 Hello, my name is Shiv Prasad, I am professor of physics in physics department of Indian Institute of Technology IIT Bombay and I have a pleasure to introduce some lectures on special theory of relativity. I do not think that as far as the social impact is concerned, any discovery of physics had such a great impact as special theory of relativity. The relation E is equal to m c square had essentially become a common name, almost everyone associates Einstein with E is equal to m c square. There have been t-shirts, there have been so many things designed with E is equal to m c square. Even if you ask a layman who would not know anything of physics, probably you would have heard of equation E is equal to m c square. That was the impact of theory of relativity. Now, in this particular lecture, I will start with an introduction of special theory of relativity. I would like to mention why it became necessary or why it become important for us to discard some of the old concepts of classical physics and eventually land up into a theory which is called special theory of relativity or to be more general theory of relativity. So, we will be discussing some of these concepts. Then, we will actually go to the formalism of special theory of relativity. We will mention what is special theory of relativity, what are the postulates and how does it change our perception of the nature. After that, we will try to work out a few examples, try to explain the intricacies of what Einstein postulated. Then, we will go to some other concepts which are somewhat more abstract concepts like concepts of four vectors and eventually land up into mass energy relationship E is equal to m c square, which as I have said just now had a very great impact on almost everyone. Finally, we would like to discuss the transformation of electric and magnetic field and that is where our force would end. So, this is in a brief the course of action that we will be doing as far as this particular course is concerned. Now, before I start, let me introduce some concepts which are sort of very common concepts in the classical physics and one of the most important thing that we have to understand is what we call as frame of reference. I would like to spend a few minutes to explain this concept because my impression has been that many of our high school children are fairly confused about this particular concept. Also, they try to mix up observations from various angles. So, let us first understand what I actually mean by a frame of reference. Well, the name sounds big, but actually the concept is very simple. Whenever we are trying to make some measurements, trying to understand, trying to speed to measure the speed or acceleration or for that matter any dynamical quantities, we should be very clear who is observing and where this particular observer is situated. So, I assume that there is a person which is sitting with a watch and some measuring instruments and has capability of measuring distances, time and therefore, speed, accelerations, etc. And this particular observer is fixed in a given place wherever we are trying to observe. Now, as we will be seeing very soon in the some of the examples, the perception can be quite different depending upon where this observer is sitting or where this observer is located. So, effectively the person who is making an observation wherever he is sitting, we call that particular place as the frame of reference. Let me just take a few examples. Let me first explain this thing little more formally. So, this is what I have written. For making observation about motion, one has to fix the position of an observer. This is what I have mentioned just now. We can assume that a set of axes to which the observer is permanently attached. We assume that there exists a set of axes. So, we know what is x-coordinate, we know what is y-coordinate, we know what is z-coordinate and he has all the methods of measuring all these coordinates. We call such set of axes as a frame of reference. Let us take an example here. I have shown a sort of a cart which can assume something like a train compartment and something which is like an observer which is sitting on the ground. This is something which is very commonly seen in everyday life. Let us assume that this observer which is sitting here is on earth and this is a train which is moving to the right hand side with a velocity v as being observed by this particular observer sitting on the ground which I am calling as s. I assume that this particular observer has a clock here. It has a measuring instrument. I assume that there is another observer which is fixed into this particular train compartment. Let us suppose he is sitting on that particular one of the seats. This particular observer also has a watch and has also a measuring instrument but he is permanently fixed in that particular reference and we assume that he has all the capabilities of measuring the distance remaining fixed in that particular frame of reference. Now, I call this particular frame s, this particular set of axes and this particular observer as the frame s. I call this as frame s. Now, as is the common perception, this particular observer feels that this train is going towards the right hand side with a velocity v. That is what we have just now told. But what will be the perception of this particular person which is fixed into this particular train compartment? This is a very common thing and I think many of us in our children, in our childhood would have asked the question to our parents when we are moving in a train. See, I see all these trains moving backwards. Why they are going backwards? Exactly in a similar way this particular person would feel that this particular observer s is moving backwards with the velocity v. He does not perceive his own velocity but he perceives that this particular observer s is moving backwards with the velocity v. So, what we are saying is that perception differs on frame of reference. So, this is what I have written. Each observer has a measuring tool and a set of axes drawn at the location and the perception of the speed is different for two observers. So, I think we should be now clear that whenever we are defining a speed, defining a distance, defining a velocity, I fix particular observer to a given place which we call as a frame of reference. And we should never confuse between the concepts of the frame of reference. We should be clear when I say that this is the speed of a particular object. Who is measuring that speed? Where is that particular observer located? Or in other words, what is the frame of reference with respect to which these measurements are being made? So, we have to be very clear. We cannot mix up the speed measured by one frame, one observer in one frame from the one which is being measured by another observer in a different frame. So, we should be clear about this particular concept. Now, once we have understood this particular concept of frame of reference, now let us review little bit the classical laws of motion and which I would like to start with Newton's law. In a normal textbook, we define, we generally give three different laws of motion of Newton. But when I say Newton's law here, I would generally mean the second law because that is what is considered as the most fundamental law of Newton, which essentially tells that force is equal to mass into acceleration. Before I come to this law, let me first go to the old concept of what is a natural state of motion. If we go to the old time, very, very old time, we always thought that the rest is a natural state of motion. This was not very difficult to understand why people thought at that particular time that the rest is a natural state of motion because we always find things to be stationary. Our observations of course at that time was fairly limited and whenever we see, for example, this particular pen, it is stationary. If I have to make it move, I have to apply a force, then only it moves. So, it appears that rest is natural state of motion because even if it moves after some time, it again stops. So, probably everything object likes to stay in stationary state. This particular observation was believed by large number of people until Galileo when we had more results of astronomical objects and we found that it seems that lot of astronomical objects, we seem to be moving at least as seen by earth or as seen by an observer on earth without apparently any force on them. So, it was postulated by Galileo and later put as the first law of motion by Newton saying that it is not the rest, but a constant velocity v, which is the natural state of motion. Of course, when we say v by v, we mean a vector v, which means the magnitude as well as the direction. So, what was postulated by Galileo and later by Newton is that it is the velocity constant velocity, which is the natural state of motion and not really the rest. In fact, rest can be perceived differently by different observers as I have just now mentioned in the given example. Therefore, what is natural is having a constant velocity and why this particular object stops for example, when I give it a force is not because that the rest is a natural state, but because of the fact that there is some frictional force on this particular paper, which makes it stop. Had this particular had this particular friction force not been present, this particular object would have kept on moving. So, this is something which was a total different or total change in our conception when we realized that the constant velocity is the natural state of motion and not just the rest. And then we come to the Newton's second law, which says that in case this natural state of motion has to be changed, it means if the velocity has to be changed either in magnitude or in direction, then a force has to be applied. So, force would result into an acceleration, which essentially means a change of velocity actually acceleration is given by define as dv dt, where t is the time. So, if you differentiate velocity with respect to time, we get into acceleration and this acceleration is given by the force which we applied to the body. So, this is what I have written in some of the next transparencies. Let us just look at this. We have said that natural state of motion Galileo and Newton emphasized that it is constant velocity and not rest which is the natural state of motion when there is no real force acting on the body. Now let me come to this concept of what we mean by a real force. In classical mechanics which actually is a result of also the Newton's third law of motion, all real forces occur in pair. Let me try to explain what I mean by this concept of a pair. If a particular body is experiencing a real force according to the classical mechanics, there has to be another body which must be exerting a force on that particular body. And the body which is exerting the force on this particular body must also be experiencing equal and opposite force. Let us consider two bodies and let us take a simple example of a gravitational force. So, when I say for example, this could be earth or this could be any other heavy object and there is a small object. We know that there is a gravitational force which pulls this particular object towards this. So, this particular object experiences a force F towards this body because of the gravity. Now what I am saying is that this cannot be a force in isolation. If this body is exerting a force on this particular body in a result as a result, this particular body will also be experiencing a force which is equal and opposite. Of course, these two forces act on two different bodies. This particular force is exerted on this particular body. The same force equal and opposite is exerted on the other body. So, there are always forces which are occurring in pair and they are equal in magnitude and opposite in direction. This is what we mean by forces, real forces always occur in pair. So, for example, if I drop a body on the earth, this is going down or this accelerating down because the force of gravitation caused by earth, then this earth is also being attracted towards this body by the equal amount of force in an opposite direction. The only thing is that earth is so big that this small force would hardly have any result or resultant change in the motion of the earth. But this body having a small mass would be experiencing a much larger acceleration. Similarly, if you would have two charges, let us say Q1 and Q2 and there is electrostatic force between them. If Q1 is attracting, let us suppose there are opposite charges, so there is an attraction force. So, if this particular charge is attracting this particular charge by a force f, this charge must also be attracted towards this particular charge by the same force f. So, all these real forces occur in pair. We do not see a real force, we just act on one body without having another body on that causing this particular force. Now, the second statement which I make is that in classical mechanics we can see that as the distance between these two bodies which are interacting increases the force on these bodies decrease. In fact, as we are quite aware that gravitational force and also the electrostatic force goes as 1 upon r square. It is proportional to 1 upon r square where r is the distance between the two bodies. So, it means if I increase the distance between these two bodies, if I make this body further off, the force of gravitation between these two bodies will go down as 1 upon r square. Similarly, if I make these charges go away, then the force on these two bodies will decrease as 1 upon r square. So, these are the two statements which I am making in this particular transparency. In this particular transparency that in classical mechanics real forces occur in pair and result from interaction of two bodies and their magnitude decays as the distance between the interacting body increases. Now, we come to the concept of what we call as an isolated object. Now, if you can imagine that I can take one particular object really far off from all other objects, then we can assume that this particular object would not be experiencing any real force to be more precise mathematically as the distance of this particular object increases or tends to infinity from all other objects, then the force tends to 0. And that is what I have written an isolated object is thus expected not to have any real force acting on it. Therefore, if there is no real force, then whatever we have said about laws of motion, then in principle this particular object should not be having any force on this. And therefore, should be moving with constant velocity, but just now we have said that the velocities depends on the perception on observer. So, we will define some frames of references which we call as inertial frames. Inertial frames are those in which this isolated object appears to be moving with constant velocity. So, let me read this particular statement. A frame of reference in which an isolated object which implies that it experiences no real force is found to move with a constant velocity is called an inertial frame of reference. Of course, it means that there could be some frames of references in which this particular object would be found accelerating. Those frame of references are special frame of references about which I would not be talking as far as this particular course is concerned, because this course is actually a course on special theoretical relativity. And the word special here means that we are dealing only with those frames of references which are inertial. It means we always assume that our observer is sitting in a frame of reference in which an isolated object is found to move with a constant velocity. Of course, again I remind when I say constant velocity, it means both magnitude and direction, not just the magnitude velocity is a vector quantity. So, let me make this statement. There need not be only one inertial frame of reference, there could be infinite number of inertial frames of references. They may measure different velocity for the same object, which I am assuming to be an isolated object. Their velocities could be different, but what I am insisting that if the frames of references have to be inertial, then all of them will observe the velocity of that particular object to be constant, not varying with time. Of course, the magnitude of the velocity could be different for different frames. The example which you have just given is assuming it is quite clear that depending upon how we are our observer sitting, the velocity measured could be different. So, what we have said that out of all these velocities, though velocities could be different, but they will always find the magnitude and the direction of the velocity to be same. Now, I make a further statement that if we have inertial frames of reference, different inertial frames of reference, then if all of them are inertial, then the relative velocity between them also have to be constant as a function of time. What it means that if I have a set of axes like this and I have another set of axes like this and there is an object, this observer S observes this particular object and finds it to be moving with constant velocity. Similarly, this observer S also observes this object and finds it to be moving with a constant velocity, then the speed of S and S, the relative velocity between them has also to be constant because if it varies, then they cannot both observe the object to be having the same, to be having a constant velocity. Now, as I mentioned that we shall assume that in this course, unless specifically mentioned observations are being made only in inertial frame of references and whenever we are talking about the force, we are always assuming that the forces are real. That is what is the typical of special theory of relativity as I mentioned that special word means it deals only with special sets of frames which are inertial frames of references. Now, in this particular transparency, I have again described Newton's law of motion which I had mentioned somewhat earlier that a force is responsible for changing the state of motion by causing the body to accelerate and this is given by this equation F is equal to ma. Remember this is a vector equation. So, if I apply a force F and M is the mass of the object, this particular object would accelerate with an acceleration A given by this Newton's law of motion. Now, I make another statement which I would like to now prove saying that as far as Newton's law of motion is concerned which I have just now written as F is equal to ma. It makes no difference where our observer is sitting, so long he is sitting, he or she is sitting in an inertial frame. What I mean to say that this particular law F is equal to ma will be valid in all inertial frames of reference. Let us just understand this particular thing a little bit more because we have just now said that velocity is different in different frames. Let us understand how it is so happens that this particular law is frame independent. It means irrespective of from which frame I am observing this law will be still valid. Let me just take an example here. Here what I have drawn are two sets of axes. One is a blue set of axes which I am calling as another set of axes which are sort of red set of axes which I am calling as it has its own set of axes. For example, this is x prime, y prime and z prime. This has x, y and z. I assume both these set of axes are representing inertial frames. Let us assume that there is an object here which of course could be stationary moving or in general accelerating or maybe even accelerating with a constant acceleration or a varying acceleration. I make no particular assumption as far as the motion of this particular object is concerned. Only thing I am assuming that the frame s and s are inertial frames. Now, let us freeze the situation at a given time and at a given time the observation is made by both the observers in s and s about their position. This observer, red observer which I am calling as frame s, observes the position vector of this particular object to be r t. This is what I have written here r t. At the same time because the origin of observer s prime is here, he measures a different position vector which I am calling as r prime t. So, their position vectors are different as we change the frame of reference. Of course, it can very easily see from the standard vector algebra that if the origin of this particular frame s prime is located at a position vector of r here with respect to the origin of s frame, then this r t must be given by this r t plus r prime t. If I have to reach from this particular point to this particular point, I can go this way and then this way and will reach this particular point. So, this vector r t can be written as sum of capital r t plus r prime t. So, this I think should be clear that r and r prime would be different when I change my frame of reference. The position vectors would be different because their origins are in general different. Now, this is what I have written in this particular transparency. This is the same equation which I have written in the last transparency. We have said that all these things depend on time because I have frozen the situation at a given time. As time changes, this position vector r will also change, this will also change, this will also change because everything is dynamic, they are all moving. Now, suppose I want to evaluate their speeds, I must measure in a given time how much position vector has changed and to be very precise, I must differentiate with respect to time. So, I differentiate this with respect to time and of course, I assume here this particular thing without making too much of comment at this particular point that the time measured by the two observers in s and s prime is same. So, when I differentiate, I write d dt, I do not write t and t prime separately of r t must be equal to d dt of capital R t plus d dt of r prime t. So, all I have done is differentiated this equation with respect to time and I can recognize this particular quantity. Once I take the derivative with respect to time, this particular quantity is nothing but the velocity of the object. So, left hand side here is the velocity of the object, here is the velocity of the origin of s prime with respect to s. So, this I am writing as v naught, v naught is the velocity of the origin of s prime with respect to the origin of s observer. This particular quantity because this observation of r prime is being made in the primed frame of reference or blue frame of reference, then I differentiate this with respect to time, I will get the velocity of the object as will be determined or will be measured by an observer in s prime frame of reference. So, from this equation I get v of t is equal to v naught plus v prime of t. The only thing you notice, I have not put here this as a function of t because just now I have said that you both these frames of references are inertial frames, the relative velocity cannot depend on time. Therefore, this v naught has to be constant. Now, I differentiate this particular equation once more with respect to time. So, I have v t is equal to v naught plus v prime t, I do exactly same thing what I did earlier, I differentiate with respect to time both these equations. Once I differentiate velocity with respect to time all of you know I will land up into what is called an acceleration of the object. So, this particular quantity on the left hand side will give me the acceleration of the object as is being seen by an observer in s frame. Similarly, this particular quantity because this v is v prime which is being differentiated with respect to time will give me the acceleration of the object as is being measured in the s prime frame of reference. I insist that this quantity will be 0 because v naught is constant as a function of time. Therefore, a must be equal to a prime. So, we have come to a very important result that though velocity is a frame dependent quantity, but acceleration is not. So, once I change the frame of reference velocities of the same object being measured by two observers could be different, but if I decide to measure their acceleration the two observers s and s prime would agree that their accelerations are same. So, now I come back to Newton's law of motion. I agreed we discussed that this a is same for all inertial frames of reference. Therefore, I need when I give the acceleration of the object I need not define my frame of reference. I can say that this is the acceleration of the object. So, long I am talking of course only inertial frame of reference. Now, if f is also a quantity which is frame independent quantity then f is equal to m a will be valid in all inertial frames of reference. The velocities could be different in different frame, but the force and acceleration would be expected to be same and therefore Newton's law of motion is equally applicable in all inertial frames. Now, so far it is good as far as all mechanical forces are concerned we are quite happy about the situation because for example gravitational force expected to be independent of force, but we have some problem when we come to electromagnetic theory. Let us see this is what I have classified as problem with classical physics. We just now said that I expect the force to be frame independent quantity, but velocity is not a frame independent quantity velocity depends on different frames even though they are inertial. Let us look at the standard expression of what we call as a Lorentz force. If there is a particular charge which is E and if it is put inside a electric and magnetic field the force on this particular charge the electromagnetic force on this particular charge which we also called as Lorentz force is given by E multiplied by E where capital E is the electric field plus V, V is the speed of the charge cross it is a vector product B where B is the magnetic field of the object. So, this force now we see it depends on the velocity it means if I go to different frame I will find the charge moving with a different velocity and therefore this particular force which I am seeing the Lorentz force would also become different. Some of you may ask this question and rightly so and this question is probably a very good and genuine question. After all what is magnetic field? Magnetic field also arises because of the motion of charge carriers. So, if I go to a different frame of reference the charges which are the motion of which is causing this magnetic field they will also move with a different velocity and therefore probably B could also be a frame dependent quantity. The answer is yes this is correct, but the fact is that the transformation as I go from one frame to another frame of electric and magnetic field is also eventually obtained only from special theory of relativity the correct transformation. So, let us leave this example as of now I just wanted to mention that this appears to be one of the problems. Let us go to a better example in earlier 19th century or even before that the Maxwell had put down a set of equations which are called Maxwell's equations. Earlier there has been some worries or doubts about what is light, but it was a set of Maxwell's equation which eventually I would say proved that light is electromagnetic wave and eventually he obtained an expression which is c is equal to 1 upon under root epsilon naught mu naught, where epsilon naught and mu naught are fundamental constants. According to him the speed of light should be given by this particular expression if it is an electromagnetic wave. The speed obtained from this particular expression matched with the experimentally evaluated speed of light. Essentially it confirmed that light is actually an electromagnetic wave. So, this is what I have written the speed of light depends on the fundamental constants c is given by 1 upon under root epsilon naught mu naught. This is what we have obtained as a result of electromagnetic theory which had found a very strong hold and strong footing in our classical physics. In fact, there are two pillars of the classical physics one was based on Newtonian mechanics and another was on electromagnetic theory. Now, let us come back to the familiar concept of relative velocity about which we have been talking quite a bit. Let us assume that there is a observer on ground and let us assume that this ground is an inertial frame. It observes two different train compartments. One train is moving to the right with a velocity v. Of course, all these velocities are with reference to this particular observer which is standing on the ground. He finds another train moving to the left with speed u or with velocity u. Now, if I change my frame of reference and I assume my observer to be sitting here on this particular train a, as all of us know, this particular person would experience as if this particular train is moving with much faster speed. And of course, if we assume that v and u are in this along the same line, this speed of the compartment b or the train b as seen by an observer a will be addition of the two. It will be v plus u. It will feel a much larger speed. This is very, very common thing. If you are sitting in a train, another train passes by your side. You find that it is moving with a very, very large speed. The relative speeds just add up. Look at a contrasting situation here in the second diagram here. Again, these speeds are with respect to an observer standing on the ground. There is a velocity v which is trying to overtake this particular train v. For example, a car overtaking another car. The velocity of this particular train or car with respect to this observer is v and of this particular car with respect to this observer is u. Now, if I fix my frame of reference here, let us see on b, I will feel that this particular a is approaching two towards b not with speed v, but v minus u. The speeds are differences. The velocities are the differences of the two velocities. This is a common concept of relative velocity. Now, let us look into a very, very similar and parallel situation which I am given in this particular figure. All I have done is that one of the compartments I have replaced by a light source. My observer is still fitted here, fixed here on the ground. He finds that this particular speed, this particular light travels towards him with a speed c, while this particular train moves to the right as seen from him with a velocity u. Now, if I assume the same thing, what I have done in a previous example, if I change my observer, if the observer was sitting here, he would find that the speed will be c plus u exactly taking parallel of what we have discussed in the previous example. Similarly, if the train was moving like this and the source to the right and the source of the light was here, emitting light in this particular direction. And if this observer finds this speed to be c and this velocity to be u, then if I change my observer to this particular compartment, he would find that the speed is c minus u. The speed will be less. So, like we do in a normal classical mechanics, the observer here, the observer here and the observer here, the observer here, the observer here and the observer here, all three will find different speeds of light. Is it correct? Will this really happen? Let us see what is the impact of this. What happens to my expression, which I have just now written? c is equal to 1 upon under root epsilon dot mu naught. Do I assume that this expression is valid in all these frames? See, remember we talked of three different frames, three different examples, person sitting on ground in a compartment here and compartment on the other side. If all the three observers, are they still allowed to use the same expression of c is equal to 1 upon under root epsilon dot mu naught? If they are allowed, we have just now seen that c is different for all the three observers. And if they are allowed to use this particular observer, this particular example, what has changed? Have epsilon naught and mu naught changed? Then do we say that this particular quantity epsilon naught and mu naught have become frame dependent? It means, if I go to different inertial frames, then I would notice a different epsilon naught and different mu naught. But remember, these epsilon naught and mu naught are related to the basic electromagnetic forces. See, remember this expression, this expression gives the force between two charge carriers q 1 and q 2 and here there is an epsilon naught. This expression gives the force between two current carrying conductors i 1 and i 2 and this force is governed by this particular constant. Now, if I say that this mu naught and epsilon naught are different in different frames, do I mean to say that these forces are also different in different frames? I have a problem, I do not understand it fully. There could be another set of argument. The other set of argument could be that the expression, which I have just now said, c is equal to 1 upon under root epsilon naught mu naught. This expression is not valid in all the frames. This expression is valid only in certain specific frame. This is the second example. See, remember in first example, we said that this expression is valid in all the inertial frames. But then I have to incorporate the fact that epsilon naught and mu naught would be different in different frames if the speeds of light measured by different frame preferences turn out to be different. Second example as I said is, I assume that no, this expression is valid only in some special frames, not in all the frames. So, then I am saying that though classically we had said from Newton's law, I have said that it makes no difference where I make observation, so long the frame is inertial. This may not be true for electromagnetic theory. Means from electromagnetic point of view, all inertial frames need not be equivalent. There is one special frame of reference in which this expression c is equal to 1 upon under root epsilon naught mu naught is valid. So, what I am saying is that the validity of this expression is only in some specific frames. In other frames, this expression is not valid. This is the second example. So, I have written here, the second option would imply that from electromagnetic point of view, there may be a preferred or a special frame in which this expression of c is equal to 1 upon under root epsilon naught mu naught may be valid. It is not valid in all other frames. I will extend this particular thing little bit more. And I say that this special frame, I can say, is probably signifying an absolute rest. By the way, the concept of absolute rest looks at least for a classical physicist, but somewhat genuine. Let us take an example. I would first say that the classical people sort of preferred this particular idea because it had some sort of resemblance, which I will just now discuss some of the things that we assume. See, like we see an ocean. In an ocean, all the fishes and all the sea animals sort of float inside the sea. I can assume that entire universe is filled with some sort of fluid. Of course, this fluid has to be given very special properties. For example, it should be essentially 100 percent transparent because we can see the light, which is coming from long distances. Probably, it has essentially zero density because we have never been able to observe any of these fluids. But let us assume that there is some sort of fluid. I mean, it seems somewhat natural for a classical physicist, which fills the entire universe. A classical physicist at that time called this particular medium as ether. So, they evolved the concept of ether. And they said that this ether fills the entire universe like in ocean, the water fills the entire ocean. Similarly, everything is filled by ether. We are actually living in an ocean of ether. And all the planets are sun, galaxies, everything move in this particular, they float in this particular ether. As I said, for classical physicists, this also found another support. See, when I talk of the traditional waves with which the physicists were very familiar. For example, let us say sound wave. When I say the sound wave has a particular speed with respect to what we talk of the speed. Who measured that speed? And it is always implied that this speed is measured with respect to the medium in which sound travels. You are all probably aware that it requires a medium for sound to travel. So, when I say the speed of sound is this much, I assume that the observer is stationary in this particular medium. And the speed of sound is being measured with respect to the observer sitting stationary in the medium. The speed of sound is given relative to the medium. But light apparently does not seem to be traveling in any medium. We see light coming from very, very far of stars. And apparently there is no medium in between. So, classical physicists did not like this particular idea. After all, the wave requires a medium to travel, sound requires in all the waves that we are familiar. A water wave, though when we talk of water waves, it is always with respect to the water. When we are talking of all these waves, probably there is a medium in which it should be traveling. It is a different thing that I am not able to see the medium. This medium must have very, very special properties. But I am not able to observe it. That is a different question, but probably a medium exists. So, the concept of ether found support from that particular concept, saying that the medium in which this light travels is actually this ether medium. And when I say c is equal to 1 upon under root epsilon root mu naught, this particular expression is true only in this ether medium. So, I can say that this ether signifies absolute rest because this is filling the entire universe, all the planets, everything, including light travels in this particular medium. So, this is what I have written here. Another support, normally waves require a medium to travel, but light seemingly did not require one while coming from distant stars. So, probably there does exist some medium, not observed till that time, because it must have some very special properties. And this particular medium could be called ether. It is this ether which can be thought to be signifying absolute rest. And the speed of light in this expression which I have written must be given only in this expression, only in this particular medium, only in this particular special frames of reference, frame of reference called ether. If I change my frame of reference, go to another inertial frame of reference. We discuss that I must measure a different c if everything of classical physics is valid. But I have said the expression of c 1 upon under root epsilon root mu naught is valid only in that special frame. I will measure a different speed. Therefore, I can talk of what is the absolute speed of that particular frame of reference. So, I can signify, I can associate a absolute speed with every inertial frame of reference. So, this is what I have written. In other frames, the speed of light would be different from c as given by the relative velocity formula. One can determine the speed by measuring the speed of light. So, if I measure a speed of light in a frame and find it to be different from this expression, I can find out what is my speed relative to ether. Therefore, I write again all the objects can be given a unique velocity which is relative to ether. Therefore, now we are authorized to ask the question, what is the absolute speed of earth? What is the absolute speed of sun? Earlier we have mentioned that using pure classical physics, velocity is frame dependent quantity. So, if I change my frame, we will change, but a will be same. But now with respect to this special frame, I can always think that I can also talk of absolute velocity. I can ask the question that the velocity of the object is how much or velocity of earth at a given time is how much relative to ether. This concept looks somewhat strange because no physical process is purely mechanical. So, when all the frames appear to be equivalent from mechanical point of view, can they not be so from electromagnetic point of view? Here we have some sort of uneasiness because strictly speaking no process is 100% mechanical. See if we are observing two bodies to collide which appear to be purely mechanical process, but the light has to go there, get reflected, come to our eye before we are able to observe it. So, there is always a mixing of electromagnetic and the mechanical concept. It does not sound 100% comfortable when we say all the frames of reference appear to be equivalent from the point of view of mechanics, but not so much from the point of electromagnetic theory. We feel some sort of uncomfort that why it should be so? What we do? We need experiments. This is a point where we are having some sort of deadlock. Those things seem to be somewhat consistent from the classical mechanics, but we still have some sort of uneasiness. Let us look whether we have an experiment, whether we can prove that ether exists. So, that is what I have written here. To be sure, we should look at the experiments which can detect the presence of ether. So, in our next lecture we will be discussing what are those experiments and whether it was possible for us to detect ether before we go ahead. Now, let me summarize of whatever we have discussed in this particular lecture. We have discussed the concept of inertial frame. We have described what I mean by inertial frame of reference. We have mentioned that from mechanical point of view, the inertial frames are all equivalent, but once I go to electromagnetic theory, there appears to be a non-equivalence. It means it appears that there is some special frame of reference which is necessary to understand the physics consistently. Then finally, we have evolved the concept of ether and we have said that a special frame of reference like ether should invoke the concept of ether if we want to understand everything from the classical point of view consistently. Though we still have some sort of uncomfort in this particular idea because it does tell that from mechanical point of view all the frames seem to be equivalent, but from electromagnetic theory they are not. Thank you.