Now, although I have said that we would like to proceed in a unidirectional fashion, remember thermodynamics is one single whole, not H O L E, W H O L E. So, we cannot stop talking about some idea, some concepts which are going to be explained. For example, we are comfortable with temperature, but in this course we have not properly defined or formally defined what temperature is, but we have sketch state spaces with temperature in it and nobody takes an objection because we generally have some idea of what temperature is about. Now, here we have the word interaction. So, let us begin with what interaction is, but before that let me say that the whole of thermodynamics is related to a scheme like this, a system A usually and a system B. Classically you may call it our system and our surroundings, but one should remember that surroundings is a word which is in colloquial English and not really defined in thermodynamics. If you have to define in thermodynamics, it is a proper surroundings is a proper thermodynamic system with which our system, the primary system interacts. So, instead of system A and system B at show near, if you want to write system A or system and surroundings that is perfectly okay or you could even say primary system and secondary system. If you want to call system B at surroundings, be sure that it is defined as a proper thermodynamic system with appropriately defined boundaries. Do not be so general as surrounding is everything that is outside the system that is to lose a definition. Now we have understood what is meant by a process. So, let us say that the two systems work together. A system A executes a process from A1 to A2, system B executes a process from B1 to B2. Well this is a short form of saying system A is initially in a thermodynamic state A1. Finally it is in thermodynamic state A2. So, A1 to A2 by system A is the process executed by system A. And why do the systems change? Well they interact with each other. There is some sort of a give and take and this give and take is known as interaction. It is actually a transaction. You go somewhere buy something and then there is a two way interaction. The grocer gives you some grain.  So, there is a material interaction, material transfer from the grocer to you and there is a crash transfer from you to the grocer. This is a sort of a once type of interaction. In thermodynamics, we consider interactions of the energy kind. So, what we are going to look at are energy interactions. And we are going to classify the energy interactions in two types. The classification of these will be the work type of interaction which is actually a primitive, but we will define it or redefine it properly. And the second one is the heat type which we will formally define using the first law of thermodynamics. So, that brings us to the first interaction which is the work interaction. Yes, I will hold on. So, the work interaction is actually a primitive for thermodynamics, but unfortunately the work interaction although is defined in other branches of physics. Generally, the idea is the work is force into displacement. You have force acting on something, I may be pressing something or even a gravitational force acting on this downwards. So, if there is a movement of the point of application of the force in the direction of the force that is the force into the displacement is the amount of work done. The directions are different. You take a vector dot product and so on. And we have worked all over for example, this bottle comes down with a mass m comes down by a height g. We say that the work done by the force of gravity is m into g into h. So, the general idea is force into displacement. In thermodynamics, we want to say that this is an interaction and according to our definition remember that whenever there is an interaction, there must be two systems involved system A and system B. A single system by itself will not have an interaction because that will be an isolated system if it does not interact with any other system. So, we have to have some sort of a proper thermodynamic definition. So, although we are not going to redefine the work, we are going to combine all the ideas of work and give a thermodynamic version or the thermodynamic definition of the work interaction. Let us see a few examples of the work interaction. Let us say some simple system. Let us say that we have a string or something like that which is our system. You could say a wire. This is our you know the Hooke's experiment I am showing. And let us say that there is a small loop. My system B is the whatever is trying to pull it. And since I am trying to pull it, there is a tension T which would be a property of the wire or the string. And under this pull, there is a small amount of displacement dx. And we will say that the small amount of work done by our system on the surroundings which is I or any machine or any entity which is pulling it will be minus T dx. T is in the negative x direction, dx is in the positive x direction. So, we generally write minus T dx. You take for example, an electric cell. The one which is inside your mobile for example, it has a positive terminal, it has a negative terminal. And this cell let it be, let that be our thermodynamic system. We connect a wire here and a wire here, positive terminal, negative terminal. And we connect something here which is our surrounding system system B. And let the potential difference from the negative potential rise from the negative terminal to the positive terminal BE. And let a charge dq be transferred completing the cycle through the other system. This could be done by withdrawing a current I for a small amount of time dt. So, in this case, you could have dw to be equal to edq. Or if you want, you can expand it as ei dt. But the main format here, just the way we had dw is minus T dx, here we had dw is edq. Continuing further, maybe we can look at liquid, system consisting of some liquid. And I put in a stirrer and try to stir it. So, as I try to move the stirrer by some d theta, the liquid is viscous and it tries to counteract by providing a torque tau opposing my movement.  We will be wondering why I am not drawing this figure. Well, I am drawing this now. Let us say we have a gas in a cylinder enclosed by a piston. So, my system is the gas. And let us say it has a pressure P. The cross sectional area is A. So, there is a force F acting. F will be equal to P into A. And there will have to be a force acting on the other side F, so as to keep the piston in its place, otherwise it will fly off. And if I allow the piston to move by a small distance dx, the work done will be force P A into dx, which is P into A dx, which will turn out to be P dv, where dv is the change in volume. So, look at these four. I can give more illustrations. For example, a spring or a liquid film stretching or all that. We have looked at stretching of a string, discharging of a electric cell, stirring of a liquid, expansion of a gas. Notice everywhere it is some P dv minus tau d theta minus T ds when e d cube. So, what we notice is each one of these interactions is of the type dw is x dy, sometimes plus, sometimes minus. So, we notice the following in this. First this plus minus is we keep a note that there is some sign convention involved, otherwise we would not be consciously writing minus sign somewhere. The second thing we look at x and y. Let us go back. Let us look at what is x is and what y is. Here we will notice that tension and length x is change in length. So, the length x or l of this thing, these are properties of over system, which is that wire or that piece of string, which is a tau. Notice T and x are both properties of the string. What type of properties? T is an intensive property, x is an extensive property. You check for yourself. If you cut the string into anywhere, the tension anywhere is going to be T, but the length is going to be that much shorter. Similarly, e d cube, you will notice that e is an extensive property. A smaller or a bigger cell of a certain kind provides you the same potential difference across its terminal. Whereas, the amount of charge or current that it can provide, so called ampere hours type of thing, that depends on the size of the cell. A big cell will have larger of q possible. So, e is intensive, q is extensive. Come to tau d theta. Here we have a funny thing. We will notice that unless d theta comes, there is no tau. So, the torque nor theta has anything to do with the properties of the system. Here we will again notice that pressure is an intensive property, volume is an extensive property. So, when you notice this, you will say x, y sometimes x is an intensive property and y is an extensive property. Again, you will notice going back to our illustrations that let me just look at 3 and 4. See, if I allow the pressure to rise slightly, so that the force on the piston overcomes from the fluid, overcomes the force on the piston from outside slightly, the piston will move to the right, the gas will expand and dw will numerically be a positive number. If I allow the gas to reduce its pressure slightly, then the piston will move to the left, dv will be negative, volume will contract and dw will have a negative numerical value. So, this is a interaction, the work interaction, which is possible both ways, positive as well as negative. Whereas, look at this interaction, tau d theta. If you start with a fluid in equilibrium, that means uniform properties, unique properties, put a stirrer in it and try to rotate clockwise, torque will opposite, the value of dw will be negative. You try to rotate it anticlockwise, again torque will opposite. In any case, the value of dw will be negative, but always the interaction will be, the energy will be absorbed. You will have to expand work in stirring the liquid. You cannot put in a stirrer there and ask the liquid to stir the stirrer, that will not do. The liquid is incapable of stirring the stirrer by itself, because tau and theta are not properties of the system. So, here we will notice that interactions 1, 2 and 4 are two way, actually interaction two depending on the type of cell could be one way or two way. The cells used in our mobile phones for example, or car batteries, they are rechargeable. So, we have a two way interaction, but the cells which you simply buy to put in your torch or your shaver, those are not rechargeable. So, they can only be discharged, they cannot be charged. So, those are one way interactions. So, here we also notice that some dw are two way interactions and some are one way interactions. And we have also seen that there are depending on the type of the system, there are different modes of interaction. This is stretching, this is electrical discharging, this is stirring, this is expansion of a fluid. Depending on whether it is a magnetic material, you will have some other type of interaction. So, there are different modes of work interaction. In thermodynamics, we have to be careful because we have to have now a definition by which we can decide whether an interaction is work or not or what component of the interaction is work. If so, what is its direction and what is its magnitude? So, what we now come to is what is known as the operational definition. The question is this. Let us say that we have an interaction between system A and system B. There is some interaction, let me simply call it I. System A goes from state A1 to state A2 because of this interaction. System B goes from B1 to B2 because of this interaction. And we want an operational definition which will allow us to determine one. Is I work or work type of interaction? If so, what is its magnitude and what is its direction? And the direction is associated with the sign convention. We will come to that. All over thermodynamics text books say that the work interaction is that part of interaction which can be converted by primitive means. That means means which need not be dependent on any thermodynamic ideas like stretching of a string, compression of a spring, raising of a weight or twisting of a solid rod. All these are sort of primitives. They do not depend on any thermodynamic entities. No temperature is involved. No entropy, no internal energy is involved. So, work interaction is defined as some interaction which can be reduced by primitive means to the raise of a weight because the idea is raise of a weight is a primitive work interaction requires work to be done. So, anything which is equivalent to raise of a weight that is mass in a gravitational field should be work interaction. Now, saying it in so many words is okay but let us see how we will implement it in an operational definition. So, let us look at the operational definition of work. Let us see what we do. We now try to implement the operational definition. Step one, what we do is we have these two systems A and B with an interaction I. What we will do is we will try to replace B. I will have this system A which goes from A1 to A2 and it will say I am doing an interaction I. But instead of B, I will try to set up another system C, say C1, such a way that the only external effect of that would be raise of a weight mass m, let us say mass m1 in a gravitational field G.  It gets raised by height h1 not lower and C1 is a primitive system that means does not depend on any thermodynamic effects. You could have ideal pulleys, you could have frictionless inclines, you could have wheels and axles, you could even have 100 percent efficient transformers, electric motors and things like that. Because we know that in principle these are possible. If you provide enough time, enough money and enough interest you can make the efficiency of an electric motor well as nearly 100 percent as possible. Because we know friction cannot really be reduced to 0 and resistance cannot be always reduced to 0. But you could try and in principle there is nothing wrong in mechanics or electricity and magnetism which prevents the efficiency of an electric machine to be 100 percent. So that is okay. This should be primitive and there should be during this interaction there should be no change of state. That means if it executes a process it should execute, C1 should execute only cyclic processes. So by that what we mean we are able to properly define and set up C1 in such a way that A believes that it is doing interaction I with B by undergoing this change of state. But that interaction can be completely reduced to the rays of a weight and then we say that if C1 can be set up then first I is work 2. Now this is about the direction. We say that A does work on B or system A does work on B or system A does work on system B. A is the donor of work, B is the recipient of work. And third one we say work done by A on B if you want you can write A B is plus M1 G H1 and work done by B is minus M1 G H1. Notice that this magnitude M1 G H1 is the magnitude of the interaction I and hence except for the direction or sign plus here and minus here there should be no difference between W A and W B. You cannot say W A has interaction I A and W B the same interaction becomes I B which is different from I A in magnitude. That is not allowed interaction cannot vanish in between. Now it may not always be possible to set up C1 like this. So if C1 cannot be set up go to step 2. What is step 2? We do the same thing but we look at it from the point of view of system B. System B executes a process from B1 to B2 has an interaction I with respect to system A. So we do not change that B does not see anything different. But that interaction we try to provide by means of a system C2 whose only external effect would be the raise of a weight M2 in a gravitational field G by a height H2 again the raise. And then we say that if C2 is equal to C1 then C2 can be set up first. I is work B does work on A and third work done by B on A is plus M2 G H2 and work done by A on B is minus M2 G. Now we go to step 2. The step 3 says is if neither C1 nor C2 can be set up. I forgot to add here that C2 again must be a primitive system fully defined no black box and no change of state. That means if the state changes if it executes a process that must be a cyclic process bringing it back to its original state. Then the conclusion is I is not completely a work interaction. This is the operational definition of work. Now if you want a slight modification to this when I say it is not completely a work interaction it does not necessarily mean that it is totally a non-work interaction. If it is not completely a work interaction then one can go back to this definition and see whether a part of I can be converted completely to work and if it can be completely converted to work we will say that component of I which can be completely converted to work either through step A or through step B is the work interaction. There may be some other part which cannot be completely converted to the rays of a weight. So that is some other component of the interaction that is not a work interaction. I will take an illustration later when I come to the end of this session or during the discussion session which would make this clear. Now there are some things here. C1 and C2 have to be primitive systems that mean contains stuff which is definable and is completely covered and handled by other branches of physics. There could be idealizations like frictionless pulleys, non-stretchable strings, non-dissipatively chargeable and dischargeable batteries, frictionless pulleys. I think I said that 100 percent efficient electrical machines, generators, motors, transformers, etc. What is not allowed is saying that it is a black box. Say that look I will set up such a system. No, you cannot do that. You have to specify it completely and second thing is in C1 and C2 no change of state is finally allowed. During the interaction there may be some local change of state but finally when the interaction is complete system B goes to B2 or system A goes to A2 you should have absolutely no change of state for C2. Third one is notice here as well as here in step 1 as well as in step 2. I have put a plus sign, I have put a plus sign here and a minus sign here. Again in the one I have put a plus sign here and a minus sign here and this is our sign convention. The sign convention here is that work done by a system is positive but mind you this is a sign convention. This is just for the matter of consistency and mutual understanding. I could have said work done by a system is negative and consistently use that. All that will happen is some form, some sign will change in the further derivations but finally in any thermodynamic problem the final answer quantitatively will be the same, qualitatively as well as quantitatively will be the same. In fact this is the first of a large number of sign conventions that we are going to come across right from here, right up to energy, right up to definition of heat, right up to definition of entropy. There are many places where we have to define direction and we will say one particular direction is positive another particular direction is negative. Why? Well we have to be consistent and let us define it that way but if you want to define it the other way well you are welcome but declare it that that is your definition and consistently use it. Do not confuse it by mixing up the two sign conventions. Now after this the next thing is because we are going to evaluate work there are also something to remember that there must be for W interaction two systems actually for any interaction two systems must be involved system A system B or you could say a donor system and a recipient system. One system by which we have to be consistent by itself cannot undergo any interaction and interaction means there are two systems involved and this is where our idea the thermodynamic idea or thermodynamic definition of work differs from some other work interaction.  Let us assume there is no air resistance or anything like that what happens? The ball goes up under the influence of gravity the speed reduces finally at the apogee it becomes 0 and then it starts falling back. If you are a good fielder you can catch it. Now here at any time there is the force of gravity acting on the ball. So there is a force there is a point of application of the force and that point of application is being displaced. If you throw a ball down from the top floor of a building just leave it falls down. You will say gravity is doing work on the ball but where is the recipient system? In thermodynamics this is not a work interaction. We will say okay the ball is your system if work is being done which is the other system which is receiving it since it is not receiving it we will say there is no work interaction involved here. However instead of ball let me hold this bottle. Now I am holding this bottle and the bottle is in equilibrium. Bottle is my system it is in equilibrium because nothing is changing it is at a given height. There is a force of gravity acting downwards and but it is not falling because the system B which is me or my hand is providing an appropriate equal and opposite force primitive Newton's third law of motion.  Now what has happened? The force acting on the bottle is downwards sorry force acting on the bottle is upwards because of my hand force acting because of gravity is downwards. So what is the interaction? The bottle has done work which is because of the force down. Who has received the work? I have received the work. If I push the bottle upwards I say I have done work on the bottle. If I push the leave the bottle slowly and bring it down the bottle has done work on but if I just leave it and let it fall down no work is done. That is the difference between the idea of work in mechanics and idea of work in thermodynamics. In thermodynamics remember this is the most important thing. Two systems must be involved a donor and a recipient. This is true not only for the work interaction but this is true for any interactions of the energy kind or any other kind. Any interaction in thermodynamics we are looking at energy interactions but any interaction work kind later on we will define heat kind. Those all will require two systems a donor system and a recipient system. The next thing is well we have defined work now we come to the next topic evaluation of work. Now remember whenever you evaluate the work done by a system notice the following systems must be involved. Two or perhaps more in case of complicated stuff. The second one is we should note that there are different modes of work and the third one is we have to take care of the details of the work and the process. By modes of work I mean the following. Let us consider a system which is made up of a fluid but which is an electrolyte. I will sketch a system for example I may have a cylinder piston arrangement. We know that those of you who use inverters which have a battery the specification is do not leave the battery set inside a room or an unventilated room because when the battery discharges or charges some gases are likely to be involved and those could be poisonous to the human being surrounding it. So you keep the battery always in a ventilated place. Let us say that there is a fluid may be partly liquid partly vapor we do not have to worry about it and there is a positive electrode there is a negative electrode. It is possible that the gas will evolve and the piston may move up and down there could be charging and discharging. So there is because of this there could be an electrical type of work because of this there could be expansion type of work ELEX and if you want you could even put a stirrer and you could have the stirrer type of work. This is an illustration where three modes are involved some of them two ways some of them one way. So whenever you evaluate or attempt to evaluate a work interaction you have to sum it up over one to number of modes. Do not ever be under the idea that DW always means PDV that is a big mistake many people tend to do. DW is always the sum of work interactions over various modes. In a special case it is possible that only the expansion work mode exists and hence you in that particular case you may have DW equals PDV. So we must check that two systems are involved here you will notice that for the expansion work to take place there must be some other fluid system or a weight or a spring must be there. For stirrer to work there should be someone to operate the stirrer that is the other system there. For charging discharging there must be for charging a cell or a generator for discharging something which runs a motor or a fan or an electric iron or a lamp or a may be a PC laptop has to be there. So any interaction will require two systems we look at the modes of work and the third one is the details of the process. Now if the process is from 1 to 2 I am just showing say x 1 y 1 again if you want you can replace this by any two properties of your convenience. Let us say this is the initial state 1 this is the final state 2 then we can say W 1 2 will be in principle integral 1 to 2 DW and this is the final and then you can depending on the mode you can write this DW expansion plus may be 1 to 2 DW electrical plus whatever you have. Let us take this for example this means integral 1 to 2 P dV. So this will be plus now the question is well we are looking at the details of process when can this be evaluated what is the answer this can be evaluated only when for any V between 1 and 2 P has a unique value. What does it mean? It means that if you are looking at the P V part of the state space if this happens to be state 1 this happens to be state 2 whatever be the process at least the P part must be uniquely defined and in that case that means the process must be a quasi static process. In that case this can be under the shown to be the area under the curve all that we know so things to remember is a component can be evaluated only when the process is quasi static. So for example in case of P V a mode of work where properties are involved the quasi static process will be shown on a state space diagram. So this is a process diagram but this is shown on a component of the state space whereas take the other component for example the stirring component when it is stirred we know that neither the torque nor the movement I am taking dw stirrer is minus tau d theta which is minus tau omega dt where omega is the angular velocity. So here the angular velocity neither the angular velocity nor the torque is a property of the system. So what we do here the process diagram is shown as a time charge against time 0 to whatever is the maximum time t process you will show tau and you will show omega as a function of time. For example the torque may vary like this this is tau as a function of time and omega also could vary something like this this is omega as a function of time but this also means that at any time t it is not exactly a quasi static process but at any time the parameters involved have unique values. If the process is non quasi static all that we will know is well this was the initial state 1 this was the final state 2 well we do not know where the system was in between. For this non quasi static process w expansion cannot be evaluated as an integral whereas here it can be integration is possible. Here because at any instant of time during the process both tau and omega are properly defined it can be evaluated but if there is a very jerky stirrer then you will not know at any instant what exactly is the omega what exactly is the tau and it is something similar to a non quasi static process you cannot evaluate it by integration not being able to evaluate it by integration does not mean that there is no work interaction there is or there may be work interaction we may be able to compute its value out using some other method for example knowing some other interactions knowing something more about the end states and so on. We will have some illustrative examples in that now there are certain standard stuff here which I will go over quickly and that all of you know how to demonstrate that w 12 by whichever mode depends on the path and other details of the process like whether it is quasi static or not. Hence dw is not exact differential hence we write integral of dw from 1 to 2 by whatever modes as w or w 12 and we will never write this as delta w 12 this is wrong never write it as w 2 minus w 1 that is meaningless just w 12 no delta nothing before that and since this dw is non exact often it is represented by d cross w or d prime w w or even some people refuse to use the d which they keep for mathematically exact differential and use delta w for us it is not necessary to use this for us we will continue to use dw but while being conscious of the fact that it is not an exact differential. My students they are all sitting there reading something and they are all solving the problems they know that if I see in an answer book something like delta w what do I do I draw a red line and I do not evaluate that answer book any further I think at least one of you would have faced that at least in the first place after that you learn but that is the way to enforce learning not only on our student but also on our self you know if I sometimes I have data went to as a what is wrong with me this is not the way to work so I will stop working there wash my face with a very cold doubt of water and maybe go and have a hot cup of coffee. Hopefully my brain will start working again now something important because we are going to come to this again and again and particularly there is a state postulate which is related to this we will see it later and that is what is known as complexity of a system we know that depending on the type of a system a system may any number of possible work some some one bit later on particularly in the light of the second law of thermodynamics we will notice that two way work modes are of significant importance. So we will say that we will define let us say n 2 w is defined as the number of variables work system now definitions if n 2 w is one there is only one two way work mode a system can have such a system is known as a simple system n 2 w is greater than 2 we call it a complex we will also notice that there is a small but important class of systems for which the number of two way work modes is 0 we will take an illustration of this soon let me call this a rudimentary system something which is still simpler than simple is rudimentary system so this is not the order you could say any n 2 w number of two way work modes is 0 rudimentary system number of two way work modes is 1 a simple system number of two way work modes is 2 a complex system some illustrations and some further sub classification for example you take a fluid a simple fluid non dielectric non diamagnetic the only two way work mode the pdv type expansion or compression so such a system is called a simple compressible take for example may be an idealized version of your battery in your mobile phone cell or battery in your mobile phone this is recharging can be charged discharged that is the only two way work mode we will call this a simple electrical system you cannot bend it you cannot expand it similarly for example a wire stretched for a spring which you can only compress or extend this will be a simple elastic you can create examples of complex systems for example I showed you an earlier illustration here let me go back here suppose this is a dielectric fluid you can charge discharge it this could be a two way work mode you could expand and compress that fluid this could be a two way work mode stirrer obviously stirring a fluid is a one way work mode so such a system is a complex system because there are two two way work modes one electrical charging discharging the second one expansion and compression you could create other complex system for example without bringing into anything you take for example a bubble bubble inside a liquid the gas inside the bubble what are the work modes it can expand and compress while keeping the area of the bubble the same by changing the shape or without changing the volume it can change the area of its surface doing surface tension type of work the surface can expand the surface can contract so when it comes to droplets and bubbles you have to consider that those systems appropriate systems as complex systems because expansion compression is possible and extension and contraction of the surface is also possible there is a pdv mode of work and then there is a sigma da type of work so that also is a complex system you could have sometimes mechanically complex systems also for example I mentioned that a stretched wire is a simple elastic system you take a spring this is not a spring and unfortunately I do not have one with me here but from your lab you can take an illustration a simple spring made up of thin wire I think you will find it inside a ballpoint pen or something you can stretch it you can compress it so fdx positive xdf negative that is one two way mode of work but without changing the length of the spring you can twist it and you can untwist it you can do work by increasing the twist decreasing the twist it is a tau d theta type of work so look similar to a stirrer but it is a two way mode of work do not be under the impression that any tau d theta type of work is one way a fluid is involved that is why a stirrer is stirring is a one way mode of work but if you have a elastic spring for example the spring which is used which is a spiral spring nowadays we do not use it but earlier our wristwatches or our table clocks used to have spiral springs now perhaps you will have to go to some old home to have a clock which needs to be wound so that is a tau d theta type of interaction which is a two way interaction so if you have a spring which can be expanded and compressed as well as twisted and untwisted those will be two simple mechanical modes but it will be a complex system because there are two two way modes of work similarly I have not taken an illustration but if you take a piece of soft tire which is magnetizable and demagnetizable then that would be and if that is the only two way mode of work then you can consider it to be a simple magnetic system but if you lay it out as a rod so that you can compress it or extend it then well the elastic work is also another two way mode of work so now you will have a complex system with one mode of work of magnetization and demagnetization may be BDH or something like that and TDX the extension and compression and if you make it into a coil and make it a spring then maybe you will have three two way modes of work so in principle you could have even simple systems simple means mechanically simple systems as thermodynamically complex systems because they have more than two way modes of work. Let me come to a situation where we have a rudimentary system illustration well I do not have you sitting in front of me so I cannot immediately ask you the illustration of a rudimentary system but I will give you an illustration of a rudimentary system take for example this forget that there is a button here and I can press it let us consider this to be absolutely solid and so rigid that I cannot twist it I cannot extend it compress it nothing any amount of force of any kind produces no displacement of any kind I cannot stir it I cannot do any two way mode of work so this is a system which is a rudimentary system okay if you say this is a useless thermodynamic system go home and from your medicine cupboard take out your clinical thermometer if you feel hot headed or if you feel feverish that is what you take out and you measure your temperature now that clinical thermometer looks like this but it is a glass with a capillary inside and a bulb of mercury connected to the capillary at one end it is a solid system can you twist it and twist it no it will break can you expand it compress it no there is no two way work mode involved but it is a good thermodynamic system because it helps us measure a property called temperature that is perhaps one of the most rudimentary systems so illustration is the clinical or in fact for any not necessarily the clinical any mercury in glass thermometer another one in your heat transfer labs you would have used thermocouples it is a bead connecting to dissimilar metals solid simple system no there is no two way mode of work so it has to be a rudimentary system no two way mode of work but useful for us for measurement of temperature bead of a thermocouple it is possible for experimental purposes a to create a system and make it rudimentary for example consider the fluid so our system is the fluid and we know that there is a two way work mode involved this is expansion work making our fluid system a simple compressible system but then all that we have to do is fix the piston sealed at one place prevent any change in volume now what is for such a constrained system is there any two way work mode possible no so this is another illustration where a simple system is constrained to behave like a rudimentary system such systems are also important particularly when it comes to 0th law and further discussions we will make use of such systems but this brings me to the almost the end of the work interaction I have finished in number 9 illustrative examples of various types of systems but exercises I will push along with problem solving in thermodynamics which will we will do at the end of today the 4th session but let me have try to have interaction with one center at least and that is home institute of technology and management center number one two five eight twisting and untwisting a spring is not a two way work mode so please clarify it sir your question was about the twisting and untwisting of a spiral spring twisting and untwisting of a spiral spring as is used in our classical wristwatches and clock mechanisms or even in old phonographs is a two way work mode because that spring can be charged or energized by doing work on it the spring discharges releases its energy and runs either the mechanism in our clockwork or the turntable in an old phonograph so a you know the winding and slow unwinding of a spring is a two way work mode of the tau d theta kind tau d theta is a detail of modeling but it is a two way work mode and if it is a axial spring it is stretching and unstretching is also a two way work mode and depending on the design of the spring you could have axial movements as well as twisting untwisting movements making it a two way work mode type of system over to you whether the two way work mode system affects the energy interaction on both the systems involved within the process or both the both the parts of the system see any interaction affects both the systems which are taking part in that interaction this will be clear when we study the first law an interaction does not go waste ever an interaction always leads to either a further interaction or the change in the state of a system a change in state of perhaps both systems which are involved so both systems the recipient system as well as the donor system do get affected so there is one more question as you are told that the pan as you are shown in your hand that you are called as rudimentary system but can you tell us that the ink is coming out of that pan when we are writing then still could we be able to call it rudimentary see the pen which I have in my hand is not a pen there is no ink in it at all it is a solid thing there may be some electronic inside it but there is only one stylus which has some sensitive issue with the laptop screen that is all I agree that if I have an ink pen then as I write it will not be I do not know whether I have to consider it is a rudimentary system or not but a plane is a fountain pen or even a ballpoint pen is a reasonably complicated system let us look at the refill as a system let us not complicate it now the refill turns out to be an open system because as the process of writing goes on some material flows out of it on to the paper there are no significant thermal interactions but if your hand is warm then it is possible that the ink in the refill will become thinner and may be more of it will flow out we can if you want we can analyze it in detail by considering the ink as a open thermodynamic system that is possible in fact you are giving me a good idea to set up a exercise in the next issue of things thank you for that and by the way I think I should take over here because it is tea time we will meet again after tea time.


Thank you very much to the organizers and to ESUS for this lovely conference and the invitation. So I'm going to talk, what? For the weather. And the weather. It's going to improve. Exactly. But you know, we don't need to go to be out. Right, there's a motivation to come inside. So I'm going to talk about Einstein equations. I'm going to introduce some geometry into our PDEs. And of course, we all heard about the detection of gravitational waves. So it was confirmed in February this year that it was detected last year in September. And I'd like to make a connection to, let's say, geometric analysis on what we know from the PD point of view, analyzing the Einstein equations. What can we say about gravitational radiation from the mathematical point of view? So basically I'm going to talk, introduce first, well, what do we mean by a spacetime? What are the Einstein equations? And so again, this will be a point of view of PD, of this hyperbolic set of PDEs from a geometric viewpoint. I'm going to make a connection with, so what do we mean by energy in GR, general relativity? So there's maybe not a straightforward notion to think of from the beginning. So we have to think about it a little bit. I will talk about what I will call null infinity. So let me just very juristically motivate this for the moment. So if you think of gravitational radiation, so they are produced during the merger of compact binaries like black holes, for instance. So there's something happening here. We have two bodies, black holes spiraling in. They lose energy into the gravitational field, so they radiate waves. And these waves will travel at the speed of light along so-called null hyper surfaces. So you can think of this as a generalization of a light cone, basically. So this will go along null hyper surfaces. Let's say when we are far away, when T goes to infinity, so we can think of where we are sitting in an experiment. So this is null infinity. So we will look at null infinity. That's where we are sitting when we're doing experiments and we are looking backwards in time along these null hyper surfaces. So we would like to understand, from a point of view of analysis on geometry, how does our space term look like out here? And what kind of information can you gain from what's happening in the past here, for instance, in such a source? I will explain this a little bit more later. And well, of course, gravitational waves will also address something called the memory effect. That's a permanent change of the space time by the gravitational waves. And we also hope that this will be detected in the near future. And detection is one thing we hope, of course, for the future, that this will be a new tool to really learn about other parts of the universe where telescopes cannot see. Okay, I'll then address some results in the asymptotically flat case. So that's where we look at galaxies, for instance, or we think we are here. The galaxy creates curvature, but far away. The next galaxy is so far away, so we can assume the space time is flat far away. Another setting is the cosmological setting. When we look at the whole space time, the whole history of the universe, what can we say there? Well, first of all, what is a space time? So we look at the Einstein equation. So on the left hand side, and I'm going to talk about four dimensions. So three plus one, three spatial one time dimension. So we look on the left hand side, so we have basically two metric objects, if you like. So we have the Ritchie curvature, the metric we are solving for the metric, the scalar curvature. And we call this gene union on the left hand side, the Einstein tensor. And on the right hand side, so we plug in the so-called energy momentum tensor. So whenever you have electric fields present or a fluid or anything else that is not given by gravity on the left hand side, so you plug it into the energy momentum tensor and you have also to supply the corresponding equations. So for instance, if you have electromagnetic fields, you couple your Einstein with Maxwell equations and you get the coupled Einstein-Maxwell system. So we are solving the Einstein equations for the metric, like to create the space time, either locally or globally in time. And well, we'll see. So it's possible now to, I mean, if you have the Einstein equations, we usually write them as, well, if you look at it this way, it looks nice and compact, but we usually write them as a system of hyperbolic nonlinear PDE. And well, if you add the cosmological term, so how does cosmology come in? Well, Einstein in the first, in 1915, had written down his equation without this cosmological term, but then he wanted to study cosmology and he was actually looking for a static universe at the time. He didn't believe in anything that's moving. The universe had to be static. And to counteract gravity, he then plugged in this lambda term, which is giving you an expansion, basically. So he got the static universe by doing that, but for all purposes now, so we think since the 1998 observations that our universe is actually expanding at the accelerated rate. This is modeled then by this lambda term. Okay. And often I will just look at the Einstein vacuum equation. So if the right hand side, your team, you knew is zero, the Einstein equations reduced to the Richie scalar of the full space time being zero. And there's already a lot of interesting information or questions you can ask about just the Einstein vacuum equations themselves. Okay. Well, first of all, in order to study anything that has to do with real physics, we would like to do a mathematically rigorous problem. Of course, you have to study the Cauchy problem. So it all starts with the Cauchy problem for the Einstein equations. And as most of the audience is not working in GR with some experts as an exception. So let me review a little bit. What does it mean to set up the Cauchy problem for the Einstein equations? So it's a bit different in various ways. So an initial data set, what is that in GR? So we think, first of all, of a three-dimensional manifold. So you can think of t equals zero.  If you look at asymptotically flat, for instance, that you're interested in. So you get a complete Riemannian metric, the induced metric on the initial hypersurface. And we specify a symmetric two tensor k. Maybe I'm going to write it down. And if you have anything on the right-hand side present, so other matter fields you also specify, of course, these equations. Now the Einstein equations couple into two sets of equations. One is the constrained equations, which the initial data has to satisfy. And the other set of equations are the evolution equations, the ones you then solve into the future. And so the constrained equations, you cannot just pick any initial data, but it has to fulfill these constraints. And only then you can think about doing an evolution problem. And then a Cauchy development of such data is given as a globally hyperbolic space-time of this sort verifying the Einstein equations and the embedding given as these things. So maybe just to say at time equals zero. So we think of some space-like hypersurface h, zero, for instance, g bar and k. And this initial data has to satisfy the constraints. And so if I have a time vector field, I can introduce that. So this k is going to be the second fundamental form with respect to t then. Well, let me first talk about asymptotically flat systems, as I just motivated a few minutes ago. So if you look at what happens, for instance, within a galaxy or a cluster of galaxies, you think the next object is so far away, you think your space-time is going to be flat very far away. So what does that mean? So we think of asymptotically flat initial data in the sense that outside of a sufficiently large compact set, so h without k is demorphic to the complement of a closed ball in R3. So and then the g bar, the metric g bar and k have to decay g bar to delta ij and k to zero fast enough. And of course, I will make this more precise. So what type of decay, what type of asymptotically flat data we are looking at. So let me first say that, well, if you look at the history of general relativity and so the Einstein equations were put out there in 1915, and it took, well, there were many, many things happening in between in math and physics, but it took until the 1950s and 60s where Yvonne Joqui-Brouillard and her and Bob Garrosh actually really solved the well-posedness for the Einstein equations. If you look at, for instance, in 1919, we have Eddington's experiment confirming the bending of light, et cetera. You have on the physics side of GR many of these big things that happened or the expansion of the universe found by Le Maître in 1927. So all he thinks to have many of these successes and mathematically could ask, well, what's been going on mathematically? There were many people actually working, like Le Raie Schauder, many people looking at the initial value problem, but it's not that easy to figure out what should be exactly the right way to put up the initial value problem. So I'm skipping all this history, but it's actually interesting to read up on that. So there's many people who contributed in many ways. So basically in 1952 that's the first rigorous general mathematical theorem in terms of well-posedness. So Yvon Jacques-Cœur-Brillard then showed that if you take hg bar k as such an initial data set, which is, well, here this is formulated for the vacuum Einstein equations, but it's been generalized to hold for most other fields on the right hand side. So then there exists a spacetime satisfying the Einstein-Vacuum equations with h into m being a space-like surface with induced metric g bar and second fundamental form k. Then the general form of what, so whenever you do solve the Einstein equations, so basically you go back to one form of these well-posedness results for these local results. So then the other formulation is given here. So Yvon Jacques-Cœur-Brillard and Bob Geralsch in 1969 then came up with this formulation of the well-posedness. Okay, so that's the first rigorous step in general, I would say. Well, let me say something about energy in GR. Well, most of us, when we think about PDEs, so often we need to find, of course, energies that have to be controlled to get control on the solutions of our equations, etc. So if you think of doing something like that in GR and you take the most naive approach, so things might fail, so what's the problem here? So one problem or interesting feature is that if I'm sitting at the point in GR, I can actually transform away the gravitational field. So just by transformation I can make it zero. So how would you define energy at the point if you can actually transform away the field there? Well, that's one problem. So you can think, well, maybe I don't look at the point, maybe I go a little bit further, maybe I integrate over a sphere, something quasi-local. Well, it turns out that's one thing we can do, but also for asymptotically flat systems, for instance, what's been understood quite well for a lot of interesting space times is you can integrate, for instance, over a space-like slice, or you can also look at what's the, if you integrate locally over a sphere and you get some mass or energy, well, expression, and you can take the limit like the Hawking mass, I'm going to write that down later, and you take the limit to null infinity, there's something else called the Bondi mass, for instance, that will show up later. So there are mass and energy definitions for certain, actually, many interesting space times that we understand. But let me give you maybe a little bit of an idea of the confusion at the beginning by citing Einstein and one of his colleagues. So Einstein formulated some sort of an energy momentum theorem very early actually which is for a closed universe. And so most of his colleagues at the time did not agree with it, and here is what he says. So here this is a citation, while the general relativity theory was approved by most theoretical physicists and mathematicians, almost all colleagues object to my formulation of the energy momentum theorem. I'm not going to go into details, but he wanted some kind of an energy momentum conservation for some closed universe, but turns out that's maybe not the right way to do it. And so this is Einstein with Schoch-Schlimmetre, maybe a side note of actually Schoch-Schlimmetre, a Belgian mathematician and physicist who derived from the Einstein equations and Slifer's observations in 1927 that the universe is expanding. It was not Hubble. Hubble had a paper two years later, and he never talked about an expanding universe. Okay, let's still stay a little bit more with energy and conservation. So what do we know? So we can look at the Bianchi identity that we know from geometry and apply it here to the Riemannian curvature tensor. And, well, we take, for instance, twice-contracted Bianchi identity and get this equation on the left-hand side of the Einstein equations. And then, of course, this implies by the Einstein equations themselves an identity of this form. So people at the beginning also thought, well, is this energy conservation yes, no, does it make sense? Well, it makes certainly sense of some sort, but to call this the energy and energy conservation is probably the wrong thing to do in GR. So maybe one more thing about that. So Neuthor's theorem, if you think of it a little bit more in a generalized way, well, what are we looking for? Within some setting of a Lagrangian theory, we always look for a continuous group of transformations which leave the Lagrangian invariant, right? So to such an object, we have then a quantity which is conserved. And in GR, the first problem, if you have a most general spacetime, you don't have any symmetries. So what are you looking for if you don't have any symmetries at all? However, nature is usually better than just the most general case. For instance, if you look at an asymptotically flat spacetime, your background Minkowski space has a lot of symmetries. And you can start looking at maybe quasi-certain isometries you can preserve basically in a way or another in the manifold or control. Well, here is maybe one more slide where Einstein, Hilbert, and Weil struggled over these energy components of the gravitational field. So they were thinking about this very hardly at the very beginning. And here's just one more quote by Hermann Weil. This is in his book Space, Time and Matter of 1921. And he looks at these components here. So he says, well, nevertheless, it seems to be physically meaningless to introduce this TIK. So that's the Einstein-Suter tensor based on some specific Lagrangian he had there. As energy components of the gravitational field for these quantities are neither a tensor nor are they symmetric. So in fact, by choosing an appropriate coordinate system, all this TIK can be made to vanish at any given point. Although the differential relations referring to the divergence of the Einstein-Suter tensor being zero are without the physical meaning, nevertheless, by integrating them over an isolated system, one gets invariant conserved quantities. So you see there was a lot of thinking reflection about what the right energy notion should be. And already one suggestion is, oh, well, maybe we should integrate something over an isolated system. So that was already a good idea then. Okay, but let's now jump to more modern days. Well, modern 61. And so if we think of one type of notion of energy and momentum, so this is the ADM energy linear and angle momentum that was introduced in 1961 by Arnawitz, Deisler and Misner. So basically, this is the definition, but the way you think of this, you integrate these geometric objects basically at the sphere, over a sphere at infinity.  So you're in one space-like slice. And you integrate basically over a sphere at infinity. And you can define then energy, linear and angle momentum in that way. So you have to say, of course, this holds for many interesting asymptotically flat systems. And so energy and linear momentum hold actually very generally. And for angle momentum, you need a little bit more of decay to be defined. So what about null infinity? Because I told you at the beginning, heuristically so radiation, we would like to understand null infinity and explain something about radiation here, extract this from our space-time. So we would like to understand the Cauchy problem and then see how the space-time, the solution looks at null infinity. So I can also foliate my space-time not only by, let's say, space-like hyper-surfaces like this. I can also foliate my space-time by null hyper-surfaces.  Of course, they have structure, but you can think of them as generalized light cones, for instance. So this is maybe, so you want, so you have this light, this null hyper-surfaces, et cetera, that I can generate like that. And again, so when gravitational waves travel at the speed of light, they travel on these null hyper-surfaces out to null infinity, and that's what we're interested in. So if you, the boundary definitions of this, so the adiom definition that you just saw on the previous slide have something like an equivalent, well, have a corresponding definition at null infinity, and we call this the Bondi definitions. It goes back to Troutman, Bondi, van der Borgen, Metzner and Sachs in the 1950s and 60s to define mass, energy and momentum also at null infinity. Well, there's some more constraints probably you have to put there to make things work. I'll say more about it. A famous theorem, of course, in GR is that mass is positive, and so this was shown by Shane and Yao in 1981, and then also by Ed Whitton separately. And then in the Bondi case, this was also studied by Shane and Yao later. So first of all, energy, what's now the real good way to talk about energy? So energy controls the curvature, first of all, but what kind of energy? So we use the Bell, Robbins and Tenzer quite often. So here's one solution to this problem. So we can think of the Bell, Robbins and Tenzer. So this is defined as follows. So this is basically, it's a four covariant tensor field, but it's basically, if you look at that, it's a quadratic in the wild tensor. So you look at the wild tensor of your space time and basically the Bell, Robinson is a quadratic in your wild tensor. It has a lot of nice properties. It is actually positive in this sense that if you plug in future directed time like vectors, so you get this positivity property and it also satisfies the Bianchi equations. So here's just a notation for what I use for the hardstools. Maybe let me point out, so this was used heavily in the work by Krzysztole and Kleineman on the Minkowski stability. So this is really the main object to control the curvature there. And nowadays, by many other people as well. So let me also introduce the current. So if K, this Q now, the Bell, Robbins and Tenzer are just introduced. So contract this now with three future directed causal vector fields and we obtain what we call the current in this case. So I'm going to call this J. That's my current. Then I can apply some diversion theorem on a bounded domain in the space time and I can also then find what is my curvature flux. So if the omega here is just the boundary of this, so if this contains a portion of a null hyper surface, so we remember that, with some affine tangent null vector field L, then the corresponding boundary term is indeed the curvature flux through this null hyper surface C. And we can write it down as like this integral. So this is the curvature flux we can define like that associated to some null vector field L. So these are some important objects to think about and often when you would like to prove, I'd say, control, Robbins and Tenzer actually controls the curvature and from there we can go on. So maybe back to the Cauchy problem for one tiny moment. So if you look at initial data of this type, so I have not yet told you how much decay you need, et cetera, for certain questions. So if you look at Christodoulou Kleinermann's Minkowski stability proof and Sipser's generalization to Einstein-Maxwell case, so the initial data has this kind of form. So you look at the initial data where for large R, the G bar has a form like 1 over 1 plus 2 M over R here, the first term, and it decays like, let's say, R to the minus 3 halves and the corresponding decay is also for K with some regularity. And in my generalization of this result, so I have less decay, so I only have a decay of R to the minus 1 half here and some less regularity. And now you can ask what do you need in order to see some interesting things in terms of radiation or memory effect. So it turns out, I mean, here you get all the information and in fact Christodoulou used exactly the last chapter of his book with Sergio to derive this nonlinear memory effect of gravitational waves. So not only you find the existence and uniqueness of solutions, you have a perfect description of how the asymptotic behavior looks like. So in my case, well, I do see some things, you can compute a few things, but in order to get the full picture, this is kind of hidden in the decay which is very low, very slow decay here. Just a comment, question on notation. Subscript O subscript 3 means that's true for three derivatives of the R piece.  So here are, let me also write down just the evolution equations, the constraints and the laps just to give you an idea. So I mean, we can then write down, we said the Einstein equations couple into constraints and evolution equations, so phi is just my laps function. So I have an equation to evolve the metric g bar given by this right hand side, I have an equation to evolve the second fundamental form Kij and so here this is actually the R on each space like slice on the right hand side. And if I choose a maximal foliation, meaning I have the trace of K0, so the constrained equations reduced to this set of equations. And in addition I have an equation for the laps of a maximal foliation here to look like that. So again, so you can put this into hyperbolic form, etc. and work with that. Now let me also mention just briefly the stability result by Krzysodulow and Kleinerman. So this was a big breakthrough of course. Question is, can you find, can you perturb Minkowski's space by just a little bit to get a global solution which is chodesically complete for all time? So, and the, well this is one very simple version of the theorem to state it fully, you need a few more pages, but you could say that every asymptotically flat initial data which is globally close to Minkowski globally close in the sense with weighted subolive spaces, etc. certain norms have to be small enough. So this gives you a solution which is a complete space time just tending to Minkowski at infinity along any chodesic. So, and interesting enough, so again, so I was just a few weeks ago I spoke to a few physicists like Stroming or at Harvard and they now talk all about the Krzysodulow-Kleinerman space time. So they are at the moment very much looking at, they are looking for solutions which are exact, there's no approximation anything. And it turns out as I said before, the last chapter of this book gives you the most precise information about some interesting physical space times.  So, I mentioned the generalization by Sipser, well in my generalizations there is, you don't really see as much of radiation, but you really read these things off that well. There's many, many people that I'm not citing everybody who have worked on related problems. So, let me just say this is actually interesting for us because we get a precise description of null infinity and that's what I would like to stress also. Okay, maybe, well, just here on the blackboard I already explained this, but we usually work with two type of foliations in this setting. So, given by a space like hypersurface, now it's called sigma here, which I call h on the blackboard, and also null hypersurfaces. So, we will be interested in outgoing null hypersurfaces and for simplicity they look just like cones on my slide. Well, one thing again to remember, I use a maximal time function, the trace of this case is going to be zero, each space like slide is a complete Riemannian hypersurface corresponding decay. And then I'm interested in the intersection of these. So, when I intersect a space like an null hypersurface, I get, of course, a second, well, an object which is different morphic to a sphere in this case. So, STU is just the intersection of these. And it will be important in what comes. Also, let me just introduce a little bit more. So, I mean, I can look at the null vector field and I'm writing null vector field L in the outgoing direction and I call L bar the one in the ingoing direction and maybe this will be B4 later and this will be B3. Then I can complement this with an orthonormal frame just on the surface S here. And I will actually decompose now my components, my curvature and my geometric components with respect to such affiliation where this is given g of L and L bar is negative 2. Okay. And well, now let's look at the Bondi mass and something I mentioned before. This is going to be interesting for radiation. So, if you look at one such null hypersurface, call it Cu in our space time and we'll let the time go to infinity. So, we would like to see what happens to local quantities which define locally what's the limit up there. So, I can define what is called the Hawking mass. So, quasi locally. So, I integrate over such a sphere basically, STU. What is the trace of chi and trace of chi bar? So, what's that? So, maybe I should introduce that as well. So, two important objects when I draw this picture again that, well, you see me draw often. So, if I have L and L bar, the generating vector fields in the null direction. So, I can say, well, okay, I have the chi of X and Y. So, if X and Y are in the tangent space here of this S, so this is going to be given by chi, nabla, XL and Y. So, it's a second fundamental form in the null outgoing null direction. And I can look at the bar version, of course, doing the same thing, but with the ingoing null direction. So, these are the corresponding second fundamental forms in the null direction. And interesting for us will be the shears. This will be the traceless parts of these objects. So, by a hat, I just use this to call this the traceless part. And then, of course, I have the trace of these objects which comes up in the definition here. So, one way to write down the Hawking mass as an integral over, of course, a local over such a surface is given like trace chi times trace chi bar. So, the trace of these objects. And, well, it's given like that. Now, questions. So, for interesting space times, let me not be too specific at the moment. So, this Hawking mass actually tends to the Bondi mass when we go out to null infinity. So, if you look at the blackboard over there or here, we go out to null infinity. So, then, this is going to what we call the Bondi mass. So, it has a finite limit for the space times we are interested in. Now, what is future null infinity? We heuristically already introduced that. So, I will call this I plus. So, this is defined to be the endpoints of all the future directed null geodesics along which r goes to infinity. And it has just the topology of r cross s2 with the function u taking values in r. So, you can think of the function u being parametrizing your null infinity. So, and our Bondi mass sits up here and is depending on u, of course. So, each null hypersurface has a finite Bondi mass. Of course, if your space time is not, let's say, nice enough, this could blow up or not be defined. But in the cases we study here that I mentioned, so this is all finite and well defined. Okay, now the Bondi mass measures... So, what does it do? So, it measures in this sense the amount of mass which is remaining in an isolated system as measured at null infinity, at the given retarded time out here. So, basically, if I say, well, out here, this is maybe cu which is hitting, well, this is null infinity for one such null hypersurface. So, I have a certain amount which we will also see the Bondi mass loss formula by saying taking the rebate with respect to this function u. So, this will give us how much radiation actually has happened. So, here it is. So, if you just look at Einstein's vacuum, we don't care about other fields, just pure gravity is acting. So, then we have the so-called Bondi mass loss formula. And again, this is something... this terminology is also used... I'm just copying the terminology of Serge's book with Dimitri. So, this is also in the setting there, derived and well understood. So, you have a Bondi mass loss formula. So, you take the derivative, and on the right-hand side you have this Xi object is actually now the limit of one of these shears. So, this shear with the depart version.  So, this is what is called Xi of u. And correspondingly, this other guy has a limit as well. So, let me just write it down, because it will show up. So, this is called sigma, then. So, these are objects at null infinity. So, I explained this already, so we have the second fundamental form. So, this is the traceless part which gives us the shears, then we have also the torsion we can identify. So, I told you that already. So, let's skip further. Now, what is gravitational radiation? Now, I gave you a setup of the spacetimes we are looking at. But gravitational radiation is now a fluctuation of the curvature of your spacetime. So, your spacetime has a lot of structure in terms of remodeling and curvature, etc. And so, when a gravitational wave travels from the source, so it's changing the curvature of the spacetime. So, it's like a wave packet that may come for, let's say, one second, traveling through. And now, we would like to see what we can learn about it. Okay, also, let me also mention the so-called memory effect for the moment. So, okay. We have heard about LIGO. And LIGO looks like an L-shape. It's an L-shape detector. So, this is the same distance like here and of a 90-degree angle. And if you, for simplicity, assume now that the gravitational wave source is coming from the third perpendicular direction. So, then it acts like a planar wave out here. So, the wave will... of course, it hits from different angles, but in order to simplify the discussion here, so let's assume it comes from this direction, perpendicular. So, what will happen is that these masses will move in the plane because it's like a planar wave when it hits our detectors on Earth. So, these masses will move and what they were able to measure in this experiment is really by laser interferometry, the distance of, let's say, mass 1 from 0, mass 2 from 0. This is where, actually, you have a beam splitter here by laser interferometry. You measure the distance. And, of course, when the curvature is changing, the space time is changing and it reflects in the displacement of these test masses. So, the test masses, you can think of, they're floating on their geodesics. So, you see what the geodesics are doing by looking at the test masses. During the time of the pass by of this wave, now you can ask what happens afterwards and this has not yet been detected but let's hope in the near future, maybe. So, what you think is, maybe in most cases, you would think, oh, well, things go back to their geodesics or as before and everything looks the same. Now, there is a prediction called the memory effect, saying, no, this will not happen. It will be that the space time will be permanently changed and the test masses will be permanently displaced for that matter. So, this is what is called the memory effect of gravitational waves and while there are various studies, many people have, in the meantime, worked on that. So, let me maybe just say a little bit about it. So, in the linearized version or linearized theory of the Einstein equations, the first people to find such an effect were Sildowic and Polnareff in the 70s and then in a fully non-linear problem setting that was Dimitri, who really used the last chapter of the Minkowski stability book of his uncertainties to, I mean, you plug in, the asymptotics also are true for large data one can show. So, he derived from all that he derived the so-called non-linear memory effect. So, people called it linear and non-linear and it was, so the first effect was supposed to be so small never to be detected, but then this effect is actually large enough it's also small but large enough to be detected, hopefully. And so, people always thought this is really a linear and non-linear thing of the same effect, this memory but it turns out, so with Garfinkel we looked at that and we found that these are two different effects which have to do with the linear one we call regular, so this has to do with one portion of the wild curvature, so the one portion of the electric part of the wild curvature changing over time and the null or formally non-linear effect has to do with fields that really go out to null infinity so things that change null infinity so these are two different things and we found this also in the linear Maxwell equations two equivalent, well not displacements but kicks. So anyway, so you can ask well what do electromagnetic fields do or neutrinos, so they will actually add to the second effect which we call null effect now and many people have worked on that, there is work by Blanchier D'Amour, Brackensky, Grishkouk, Thorne and many people I'm actually probably missing but it's interesting that recently froming around collaborators kind of also took up this work on memory and so they have the idea that the memory effect is actually part of a triangle where they look at water identities and BMS super translations and now a lot of people are looking for memory effect in other field theories in ADS CFT and anything you can think of so it seems that there is something interesting but maybe not, it's just the beginning of understanding of what this means in other theories okay, well there are two types of memory and let me now come back to the Riemannian curvature if you think of, let me decompose this a little bit and kind of lay open the structure of radiation I would like to show you again, three denotes just E3 is just the ingoing null vector field right down here and E4 is the outgoing null vector field so if I'm just in the Einstein vacuum equation so the Riemann curvature is my wild curvature and I decompose here the curvature components with respect to this foliation now the most interesting part to remember is this alpha bar so there's a part which goes like 1 over r the tau minus you can think of like u it's 1 plus u square root so there's a part that goes like 1 over r and this will be the interesting part for us for radiation so and again you can look at different space times and see what happens with this so if you stay within the so-called Christodulo-Kleinemann space times you will see well the alpha bar part has a limit at null infinity I call this capital A of u and this is a symmetric trace free to covariant tensor field at null infinity other components also have certain behavior I'm not going to look at that but if you change the type of space time and you do another Cauchy problem to really understand fully what's happening at null infinity so then you can do more general settings so with Garfin-Klevy also came up with a different method so the most rigorous way to study that is of course to do the Cauchy problem for each set of initial data you have well this can be cumbersome if you're just interested let's say in the radiation so we have another method which is an approximation by perturbing the wild curvature which is Gaetschen variant to actually for more general space times also look at radiation at null infinity okay maybe here is just one short note on one of these methods so Christodulo-Kleinemann actually introduced an interesting set an interesting theory looking at let's say elliptic equations on such a surface but then propagating along null or space-like directions so if you look at the trace of chi with the parameter s in the null direction so this is given by on the right-hand side we have some the shear squared we have a trace of chi and then this is now just anything if you plug in a null fluid so this comes from the energy momentum tensor on the right-hand side and this comes if you have Einstein-Maxwell equations this will be a component of the electromagnetic field so in general when we look at the Gauss equation how do these things look like so I can write down the curvature of this intersection I call stu in terms of as follows I have the trace chi chi bar on the right-hand side I have here the product of the shears plus w I call just a component of the wild curvature and contribution from t so I have either quadratics in shear or trace of chi one pure component of wild other than the alpha bar alpha bar is the one with least decay or contribution from the t on the right-hand side so and if I look at the null-kudatz equations in this notation so we can define what we call a mass-aspect function or its conjugate so c does just torsion so this is given again of this structure and we can write this with help of the Gauss equation we can actually write this mass-aspect function like this and now if you look at the null-kudatz and the corresponding conjugate kudatz equations so they have a form like that so you have either quadratics of Ricci coefficients on the right-hand side or the derivative of one of those here's another quadratic or a pure component of curvature but not alpha bar not the worst decay and contributions from t if any so I already told you about the limits so the limits again so chi hat and chi hat bar the shears have limits like on the left blackboard here at null infinity now the structure let me introduce a little bit more about the structure so if psi is a component of second fundamental form or torsion phi is a component of the wild curvature and t just energy momentum so then you can write down and n is just now a normal to s into the space like slice so you have a structure of the equations when you look at the propagation here for chi hat so you have a quadratic again in psi or a derivative of such a psi term and we have just curvature or t itself and eta hat is actually the chi bar over there, chi bar hat so similar for the other equation if you look at this type of equations and take limits now I would like to understand what happens at null infinity so let's just multiply by the r that is needed and then we see on the right hand side so alpha bar has a limit which is called a so we get for these equations we get limits actually for many different space times this is actually true that the behavior at null infinity so we have this behavior between the shears so we take a derivative with respect to u and one shear here is related to the curvature at null infinity which is the one part of the curvature now energy radiated so we can say well in the pure Einstein vacuum case so the energy radiated is integrated from minus to plus infinity let me show this here so if you are in pure Einstein vacuum you only have this shear part but if you add electromagnetic fields you have a contribution from the electromagnetic field or one portion and if you have let's say neutrinals that you can model by some null fluid you would also get a positive contribution to this energy which is radiated away and this is so we integrate this from minus to plus infinity at null infinity okay here is maybe a theorem just generally I mean in the Einstein vacuum case you can just forget about the s but so what happens if you have different fields in the Einstein equations so we can say that this sigma so the shears of this chi hat over here so this is the limit of this shear so it has limits at plus and minus infinity for u so this has limits and this will be related to this permanent change of your space time after gravitational wave has passed and well s is now be any tensor of function which depends on the fields with the right decay and t will denote any lower order components of the stress energy tensor well what the what the theorem says well we also have some function phi which is the solution of the following so we have f minus its mean value f bar over a sphere at infinity and this strange notation means this is just at the sphere at infinity and then this difference sigma sigma minus is given by this equation so in other words the radiation here or what is radiated away this energy comes into the difference of this shears at null infinity and this is directly related to some displacement of test masses so maybe let me skip the ideas of the proof and do something else so you can prove this you need some to investigate some hot systems locally and then go to null infinity and take the limits of these and see what happens out there well now it can be shown that for now we are still in asymptotically flat space times so the permanent displacement if I write this down again you remember from before so maybe here so if you have test masses of this type and the gravitational wave comes and travels through so now the claim is and this is not a Laplacian this is just a delta the claim is that the permanent displacement is given by this right hand side and you see this right hand side is this difference of the shears at null infinity that I just developed in the theorem and in the theorem is linked to the energy irradiated away which is then sourced in the pure gravitational case it's sourced by the limit of this Xi so of this null second fundamental form here and if you have extra fields like electromagnetic or neutrinos they will actually add to that okay and so there is also something called the ordinary memory that's what people thought was the linear memory before so and this is now a completely different thing so the null memory is what the big portion of this permanent displacement actually is and well maybe here very briefly how does it actually now relate to experiment so we have well by really starting the Cauchy problem and looking at the null infinity so we can really derive a lot of geometric or analytic information but how does this now reflect anything in the experiment well if I set up an experiment so let's think of three geodesics in spacetime and denote them by gamma zero one and two so just geodesics on which my particles are floating and well T is my future unit time vector field sitting at gamma zero and then I'm looking at what's happening so I look at these geodesics I introduce an orthonormal frame field so I can construct the frame field along the gamma zero geodesic and well I can then set things up so that I can measure nicely what's happening with the distance between one and zero and two and zero now I can well under certain circumstances I can replace a geodesic by the Jacobi equation and I have two derivatives on the left hand side which give me a Riemann curvature component on the right here interacting, contracting with XL and interesting enough so depending everything is really sitting in this Riemann curvature component so you really need to understand the Cauchy problem how your spacetime looks like to really understand what's happening here and the interesting thing is maybe let me also add now a null geodesic or a null fluid so basically your team you knew is some positive function here K i, K j or K is a null vector and looking at the twice-contructed Bianchi identities if I plug in a null fluid so we have shown that well the null fluid will also contribute the Einstein equations for a null fluid reduced just to this equation here so the spacetime which is given by some constant times the energy momentum tensor where the null fluid is sitting now if you look at the portion of this null fluid which has let's say the right decay behavior so it decays like 1 over R squared and something in U and we like to understand how is now the Riemann curvature related to this portion of the null fluid which comes in through the right side of the Einstein equations well we know that the Riemann curvature can be decomposed into the traceless part which is the wild curvature and we have Ritchie curvature and scalar curvature here and so well if you plug everything in and we look at the let's say worst components we find zero is just a T element so we find that well if you look at that well we have some component in the pure wild curvature but also through the Ritchie coming from the null fluid if you look at the Ritchie component we go and look at the corresponding Einstein null fluid on the right hand side so which component we plug in the worst components and we use null foliation again and when we plug everything in and go back to the null L and L bar notation we can see well there's two things that happen first of all I have this alpha bar curvature which is going like 1 over R I can define the limit and write it down like this and I have a component of the null fluid which goes like 1 over R squared so we would say well wait a minute so we have 1 over R in the curvature and 1 over R squared here so it does not contribute so this is lower order so it's true for the instantaneous displacement so when the gravitational wave is traveling through so this is indeed lower order for that but it turns out that for the cumulative effect afterwards the memory this is actually of the same order how does that work so let me write down okay I have maybe two more minutes so let me write down the second derivative here in the Czechubi equation again on the right hand side and now this is the notation for the curvature component at null infinity that 1 over R component so now we have this nice relation between the shear and the curvature and we know also that the lemus of this psi goes to 0 when u is very large so this means that if you have test so here we have just if you plug this in this means that if you integrate here once or you substitute this means that the velocity will be 0 after the gravitational wave train has passed so something happens but for large u the velocity will be 0 so they go back to rest I substitute twice here the shears and plug that into this equation and when we take the full limit I get exactly what I told you before namely this displacement here which is this permanent displacement is given by the difference of these shears which again behind the right hand side is this energy which is radiated away so what exactly contributes from the spacetime what exactly contributes to this to this displacement okay maybe I should say one more word just at the end so with Garfinkel we studied also cosmological spacetimes the question is if you are in a cosmological setting are these things still there how do they interfere how would for instance the cosmological constant come in so does this play any role so we look at Friedman-Lehmann-Robertson-Walker plus dositor so with positive cosmological constant and we found for the dositor spacetime so we can write down the metric like that so the dositor spacetime models the basically inflation period of the universe so that's one way to think about it so in that case we found in dositor that well there is a factor 1 plus r h 0 the Hubble radius which are the Hubble constant so which is giving you an enhancement of this memory effect so the memory effect is multiplied by this factor so it becomes bigger something similar maybe I jump through that to the very end but something similar you can say for the FLRW case so this is working progress we're writing it up but for FLRW which is modeling basically our present universe so you can also see that this memory will be enhanced by such a factor so maybe I stop here thank you does the error is there a lasting effect on the area of the triangle pardon do a triangle so first of all let me maybe make this a little bit more precise so if oh yeah exactly so if I have just a circle and the wave is hitting from here so what it does basically it's kind of stretching and squeezing it right so it's like a planner wave so this is what it does and you can think of this like well at pick points on that and now the claim is that with the memory so you would have this has a certain well displacement in this direction and this is by 90 degrees in the other direction so this will be permanently displaced and then you can actually compute what the angle would be but what the area would be but it's not clearly bigger or smaller is it? Not really I think maybe you could compute that so what means after LIGO they saw that it saw their one signal and now they have to recalibrate their machine well that's actually the problem so this memory is at low frequency and also what they saw in let's say if you look at the strength of the signal they saw let's say they dealt the lambda over lambda is like 10 to the minus 21 of what they saw now there's a paper proposing to to measure memory with LIGO that just came out a few weeks ago so they propose something like at the order 10 to the minus 22 and also all the noise that you can think of is at low frequency for LIGO so it's probably not the best way to look for it but nevertheless so some people are proposing to do that and they have some kind of a filter they put there so I don't know they are improving the instrument Yeah right absolutely I mean the place to look for that would be the space project that NASA and ESA has been working on and I think the Pathfinder was launched by the European group so in space this would be easy well easy technical details now modulo those out so this would be really easier to see there also with the frequency and no noise Could you say a little bit more about what you meant but that there is a memory effect even for Maxwell's theory? So actually we thought about we looked at the pure Maxwell equations which are linear of course and there's some if you take let's say Jackson and take one of the formulas play around so you get easily what is called the linear what we call the ordinary memory and this would not be a permanent displacement but you could think instead here these masses are not charges just test masses following the geodesics but what you would do for the Maxwell cases you have charge test masses and you would get an overall velocity after the passage of an electromagnetic wave so a kick so we found the ordinary thing and we also found a null kick which is then again bigger and corresponds to the null memory here actually and that's for the pure classical Maxwell equations of course you can now ask what about QED well one thing seems to be there the other we don't understand yet okay some more questions well if not let's find the speaker again thank you