 So, let us begin. So, we start with recapitulating what we did in today morning. We discussed Michelson-Modell experiment and you know said that it indicates that ether does not exist. Then we discussed the postulates of special theory relativity as we are given by Einstein. Then we discussed Galilean transformation and emphasize that this Galilean transformation would yield to a velocity transformation which will not satisfy the postulates of special theory relativity. Therefore, we need another transformation and that is what we discussed. Remember we have to think totally afresh because this is not possible through our classical ideas. Einstein told that after lot of thought he eventually thought that it is time which is the suspect. The equation that t is equal to t prime which we had written in Galilean transformation about which we had mentioned earlier that this is not formally talked about it in the classical mechanics. He thought that this is this equation which is likely to be suspicious. So, how let us try to put some arguments. Generally, when we say about time we talk about the simultaneity of two events. For example, if we say that the train leaves that particular station at 10 a.m. essentially it means that train leaving and clock showing 10 o'clock are two simultaneous events. If I say my class starts at 3.30 or class tomorrow starts at 9.30 essentially means when my watch shows 9.30 and when my class starts they are simultaneous events. So, this is related to the concept of simultaneity. Now, what I want to show first that if we take the second postulate if we agree on second postulate which we agree now then the simultaneity is relative. It means two events one event for example watch showing a particular time the second event for example the car leaving or train leaving. If they appear at the same time in one frame of reference in a different frame of reference they may not appear to be simultaneous they may not appear to be occurring at the same value of time. By simultaneous we mean that they occur at the same time as determined by one observer. So, this is the first thing that I would like to show and for this I take a very very simple example to show how this is possible. So, let us assume that there is a running train compartment which you can always assume of course we are always assuming that earth is a inertial frame of reference in all these arguments. So, you know just to make our life simple and I mean our imagination simple and let us assume that in this long compartment there is a longish compartment assume that there are no seats just for sake of argument a person is sitting exactly in the half way or standing exactly in the half way of the compartment then sitting in that particular compartment that particular person throws two walls at the same time. So, person is standing there he has a watch or he or she has a watch and from that watch he finds out that he throws two walls simultaneously you know you trace one like this another like this one in the direction of the motion of the train with respect to earth another in a direction opposite to it. And let us assume that the speed is same as measured by him u prime this is what I am calling it. So, let me repeat an observer is exactly half way in a running compartment of length L he throws two walls at the same time t prime is equal to 0 with same speed u prime as measured by him one towards the front wall another towards the back wall when I say front wall I think it is clear that front wall means if I am sitting in the compartment the wall which is moving further another which is moving backwards let me show it by fear something like this. So, assume that is a longish compartment you know just for picture I have drawn it to be very big person. So, this is the way it is moving with respect to earth and this is an observer with sitting in earth which I am calling is s frame this observer I am calling is s prime frame now this observer here sitting here throws one ball like this which is in the direction of velocity this I am calling is the front wall this I am calling is the back wall. So, according to this particular person this ball is thrown in this particular direction with a speed u prime and this ball is thrown backwards with a speed u prime now eventually this ball goes and hits this particular wall this ball goes and hits this particular wall. So, let us call these events as I say in special theory of relativity our ideas are better described in terms of events in fact I also tell students that you know many times when we do special theory of relativity we realize how clumsy or vague we were in classical mechanics we have our thought processes are mixed up in special theory of relativity you require a very very clear cut thought processes then only you can understand these particular problems in a better way. So, one particular person throws a ball in this particular this person throws a ball in this particular direction and this ball hits this wall. So, let us call this as one event and another event when this particular ball comes and hits this particular wall. So, we assume that it has enough velocity that hits this wall before it falls on the ground. Now my question is that these two events are they simultaneous in this frame are they simultaneous in this frame. So, we describe two event event one the first ball reaches the front wall event two the second ball reaches the back wall. I just now told what is front wall what do I mean by front wall and what do I mean by back wall. Now question is are events one and two simultaneous implying do they occur at the same time. Okay let us first try to work out this particular problem with a spring frame in mind that is the person who is sitting in the train. Now person sitting on the train each ball has to travel a distance of L by 2 because he is sitting in the compartment or rather standing in the compartment this particular total length is L that is what we have said. So, this assume I mean do not go by by long hand the hands of this particular person essentially we assume that this ball is thrown here and this ball is thrown here and this particular ball has to travel a distance of L by 2 this ball also has to be traveled a distance of L by 2 in this particular direction horizontal direction. But your component does not matter we are looking only the horizontal component and horizontal distance that it travels will be L by 2 and this travels L by 2. So, total time taken for this ball to travel up to the wall will be this length which is L by 2 divided by U similarly here or other U prime similarly the time taken by this particular ball to travel this particular distance will be L by 2 divided by U prime. So, this is the time for event one which is T1 prime is equal to L upon 2 U prime T1 prime I am saying because prime I have put because this is time measured by S prime frame of reference okay and T1 I have said because this is for event one. Similarly, for time for the second event again this ball has to travel a distance of L by 2 and according to this particular observer this particular ball also travels with the speed U prime. Therefore, the time when even 2 would occur assuming that this is thrown at let us say 0 time T2 prime will be equal to L upon 2 U prime because time for these 2 events are identical hence this particular observer sitting in the train will conclude that these 2 events are simultaneous. Now, let us try to observe exactly this phenomenon with respect to observer who was standing on the ground when the observer was standing on the ground according to that particular person this ball these balls were not thrown by same velocity as U prime. In fact, one ball is thrown with a larger velocity another ball is thrown with a smaller velocity. Now, let us look at same events in S frame the speed of the first ball using inverse velocity transformation because the velocity which has been given was given in S prime frame of reference. So, I have to find out velocity in S frame for that I will use inverse transformation which I have told you okay or otherwise if you are very familiar with the relative velocity concept this is the standard classical relative velocity concept. The velocity of one of the particular ball will be U prime plus V for another ball it will be minus U prime plus V. So, so according to this observer this particular velocity will get added up. So, this will actually travel with a larger speed while this particular ball because this is going in this direction and this velocity gets added up which is in opposite direction okay. So, essentially the velocity will get subtracted. So, according to this particular observer that particular observer will feel that this particular ball is actually not thrown with a speed U prime but thrown with the speed of U prime plus V okay. Similarly, in opposite case it will appear that this ball particular ball is thrown with a small velocity alright but as the ball moves towards wall observer here would notice that this particular wall is also moving to the right hand side okay. See when observer sitting here according to him this wall travels along with him alright this ball also travels with him while according to this particular person the entire compartment is moving. So, when this particular ball was thrown the carriage was here by the time this particular ball reaches here the carriage has moved forward it will be somewhere here okay. So, this particular ball though thrown with a larger speed will have to travel larger distance before it hits the ball. On the other hand as far as this ball is concerned this ball is thrown with a smaller velocity but on the other hand as the ball is approaching the wall is also approaching towards the ball okay. So, when actually the hit takes place wall will be somewhere here and therefore this particular ball would have to travel a smaller distance but with a smaller velocity and if I can do a small calculation here which is very simple calculation and you will find out that according to the observer as also this particular ball will hit at the same time as this particular ball will hit this particular ball the two events will also be simultaneous here except that his interpretation of this particular experiment in terms of velocities will be different. According to this observer velocity of this ball is larger than the velocity of this ball but this ball travel larger distance this ball travels a smaller distance and then still the events are simultaneous. So, the speed of the first ball using inverse velocity transformation will be this the speed of ball the second ball will be this particular thing because now the speed is in opposite direction okay. We calculate the total distance the first ball will have to be travel let us suppose t1 is the time when it hits the wall then the total distance that this particular ball has to travel will be l by 2 which was our original distance plus this much distance the carriage has travelled in time t1. So, the total distance travelled will be this much and this distance is travelled with the speed of u prime plus v while in this particular case the total distance travelled will be v t2 minus l by 2 with the speed of v minus u prime okay eventually you can calculate from this particular thing you will find out you just take t1 on this particular side calculate t1 and you can find out that t1 will turn out to be because v t1 will cancel from this thing you will find that are exactly the same expression t1 is equal to l upon u1 u prime remember I have not put prime here because this is the time which is measured according to the observer which is s observer which is standing on the ground. So, t1 and t2 will also be turn out to be l upon u prime so both of them will agree that the time for the event 1 and 2 are identical and not only this the time value that they will find out will also match with the time which is which s prime has observed. So, not only t1 is equal to t2 but it is also equal to t1 prime is also equal to t2 prime. So, this particular times match okay it is nothing surprising in classical mechanics has been designed as I have been saying with this particular concept that t is equal to t prime okay both the observers will note the same time. So, whatever method you adopt to measure time okay you always get the same values of time. So, if the two events are occurring at the same time in a given frame of reference in a different frame of reference also they will appear to be occurring at the same time. Now I change this thing now I assume that there is a special theory of relativity and let us assume that the ball that we are throwing is replaced by light sources. So, I have two light sources one here and I shine two lights one in this particular direction another in this particular direction and my events are same except that now instead of ball reaching it is a light which is reaching let us assume that second postulate of special theory of relativity is correct. Now imagine that the observer shines light instead of throwing balls which travel both to the front and back directions wise just have one two let us say small touches when you know just put on the right hand side and one on the left hand side and let us find out the time when this light reaches this particular wall and the time when it reaches this particular wall. The event one light reaches the front wall event two light reaches the back wall. So, this is what I have done instead of ball I have put a source of light here which goes like this which goes like this all other things are identical nothing has changed except that instead of ball I am just using a source of light and it is light which is travelling in this particular direction and light which is travelling in this particular direction and I am assuming of course that special theory of relativity is correct. Now let us look what happens as far as S prime observer is concerned. As far as S prime observer is concerned nothing has changed except that instead of u it is c. So, as far as the time of the event number one is concerned again the light has to travel a distance of l by 2 it will travel with of course much larger speed which is c but time will be given by l by 2 c while this particular light also travels backwards with the same velocity c and it has also to travel a distance of l by 2. So, as far as this event second event is concerned that will also occur at the same time l by 2 c. So, according to this particular observer these two events will occur at the same time. So, along this particular ball has been thrown exactly from the center of the compartment. So, S prime time t 1 prime for event 1 will be given by t 1 prime divided by l by 2 c time t 2 prime for event 2 will be given by t 2 prime is equal to l up all 2 c. So, rather than u prime I have replaced this by c nothing else has changed. Hence the two events are simultaneous in this frame as before. Now let us look with respect to the observer which is here. Now things have changed as far as this observer is concerned if second postulate is correct. Because according to this particular observer now unlike the case of the balls which I mean which I or unlike the case of classical mechanics this light will also travel with the same speed while this light will also travel with the same speed c because just now we have said according to second postulate speed of light is same in all the frames. So, whether the light is thrown from this particular source or light is thrown from this particular source does not matter. So, as for this observer is concerned because this is light, light will always move with the same speed c. So, now this particular light will move with the speed c here also backwards it will also move with speed c. But this light as we have said will now have to travel much larger distance to reach here while this particular light has to travel much smaller distance but the speeds will be same. So, this particular observer if the second postulate is correct would notice that the light would reach this wall first while light will reach this wall later because this light has to travel a larger distance this light has to travel a smaller distance but the speeds are same. So, according to this observer these two events would not occur at the same time. According to S prime the two events are simultaneous this is because S in S prime they like I mean this I am talking about the train frame of reference the light covers the same distance in the front direction as well as in the back with the same speed. According to S which is on the ground observer the speed of light is still see in both the directions according to the second postulate but it has to travel a larger distance to reach the front wall then the back wall conclusion there is a simultaneity of two events in S prime but not in S. So, looking at these two events we can conclude very easily that the two events appear to which appear to be simultaneous in one particular frame of reference will not appear to be occurring at the same time in a different frame of reference. So, simultaneity of two events thus depends on the frame chosen thus time may have to be taken frame dependent if we have to make C frame independent. So, if we believe that C has to be frame independent probably the thing that we have to look is that time also has to become frame dependent quantity and this is what was a shocking conclusion of shocking thought of Einstein which as I say even shocks today even after 110 years of validity whenever I teach to first year students this statement appears to be exceedingly shocking because I mean its outcome is also equally shocking. So, remember now we are back into dark because we do not know what are my transformation equations because now I have to allow a frame dependent time. So, this is what I am calling as a Lorentz transformation. I have chosen to write the transformation equation in this particular form. One can give certain amount of logic to come to this type of equation for this I would suggest that you know you could look at the book of Resnick. There is a book a small book by special theory of relativity by Resnick in which he has discussed detail in detail how these equations can be thought to be on this particular form which are sort of very interesting and physical arguments look at this particular thing. But let us at this particular stage start from this particular thing. So, remember if Bxx would have been equal to 1 then X prime would have been equal to X minus VT which was the Galilean transformation equation. Y prime is equal to Y, Z prime is equal to Z are anyway the same equations as in Galilean transformation. A totally new equation that we have entered is I am making T prime also to be dependent on X and T alright. So, while earlier in the Galilean transformation we just had T1 is equal to T here we have T1 also dependent on X and so basically we have introduced 3 constants one is what I am calling as Bxx which is a constant which is now multiplying X minus VT. Had this constant been 1 then this constant this particular equation would have been same as Galilean transformation. If this constant VTX would have been 0 and VTT would have been equal to 1 then this particular equation would have been as Galilean transformation. So, I am not giving any argument to see how these equations have reached this particular form. As I say if you are interested you can look into the reasoning book. But we start from this particular point. So, I have made I have introduced 3 different constants one is Bxx which is now multiplying X minus VT then I have made T prime to be dependent on both X and T and these constants BTT and BTX have been introduced. Now we make it is easy to see that these equations satisfy the condition of time being 0 and the origins of the two frames considered. You know this is what we have already said that we always started these transformation equations with a special type of frame of references in consideration where relative velocities are only along the X direction Y and Y prime are parallel Z and Z prime are parallel and also the time starts getting measured from the time when X was 0 when the two origins coincided it is very easy to see that these equations will satisfy that particular condition. So, at T is equal to T prime is equal to 0 now let us assume at that particular time the origins of both the frames had coincided a spherical wave front is emitted from the origin a spherical light wave is emitted from the origin. So, you can assume that there is a point source which is emitting light in all possible directions. Now it does not matter where my light source was existent whether it was existent at the origin of S or it was at the origin of S prime or it was neither at the fixed at to S frame not to S prime but was fixed to some other place. The only thing I am making is that when the two origins were coincident at that time from that particular point of origin a light source emitted light because as far as light is concerned unlike the classical mechanics when I am trying to say I have to say where my observer is fitting if I am throwing a ball with a particular velocity where is that observer this particular observer is throwing ball from the train or is throwing ball from the ground while as far as light is concerned we need not answer this question because irrespective of where from which particular point light is thrown or light is emitted okay this will always travel the speed C irrespective of from where are you observing in all inertial frames the speed of light is going to be C and it is going to be C in all the directions. Now let us imagine that I am sitting on the origin of S now as the time progresses I mean the person sitting on the frame of at the origin of S he would feel that light has started from his origin his or her origin and as the time progresses it emerges in a spherical wave front. So let us suppose I am in frame S at t is equal to 0 I find a light source here which emits light as the time progresses according to this observer sitting here this light will emerge in all the directions because this is a point source light will travel in all the directions and after a time t this particular light will be situated at it is at this sphere this will emerge in the form of a sphere and this will occur at the time of time is t this radius will be equal to C t. So this person will feel that this is emerging in the form of sphere and because in all the directions it has to travel with the same speed C okay so after time t this particular light will emerge as a spherical wave front which is like here alright the radius of which will be equal to C t. Now if I look at another observer which is S frame of S reference that particular person also feels that the origin was light was emitted from its origin and in his frame also the light emerges exactly in the same fashion with the spherical wave front after time t prime because I am talking of S frame of reference after time t prime this will also be here in a spherical wave front because according to him also the light travels with same speed C in all possible directions. Now during this particular time t the origin of this and origin of this are no longer coincident but still the observations if we believe in the second postulate of special theory of relativity will be identical for both these observers according to one after his time t this particular light will be found in a spherical wave front similarly according to this particular person also in his time t light will be found in this particular hemispherical wave front with a radius of C times t prime remember C is same. So this is what I am showing in the other transparency at t is equal to t prime is equal to 0 a spherical light wave is emitted from the origin the observers in both S and S frame will find that the spherical wave front is emerging from their respective centers with the same speed C this is the picture which I am drawing here according to this observer S this particular light emerge from here and after time t will be on this sphere according to this observer the light emerge from here at t is equal to 0 and after a time t this will be found here. All right now during this time this has moved from here to here the origins are no longer coincident but nevertheless both will conclude that light is being found in this particular form of a sphere at this particular time remember the time as per AS will be different from time of S prime because we have assumed that time itself is a frame dependent quantities very very shocking and very very different thought as I said you know this as I say is fairly shocking. It means if I ask them to write the equation of the sphere over which the light will be found both the observers will write this equation which is exactly the same equation x square plus y square plus z square is equal to c square t square another observer will also write the same equation except that instead of his x or her x it will be x prime this will be y prime because whatever see an observer has to be consistent in his or her own frame of reference. So, when I he or she is writing the equation he will write according to his x prime and according to his time. So, according to observer S prime x is x prime actually what is measuring is y prime and what is being measured as z coordinate is z prime and what is the time which is measured measured is t prime. So, both of them will write exactly the same equation of motion is the same equation of the spherical wave front except that here there is no prime here there is prime. Now, whatever I had written as a trial the transformation equation relating x prime to x and t rating t prime to x and t will substitute this and try to find out the values of these three constants. So, this was my equation these were my trial you know transformation equation for which I had not show anything other than saying that you know there is certain physics logic which can be given here substitute this particular here. So, instead of x prime I write this particular thing y prime and z prime no problem because they are exactly they are same as y and z for here t t prime I write c square is equal to this particular thing c square multiplied by t prime square which is the square of this particular quantity. Then we try to reorganize the terms. So, we write this as this equation take all the terms which are containing x square then take all the terms which are containing x t and write and take all the terms which are t square. So, this equation is if I use this particular type of transformation equation these equation can be changed in the form of this particular equation this equation has to be compared from what the person would have been formed what would have person would have in the frame s would have found as the equation of the wave front all right. See these transformation equation are valid only if these two equations are consistent because from the information see what do the transformation equations do you have information in one particular frame of reference and I want to transform and find out the equivalent information in another frame of reference. So, if I have found out x y x prime y prime z prime t prime in s prime frame of reference I would like to find out in x y z I mean the same things x s frame of reference x y z and t and if these are correct transformation equation this equation should be consistent with this equation what it means that this particular quantity should be equal to 1 and because there is no x t term. So, this particular quantity should be 0 and this particular term should be equal to c square. So, I will be getting three equations and there will be three unknowns. So, you can solve these unknowns of course, one has to be little careful about the signs of these things but if you take care of sign probably you have to give some physics arguments there then you use actually these three equations and solve for them these three equations three unknowns can be determined one has to take care of sign once you know you get this particular equation alright. So, basically I have achieved what I am calling as the new transformation equation which is called Lorentz transformation. So, this becomes my Lorentz transformation this x minus v t is actually multiplied by now 1 1 1 minus and root v square by c square this t prime now you have t minus v x upon c square and further multiplied by 1 minus v square by c square these are very very common or other standard notations that we use as in relativity because the equations as you see turn out to be fairly complex often we write beta is equal to v by c once we write beta is equal to v by c I can also write gamma is equal to 1 upon under root 1 minus beta square. So, these equations when we normally often always remember these things and this equation can be written as x prime is equal to gamma x minus v t y prime is equal to y z prime is equal to z and t prime is equal to gamma t minus v x upon c square alright. So, this is the new transformation equation that we have obtained using whatever logic we have which are called Lorentz transformation. First of all in the classical limit when you have v very less than c we have very less than c then this beta will be very very small. See if you are talking about let us say even point let us point 1 c which is not by that moment. So, to say so small speed point 1 times c then the beta square here will be equal to 0.01. So, this factor will give you only about 1 percent connection and if beta is still smaller you will find that this gamma will be very close to 1. So, when you have v very very smaller in comparison to c then this gamma will just be equal to 1. This gamma will be equal to 1 and if v is very very small in comparison to c square this particular factor will be essentially 0. If that is so this equation goes back to Galilean transformation. So, in that case in what we call as a classical limit these equations will just move back to Galilean transformation and I know that Galilean transformation works when I am talking of let us say typical velocity whenever I was throwing the balls Galilean transformation I know works. If I have to describe the motion in train I know that relative velocity formula does work and that is because of the fact that when v is much smaller than c these transformation equations remain same as Galilean transformation. So, unless this particular factor gamma turns out to be fairly large you will not see the relativistic effects. So, only when this gamma or v is becoming fairly close to c that you will start seeing these relativistic effects. So, that is the reason these relativistic effects are not very commonly observed but they are important to be understood if you have to understand the way nature works. So, this is what I said in the classical limit that is very much less than c the Lorentz transformation reduces to Galilean transformation. What will happen if v is greater than c? If v is greater than c then I know that this particular factor will become a imaginary because if v is greater than c beta is larger than 1 this factor is imaginary it means x prime would be imaginary. So, if these equations are correct okay x prime will give you an imaginary value which as I mean we know that this is not possible for them to have imaginary value. In fact, this is one of the region I had told today morning about the same about you know the causality that was another argument but this is the one argument because of which Einstein said that the speeds greater than the speed of light should not be allowed. So, particle will not be able to travel with the speed greater than the speed of light. So, that you know x prime always remains real t prime always remains real otherwise this gamma me imaginary will also may also make t prime imaginary okay. So, we can always write inverse transformation this I think we should not discuss because all I am doing is that replacing x by x prime y by y prime z by z prime and t by t prime and changing v to minus v. Now, let us discuss what we call as velocity transformation. Now, it means we are looking at the new relative velocity formula. So, now you assume that we know the velocity component of a particular particle we know we have measured the velocity component okay. Well at this particular moment let me be very clear in fact I have told this particular thing in morning that because we are going to have many velocities. So, let us be very clear that there is a particle this particular particle moves with a particular velocity let me just write it here. So, if there is a particular particle this is moving with a particular velocity. Now, if this particular particle is being observed by a frame s I call this speed as u or this velocity as u if it is being observed by an frame of reference s prime same particle in general the velocity will be different. So, I will call this as velocity as u prime and the relative velocity between the two frames I call as v remember v is always constant u and u prime need not be constant because the particle could be accelerating all I want is that my frames of references are inertial but they could be observing a particle which is under acceleration. So, the component of the velocity of a particle are measured in s and we want to find out the velocity component of the same particle in s prime frame of reference. So, that is what will be called velocity transformation or in other words the relative velocity formula. So, this is what I am using as my notation v is the relative velocity between the frames which has to be constant as a function of time because I am assuming that the two frames are inertial. u prime is instantaneous velocity because u also in principle can change the particle could be under acceleration. u prime is the instantaneous velocity of a particle in s which need not be constant as I said u prime is the instantaneous velocity of the particle in s prime frame of reference which also need not be constant. So, u symbol has been reserved for the particle velocity and v symbol has been reserved for the relative velocity between the frames, relative velocity between the frames. Now, we have been talking about the events. So, let us talk about two events by which we can use Lawrence transformation to find out the velocity. So, let us suppose this e1 is event one that I person sitting in s frame let us suppose I am sitting in frame s I measure the particle according to my watch the time was t1 and I find to be the particle to be at a position x1. Then at a later time t2 I find the my particle to be at the time x and the position x2. So, these are my two events. Now, there is another person one of my friend who is also sitting in a different frame of reference in s prime when I make a measurement and find the position of this particular particle he also finds the position of the I mean he also notices this particular event and according to that particular event this particular x1 would have occurred at x1 prime because the particle could have been located at a different position according to the observer which of course is also classically true. See if I am observing something at the origin of my particular frame of reference same observation made by another person who is sitting on the train may be at different value of coordinate because that particular train must may be much ahead that origin must might be much at a much larger distance it depends on what time this particular position was observed. So, according to that observer this particular particle this particular event occurred at a position x1 prime and now also a time t1 prime. Even two would have occurred at a different position let us call it x2 prime and would have occurred at time t2 prime. Now, both of us would calculate the velocity of the particle. So, how we will go about it we will take the difference of x2 minus x1 ensure that t1 and t2 are very, very close to each other then or I will put the limit in the limit of the difference of this time becoming 0. Take difference of this take difference of this divide and then I will calculate the speed or the velocity of the particle or I am talking of the x component. So, I will calculate the x component of the velocity also I could have observed the y component that will give me the y component and things like that. Only thing I would like to insist on this particular thing that if I am sitting on my frame of reference and if I have to calculate the speed of the particle I am going to use the positions as determined by me and the time as determined by me. I cannot take somebody else's position and take my time or some my position at somebody else's time I have to be consistent in my own frame of reference. All right. So, if I have to calculate the speed of the particle I have to observe in my frame of reference what is the distance established by the particle in a given time as the with the time as being measured by me. While another person sitting in a different frame of reference that person has to calculate the coordinates in his frame of reference time difference in his frame of reference and then take difference of coordinates and different of time in his frame of reference and calculate the velocity. So, this is what I have written even if the velocity of the particle is not constant delta x delta t which will be given by x2 minus x1 divided by t2 minus t1 in the limit delta t delta t is remember t2 minus t1 tending to 0 would give the instantaneous velocity of the particle in s that is what we do always calculated as this instantaneous velocity. If the motion is in three dimension in general ux will be given by delta limit of delta t tending to 0 delta x by delta t if I am talking about the y component of the velocity this particular particle may not exactly be moving in x direction may be having some sort of motion in some arbitrary line. All right. Then I will find out in same time delta t or in a particular time delta t how much this particular particle has travelled along the y direction take delta y difference similarly take delta z difference let this time delta t tend to 0 this will give me x component y component and z component of the velocity of the particle as determined by the observer in s frame of reference. Similarly, looking at the same particle in s frame of reference we can define ux prime is equal to now remember this is x prime because the velocity is being calculated in s frame of reference this will be delta x prime divided by delta t prime. See remember we do not have a luxury of taking t prime equal to t because now we are dealing with Lorentz transformation and it is this delta t prime which should tend to 0. So, in the limit that delta t prime tends to 0 delta x prime divided by delta t prime will give me ux prime similarly ui prime and uz prime note that like displacement the time difference has also to be measured in one s own frame of reference I cannot put delta t here or I cannot put delta x here but delta x prime has to be in prime frame of reference delta t has to be in prime frame of reference. So, now we know in fact I can write this particular Lorentz transformation what we call as a differential form. So, let me just spend one minute to sort of explain this particular part see I have written x is equal to gamma x minus x prime is equal to gamma x minus v t. So, let us suppose this correspond to the event number one. So, I write this as this then let us say for the second event I can write exactly the same thing gamma x 2 minus v t 2 now I take the difference of the two I take delta x which is the difference of x 2 prime minus x 1 prime. So, this will be delta x prime gamma will be same because it does not depend on the coordinate or the time it depends only on the relative velocity between the frame x 2 minus x 1. So, this becomes delta x v is again constant because this is the relative velocity between the frames v t 2 minus t 1 delta t. So, this is what we call writing this particular Lorentz transformation in the differential form. All right. So, this is what I have written in this particular equation delta x prime is equal to gamma delta x minus v delta t delta y prime is equal to delta y delta z prime is equal to delta z. Similarly, I have written delta t prime is equal to gamma delta t v delta x upon c square. So, basically in the same Lorentz transformation equation below ahead of all the variables I have put there in us. Now, I divide in order to calculate the velocity transformation I divided delta x prime by delta t prime. So, I divided this quantity by this quantity gamma will cancel it. So, it will become delta x minus v delta t divided by this particular quantity delta x minus v delta t divided by delta t minus v delta x by c square. I divide numerator and denominator by delta t. So, this becomes delta x by delta t because I have divided by delta t. So, this becomes just v. So, this becomes delta x minus delta t minus v I have divided by delta t. So, this becomes 1. So, this is 1. Then we have v upon c square delta x upon delta t. Now, in the limit delta t tends to 0 in the limit delta t tends to 0 because delta x is also like 10 to 0. See if the time difference is very small the two particles are going to be very very close to each other. So, delta x is also going to be 10 to 0 and therefore, delta t prime is also going to be 10 to 0. So, let us take the limiting case when delta tends to 0 at the time delta t prime will also 10 to 0 and taking that particular limit this will be this force will be u x prime this will be u x v always is same minus divided by 1 minus v upon c square and this will also becomes u x. So, you have u x prime is equal to u x minus v divided by 1 minus v u x upon c square. So, you see in the case of in the classical mechanics the velocity transformation yielded u x prime only equal to u x minus v, but now we have also something in denominator. Let us look at u y prime which is delta y prime by delta t prime. The thing is that delta y prime is equal to delta y, but delta t prime is not equal to delta t. Therefore, delta t prime I have to substitute as gamma delta t minus v delta x upon c square. We do the same thing eventually you will find that u y prime will be equal to u y gamma 1 minus v u x by c square very, very interesting result. In classical mechanics because the relative motion was only along the x direction the y component of the velocity would not change it is same in the two frame of reference, but according to relativity that also changes. So, if the relative velocity direction is along the x direction the y component of the velocity will also turn out to be different in the two frames not just the x component. In the classical mechanics it was only the x component was changing the y component was not changing here even the y component will change and even the z component will change exactly by using the same arguments. So, this is what we obtain from these particular results this is what we call as the velocity transformation. So, this is what is becoming of my velocity transformation which is u x prime is equal to u x minus v divided by 1 minus v u x upon c square. Then I can write u y prime is equal to u y divided by gamma 1 minus v u x upon c square. Similarly, I can write u z prime is equal to u z divided by gamma 1 minus v u x upon c square. There are certain things which I would like to mention about this particular velocity transformation I will probably just give you one or two simple examples. Of course, you have inverse velocity transformation which is just as I told you the way to obtain the inverse velocity transformation. Now, it can be seen from this equation which I am not showing it what I would suggest that you can do you can take the direct transformation take u x square plus u y square plus u z square. We try to express this in terms of u x square plus u y square plus u z square. You will get an equation from which you can show this particular thing very easily that if in one particular frame of reference the particle velocity is less than c in every other frame of reference it will turn out to be less than c. And if the velocity turns out to be equal to c in one frame of reference in all the frame of reference it will turn out to be equal to c that is what we are looking for what we are looking when we are talking about the velocity transformation that velocity of light should become frame independent and that is what you will see will come out to be in this particular case that if u is equal to v u is equal to c then velocity will always turn out to be same in all the inertial frame of reference. This also implies this particular statement which I am writing in this particular thing which is probably a sort of a not necessary statement but lot of students ask because people who are happy with mathematics they always write to write like this these equations will also show that if at all it so happens if at all it so happens that the particle velocity turns out to be larger than c because we have no evidence of doing that thing then in if it happens in one frame of reference in all other frames of reference you will find that you will turn out to be larger than c so long the relative velocity between the frames is less than c. This statement if you want you can ignore what is most important is this particular consideration is that if velocity is less than c you always turn out to be find out to be less than c in all other frame of reference and if it is equal to u it is equal to c in all other frames you will try to you will find it it turns out to be equal to c. Let me just give you one example for which I have not made the transparency just to make it sort of let us assume let us try to solve a simple relative velocity formula using relativity this is what I have mentioned earlier in the morning let us assume that we have a particle which is traveling with a speed of let us say 0.8 c along x direction and there is another particle which is traveling in this particular direction let us say with a speed of 0.6 c. Now let us suppose both these speeds are being observed by an observer which is standing on the ground here all right now this particular observer observes both these speeds so according to this observer in the ground a particular particle moves with the speed of 0.8 c on the plus x direction and a particle moves left hand side with a speed of 0.6 c. Now my question is that if another observer would have been standing on or sitting on this particular particle what would have been the speed of this particular particle seen by this particular observer sitting here. So, if there is another observer sitting here which is on this particular particle what would have been the speed of this particular particle let us call this particular particle a would have been seen as far as this particular observer is considered. Classically this will turn out to be equal to 0.8 plus 0.6 which will be larger than speed of light what I want to show that according to velocity transformation this will turn out to be less than c. So, what I will do I will first write what is the velocity of this particular particle as seen let me call this observer as s this observer as s prime. If this observer is s then the speed of this particular part this is the speed of this particular particle and this ux remember this is s prime so I am not putting this s frame so I am not putting any prime so ux will be equal to minus 0.6 c all right because this is moving in minus x direction and it is with the speed 0.6 c ui and uz are equal to 0 of course we need not bother about it because if they are 0 they will remain 0 in other frame also. Now, I want to find out what will be the speed of this particular particle if my frame is changed from this s to s prime if I do that particular thing then I am transforming from this s frame to another frame which I am calling as s prime the relative velocity between these two frame is 0.8 c let us assume that the velocities are constant just for easiness it means my v will become 0.8 c. So, remember this particular frame of reference is moving in plus x direction relative to s so my velocity v will be given as 0.8 c all right now what I have to find out is what is the velocity of this particular particle seen in s prime frame of reference it means I am interested in finding out what is ux prime all right now classically as I have said ux prime would have been equal to ux minus v but now relativistically it should be this divided by 1 minus ux v upon c square all right now let us put substitute the numbers on the top if I substitute the numbers on the top ux will be equal to minus 0.6 c v will be equal to 0.8 c minus 0.8 c this would have been equal to minus 1.4 c this would have been the classical result that of course relative velocity would have been in this particular direction negative x direction because the observer sitting is here so particle to him also would appear to be moving in minus x direction this is what one would have obtained classically but now we have also a denominator this denominator will make it 1 because ux is negative 0.6 c so this negative sign will become plus ux is 0.6 c v is equal to 0.8 c divided by c square all right let us calculate this number this is minus 1.4 c all right now this is 1 plus c square will cancel 0.8 into 0.6 this will be equal to minus 1.4 divided by 1.48 c as you can see very clearly that this is less than c so according to this particular observer the velocity of this particular particle will be appearing as minus 1.4 times 1.248 c all right so this is what is basically the implication of special theory of relative velocity transformation that you will find out that these relative velocities will always turn out to be less than the speed of light I think I will stop here so we have another five seven minutes maybe 10 minutes 15 minutes we can ask questions I have not covered too much of portion because thinking that is little difficult portion so I was not putting too many things hello good afternoon sir good afternoon yes sir as we know the phase velocity of de Broglie wave is greater than c yes that is not valid the Einstein second postulate that c is ultimate velocity I have already told you this number of times that phase velocity has no physical significance okay it is only for mathematical thing okay what is important is group velocity and group velocity will never exceed c physical velocity is purely of mathematical pleasure it has nothing to do with the reality so I mean it does not matter about the phase velocity you know so I mean the information cannot be sent through the phase velocity information can always be sent only through the group velocity the energy transfer will take place only with the group velocity so that phase velocity has absolutely no consequence it does not violate special theory between any sense please go ahead sir anyone sir anyone has tried the Michael's Mollie experiment with different parameters and and get another different friendship because my point for the notes sound very good yeah no you are right see think is that I would suggest that you know you look at a book which I had mentioned in the morning I am again mentioning the book name of the book it is book by Resnick see this it is a small book this Resnick is same as this or no the famous book by Resnick and Halliday for relativity okay so this is a book which is a small monograph book now I am very very thin book now it contains only special theory for relativity nothing else in fact to look at the historical aspect this is very very nicely written book in fact he has given a table in that particular book in which he has said what have been attempts of repeating Michael's and Mollie experiments with better and better accuracies so you can see in that particular table that this particular experiment was repeated later with much better accuracies okay where you are very very sure that you know you would have observed things free shifts if they were present but the fact is that the result had always been negative in fact in one of the transfer I have been mentioned that thing that particular Michael's and Mollie experiment had been repeated number of times okay later with much better accuracies and still the result was always negative but I would say is that you look at this particular book by Resnick in which he has actually tabulated and he has given I am not sure whether he has given the references but he has definitely mentioned who are the persons who have attempted which year they were attempted and using what type of parameters they have been attempted and how the results have always turned out to be negative. Sir one more question please. Yeah please go ahead sir we have u dash equal to my u x minus v upon 1 minus v upon c square into v x yes and and u u equal to u x dash plus v upon 1 plus v x v upon c square or plus u x yeah sir if we take sir if we take c in both the cases yeah so where we have to change the sign of c and where not see for example let us just look at this particular case what I have said if this particular thing would have been just equal to c for example if this is 0.8 c and this particular thing would have been actually equal to photon which is equal to c u x prime is equal to u x minus v upon 1 minus u x v upon c square this is the equation which I had written now my parameters are u x is equal to 0.8 c not 0 because u x is this particular particle velocity which I am taking as c so this will be equal to minus c and v will be equal to 0.8 c so let us substitute in this particular thing here so u x is equal to minus c so I am writing minus c v is equal to 0.8 c so this becomes 0.8 c this I am writing as 1 minus u x which is minus c so this becomes plus into c into v is 0.8 c divided by c square so if you look at the numerator it will be minus 1.8 c and denominator the c square will cancel will be 1.8 c so this will be equal to minus c so you can see very easily that if this particular particle was traveling with the speed of light then this particular observer with the ground though he observes speed of light this particular particle will be traveling with the speed of light this particular observer will also find exactly the same thing that this particular particle is traveling with speed of light. So this is what we expected that if the light is traveling in one particular frame of reference with the velocity c in all other frame of reference it should travel with the same speed. Good evening sir. Good evening. My question is how can we link relativity with Doppler effect especially in light? Well that Doppler effect I will probably be mentioning little briefly see all that particular thing that we are talking is only about the velocity being same I have not talked about the frequency being same. Now as far as Doppler shift is concerned there is a very simple way of describing when we talk about what we call as energy transformation unfortunately I will not be able to cover energy transformation too much because you know see I will also say that if you are interested I have 24 lectures a series of 24 lectures on YouTube on special theory of relativity which you can probably download not download you can even watch from YouTube there I have given many more details but basically thing is that you can treat a photon as a particle with 0 rest mass m not is equal to 0 and energy equal to h nu and momentum is equal to h nu by c this of course I have talked also about the quantum mechanics. Now you use energy transformation that once you are going from one particular frame of reference to another frame of reference how the energy would change was the energy changes essentially it means the frequency changes which means there is a Doppler effect okay there is only one important thing in relativity that you also get Doppler what we call as a transverse Doppler effect okay in which the relative velocity directions is perpendicular to each other but you know that is actually a matter of detail but having Doppler effect is just I mean a transformation of energy the energy of a photon like for energy of any particular I mean let us why not like let us look at classically a particle moves with one particular velocity even in classically okay if you look at another frame of reference its velocity becomes different velocity is observed to be different so energy gets changed they they would not agree with the value of kinetic energy energy does not get changed energy is same in a given frame but if you go to a different frame of reference the value of energy of the same particle will turn out to be different now all that I am doing that instead of this particular particle I am talking about photon so if this photon is being observed in two different frame of reference and if the energy is that say E in one frame of reference in another frame of reference energy will turn out to be E prime and if it turns out to be E prime essentially means nu becomes nu prime all right which means there is a change of frequency which is Doppler effect is Neeta Subash engineering college I what I was asking is that whether interference is possible between matter waves and infinite waves like electromagnetic waves no no no no no no no see then we do not talk of interference we can always talk of normal scattering so if for example if you are hanging like for example when we talk about the Compton fact which I will talk briefly tomorrow okay so if there is a particular particle and a particular now let us say electromagnetic wave which comes which is also of the form of proton okay then of course we will have the normal scattering processes okay but the typical type of interference when I write y is equal to y1 plus y2 where y1 represent displacement of electromagnetic wave and y2 represents the probability wave they will not interfere in that sense well another question is like Einstein's second postulate about this special field of relativity yes that the speed of light will remain constant in all inertial frames of reference that's right then like it was just a postulate that the speed of light will remain constant but as per the other material particles and velocities and forces and all these things the position of the particles for all these things the these will change as per the Lorentz transformation rules yeah as we know and you have discussed the velocity is all distance change and Einstein only postulated that velocity of light will remain constant in the different frames of reference yeah but why this type of postulate like why the velocity will remain for these waves velocity but for light waves the velocity will remain constant but because we know light is a wave it is electromagnetic wave and the question is whether for the other electromagnetic waves like other manifestations of electromagnetic waves like gamma rays no no it's true for every way it's true for every electromagnetic wave yes and if we come to the other kind of waves like sound wave that is a mechanical wave no no whether any postulate can be done or Einstein has not postulated anything about this see remember the problem whether that anything can be said yes sir anything can be said about these waves mechanical waves like sound waves or other yeah I have understood your question I have understood your question okay let me try to answer this particular question see thing is that see the problem that was faced was only with electromagnetic waves because electromagnetic waves do not require a medium to travel sound requires a medium to travel okay so the hypothesis was only about the electromagnetic wave now whether the electromagnetic waves are gamma rays or x-rays or microwaves or radio waves it does not make a difference okay these are all electromagnetic waves okay also let me add which I should have added which I probably did not specifically mention that when I am talking of the velocity of light I am always talking of velocity of light in vacuum okay if you are talking of velocity of light in a medium okay in that particular medium the velocity can be different okay turns out to be generally turns out to be smaller so when I am talking of the velocity of light I am always talking of velocity of light in vacuum and they refer their reference which Einstein made in the second postulate was only for electromagnetic waves not for sound wave not for water waves anyway sound wave does not travel with that speed anyway the water waves do not travel with that speed so it was only because the problem was faced only when we are talking about the electromagnetic waves because when we are defining see sound wave when I am saying sound velocity okay sound always requires a medium to travel and when I say that this is the speed of sound okay have no ambiguity this speed is with respect to the medium in which it is traveling okay so if you are talking of water waves I am talking of water waves the speed of water wave with respect to the medium in which it is traveling which is the water medium okay problem was with electromagnetic waves but because when I say that this is the speed of light with respect to what okay now if I say medium I do not see a medium that is why people thought that okay there may be a medium which I do not see it that is why they started calling it ether and that is why Einstein made a bold statement saying that you do not require a medium and the velocity of light whether you have medium or not okay it will always be the same okay that particular condition is not meant for any other type of waves so it is only for the light okay by light I mean in general any electromagnetic wave whether it is gamma ray x-ray electromagnetic radio waves microwave does not matter for all of them okay velocity will be same as given by 1 upon under root of epsilon or not may not and independent of the frame of reference so good evening sir good evening my question is yes does quantum uncertainty apply in relativistic effects see the thing is that you know I mean the traditional classic quantum mechanics does not take relativistic effects into consideration okay so I have mentioned that particular thing that you know when you are talking about you know quantum mechanics it is always a classical particle so relativistic effects we are never considered unless you talk about really the Dirac equation and you want to talk about relativistic quantum mechanics which is a sort of a fairly advanced topic so I would at this particular moment I will like to keep these two things separate that when I am talking about electromagnetic waves I am talking of wave particle loyalty which is for the classical particle which is traveling with speed much smaller than the speed of light so one more question yeah please go ahead why traveling at light speed is impossible is it possible in future see the thing is that according to a postulate a special theory of relativity any particle with has a non-zero rest mass cannot travel with the speed of light or greater than the speed of light okay because the mass would turn out to be infinite which does not make any sense all right so therefore according to the postulate of special theory of relativity only those particles which have zero rest mass can travel with the speed of light okay one example of which is photon for other particles it is not possible good evening sir good evening how is time dilation invariant of observer oh no see time dilation in fact there are two things which I have not talked about is what what what I which is generally very very popular because I mean normally I would have like to talk about it but we do not have any time with only four lectures given for relativity it is not possible to talk about there are two aspects which are always talked about what is called length contraction another is time dilation the main reason I do not did not talk about these particular things which are actually very important concepts is that you know they I mean these are often very much misunderstood then understood in fact my close friend who will give a talk tomorrow who is a specialist of relativity which I am not okay in fact he always mentions that you know length length contraction is better not to be taught because this only confuses the students it never you know sort of they never use it properly see thing is that we define certain things which are called proper length and we call a proper time interval okay so because your question is about time dilation so I would like to mention about this particular aspect only see if two events occur which occur at the same position this is very very important that is what I say people generally ignore this particular thing that if you have any two events which are occurring at the same position for example something happens sitting here in my frame of reference and in my frame of reference again something else happened exactly at the same point all right for example I watched I I started class at this particular moment of time okay and I ended the class exactly sitting in the same place or in other words for example if I am sitting in a train and I want I see one end of the platform coming to me and I see the another end of the platform coming to me both are occurring at the same position as far as the observer in train is concerned because that observer is sitting there and find he will have noticed as if the platform is moving backwards okay if the two events are occurring at the same position then the time interval between them is what is called a proper time interval so proper time interval is the time interval between the two events provided they occur at the same position this is very very important because this is the aspect which people tend to ignore when they I mean they treat this as a formula and whenever they see time interval they will always dilate it which is not correct similarly when they have see a length they will always contract it okay similarly there is a length also which is a proper length which is defined but let us not talk about this so we have two events which have taken place in a given frame of reference at the same position all right and the time interval between them is let us say tau then if I if anybody else observes the same two events okay in a different frame of reference then that particular person will observe that the time interval between these two events get enhanced by a factor of gamma okay gamma I have already defined okay 1 upon under root 1 minus u square by v square by c square so this is what is called time dilation so this particular time interval between the two events remember two events which are occurring at the same position okay this will not happen if they are not occurring at the same position if one event occurs here another event occurs somewhere else okay then the time interval between them will not be related by this particular type of time dilation it is related to this type of this particular thing only if the two events are occurring at the same position then if the time interval between them is tau then in any other frame of reference this particular time interval between same two events will turn out to be gamma tau this is what is called time dilation yes Gyan Mani college good evening sir yeah please go ahead our questioner yeah is there any formula to relative transformation for four-dimensional vector in space see the thing is that you know we talk on relativity about the four-dimensional space which is also called Minkowski space okay in which time with a sort of imaginary number of course there are many various ways of defining Minkowski space okay we talk about that four dimensional and then what is important in that Minkowski space that all the transformation equations can be combined to one four by four matrix so that makes the life somewhat easier that makes the transformation loss somewhat easier and this is what we generally call as a four vector and all the four vectors if they have to be called Minkowski four vector then they should always obey a particular transformation equation all right so sometimes we do talk of the four-dimensional spaces but as far as this particular thing is concerned because we are not going to that much detail I will not be talking about the four-dimensional spaces I will be only talking about the three-dimensional spaces I mean I will only be talking about individual transformation so I will treat the transformation of coordinate separately and transformation of velocity separately I will not even be talking about the transformation of energy so I would have like to talk about it okay definitely not about the force definitely not about the acceleration and things like that yeah you have one more question yes yes will I institute of technology sir please explain the physical meaning of equation t dash equals to t minus v x upon c square yeah upon under root minus v square upon c square is it merging of space and time well you know I do not want to talk of any of those things as of now let me just write this transformation equation which you have written which you have mentioned t prime is equal to gamma t minus v x upon c square I mean at the moment the only interference inference which I would like to give you this equation that if a particular event occurs at a particular time t and at a particular value of x in a given frame of reference then that event will appear to be occurring at a different time t prime in a different frame of reference provided they have synchronized their watches okay by this particular formula so as I said this particular time depends not only on the time but also depends on v this is v multiplied by x okay this is not v x but v multiplied by x okay so it will also depend on the position in which this particular event is occurring so that is I would like to say I would not like to talk about you know because I am not talking about those energy other and x y and time diagrams you know which makes I mean as I say I mean relativity is a very very difficult topic and if you want to talk about more details you know we can keep on talking forever so I mean the idea here was to just to give you a sort of a flavor of special theory relativity and in that particular thing at this moment I would like to take only this particular thing that time is no longer p I mean t prime is not equal to t but it depends when if the event takes place at a given value of x and a given value of time then this becomes the time as seen in a frame of reference s prime or questions yeah please go ahead so yeah we have come across a situation where a charged conductor is kept in a frame of reference which is stationary yeah another observer in a frame of reference which is moving yeah now he will he observe any magnetic field that's right that's right see think is that okay okay then the observer in the fixed frame of that's right will he get the effect of the magnetic field let me sort of try to explain this particular thing see the thing is that you know once we are coming to the transformation we are going from one transformation to another transformation be a normal the course on relativity or the first level course on special theory of relativity will normally end when we are talking of electric field and magnetic field transformation this was I mean what is actually one of the most interesting aspect of special theory of relativity is that we found a problem which that of not matching electromagnetic concepts with the concept of classical physics but all the reformation that took place as a result of special theory of relativity actually turned out to be on the reformation on the mechanics aspect the Maxwell's equations are valid also under the special theory of relativity that's what is a very very interesting aspect now all that has become different is the interpretation of electric field and the magnetic field a field which is called purely electric field may be mixture of electric field and magnetic field in a different frame of reference okay what we are calling a particular field as a magnetic field may turn out to be mixture of electric field and magnetic field in a different frame of reference so we use a particular transformation equation which again as I say is slightly beyond the scope of this particular course but I will refer to some other textbooks okay I have also given some as I was telling I have also given some lectures on YouTube which are available on YouTube on special theory of relativity you can see the details these equations will tell you that if I measure electric field and magnetic field of at a given particular point what will be the electric field and magnetic field at that particular point as seen by an observer in a different frame of reference so it should be sort of very very clear for example if I am talking of a velocity okay and if there is a magnetic field here okay if I go to a frame of reference in which the velocity of the charge is V the force is V cross V which is the magnetic force but if I go to a frame of reference in which this particular particle is stationary okay obviously the magnetic field would not produce the same force however of course force also transform okay in fact if you know what are the forces on a particle in a given frame of reference in a different frame of reference the forces will turn out to be different all right using the but but the force will still be present okay now what will happen in that particular frame of reference this particular force will turn out to be an electric force all right so the things that magnetic field and electric field will transform in such a fashion that the force transformation becomes valid so what we are talking about pure electric field or pure magnetic field that loses significance in special theory of relativity okay you are talking of electromagnetic field so it could be partly electrical one particular frame of frame of reference partly magnetic it could be other way when it could be different way the weightage of electric field and magnetic field could be different when you are going from one particular frame of reference to another frame of reference mutila lehru yes MNIT yes sir good evening sir good evening yes sir my question is what type of fringe shift is expected in michaels and molly experiment how we can explain those fringe shift to a student see as a student asks what type of fringe shift how can we explain them see as I told you that you can make a rough calculation we by taking those distances and that free shift turns out to be of the order of 0.4 fringe shift so as you keep on moving you will find that this particular thing whatever is the width of the fringe shift will move half the way approximately okay but as I said this course as somebody had just now pointed out that this appears to be too small so has this experiment not been repeated so I answer is that this particular experiment has been repeated multiple times with better accuracies okay and you can see these fringe shifts you know when you are sure that fringe shifts are much larger than that that thing and that has also never been observed please explain the evolved sphere see it is again something about the reciprocal lattice space okay I roughly remember so I can probably tell you that you know see you talk of reciprocal lattice let us suppose this is the reciprocal lattice okay see this has the same dimension as the wave vector k so you know what is your lambda wavelength of the x-rays you know the lambda then you know what is magnitude of 2 pi by lambda so this becomes equal to the magnitude of k okay in fact some people had asked question I am not sure whether I have answered this particular thing that see ijk directions are same and direct lattice as well as in the reciprocal lattice space so ijk what I am calling is x direction is also being called as x direction in the reciprocal lattice space so this x direction i direction is same both in reciprocal space and direct space so once I know in which direction my x-ray is coming is being incident with respect to your direct lattice okay then you know what is the direction with respect to which it is incident in the reciprocal lattice so you know what is the wave vector let us suppose we now sort of write a wave vector which ends the tip of which ends at one of the particular point one of the reciprocal lattice point if tip ends here of course this lambda could be any arbitrary value so this end may not start from any of the reciprocal lattice point all that I want that the tip of this particular vector should end at one of the lattice points okay then you draw a sphere in the reciprocal lattice space okay I have drawn it very very small but you know normally this k will be much larger it draws a sphere I am sorry my drawing is very bad and let us assume that my lattice points are very very closely spaced now if it so happens that if one of the reciprocal lattice point lies exactly on this sphere this is what is called evolved sphere if there is a reciprocal lattice point which is exactly lying on this particular case then you can show that the condition of the Bragg condition because then this is a proper g then delta k will turn out to be equal to g so the lowest condition gets satisfied with respect to this particular sphere so basically this is sphere which you draw which has a radius equal to the wave vector of the x-rays it has the direction which is the same direction as that of incident x-ray so basically you are drawing a wave vector in a way that it terminates this tip lies on a reciprocal lattice point then you draw a sphere in this particular reciprocal lattice space with the same radius as the length of this particular wave vector then if it so happens that one of the point any point on reciprocal lattice exactly falls on this sphere which you call as evolved sphere then corresponding to that particular reciprocal lattice point there will be a Bragg reflection see basically things are that what we call as hkl planes in direct lattice in lowest condition they become gha plus kb plus lc so in lowest condition we talk of hkl which are which which are appearing in this particular equation okay and these are related exactly to hkl plane so what I am talking is hkl plane in Bragg's condition becomes an appropriate reciprocal lattice vector in the lowest condition so this is roughly speaking about the evolved sphere why my collection Morley performed experiment on the basis of only ether medium why they are not performed any other medium no no no see he does not know whether this medium is existing or not see that's that was the thought people thought that there is a medium ether medium which is existing there at that time okay this was the normal thought process at that particular time imagine yourself in that particular you know late 19th century or earlier 20th century that was the time when people were thinking that ether exist okay so he thought if really ether his idea was to test whether there is ether or not where light is actually a frame dependent quantity or not okay it means sitting on earth if I measure the velocity in different directions well whether I will find them to be different that was his idea and that was based on the fact that people thought that there is a ether medium all right okay now we know that the term does not exist and in fact light irrespective in which direction you measure will always give you the same value see good evening sir good evening sir I have a question related to the Einstein's mass energy relationship derivation that is equal to mc squared while deriving this equation he considered that the total energy of a particle is equal to the sum of relativistic kinetic energy and the rest mass energy of the particle that's right and he taken that the rest mass energy is in the form m naught into c squared that's right before he arrived this equation is equal to mc squared that's right how how do you put at the rest mass energy is equal to m naught c squared okay about this actually I am going to talk tomorrow tomorrow in morning lecture but because anyway you ask the questions so let me try to answer it first of all I would disagree with you that Einstein derived is equal to mc squared Einstein did not derive okay he proposed this particular thing to write in this particular fashion that was a proposal in fact what we will be seeing in fact I wish I could have time to discuss this particular aspect of momentum conservation and energy conservation in more detail okay the thing is that in relativity energy and momentum both are always conserved in any any particular process okay the only thing he enlarged the definition of energy okay so he thought he he what he proposed is that what we classically called as the energy classically energy was only potential energy kinetic energy that was the type of energy that we were talking about in the classical mechanics okay he proposed a totally new form of energy when I am writing e is equal to mc squared this e has no classical analog okay if you go into the limit v much less than c okay let me try to tell what that is what I am going to cover tomorrow see only I have discussed Lorentz transformation next thing that I am going to discuss is that Lorentz transformation is not enough because it is not valid with the first first positive of relativity then you require some new definition you require new definition of momentum and you require a new definition of energy and that is how he landed upon this equation because he had a strong belief on in what what what what what he has said in his postulates okay so he proposed a totally new form of energy which has no classical analog and that he proposed it to be equal to rest mass energy plus all other forms of energy potential energy kinetic energy alright and this rest mass energy is supposed to be involving almost everything for example classically I mean mass is just mass but for example if you heat the material okay the particular material has now bigger in larger energy essentially it will mean that mass goes up goes up if you have one I mean if we take a classically inelastic collision when this particular particle comes and this particular particle comes and hits and they can let us stuck together this is a very standard completely inelastic collision in classical mechanics we say that the energy is lost but when we say energy is lost actually energy is not not lost it is only the mechanical energy which is lost okay rest of the energy is converted into some other form of the energy now in relativity you will still consider easy is equal to mc square but then what you will find that the more mass when these two particles get stuck to each other will be larger than m1 plus m2 if this was the mass of mass m1 this was our mass m2 rest masses when they are combined the rest mass will be different from m1 plus m2 it has become larger than m1 plus m2 because that energy has essentially been used to enhance so basically mass is energy okay sir one more thing sir yeah please according to Newtonian mechanics the space mass time all are absolute and aspect theory of relativity they are all relative quantities yeah can you elaborate this sir I mean this is what I was trying to essentially mention that you know I mean I am not sure what you mean because these statements are sort of strong sent statements I am not sure whether I mean the way one would like to understand these statements okay let me put it like that I mean in this particular case what I essentially try to mean is that when see x being different from x prime was there in classical mechanics also okay what is more important here is to realize that time is also function of x and time which is something which is very very shocking so if two events are appearing to be occurring at the same position okay I have same time in a given frame of reference they may not appear to be occurring at the same time in a different frame of reference that is what is I would call is the most you know sort of important idea of the special theory of relativity okay we will close now it is already 5 o'clock you can probably post the questions if you have any more question tomorrow will be the last day so I will not be able to attend the validity function because I have to go somewhere but in morning session I will come and give my lecture and that will be the end of it and then I will say bye bye to you okay