 It is Deep Chatterjee from the department of physics IIT Roorkee and I shall be talking on the special theory of relativity. Well, the plan for the next series of lectures is something like this. I shall talk on how relativity arose while reconciling the laws of mechanics and electrodynamics that we more precise reconciling the transformation laws of mechanics and electrodynamics. In that context I will be talking of Galilean and Lorentz transformations moving over to the all important postulates of special relativity. And then I shall of course go over to the consequences which are quite interesting. We will be talking of length contraction, time dilatation, mass energy equivalence and this is one thing perhaps many of you are quite familiar with equal to m c square. Let us try to see we will try to see at a certain point of time how it all arose. And in explaining all these things what I shall do is that as and when necessary we will talk of certain problems. We will try to do some problems so as to illustrate the principles involved. So, let us start at the beginning. Mechanics this f equal to m a this perhaps is t most famous equation in mechanics if I may say so. You all know that if force of is applied on a particle of mass m it is going to accelerate with acceleration a I mean everybody knows this and this you can even tell me that this is actually Newton's second law of motion. But what is assumed here apart from of course in this particular case that we treat m that is the mass of a particle that is always constant. Well if you have when you can argue that if you have a variable mass you can talk of force as rate of change of momentum. But let us stick to this form of Newton's second law for the above. But what is important here is that we always assume here that somehow we have a frame of reference where we are able to measure this acceleration. Now frame of reference is actually very fancy name for simple coordinate system I mean coordinate system you know simplest one of course is the Cartesian coordinate system. If you see where the walls of this room meet I mean the room you are in if it meets with the floor and then you see the axis of the coordinate system. So, we are quite familiar with the word coordinate system so that is a fancy name frame of reference. Now another thing is we sometimes hear of this word inertial frame. So what is an inertial frame? Well a very solid definition is an inertial frame is one in which Newton's laws of motion are valid. Now Newton's laws of motion are valid in inertial frames. So what does this explain where explain a person explain and tell an engineer or a scientist where to look for this inertial frame. We need to be a little bit more precise than this idealistic definition and in doing that in defining that we can say that it is a frame whose coordinate axis are fixed relative to the average position of a fixed star fixed star in space of course or it is that frame is moving with an uniform linear velocity that is a constant velocity relative to the star. Of course there should not be acceleration of this star otherwise the definition is not valid. Well once I have said this and you realize that any frame or the earth itself the earth itself is revolving and it is there is day and night there is this rotation due to which you have of course day and night and revolution around the sun that is the that gives the change of season. So there is some amount of acceleration so basically any coordinate system that is to its surface is non inertial. But for many purposes this acceleration is slight and then by many purposes I of course do not mean for all purposes. If this acceleration can be considered slight here this particular frame that is the frame on earth can be considered inertial. A little better would be a non rotating frame with with a origin fixed at the earth center and axis pointed towards the fixed star so that is we can say it is approximately inertial. Even if you have this frame to be fixed at the center of the sun for example that will be more inertial than compared to the frame I just described. But having said this we should be clear that non inertial frames are actually quite common in in mechanics must have heard of Corioli forces. So these are non inertial forces okay. So let us try to have a visual explanation of what we have been trying to say in more definitive terms okay. So here we have frame S okay so this is three dimensional Cartesian system right handed Cartesian system. I have not written the z axis here but you can all guess and you can all you can all figure out that z axis here actually points outside the screen okay. So this is frame S and then we have another frame S prime let us say and then this frame S prime is moving with a constant velocity or let us say a uniform velocity P along the common x x prime axis okay. Now the coordinates here are the x prime y prime and z prime and here z prime again points outside the screen. Now at the beginning at the very beginning you know let us we have two observers who are let us say at the origins of both these frames and at the beginning both these frames coincide that is S and S prime they have a common origin at the beginning at time t equal to and t and t prime equal to 0. So I will talk of this coordinates and time a little bit later. So how to do that I mean it is simple I mean you start with two observers are there when they look at their watches and say that okay fine. So our watches agree and this is the time we set as t equal to or t prime equal to 0 and then S prime frame starts moving with uniform velocity v with respect along the common x x prime axis. So down the line if you have a point P which has coordinates x y z and then it is measured at a certain time t remember clock is moving I mean time clock is working clock is moving I mean that time is flowing at a certain time t. So at that same instant let us say observer in S prime measures the coordinate as x prime y prime and z prime okay. So how are they related well you can say that it is simple actually. So how are they related x prime is related by this relation x prime is equal to x minus v of t v and t y is equal to y prime and z equal to z prime and it is important that both these time coordinates agree okay. So that is we this point is measured at the same instant of time okay. So they all started with synchronized watches. So t prime equal to t here okay. Now this transformation we call this transformation as Galilean transformation okay. Let us delve a little bit further how are the velocities if you measure the velocities in both these frames how are they related well that actually will be given by the Galilean velocity addition formulae. So let us see how it can be derived we have this equation x prime equal to x minus v t. See that now you differentiate x prime with respect to the time in its own frame. So that is dx prime dt prime and on the right hand side where you are going to have dx by dt prime minus v dt dt prime okay. And now you realize that on the right hand side you have dx by dt prime. Now x is the coordinate in the s frame but t prime that is the time that is measured in the prime frame that is s prime frame. So you need so if you need velocities so you need coordinates of the same frame okay. So coordinate and the time of course in the same frame I should say. So the third step clarifies how to do that so you have dx dt and then you take this dt dt prime minus v of dt dt prime okay. So then you realize that dx prime dt prime that is the u prime that is the velocity that is being measured in the primed frame okay. Now you realize since t is equal to t prime in Galilean transformation so dt by dt prime that is equal to 1. So you realize and then dx dt that is u and so dt by dt prime that is 1 so minus v of minus v. So you have u prime is equal to u minus v. Now I mean there is a subtraction sign here so don't be bothered about that when I use the word addition formula because you can very well write the velocity which is in the s frame in terms of the s prime frame by saying that u is equal to u prime plus v and what is v by the way so it is just the velocity with which the s prime frame is moving uniform that is uniform velocity with which the s prime frame is moving with respect to the s frame along the common x prime axis okay. So that goes for the velocity addition formula. What about the acceleration? Well let us start with what we have obtained for the velocity so that is u prime is equal to u minus v. So if you differentiate this once again so you are going to have du prime by dt prime is du dt of course you know how to get this now so this is du by dt you know how it will be so if you take du by dt prime and then you have dt prime or dt by dt prime that is equal to 1. So the acceleration is going to be the same in both frames so these frames are moving with uniform velocities with respect to one another. So what we have is acceleration is being unaffected by if you have frames which are moving with uniform relative velocities okay. Now on top of that if you consider that mass is unaffected by motion of reference frames you come to an very interesting conclusion we see that the form of Newton's second law is valid. Well actually Newton's second law is valid in both these frames in both these inertial frames okay. So what does this mean? Well this means that by doing experiments entirely in one of these frames you cannot distinguish it from the other okay. So if you are doing experiments entirely in one of the frames so and this frame is moving with an uniform velocity v with respect to another frame you will you cannot distinguish this particular frame from any other inertial frame okay. So by mechanical experiments alone that is what I am going to say here. So what you can ask so what happens is that since Newton's laws of motion are being valid are valid in these two frames so are equations of motion okay which are derived from them and consequently the conservation laws. So you are going to have the conservation laws same in all these inertial frames okay. So if you do your mechanics in one of these frames and you derive a conservation law you can be rest assured that in another inertial frame it is going to be the same it is going to be valid okay. So we could say that the laws of mechanics are being invariant in all inertial frames okay. So this is an important conclusion. So next we move over to what is going to happen in electrodynamics okay. So what is electrodynamics? You see it is an interesting thing. So here you have what does it give you? It gives you that if you have a changing electric field you are going to have a magnetic field then if you have a changing magnetic field you are going to have an electric field okay. Now we ask this question that is electrodynamics or the laws of electrodynamics invariant under Galilean transformation remember laws of mechanics they were invariant under Galilean transformation. So we ask another branch of physics electrodynamics are the laws there invariant under Galilean transformation? Well for that let us see what those laws are okay. The basic laws they are the encapsulated all in Maxwell's equations well divergence of E that is equal to rho by epsilon 0 and E as you know is the electric field rho is the charge density and epsilon 0 that is the permutivity of free space. And then the curl of E that is minus del B del T P is the magnetic field and then the divergence of B that is 0 the curl of B that is mu 0 J mu 0 that is the permeability of free space and J that is the current density plus of course mu 0 epsilon 0 del E del T. Now do not be bothered too much about this mathematical details so what do they stand for I mean that is what I have written on the right hand side of these equations. The first one is actually Gauss's law well so it is a very important law it actually allows you to calculate the electric field if you have symmetries in the problem symmetric the charge distribution that is okay. The second one curl of E that is equal to minus del B del T that is actually Faraday's law and I am sure you are aware of it because had this law not been there I mean we would not have motors you electric motors that is you must have heard of Faraday's important experiment in which you had this he was moving as a magnet within a solenoid and then he detected an EMF within the leads of the solenoid okay. Then the third thing I mean third equation that I have written here when should not be talked as a third law that is divergence of B equal to 0 which well does not have a law which does not have a name as such but its physical implication is that there are no magnetic monopoles like that like you have a you have a positive and negative charges you do not have charges in magnetic poles in isolation okay you do not have magnetic monopoles. The curl of B that is mu 0 j plus mu 0 epsilon 0 del E del T that is actually a curl of B is equal to mu 0 j that is actually Ampere's law okay and then added to that is Maxwell's connection well that Maxwell's well Maxwell's corrections are actually quite important you are going to see later on because it had rather quite rather very interesting implications in showing that these Maxwell's equations well by the way so what you see here is all E and B are coupled these are coupled partial differential equations. So well these corrections were important because he was able to show that well he put he introduced this concept of displacement current and then was able to show that these equations when written in terms of only E or only B could be framed in terms of the wave equation more of that a little later okay. So our Maxwell's equations invariant under Galilean transformations under the transformations mechanics is invariant on so what are these transformations once again. So you have an s frame frame moving with an uniform velocity v along the common x prime axis okay again z and z prime this axis are moving are actually out of the screen okay they are pointing out outside the screen and this transformations x prime is equal to x minus vt y prime is equal to y z prime is equal to z and t prime is equal to t. So our Maxwell's equation invariant under this the answer is no okay they are invariant under a different transformation Maxwell's equation are not invariant on the Galilean transformation but Maxwell's equations are invariant under Lorentz transformation okay. So what is that so again we have the frames s prime moving the uniform relative velocity v along the common x x prime axis but here we need x prime to be given by not only x minus vt divided by root over of 1 minus v square by c square okay and of course here y prime is equal to y z prime is equal to z remember we are moving along common x x prime axis. What is interesting here is that see that the times are not matching in Galilean transformation we had t is equal to t prime but here t prime is equal to t minus vx by c square divided by root over of 1 minus v square by c square okay. So this v is actually the velocity with which the frame s prime is moving with respect to the s frame okay. What is this see here well you have guessed that is the speed of light but all of a sudden how come this speed of light is there so remember this is the transformation under which Maxwell's equations are invariant okay. So do we see that is the velocity of light or I am sorry the speed of light in vacuum explicitly in Maxwell's equation I mean on the left hand side I have written that once again just for your convenience well it is not present explicitly. So we ask this question so where is this coming from okay so is c in grain somewhere within Maxwell's equation itself okay. Now for that what you have to do as I said is that Maxwell's equation these are coupled partial differential equations and if you uncouple them okay there is a price to pay. You see that you have a second order equation then okay so you have del square E is equal to mu 0 epsilon 0 del 2 del t square of E and similarly for the magnetic field also you have the Laplacian I should say del square B or the Laplacian of B that is mu 0 epsilon 0 is del 2 V del t square okay. Now this has an uncanny resemblance with the wave equation you know waves water waves some waves this kind of waves so it is wave equation here so the Laplacian of F that is equal to 1 by V square of del 2 F del t square okay so t is the time here and what is V? V is the velocity of the wave. Now you see in these two sets of equation if you compare Maxwell's equation of V and E with the wave equation what you are going to see is that this term mu 0 epsilon 0 can be compared with 1 by V square okay so which means that if it I mean since it resembles the wave equation mu 0 epsilon 0 somehow has some sort of relation with velocity okay it actually you are going to see that 1 by mu 0 epsilon 0 is does indeed turn out to be and 1 by root over of mu 0 epsilon 0 does indeed have the dimension of velocity it is actually 3 to 10 to the power 8 meter per second okay so more on that value later on when you put in their values okay but also from physical principles in hindsight you can also check that mu 0 epsilon 0 should have the dimensions of 1 by velocity square well how to do that well well check anyone of Maxwell's equations in E or V check the first one the Laplacean of E is equal to mu 0 epsilon 0 del 2 E del t square now this Laplacean of E Laplacean how does it look like del 2 del x square plus del 2 del y square plus del 2 del z square that kind of a thing so it has a dimension 1 by length squared okay so on the left hand side I mean for the moment look at this operator that is that that is more important now because E and E so the E and E they have the same dimension so what we need to do is to balance the dimensions of rest of the operators and rest of the things here but on the right hand side we will be concerned with mu 0 epsilon 0 del 2 del t square okay now you have a t square in a denominator here so which means that that is time squared okay so on the left hand square you have on the left hand side you have 1 by length square and then on the right hand side you have 1 by time square okay so what should be then the dimension of mu 0 epsilon 0 so that you have this entire thing mu 0 epsilon 0 del 2 del t square to have the dimension of length square okay well it has to have then the dimension of 1 by velocity square okay so then in hindsight we can actually we actually can figure out that mu 0 epsilon 0 should have the dimension of 1 by velocity square well similarly that is the same thing the same conclusion you realize from the second equation Laplace-Saino B is mu 0 epsilon 0 del 2 B del t square okay now on this value 3 into 10 to the power 8 meter per second okay and you might have already guessed that this number is actually the speed of light okay so you see speed of light is actually ingrained within Maxwell's equation itself and then del 2 Laplace-Saino E is actually equal to Laplace-Saino the electric field and then Laplace-Saino of the magnetic field you see that it is 1 by c square and then here on top you have del 2 E del t square and for the magnetic field del 2 B del t square okay now since it follows the pattern since it follows the form of the wave equation okay so Maxwell concluded that then light must be an electromagnetic wave okay now this had a profound significance because light electromagnetic wave and you see that I have written wave in in in italics because in those days well in the 19th century actually people thought that waves actually require material medium to propagate well why was that the the the reason that okay you have water waves which require water to propagate you have sound waves you need medium you need air or even sound waves can travel through another material for example through metal but in any case you need a medium to propagate so so the reason that perhaps they also light also should require a medium to propagate so and then they just name this medium as ether or actually they used to call it the luminiferous ether okay and the further reason that as light can travel through vacuum then vacuum must contain this medium of light which is ether so vacuum is full of ether okay that is the medium of light okay now like every assertion in physics and even if you make a theory it has to be proved it has to be validated by experiments and so that is the challenge that confronted faces at in those days in the late 19th century is to detect ether and its properties okay so they were thinking of a possible experiment in which to measure the speed of light in different inertial frames okay and to see if this speeds were different in in in this different systems okay now in case they were different they were going to look for evidence of special frame where that is the ether frame and that that is going to be a preferential frame where the speed of light is c itself that is c 3 into 10 to power 8 meter per second that is the speed of light in vacuum okay so they were they were looking for an ether frame and this experiment remember was to be done on earth so sitting on earth they were supposed to detect ether now consider the fact that earth is in motion okay so if an experiment is been done on earth and then earth is in motion so you should be able to detect an ether wind in a sense quote unquote an ether wind okay and then the magnitude and direction of ether of this ether wind would vary with season and of course the time of the day because of rotation of the earth okay so the point was so the suggested experiment was to measure the return speed of light okay so going and coming back okay since ether was always there in an ether frame in in different seasons and in various times of of the day okay why because if earth is moving relative to the ether frame the return speed of light would be different and this difference could be detected then and that would be a test for the presence of ether remember by devising such an experiment was indeed very difficult okay so but but there were smart people there were there were there were Michelson and Morley who in the later part of 19th century they devised an interesting instrument they devised actually they devised an interferometer which goes by their name now so that had a light source okay so light is emitted from the source it comes and hits the semi-silvered mirror that is actually a beam splitter okay so then you have mirrors on two sides perpendicular and parallel to this light source and then and detector on the other side like it is shown here so what happens is that light comes and hits this semi-silvered mirror it splits into two parts okay goes to the mirrors is reflected back okay so there you see the signs a little bit different symbols for these reflected rays and for this reflected ray then recombines it goes to the detector and there will be constructive constructive and destructive interferences due to which there will be a fringe pattern at this detector okay now remember this experiment is being done on earth okay now as earth is moving in the ether frame okay and so if this flow of ether is parallel to one of the one of these beam directions let us say paddle to the going from if the direction of ether is from light source towards the mirror on your right hand side and then what will happen is that the return speed of light will be different from the return speed on the perpendicular to the ether why because if it is parallel to the flow of ether and then once it goes parallel it is towards its flowing width you know in the direction of ether but when it is being reflected black it is opposite to the flow of ether okay so there is going to be a difference in time of the return speed of the return of light in both this axis and what you are going to have is that this difference is going to cause a shift in the fringe pattern at the detector okay so the expected result was that there would be a fringe shift at the detector which would confirm the presence of ether okay but surprise surprise the actual result was that although this was done different season different times of the day no discernible fringe shift was observed I mean you could have argued that maybe a more sophisticated instrument or later on they could have rechecked it was checked even by other people and also by more sophisticated equipments and there was no evidence of this ether frame well jokingly of course sometimes people call this is the most famous field experiment okay so there was no ether now towards the end of the 19th century on the other hand Albert Einstein was also very concerned and he was also concerned on a different thing he was concerned that the laws of classical mechanics and the electrodynamics we are not following the same transformation laws they were following Galilean and we are all in transformation laws okay so this was quite troublesome to him he being a theoretician so he isn't that does it mean that an inertial system which is actually indistinguishable by mechanical experiments remember we saw earlier that with the help of mechanical experiments you are not able to distinguish between inertial systems different inertial systems because Newton's law is going to be valid in each one of them in the same form so does it mean that okay I mean with mechanical experiments it is not being possible but by other means by other electromagnetic means or maybe optical methods can you then distinguish between inertial systems that to Einstein was a very worrisome thing because here you have then different branches of physics following different transformation laws okay now he reasoned that this need not be so this that there is there is somehow there is a there is a problem somewhere so he figured out that actually it is the Lorentz transformations which were more general than the Galilean transformations we will put the words more general in italics here so that means that I will explain that a little bit more later on and he talked of the need to modify mechanics the laws of mechanics accordingly so that electrodynamics and mechanics follow the same transformation laws okay now to do this Einstein had to make two important assumptions okay they are actually the postulates of special relativity so the first postulate that is the principle of relativity so which tells us that the laws of physics are going to be the same in all inertial frames okay so there is no there should not be any preferred inertial frame okay there is no preferred inertial no preferred inertial frame exists okay and then the second postulate which says which talks of the constancy of the speed of light the second assumption postulate and the speed of light in free space it has the same value c in all inertial frames okay now with these two postulates Einstein started his calculations and let us go let us check a little bit more on let us let us take this idea little bit more on the second postulate so here we have two frames s and s prime moving with velocities moving with velocity v with respect the s prime frame is moving with velocity v along the common x x prime axis then and then of course at t equal to t prime they started so they coincide and we considered a ray of light starting from a common origin and reaching point p okay and then let us measure the distance o p and o prime p in both these frames so what would an observer in s frame measure o p as and what an observer in s prime frame measure o prime p as okay so the distance y is that would be o p that would be x square plus y square plus z square so that is equal to c square t square remember c is the speed of light and then o prime p that is x x prime square plus y prime square plus prime square that is equal to c square t prime square now notice of course so getting to the second postulate we have taken the speed of light to be the same in both these frames okay now you are assured that x square plus y square plus z square now you you subtract out c square t square okay it is going to give you zero and a similar thing is going to happen if you subtract out c square and t prime square from the primed from x prime square plus y prime square plus x prime square that is that two is going to give you zero so now the same thing is going to happen if you go to another frame moving with certain other velocity v prime let us say or v double prime let us say okay so the distance there could be x double prime square plus y double prime square plus z double prime square and if the observer there has measured time t prime minus c square t prime square remember that this speed of light is taken the same in all inertial frames here but we point out that the quantity x square plus y square plus z square minus c square t square is an invariant quantity okay so that is the thing that is not changing now for this invariant quantity so which of this transformations Galilean or Lorentz preserves this invariance okay the answer is on you can actually check this out you can put x prime is equal to x minus vt x prime y prime is equal to y z prime is equal to z t prime is equal to t and check if this invariance is preserved you are going to see that it is not so it is only the Lorentz transformation which is going to preserve this invariance okay now let us see what that is so well we have we have been introduced to Lorentz transformation before well I have been talking about the laws of electrodynamics so but let us write it down once again so that x prime is equal to x minus vt by root over of 1 minus v square by c square y prime is equal to y z prime is equal to z and t prime is equal to t minus vx by c square root over 1 minus v square by c square of course I mean if this looks a little bit more complicated sometimes people actually write it in more compact form by taking this ratio v by c as beta and then writing gamma as 1 minus by 1 by root over of 1 minus v square by c square which is the same as 1 by root over of 1 minus beta square so Lorentz transformations can be very concisely written in terms of in this fashion written in this white box x prime is equal to gamma times x minus beta ct okay so y beta c because you see beta is equal to v by c okay and here we had x minus vt so we had to write beta c here okay so the y's and z's are the same here but it is very interesting to write the time coordinates so if you multiply that by c so it has the dimension of length again so ct prime is equal to gamma of ct minus beta x have you noticed one thing it is that notice the thing for this x prime the transformation equation for the x prime and the ct prime okay see that x prime and the last you have beta times ct okay but when you have ct prime you have in the last you have beta times x okay and then so you see it looks very symmetrical so when you have the length coordinate you have the time coordinate and when you have the time coordinate you have the length coordinate now this is something new this is something new to us this is something which is not natural to us I mean we are quite used to Galilean transformation in our real life okay but here what you see is that in this time coordinate you have some you have you have the length coordinate as well okay so naturally this is going to have consequences and we are going to check all these things in subsequent lectures okay just one more thing we we were actually talking of all we were actually always talking of what is the quantity in the s prime frame in terms of quantities in the s frame so we can also talk of the opposite thing so what is the I mean how can you say what what are the quantities in s frame in terms of quantities in s prime frame for that it is very easy to I mean it is it is the same situation if you consider s s frame to be moving with a velocity minus v with respect to the s prime frame in that case you can simply write down x is equal to gamma times x prime plus beta c t prime of course y and y prime are the same z and z prime are the same and then c t is equal to gamma times c t prime plus beta x prime now what we have done here as I said was to express the quantities in in in s frame so we want to understand calculate quantities in x frame and in the s frame in terms of quantities in the prime frame okay so like we can take a break here and in the next talk we are going to focus on again on the postulates of relativity and then the consequences of the Lorentz transformations where we are going to carry our discussions a little more okay so to summarize what we have been doing today we have been looking at the laws of mechanics and electrodynamics and we saw that actually they were not that transformation laws of mechanics and electrodynamics were not the same they were the Galilean and Lorentz it was Einstein was so concerned and then he he he showed that you take he took he actually showed that the Lorentz transformations were actually the more general transformation laws and if you take that you want to you need to change mechanics okay so these are certain things that we will be considering in our future talks okay thank you.