to r squared. Now, I go and do an experiment and find that Q is proportional to r to the power

I want you to think over lunch about why this should be so. Well, every all the water that’s in length V is going to flow out in 1 second. If the velocity is V. Multiplied by this, so the volume of this water is pi r squared V. That’s the volume per unit time. There’s only one possibility that the reason why it cannot be it’s not proportional to r squared, and that’s because this V cannot be uniform. This V cannot be uniform, right? This V is going to be different. The center of the pipe it’s going to flow faster, and the edges of the pipe it’s going to flow much slower. Due to what? Why is that happening? So, you have this pipe flowing. This is the axis. So, there’s a very high velocity here. This is lower velocity here. This still lower velocity here, etc. Till at the edges there’s zero velocity. What’s that due to? Viscosity, exactly. So, we’ve neglected viscosity. So, therefore, there’s something wrong with this calculation. The correct answer is this is proportional to the average value of this V. And this average value must somehow be proportional to r squared, and that will give you an r to the 4. Try to find this average velocity dimensionally by dimensional analysis. What can it possibly depend on? It should depend on r itself once again for the simple reason that you can see that if you had a thin pipe, the flow would be much faster. And if you had a thick pipe, the flow would be much slower. With the same pressure head. But, we are a little stuck. Because the length of this pipe will now start mattering. There’s a radius here. And dimensional analysis will fail. Because whatever you answer you give, I’ll say I’m going to multiply it by some function of r divided by L, which is a dimensionless quantity. So, the moment you have two parameters which have the same physical dimensions, elementary dimensional analysis is going to fail. Because you can multiply by an arbitrary function of the ratio of these two parameters, which is dimensionless. How do you overcome that difficulty? Think about it, and then we’ll discuss this. Okay. How are you going to do that? It has to be done scientifically, right? What are the terms? Yeah, yeah. The L and P are L and R are independent parameters. I can have a short wide pipe, a narrow long wide pipe, and so on. I can They’re independent parameters. Yeah, one is not related to the other. They’re independent parameters. So, there’s a clever trick to get rid of this dependence. So, you have to tell me think about it. It’s very simple. It’s a physical principle, so just think about it. Very mundane problem. It has to do with water supply and so on. By the way, the fact that it’s going to be the fourth power of the power radius means that you get we change the radius a little bit, you get a huge amount of water difference. They solve the lay corporation connections and so on do so at midnight, you know. They say because the traffic should not be affected by the pipe work and so on going on and so on. But, actually what they’re doing is quietly changing a half-inch pipe sanction to a 1-in pipe. And then they get 16 times that much water. So, there’s a lot of businesses involved here. So, the contractors know this. They know this very well, except they don’t know this dimensional analysis. They’re not bothered about that. But, they know that if you take a sanction for a 1-in pipe and you change it to a 1 and 1/2-in pipe, you are going to gain a huge amount of water at someone else’s expense. Good to be aware of it. Okay. Okay, let’s stop here today.

[music] Swayam Prabha, digital India, educated India. [music]