 Hi, I'm Zor. We'll continue Zor education. Today we will continue talking about conservation law in theory of relativities. That's a continuation basically of the lectures which I was talking about before. So basically it's a course. You cannot really consider this particular lecture just by itself. It's all the continuation. So I strongly recommend you to go to the place where the whole course actually is represented which is unizord.com and this course is called Relativity for All. Now on the same website you can find prerequisite courses which are mass for teens and physics for teens. Obviously we cannot study theory of relativities if you don't know the basic classic physics and significant piece of mathematics like calculus at least and vector algebra. Now the website unizord.com is totally free. There are no advertisement. You don't even have to sign in. There are some functionalities, advanced functionalities for teaching basically where you might need to be like signed in. But that's there is a relationship between the student and the teacher. But for self-studying you can just go straight to the course and study as much as you can. Now every lecture on that website including all the lectures about theory of relativities is supplemented with very detailed notes. It's basically like a textbook. So consider you have a textbook but for every chapter there is not only the written piece but also there is a lecture which basically explains the same thing. So I strongly encourage you to use the website and read all the notes for each lecture because in some cases I might actually be a little bit more detailed in the notes than during the lecture and just to save some time in the lecture. Okay so today we will continue talking about conservation law as I was saying and the previous lecture was about kinetic energy and this lecture would be about total energy which any object possesses including the famous Einstein's formula is equal to mc square. Okay so let me start from the results which we have obtained in the previous lecture. Obtained relatively simply but looks complicated. So if you have an object of so-called rest mass m0. So that's the mass which we measure in the reference frame associated with this particular object. So the reference frame where the object is at rest. So we measure its mass. Some whatever we can for instance using the gravitation or some other way. So we measure the mass it's m0. Now let's assume that now the object is moving in that same frame. So it's no longer at rest in the frame. It's moving with speed u. Then we have come up with a formula for kinetic energy of this particular object. It's equal to m0c square divided by square root 1 minus u square c square minus m0c square. Now this one is less than one generally speaking because this is the speed of object. This is the speed of light which means that the whole fraction is greater than m0c square. We divide by something which is less than one sorry which means it's increasing. So that's why this is the positive or at least not negative. Let's put it this way. Only if u is 0 we have this 1 and we have result is 0 which is obvious. If speed is 0 you have kinetic energy 0. That's okay. But formula looks quite complicated quite frankly. Remember the formula in classical mechanics. One half mv square remember this or mu square in this particular case. Okay. So that's much more complex and well there is a justification for this. The justification was that the speed of light c is constant in every frame in every inertial frame. The speed of light is the same and to preserve this particular postulate which we have accepted because based on experiments basically we have come up that the formula must be a little bit more complex. We also talked about in the previous lecture that if u is really small we basically change this relatively weird expression to Taylor series and if we will approximate with only a couple of members of Taylor series and do all the calculations we will have the classical formula one half of mu square. So it corresponds to the classical mechanics when u is really small when the speed is small. However there is still some explanation obviously needed because formula looks weird. But let's just think about this. Now k is kinetic energy. Well which means this is supposed to be energy right? Now if we are using plus or minus between two different things in physics they must be of the same type. Now in mathematics we can add, subtract any kind of numbers like pi and one half. Well pi plus one half is another number. But in energy we cannot combine let's say meters and seconds and add them together. It doesn't work this way. In physics we have to, if we are adding something it should be comparable things. So this is energy and this is energy. There is no other way. If this is energy then this must be energy. Okay. Question is what kind of energy? Let's just change the formula a little bit. m0c square divided by square root minus c square c square is equal to mc square plus k. So this is energy which is independent on any kind of reference frame you are measuring this energy. Because this is the rest mass which means it's always within a particular reference system which is related to this particular object. The object is at rest. So whenever we go to another reference system or this particular object starts moving whatever it is, this is a constant. Now this thing depends on the speed. And as speed is increasing the u over c is closer and closer to one. The whole thing is closer to zero which means that the whole thing is increasing to basically to infinity. Because if u is closer and closer to c the whole, this thing which is energy again, we have agreed that these are all energies so this energy is increasing to infinity. Well because the k is increasing to infinity and this one stays constant. Now what Einstein has basically proposed and it was confirmed in many experiments and it was theoretically justified in many different ways. He has assumed that this is an energy. You know it's supposed to be an energy but what kind of energy? It just stays with the body as a constant basically. So he assumed this is energy which is concentrated in the body just based on its mass, rest mass. So this is an amount of energy concentrated in the body just because it has a mass. Now as the body starts moving with certain speed we add kinetic energy to it as well and that's why this can be considered as a total energy. So the total energy is combined from the energy of the object at rest and it's kinetic energy. Now there is another type of energy potential but that's if we are in some kind of a field like gravitational field or I mean that's supposed to be added as well to a total energy but right now we're not talking about this. We're talking about vacuum, we're talking about only the body, the object and it's moving in vacuum, just that. So again on the left side we have a total energy which object possesses and on the right side we have basically split it into the inner energy the object has. This is a rest energy concentrated in the mass which is at rest which is constant and kinetic energy based on movement. Now if you have an object and if you are in some other inertial frame moving relative to this particular object which is at rest in its own rest reference frame well the speed of this object relative to the moving frame obviously is not zero which means the object is in possession of certain kinetic energy in the moving system. So again one system object is at rest but then another system which is moving relative to the first one and that's well since the object has certain speed relative to the moving frame right is moving frame doing this so they are mutually moving against each other which means it has certain speed which means in that frame the total energy would be greater because the object would possess in that moving system certain speed therefore kinetic energy. So what's remain as a constant is inner energy which is concentrated in the object and kinetic energy depends on how it moves. Okay now that's quite an interesting actual observation it means that total energy depends on the system where you measure it right you measure it in a system where object is at rest and you have just this one you measure a system you measure the energy in some other system which is moving relative to this one and it has a speed component and that's why it's kinetic energy is supposed to be added so the total energy depends on the on the frame which where you're basically measuring this energy that's very important. Okay so let's just leave it as this and what I would like to do is you remember the Lawrence factor gamma now using this we will see that total energy is equal to this one which is gamma m1c squared. Now this is the formula for a total energy. Now whenever you're talking about mc square you really either imply the rest energy and that's basically something which people used to see in some textbooks they used to see this formula so that's not exactly correct because in this particular case m is the combination of gamma factor and rest mass but usually the more I would say more correct way is to represent it this way then you know that this particular coefficient is constant which is depending on the speed and the rest mass but people kind of used to see this one but that's assuming that instead of this mass you really mean this one. Sometimes gamma times m0 is called relativistic mass but I think we're talking about that some people are against using gamma times m0 as relativistic mass well it has certain negative implications but some people still do this and they call this formula correct where m is relativistic mass not rest mass as in here so basically sometimes the gamma times rest mass is replaced with just m implying relativistic mass. Okay so we have this now what happens is that if the body is at rest in some system well it possesses certain amount of energy which is m0 times square it's at rest no speed now what if this body is for whatever reasons explodes now it's no longer at rest its pieces are going all the way right so what happens in this particular case with the total mass so you have to have certain mass then it splits into little pieces and going into different directions what if you will accumulate these pieces and measure the total mass well this formula basically tells that the total mass should be less because certain amount of initial mass is converted into energy let's just calculate what happens in this particular case so let's consider we have an object of mass m and it splits into two pieces m and m which are going into well I probably should draw it differently for simplicity I would say it goes let's say up m and down m okay so these are two equal pieces into which the object was mass capital m this is the rest mass so it sits and then all of a sudden it splits into two halves which are running in both ways with speed u and minus u opposite speeds so my question is let's just calculate what happens with masses is there a difference according to this formula is there a difference between some of these masses two lowercase m's and compare with the capital m do we really have a loss of mass in this particular case which is converted into energy well let's just calculate very simply initially the energy of the object at rest was m times c square capital m is mass at rest so the total energy was this one now this energy must actually be preserved because we are assuming that the conservation of energy is a law right so supposed to be equal sign now we have assumed that the masses of these objects and we're talking whenever you're talking about mass let's just agree that we're talking about rest mass okay so the rest mass of these two pieces is lowercase m which means that it possesses the energy to mc square right plus kinetic energy now what's kinetic energy of each one well mc square divided by square root of 1 minus u square c square minus mc square right that's the formula for kinetic energy again lowercase m in this particular case is rest mass of these two pieces on which our object split and this is the formula for kinetic energy and two because there are two pieces and the same thing here so these are energy at rest this is kinetic energy so energy at rest plus kinetic energy of two pieces must be equal to initial energy concentrated in our object right but what do we equal to mc square will cancel out so we have two mc square divided by square root 1 minus c square c square okay now c square is also canceling out and we have the final formula which is m equals 2m divided by square root 1 minus u square c square now this is obviously it obviously means that sum of two pieces is less than initial mass of the object you can just change it slightly to m divided by capital m is equal to square root of 1 minus u square c square right that's the same thing put m here and square root there so this is less than one it's one only if u is equal to zero so there is no speed if we just represent our object as two halves which are not moving then we will have the sum of masses of these two pieces is exactly equal but if they are moving if u is greater than zero you have this less than one which means we are losing certain amount of mass to convert it into kinetic energy now that was something which was the beginning of nuclear bomb because what what actually happens in nuclear bomb we have certain amount of energy released when the nucleus of uranium 235 or plutonium 239 is split into other elements much more complicated than in our case when we have just two equal pieces there are many different pieces but in any case they're all going into different directions so they all have kinetic energy and that kinetic energy is released just because our one particular nucleus is split into few parts so this was the beginning of this Manhattan project in the United States and the development of atomic bomb in all other countries so that's very very important thing and what's also very important is that well obviously it depends on how much mass we are losing but it looks like in certain cases we are losing significant amount of mass and calculated multiplied by c square gives you a huge amount of energy that's the source of energy which is really very very big okay now what remains to be done two things first of all I would like to come up with very important equation between impulse or momentum of moment momentum and energy okay now you remember that momentum and again that wasn't one of the previous lectures momentum was u is speed m0 is rest mass of an object so its momentum is basically slightly more complex than classical momentum classical momentum is just mass times speed in this case it's not just mass but mass at rest because as we know it's all different whenever something is moving time speed and then we have to again this Lawrence gamma factor okay now as far as energy is concerned we know what the energy actually is the full energy is m0c square divided by square root the same square root now believe me or not and you can actually do it yourself and in the notes I did it explicitly you can actually have this particular expression so if you will just do this whatever I just did you will see that this equal to m0c to the fourth I don't want to do this algebra right now it's really simply a couple of lines of code but I know the result and if you would like to see these couple of lines they are represented in the notes for this lecture on unisor.com so what does it mean that's very very important remember we have spent a certain amount of time and again in my notes I have a full proof that P which is the momentum is invariant if you go from one reference frame to another now m0 is constant that's where every object is basically constant it's mass in the rest frame c is constant which means that you have e square equal to p square c square plus m0c to the fourth now this is invariant if you go from one reference frame to another inertial reference frame which means e is invariant as well so first of all this is called the energy momentum equation it's a very useful one and it basically proves that total energy not just kinetic energy or potential energy whatever it is now we are talking about total energy which is this one it is invariant whenever you go from one reference frame to another inertial reference frame the amount of total energy is always the same that's very very important okay now what's interesting from this is what if m0 is equal to zero for certain objects I should say the word object is not maybe a good one consider light light has speed but it doesn't have mass there is no rest mass for photons if you remember from our electromagnetism course there is a concept of a photon which is an elementary piece of light if you wish it's the beginning of the quantum mechanics but in any case photons have no mass no rest mass sorry however they do have speed but in this particular equation if this is zero what do we have we have e is equal to p times c so energy of light and its momentum are related in this particular kind of relationship or if you wish momentum of light is equal to energy divided by speed of light and momentum is very important because when light bomb bards let's say some kind of a metal plate it hits electrons and they're kicked out if you remember this this is the effect which was actually Einstein who was one of the first who studied this particular thing photoelectric effect right so that's what the impulse is coming from by the way the same formula can be obtained from classical electromagnetism and Maxwell's equations etc using something which is called pointing vector there is a pointing the name there was a guy so what's interesting is that from this equation we can actually go all the way to this and prove this just algebraically so that's just another approach to this and another confirmation that using this theory we have come up with a formula which has been known actually since the end of 19th century before before Einstein before theory of relativity this formula was known for light for electromagnetic postulations and again if you start with this one then you can actually go all the way there we don't do this but if you wonder is something some information on the internet how to approach it so these are basically all the little things which I wanted to talk to you and the most important result of this lecture is this where gamma is Lorentz gamma factor so that probably would be I do recommend you to read the notes for this lecture by the way this kind of algebra I put in the notes so it's you know it's I think it's more satisfactory if you will see it with your own eyes rather than believe in me that I just said you can do it very easily and it is very easy it's just a couple of couple of lines okay that's it for today thank you very much and good luck