History of Special Relativity (Part 1) - Galilean Invariance & Maxwell's Equations

in 1905 Albert Einstein first presented his two postulates also known as the special theory of relativity they would later on make predictions about the nature of space-time mass energy which at that point of time and even today seems very bizarre and non common sensical in nature however they would end up creating a very successful domain of modern physics now these ideas they did not culminate in just one day there is a certain history behind it in this video and even in the next video I want to talk about the ideas the inconsistencies and the major issues which was being faced by physics at that point of time which led to the birth of special theory of relativity I will discuss these topics in two videos in the first video which is today I will talk about the Galilean transformations then I'll talk about the Newton's laws of motion and how they are invariant under Galilean transformations when you go from one frame of reference to another then I want to talk about the Maxwell's equations and how the Maxwell's equations predict the existence of electromagnetic radiations then I want to talk about how the Maxwell's equations are not invariant under a similar Galilean transformations when you go from one frame of reference on to another so at that point of time two major domains of physics one the Newton's laws which successfully explained mechanical behavior another the Maxwell's equations which explain electromagnetic behavior showed a certain inconsistency under Galilean transformation I'll end the video by talking about how to make sense of these inconsistencies in the second video I want to talk about the concept of ether and absolute frame of reference then I will cousin detail the michelson-morley experiment which successfully failed to prove the existence of an ether and then we will carry forward from there so let's begin before Einstein the concepts of space and time were described by Galileo and Newton now the Newton's laws of motion are valid in the inertial frames of reference only now what are inertial frames of reference these are those frames of reference which are either at rest or moving at uniform velocity to put it simply now here I have two inertial frames of reference s and s - now let's suppose for the case of simplicity that both these two inertial frames of reference are aligned in exactly the same way that means the accesses are parallel to their respective accesses means the X is parallel to X dash axis Y is parallel to y dash axis and Z is parallel to Z dash axis respectively in three-dimensional space also less su for the case of simplicity is that at time T is equal to zero both these coordinate frames of reference coincide that means oh and OH - coincide at time T is equal to zero but one of the frames of reference as - Ramel reference is moving with uniform velocity V along the x axis so after some point of time this second frame of reference will have moved ahead with respect to the first frame of reference now if I have two observers both in each frame of reference and they have synchronized clocks and synchronized measuring rods and they look at a given particular physical event then their measurements of that physical event will be related by what is given known as the Galilean transformations in the Galilean transformations the position coordinates in a in that direction which is perpendicular to the direction of motion will remain unchanged and the position coordinate in the direction where the motion of the frame is taking place X - will be equal to X minus VT where V is the velocity and time is a time period which has elapsed now in Newtonian physics the concept of time is absolute throughout the universe in the sense that if there are two observers and they are both moving to each other with certain velocity then if they measure the time period between two physical events it doesn't matter how fast one is traveling or how slow one is traveling or whether or not both of them address they will both measure the time period difference between two physical events to be exactly the same in in a way it is saying that time flows in the same manner for all observers in the universe now the Newton's laws of motion are valid for both these two observers that means the nudists laws will retain their form and expression for both these two observers even though one is at rest and the other is in motion let's see how that happens so let's suppose in the ass - frame of reference the observer is looking at a physical event and he concludes that the Newton's law expression is given by F dash is equal to M a - where F dash is a force then he measures in his frame of reference a dash is the acceleration and he measures in his frame of reference and for the sake of simplicity let's assume that this acceleration is only happening in one axis which is deep X dash axis now if you substitute a dash with this d2 X dash by DT - - then you can use the Galilean transformations X dash is equal to X minus VT where it is a constant and T is equal to T dash in the derivatives and you can simplify it in this particular manner to md2 X by DT - which is nothing but M a that means f dash is equal to Ma dash is equal to M a which is nothing but the force in the first frame of reference now what does this mean this means two things that both these two observers looking at the same physical event will conclude that the forces involved are exactly the same they will also conclude that F is equal to Ma and F dash is equal to MA - hold true in their own frame of reference that means the Newton's law the form of the Newton's law remains in and for either of those two observers this is what it means that the Newton's laws of motion are invariant under a Galilean transformation when you go from one inertial frame of reference to another what does this signify it signifies that the Newton's laws of motion remain the same they are invariant they retain their form and expression across all inertial frames of reference this means that these physical law is universal in nature for all inertial frames of reference this is a good thing because observers across an infinite number of inertial frames of reference can use this universal law to explain mechanical behavior it also suggests that there is no absolute frame of reference as far as Newton's laws are concerned all inertial frames of reference are equivalent in nature now can we make the same conclusions for the laws of electromagnetism um and more specifically the Maxwell's laws let's get into it these equations describe the behavior of charges at rest charges and motion electric fields magnetic fields now here I have written the Maxwell's equations in vacuum where there is no presence of charges or electric currents the first and second Maxwell's loss tells us about the divergence of the electric field the divergence of the magnetic field and the second and the third and the fourth equations tells us about the curl of the electric and the magnetic fields respectively now I'll not go into detail about the physical meaning behind each individual equation rather what I'm interested in is whether or not they retain their form as you make a transformation from one inertial frame of reference to another in the same way and Newton's laws did but before we do that let's look at a very simple prediction of these Maxwell's laws of equations so if you look at the left here I have taken the curl of the curl of an electric field now if I take the curl of a curl of an electric field the curl of an electric field is given by the second Maxwell's equation so I have written minus DV by DT in the right hand side now if you're familiar with a vector calculus del operator simply is the derivative with respect to XY and z coordinates so the curl of the curl of an electric field or the curl of a curl of any vector field can be written as the gradient of the divergence of the vector field minus the square of the del operator of that particular vector field if I do that and the right-hand side I get minus del by Del T of del cross B now the gradient of the electric field is zero as it's shown here in the Maxwell's equations so I take this out I'm left with minus del square e is equal to minus D by DT of del cross B which is nothing but the fourth Maxwell's equation if I substitute these values make certain rearrangements I get this equation del square by Del T square of e minus C square del operator square is equal to zero similarly I can also substitute the values of B here so del cross del cross B and I can make similar sorts of substitutions and come up with this equation that I've written here now what is the meaning of these two equations these equations are nothing but the wave equation which is similar to Del square Y by Del T square minus C square del square LX square is equal to 0 because the del operator here simply signifies that it is in three-dimensional space it is the second order space derivative having both XY and z components now these equations predict the existence of electric and magnetic disturbances which travel through vacuum and the speed given by see now as you can already see here that the Maxwell's equations already tells us that whenever there is a time derivative of magnetic field taking place it will always lead to the presence of an electric field and when there whenever there is a time derive a different electric field at existing it will always lead to the presence of a magnetic field so how do these disturbances travel through space if you have an electric field which is changing with respect to time then that electric field will lead to the presence of another magnetic field which is also changing with respect to time and that magnetic field will lead to a subsequent electric field which is changing with respect to time now these disturbances in the electric field and the magnetic field sustain itself 1 on to 1 and then these disturbances travel through vacuum and the speed given by see now these electromagnetic oscillations or radiations is nothing but light itself the light that helps us see the world around us the light that reaches our eyes and gives us an image and it helps us in mapping the world that exists around us now when we talk about light and light which is traveling with the velocity of C there is one question which is very important and let me mention a few notes here before we proceed further now the speed of light or the speed of these electromagnetic disturbances as they travel through vacuum or as they travel in empty space is given by C which is numerically equivalent to approximately 3 into 10 to the power 8 meters per second now what is this velocity with respect to because if you look at the Maxwell's equations the Maxwell's equations introduces this constant but it does not tell us about the nature of the source of these disturbances if the source which caused these disturbances was moving at a certain relative velocity would that constitute or would that change the velocity of the effective radiation that was emitted by the source what I mean to say is that what is the velocity of the light with respect to now at that point of time whenever we studied waves we usually studied waves with respect to some sort of a medium for example if you look at the waves propagating on the surface of a pond now you can calculate the velocity of those waves moving on the surface of a pond with respect to that point itself if you look at longitudinal oscillations in air which constitute sound then you can look at the velocity of those sound waves with respect to the propagation medium of still air now if you look at electromagnetic oscillations or light traveling through vacuum what is the propagation medium now at that point of time it was assumed that there was a propagation medium called ether now eta is some sort of a invisible absolute medium which permeates all of empty space and electromagnetic disturbances travel through ether now an important question arises so the velocity of light if it is with respect to that ether then the velocity of light should change when we look at it with respect to some other inertial frame of reference now does it suggest the existence of an absolute frame of reference does it mean that the conclusions that we derived from Newton's laws that there is no absolute frame of reference all inertial frames of reference are similar is not correct so to prove that let's first write down the Maxwell's laws of equations and try to transform them from one inertial frame of reference to another and then see if they retain its original form now to transform the Maxwell's equations from one inertial frame of reference to another first we have to do two things since the Maxwell equations consists of both electric and magnetic fields and they also contain derivatives first we need to transform the electrical magnetic field from one frame of reference to another and then we need to transform how the derivatives change when we go from one frame of reference to another so now what does it mean it means that let's say for example if I have a charged particle which is at rest with respect to me then that charged particle is going to have an electric field but it is not going to have a magnetic field but if there is another person who is moving with some relative velocity with respect to the charged particle then for that person the charged particle has umbrellas relative velocity so the charged particle is going to show not only electric field but also magnetic field so it this means that the electric field does not necessarily they retain its autumn value when you look at it from two different inertial frames of reference so we need to see how the electric and magnetic fields transform as you go from one inertial frame of reference to another now to find this out we make use of the invariance of the Lorentz force now if you remember when we were talking about the Newton's laws of motion we saw that when two inertial frames of reference and looked upon a given physical event the forces were exactly the same that means F is equal to F - and we are going to make use of that so let's imagine that we have two frames of reference s and s - and there is some charged particle Q present in the frame of reference and in the charged particle who is moving with respect to let's say the x axis with some velocity let's suppose V Q then what is the force experienced by the charged particle due to the presence of two electric and magnetic fields in either frames of reference now since we are talking about three-dimensional space just so that we simplify these calculations let's take a very particular example so let's suppose that we have an electric field which is going in the y axis and we have a magnetic field which is going in these z axes and let's also suppose for the case of simplicity that the electric and magnetic fields are only functions of X and T now in that case if a charged particle is experiencing an electric magnetic field then the force experienced by the charged particle is given by F is equal to Q E Plus V Q cross B if I write the components separately I can find out that the charge experienced can be written in this particular manner which as it is seen here is only existing in the y-axis now this is the force experienced by the charged particle in the s frame of reference what about the force experienced by the charged particle in the ass - frame of reference in this case I write that the electric fields are a dash and B dash and the V q dash is a velocity of the charged particle in the ass - frame of reference now if I use the Galilean transformations by taking the time derivative of x - is equal to X minus VT I can write the velocity addition theorem which simply states that V q dash is equal to V Q minus V where V q dash is a velocity of the charged particle in the ass - frame of reference V q is a velocity of the charged particle in s frame of reference and V is the velocity of the ass - frame of reference with respect to the a stream of reference in the x axis so if I make this substitution in v q - here if I make this substitution here so and since I've already applied the invariance of the Lorentz force law then I can separate these components of E and E - and b and b - and then I come up with these two transformations so this is a transformation when I make the Tran substitution in of v q - and this is a transformation then i'll obtain if i make the substitution of v q so e is equal to e - plus v b - b is equal to b - so that means when i go from one frame of reference s to s - then the electric fields and the magnetic fields measured by two observers in these two respective frames of reference can be given by these two transformations now let's take the case of the fourth Maxwell's equation which is the curl of the magnetic field now if I want to show that the all the Maxwell's equations are invariant under a Galilean transformation then I will have to prove it for all each individual case but if I show that even in one case is not invariant then I can automatically conclude that the in general the Max's equations are not invariant so let's start with the most simplest case let's take only one Maxwell's equation and let's simplify it so that we only look at the components in the one axis so as to make our calculations a little bit simpler so I take the matter fourth Maxwell's equations which is the curl of the magnetic field is equal to 1 by C square del Y by Del T now I have taken a very specific case where I have taken the electric field to be in the y axis and the magnetic field to be in the z axis if I apply that case here so del Y by Del T simply is Del E by Del T in the Y axis and the curl of the magnetic field simply transforms into minus del V by Del X in the y axis so this Maxwell's equation in for this particular case of electric and magnetic field transforms into del V by Del X is equal to minus 1 by C square del Y by Del T now what we want to see here is that if so since this is here we have B e X and T that means this is the Maxwell's equation in the S frame of reference so this is the Maxwell's equation which is being observed by the observer in the s frame of reference now I want to transform the B and the E to be - Andy - and I also want to transform from X T to X dash and T - so to make that transformation first we'll have to look at the transformations of the derivatives so to look at the transformations of the derivatives we use make use of the chain rule so I write del V by Del X as del V by Del X - del X - by Del X plus del V by Del T del T - by Del X because bxb is only dependent upon X and T and obviously it's also dependent upon x - + d - similarly I also write del V by Del T as del Y by Del T - del T by T - by Del T plus del e by del X - del X - by Del T now I can automatically obtain and these there are partial derivatives by looking at the Galilean transformations so the Galilean transformation tells us that X Dash is equal to X minus VT so what is Del X - by Del X if look at the equation it will automatically come out to be one so del since X dash is equal to X minus VT Del X dash by Del X will automatically come out to be one similarly del t dash by Del T will come out to be one because T dash is equal to T and del T dash by Del X will come out to be zero because T and X are independent variables similarly del X dash by Del T will come out to be minus V because X dash is equal to X minus VT so if we take the time derivative you end up getting minus V so if I substitute these four values in these particular expressions then deep first of all del V by Del X becomes del V by Del X dash now as I have already proved that as we go from s to s dash frame of reference B is equal to B dash so I can substitute B dash here so I end up getting del B dash by Del X X dash so that means del b by del x is equal to del b dash by del x dash now if we look at Danny by Del T here and if I substitute these values then I end up getting del V by Del T dash minus V del Y by Del X dash now let's put the values of the transformation of e from one frame of reference to another in this equation and see what we kept so now is substitute E is equal to e dash plus VB in the equation that we just now obtained then e by Del T is equal to Del E by Del T dash minus V del V by Del X dash if I make these substitutions then I get this particular expression so now if I write the Maxwell's equation that I started with which was del V by Del X is equal to minus 1 by C square del V by Del T I can write del V by Del X as del B dash by Del X dash as we just now saw and if I write del u by Del T this entire expression here then I see that the nature of the expression is not exactly the same as this first expression so basically the fourth Maxwell's equation in one dimensions in the s frame of reference is this and this similar Maxwell's equation in the S - frame of reference this they are not the same I mean they'll be - by del X Dash is equal to minus 1 by C square del e - by del t - but there are these extra terms here so as you can see the Maxwell's equation has whatever form it has in the a stream of reference the in the as frame of reference the form that it has in the as - frame of reference is not the same now this constitutes a very big problem because it suggests that if there are two observers in two different inertial frames of reference and they are both studying some sort of an electromagnetic event then both of them will have to use two different forms of Maxwell's equations that means if there are an infinite number of inertial frames of reference there are going to be an infinite variations of the Maxwell's equations this gets rid of the idea of the universality of physical laws this is a big problem because all these observers who are in different frames of reference studying the same phenomena cannot agree upon the basic laws using which they are studying this phenomena so the fact that the Maxwell's equations are not invariant that means they do not retain the same form and expression when we go from one frame of reference to another constitutes a huge problem in physics now what could be happening here there could be many explanations for this maybe the Maxwell's equations are wrong now it does not seem likely because Maxwell's equations are some of the most successful equations coming from a very successful domain of physics which is there of electromagnetism it successfully explains the behavior of stationary charges currents magnetic fields and the existence of electromagnetic waves so this option does not necessarily seem to be very attractive maybe the Maxwell's equations are correct but they are only correct in some prefer frame of reference now as I mentioned earlier that the velocity of light could be with respect to some preferred absolute frame of reference which is the medium of ETA maybe the Maxwell's laws of electromagnetism are only valid in the dis preferred absolute frame of reference and we need to modify the Maxwell's equations a little bit so that they remain invariant as you go from one frame of reference onto another or maybe the Maxwell's equations are in fact correct and are invariant when you go from one frame of reference to another what is wrong here are the Galilean transformations themselves now I will talk about this in more detail in the second video where I want to discuss the concept of ether and the experiment performed by Michelson and Morley where they tried to determine the changes and the velocity of electromagnetic radiation as it is traveling in two different directions with respect to the ether so to give a summary of what we have discussed in today's video is on one hand you have a very successful domain of classical physics which consists of Newton's laws or some equivalent formulations and these Newton's laws are same for all inertial observers in inertial frames of reference that means these Newton's laws cannot distinguish between two different inertial frames of reference there is no absolute frame of reference whatsoever and these Newton's laws are universal for all observers in all inertial frames of reference on the other hand you have these Maxwell's equations of electromagnetism which undergo change the moment you go from one frame of reference on to another that means they are not invariant under Galilean transformations they also seem to suggest the existence of an absolute frame of reference so these two very successful domains of physics seem to show a certain inconsistency when it talks about the idea of physical laws being universal across all inertial frames of reference and the Secession of absolute frames of reference what is happening here so in the next video we'll talk about the concept ether and in the third video I'll introduce the final suggestions coming from Albert Einstein himself any spots of special relativity that's it for today thank you very much you

Channel: For the Love of Physics

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Keywords: history of relativity, history of special theorycrelativity, history of special theory of relativity, non-invariance, invariance of newton's second law, form invariance of newton's second law, galilean relativity vs special relativity, invariant under galilean transformation, galilean invariance, galilean relativity, galilean transformation, inertial frames and galilean invariance, galilean relativity principle, history, relativity, special theory of relativity, Maxwells equations

Id: P79Ur8LVcy8

Length: 27min 50sec (1670 seconds)

Published: Sun Mar 25 2018