Special Relativity in Electrodynamics

hello and welcome to this final lecture in this virtual lecture course on electromagnetism I'm dr. Andrew Mitchell and in this lecture we'll be talking about relativity in the theory of electromagnetism this lecture will really be the culmination all the different topics we've been discussing in this course bringing them together into one unified framework we'll be using many concepts from special relativity and in the first part of this lecture I want to do a bit of a revision and a reminder about concepts in special relativity in particular the concepts of for vectors for vectors are objects which transform under the full symmetry operations of the Poincare group so not just translations and rotations of our coordinate system but also Lorentz boosts along a given direction meaning a relative velocity between different international reference frames so this is a wider kind of symmetry which can be incorporated into our description of physical phenomena and in particular here into the description of electrodynamics for vectors can be contracted into so called Lorentz scalars which are relativistic invariants they're the same in all reference frames and this is something we want for the principle of relativity so we'll be describing all of these important concepts at the beginning part of this lecture and then in the second part of lecture applying them to Maxwell's equations and the theory of electrodynamics in particular we'll be able to derive a relativistically invariant or covariant form of Maxwell's equations involving the potentials but these equations will feature four vectors they're the four actors were interested in in electrodynamics will be for example the four potential which has an in its components the scalar potential but also the vector potential in one single object we'll also look at the for current which contains both the charge density and the current as its components again combined into one object so relativity allows us to to combine the different aspects of electrodynamics both the electric side and the magnetic side on an equal footing in single for vector equations of course Maxwell's equations are actually already incorporating the effects of special relativity but when we formulate it in terms of four vectors will see a deeper structure and we'll see that this has much much greater power we'll also see and use the idea of gauge transformations of the potentials and we'll see how we can perform a Lorentz transformation between different reference frames and although the equations themselves remain the same the kind of physical effects may have different interpretations so to one observer we may regard a certain phenomena as being a magnetic effect but to a different inertial observer that same physical thing might be regarded as an electric effect so this deeper understanding of the phenomena are revealed by this relativistic formulation finally I want to talk about Lagrangian formulation of electrodynamics and this really draws very heavily from our relativistic formulation we'll see that with some grams and overarching principles of locality of Lorentz invariance of gauge invariance we can cook up a Lagrangian for electromagnetism this Lagrangian can then be used with the principle of least action and from the Euler Lagrange equations of motion we'll see that we recover Maxwell's equations so this draws together and unifies everything that we've been discussing into one very powerful framework and that's the topic of this lecture and it's the last topic in this course so let's get down to work so in this lecture will be discussing relativity that is to say Einstein's special relativity in the context of electromagnetism so this relativistic formulation of electromagnetism is somewhat technical however it does reveal the deeper structures to the theory of electromagnetism particularly in the dynamical case we'll see that it's extremely powerful and a wonderfully elegant theory in particular allows us to change inertial reference frames and consider the effect on the electrodynamics and what we'll see is that electricity and magnetism are intertwined and that's indeed the concepts of electricity and magnetism as distinct phenomena really depends on the reference frame so before we get started I just want to mention that in this lecture we're going to use the minus plus plus plus signature the meaning of that will be explained in due course okay so before we get into the discussion of relativity as it applies to electromagnetism let's review and revise some features of special relativity and in particular the concept of four vectors of course there are many things to be said about this and that's the focus in fact of the entire course on Pascal mechanics and relativity however and what I want to do here is just sort of summarize in a concise way the technical details so I won't provide an explanation for this I won't provide a derivation I will simply just present the in the simplest possible way all the things that we're going to need the technical and mathematical machinery behind for vectors and special relativity Lorentz transformations contractions Lorentz scalars and all this good stuff so the discussion of special relativity and for vectors begins with the concept of a spacetime events so a spacetime event occurs at a particular point in space meaning we have to specify x y&z coordinates in 3d space but also at a particular time so altogether we have four components one for time and three for space this is in in a so-called space-time event for vector this full vector is denoted by X superscript mu this Greek index mu runs over the four components of this four vector and by convention the time component is called the zeroth component so x0 is CT whereas these three spatial coordinates the space-like components and they essentially form the usual position vector and so we have X 1 is X capital X 2 is y Capital X 3 is said and of course this can also be something that can be just written as a usual three dimensional standard vector ah notice also that the time component is multiplied by the speed of light and this is this can be thought of as kind of a units choice we want in this four vector all of these components to have the same units so if we're going to put time in there we need to multiply it by the velocity and the velocity we choose is the speed of light choosing the speed of light of course is a particular choice of units and they're the sensible of units to choose in special relativity because there is certain there is a certain symmetry which connects the space and time light components that's the Lorentz symmetry so we want to choose basically the same kind of units and it turns out that the units we need to choose dictates that we use the speed of light then to use anything other than C for the velocity would be equivalent to saying let's measure the X direction in meters and the direction in feet or something like that let's choose something that's actually the same on the same footing for all four components of our four vector now this is a so-called contra variant for vector and there's an alternative type of vector called a covariant for vector which is very closely related in special relativity this takes a very particular form and it's extremely simple we just see that the time like component the zeroth component of the four vector picks up this additional minus sign there is a deep more technical distinction between contravariant and covariant for vectors but I don't want to get into that here also notice that if we were to go to general relativity the difference between contravariant and covariant for vectors would be different so here let me just specify that what we're talking about is special relativity so again the only difference here in special relativity between the contravariant and covariant for vectors is that the time component gets an extra minus sign and this is all we need to know for this particular course of course we could dig deeper into the subtleties but we won't do that in this lecture so we have four vectors with an upper index called contravariant full vectors and four vectors with a lower index called covariant four vectors and we can actually convert between a contravariant and covariant four vectors backwards and forwards by so-called lowering the index or raising the index and that is done in the following way we can lower the index using this account expression in which we contract the contravariant four vector here with this minkovski metric tensor 8 mu nu and I'll explain what that means and how to deal with it in a moment but before I do that let me just say that raising the index has a similar kind of structure we can get a contravariant for vector from a covariant for vector similarly similarly by this kind of contraction so there's two things we need to know here first of all what does this notation mean and secondly what is this metric tensor Aitor so here we actually used the famous Einstein summation convention which tells you that whenever you see the product of two four vectors with the same index repeated one of them a lower index and one of them an upper index it's implied that were actually supposed to sum over the four values that this index can take so for example what is actually meant by this lowering the index equation here is actually the sum of new goes from zero to three of a turn mu nu X nu and that's because on the left hand side here we have the product of two terms one of them has a lower index nu one of them has an upper index nu and this is a repeated index one of them lower on one of them upper and so it's implied that we sum over the four values that nu can take a new runs from zero to three and so in this expression we sum over nu and we'll we do that for a given mu and mu is something that's left undetermined so that's still a variable and index and that is the only index that survives on the left hand side of the expression on the left hand side we cannot have a new because new is being summed over we just have the MU on the left hand side and likewise for raising the index we see that there's a repeated new one upper and one lower and so this implies that we sum over new leaving just this mu left over which ends up on the left hand side so of course here we've been talking about four vectors which the event four vector is denoted with this capital X and a single index mu or nu some other Greek letter perhaps which can either be upstairs or downstairs but I've sort of swept under the rug a little bit the idea of this objects this ATAR this has two indices and so it's a tensor rather than a vector but the same logic applies so this 8 a mu nu is so-called Minkowski metric tensor and describes the geometry of flat 4d space-time if we were going to general relativity then this would take a more general form it would not be flat space-time but curved space-time but in special relativity that we're going to talk about today in the context of electrodynamics and we're just going to use the flat space-time in Coffs key metric tensor and for the purposes of this lecture and you can of course delve into this in more detail and understand the subtleties but for the purposes of this lecture we can just understand this metric tensor which is something with two indices as being like a matrix which has two indices for the rows and columns of the matrix and specifically in this particular signature we have the following so we can regard this as just a diagonal matrix with minus one as the zero zero component and then plus ones for the one one two two and three three components and everything else is zero and again this relates to what I said at the beginning of lecture that in this lecture we're going to talk about the physics in minus plus plus plus signature you can see what that means now this signature is referring to the diagonal elements of this metric tensor we could also have chosen a different signature another popular one is plus - - that's just like an overall minus sign and doesn't really make any difference however we have to have a convention and stick to it and the one we're going to use is this convention and so we can see that this is almost the identity matrix but we have a negative sign in the zero zero entry which encodes the difference between covariant and contravariant for vectors so when we raise or lower the indices this extra minus sign has to crop up and that's encoded in this metric tensor notice also that's the metric tensor with the upper indices is just exactly the same as the metric tensor with lower indices and that's basically all we need so when we think about raising or lowering the indices we're actually just doing essentially a matrix product of a column a four-dimensional column vector with this four by four matrix a turn on a sort of technical or mathematical level that's the way we can view the the algebra here but in the end because ATAR is a diagonal matrix which is so simple actually it turns out that the difference between the contravariant and covariant vectors is really just this minus sign on the time line component so having defined now our contra variance and covariance for vector for the event in space-time we can ask a deeper question which is what is really a for vector is it just an array of four numbers well no it is something very specific a four vector is an object that transforms under Lorentz transformations in fact it transforms under the full symmetry operations of the Poincare group meaning that it's translations in space rotations in space and also the Lorentz transformations so just as we would define a vector in normal three-dimensional space as something that transforms under translations of our coordinate system and rotations of our coordinate system likewise for vectors are very specific objects that transform under these symmetry operations as well as the Lorentz transformations and of course Einstein showed that this four vector event that we considered on the previous slide is such an object but that does not mean to say that any old array of four numbers or four quanta physical quantities would constitute a four vector they have to transform in the proper way and there are actually many different kinds of four vectors that appear in the theory of electromagnetism and there are also different for tensors that we will consider and so we need to understand the transformation properties of these things to confirm that these things are indeed four vectors so what is that Lorentz transformation in special relativity well the Lorentz transformation Arap allows us to change our inertial reference frame we go from a particular reference frame to a different reference frame related to the previous one just in terms of a constant relative velocity converting from one reference frame to another is done using the Lorentz transformation as Einstein taught us and Lorentz transformation of four vectors takes the following form so let's try to unpack this equation on the right hand side we see a contravariant four vector in a particular inertial frame which alcohol s here we're doing this example for a contravariant four vector which is the space-time event four vector but we'll see that this same transformation equation applies for any four vector we see in here a repeated index nu of two objects one with a lower index one with an upper index and so here we have the Einstein summation convention and that is a sum over nu equals zero one two and three on the right hand side we see the contravariant for vector in a different inertial frame which we'll call s primed and the facts are calling this s primed is denoted here by the fact that we have a prime symbol on this contravariant full vector that tells us that within the primed reference frame and finally what is this object here so this requires a bit of further explanation this is the so-called Lorentz transformation tensor and for for the purposes here we can just regard these things with two indices as a matrix with the mu here denoting the Rose ends the new denoting the columns now this object looks a little bit funny because whilst we have two indices we see one of them is upper and one of them is lower but don't be fazed by that for the purposes of this lecture we can just regard this thing as a matrix given that mu and nu these Greek indices run over four components zero time and one-two-three-four space this object is a four by four matrix or it can be viewed as a four by four matrix again there's some subtleties behind the scenes here that I I don't think we need to get into for this discussion so what is this four by four matrix is this object it is something that has a block diagonal structure we can sort of imagine dividing this up into these blocks and the non-trivial part is occurring in this 2 by 2 block in the upper left-hand corner the 2 by 2 block in the lower right-hand corner is just the 2 by 2 identity matrix and is kind of passive so some words of explanation are in order here in particular this is a transformation for a Lorentz boost along the x axis and we can see that because the time like components of this matrix which are the zeroth row in the 0th column are getting scrambled up with the x-direction space like components which occur in the one column and the one row that's what's happening here in this in this 2 by 2 block of this matrix transformation so when we talk about a Lorentz boost what that means is that the 2 inertial frames s and s prime are related by a constant relative velocity along the X direction can we come to know to that VX it has to be a constant relative velocity because they should both be inertial reference frames and therefore if s is inertial then s primed is also inertial so this is a specific transformation for a relative velocity along the X direction and that's all will actually consider in this lecture but please note that if you considered any other and direction that's simply related to this matrix by a trivial rotation of the coordinate system so all of the non-trivial stuff is contained already in this transformation we can always just consider a relative velocity along some other arbitrary direction by performing this transformation and then doing a subsequent rotation the other thing we need to know to be able to understand this transformation property is what these parameters are in here gamma and beta so this combination here is minus beta times gamma this is the same gamma so we need to know what beta and gamma are so beta is there for the relative velocity between inertial frames s and s Prime's in units of the speed of light and here we're talking about velocity along the X direction relative lost along the X direction gamma is the so called Lorentz factor which is 1 over the square root of 1 minus beta squared where this is the same beta that I just defined so if we know that the relative velocity between s and s prime we can work out beecher and gamma and then of course we just put them into this expression for Lorentz transformation tensor which as I say can be viewed as a 4x4 matrix one question that might arise is which of these two indices here the MU and the new refers to the rows and the columns of this matrix and the answer is that it doesn't matter because this matrix is a symmetric matrix so whether we choose rows or columns for the first and second indices here doesn't matter so just don't worry about that so here is that information summarized again let's just do a quick example of this suppose that I want to know the x coordinate in the primed reference frame that is equal to the 4 vector with the 1 components in the primed reference frame which can be related by the Lorentz transformation to the 4 vector in the unprimed reference frame in this way we're in this equation I simply used that because I'm interested in the X component that's the one component of the four vector and that means mu is equal to 1 and therefore on the right hand side instead of writing mu R just write 1 but we still have this sum over nu which is basically a matrix product and I just sum over the 4 values of MU in that expression from 0 to 3 and in this matrix I pick for example I say that this first index is let's say it's the row so it's this row here the first row remembering that the numbering starts from 0 this is 0 1 2 3 so the way to read this is that this is the first row and then we sum over the 4 values for the column index with this new here 1 2 3 4 and so the way we unpack this is to say first of all it's lambda 1 0 times X 0 and so on lambda 1 1 X 1 lambda 1 2 X 2 and lambda 1 3 X 3 these last two contributions are 0 because lambda 1 2 featuring here is this element which is 0 and lambda 1 3 is this element which is 0 in this expression we have that X 0 is of course CT and X 1 is of course just X in the unprimed reference frame and so putting it together we learn that X prime is equal to CT times lambda 1 0 which is minus beta gamma plus-x lots of lambda 1 1 which is this term which is gamma all cuts another way X crimes is equal to gamma into X minus V T and I got V T here because I'm multiplying C by beta beta being V over T and of course that's what we know from the Lorentz transformation and you can work through the other equations very similarly and convince yourself that this is correct now if you don't like all of this index notation you can actually just think of this as a straightforward matrix product regarding these various objects for example this lambda thing just as a four by four matrix and then the four vectors simply as a sort of four by one if you like column matrix so that really simply just looks like this so here I have the X mu primed here I have X mu and here I have a lambda Lorentz transformation matrix so if I want the X primed element that is the second element in here the second row and so by matrix multiplication I multiply the second row of this matrix by this column and indeed and indeed this is the term that I generate so you can think about this in terms of matrix products if you want to do then of course what I've written down here is the Lorentz transformation for the contravariance and for vectors so what is the corresponding Lorentz transformation for the covariant for vectors well the easiest way to do this is simply to relate the covariant for vectors to the contravariant for vectors using the raising and lowering operators that we already discussed and then simply use this transformation so what do I mean by then I simply mean that if I want to know X mu which is covariant for vector in the primed reference frame I can just relate that to a contravariant for vector in the primed reference frame so all this is in the prime reference frame and actually this is just equivalent to and lowering the index of X new in the prime reference frame and that's because this minkovski metric tensor does not depend on the reference frame so this there's no effect of having this inside this this primed thing here that's the same in all reference frames and now I have this object which is a contravariant four vector in the primed reference frame I can now just go ahead and use the rents transformation expression for the top of this slide except I just have a pre factor of beta mu nu sitting out the front also have to just be a little bit careful because I've already used the index nu for the lowering of the index so here I'm going to use another Greek index Sigma it doesn't matter which index I use because according to the Einstein summation convention this is being summed over the new is being summed over the only thing that survives is mu and that's the thing that appears on the left side of the expression so here I have a Lorentz transformation of the covariant for vector in the primed reference frame then it relates to a contravariant for vector in the unprimed frame so if I wanted to find a full expression relating the covariant for vector in the prime frame to the covariant for vector in the unprimed frame there is one more step that I need to do which is to lower the index on the right hand side so here I have a 2 mu nu lambda nu Sigma and then here I want to lower this index and so I would have Sigma let's call it gamma and X cama so this looks a little bit complicated but it's really not so bad I just have these three products of things nu is being summed over Sigma is being summed over gamma is being summed over on the right hand side I have the covariance for vector in the unprimed frame and on the left hand side I have the covariant for vector in the primed frame instead of writing out this triple product here because these are all constant matrices if you like and I know what they are I can just do that once and for all and just work it out and just by using the usual raising and lowering operators and I know that this whole thing is basically equivalent to this thing where the which the index which is up here in the index which has been lower is has now been swapped and you can just plug that in and work it out and you can find that's without too much effort that this object which is a four by four object is very similar to the previous one except without the minus signs on the 0 1 and 1 0 elements so it's not so mysterious so what I mean in the end just going through this algebra is this expression which then relates a covariance for vector in the prime frame to a covariance for vector in the unprimed frame so hopefully it's clear how to apply the Lorentz transformation for for vectors and also use the Einstein summation convention and raising and lowering operators from these examples so now that we have the event for vector X mu what other kind of four vectors can we generate well one thing that we can do straight away is to define a four gradient and the four gradient is defined as the derivative with respect to the components of a four vector and notice that the one the mouth written down here on the left hand side has a lower index mu and therefore this is a covariant four vector the definition of that is the derivative with respect to the contravariant event and the simple rule of thumb here for remembering this is that here we have an index which is lower and it's lower because we see on the right-hand side that's the index is in the denominator here of course there's a proper reason for that but I won't go into it here so just to spell out what this thing actually means therefore it would be 1 over C of D by DT and then D by DX D by dy D by if you said and again we can separate that out into time like component and space like components the vector the three-dimensional vector of the spatial derivatives is of course this gradient operator intonated business with this standard novel assembler symbol that we've been seeing many many times in this course on electromagnetism so this for gradient involves the derivatives with respect to time and also with respect to space and similarly we can define a contravariant for vector which in the expected notation would have the index upper this means the by the X with a lower index slightly confusing Klee and that means as you might expect that it has a minus sign on the time like component of the vector and it follows the same rule that we can raise and lower the index to go from the contravariance to the covariant for back soon so just to spell it out using the Einstein summation convention I can write this because here again I have the product of two things one with an upper index one with a lower index and that index is repeated and so I sum over nu and I'm left with mu which appears on the left hand side and likewise I can convert the other way using this expression so here we have introduced the the four gradient which is certainly a four vector if the event is a four vector because this four gradient just involves derivatives with respect to the components of the eventful vector so here we've generated a new four vector it's called the four gradient we can also do a Lorentz transformation of our four gradient and we do that in exactly the same way as we did for the Lorentz transformation of the event for vector namely let's take the contravariant Grady for gradient in the primed reference frame that's equal to this expression which involves the contravariant for radiant in the unprimed reference frame and again remember that the Einstein summation convention here employ implies doing the sum over the new variable here leaving the new variable alone on the left hand side and if I want to know the Lorentz transformation of the covariance for gradients which is given by the gradient in the prime reference frame with the lower index I can simply lower the index from the previous expression where here I'm using the Greek index Sigma to be summed over which can then be written out in this way and this relates the covariance for gradient in the primed reference frame to the contravariant for gradient in the unprimed frame but if I wanted to express the covariance for gradients in the prime frame to the covariance for vector in the unprimed frame then I have to again do a lowering of the index on the right hand side which yields this and precisely as we saw before from the Lorentz transformation if the event for vector I can absorb all of this stuff into the definition of Lorentz transformation matrix which looks like this one with the mu index lower and one with the new index upper so here we've discussed two different types of four vectors one is the full vector for the event and the other is the four vector for the gradient we've discussed raising and lowering the indices between contravariance in covariance backwards and forwards and also the idea of doing a Lorentz transformation between a different inertial reference frames the last very important thing I want to talk about before we get back to electromagnetism is the concept of a Lorentz scalar Lorentz scalars are objects which are the same in all inertial reference frames they're relativistic in and they can be obtained by doing a generalized inner product which is something like the usual vector dot product between one contravariance and one covariance for vector so let's see what this means let's do an example using the event for vector and we form a product which is the equivalent of a dot products for four usual three vectors one with the covariant for vector here and one with the contravariant for vector but notice importantly we have the same Greek index one lower and one upper and so this implies the Einstein summation convention over mu over all the four components from 0 to 3 and the only difference between the contravariant and covariant for vectors in special relativity is the minus sign on a time line component and so when we perform this product we will get minus C squared T squared the minus precisely coming from this difference between the contravariant and covariant for vectors plus x squared plus y squared plus Z squared and this object is kind of like the usual Pythagorean theorem except in four dimensions with the fourth dimension being time and entering with this funny minus sign let's do another example we could look at the contraction so-called of the gradients again this implies the Einstein summation convention I won't bother writing it out again it's always the same and it's exactly the same as before we can this extra minus sign on the timeline component and plus signs on the space-like components and in this case because this is an operator we see that this object is actually the sum of these second derivatives where the time line component has this minus 1 over c-squared part and this is exactly the definition of this stalin version the so-called box so we call this box squared where just for emphasis these three second derivatives with respect to the spatial components are of course the laplacian so this object is a Lorentz scalar this box squared operator is the same in all reference frames and indeed contracting to four vectors in this way always yields something that is a relativistic invariant something that is the same in all reference frames and that is the important and powerful point about these objects so let's prove that important property to do that let's consider this object in the primed reference frame now we can individually do a Lorentz transformation on both of these objects into the unprimed reference frame and we'd obtain this where the first factor here which is the covariant for vector in the primed reference frame is given by the Lorentz transformation of the contravariant for vector in the unprimed frame here and the second factor for the contravariant for vector in the primed reference frame is given by Lorentz transformation in terms of the contravariant for vector in the unprimed frame so just sort of tidying up the algebra of it here we can sort of pull out all of these transformation matrices like so and then we're left with two contravariant for vectors in the unprimed frame themselves and we can just go ahead and work out what this object is now leave that to you as an exercise but it can pretty easily be shown that when you work all of that out you actually get a tur being careful with the indices Sigma Lambda so all of this is a toe Sigma Lambda of X Sigma X lambda so what does this mean we have a repeated index of Sigma here one upper and one lower and we also have a repeated index of lambda one upper and one lower and so what that means that we sum over everything another way of putting it is that we can rapidly is that we can actually lower at the index of this X Sigma here and we'd obtain simply X lambda as a covariance for a vector and X lambda as a contravariant for vector and we still have one sum to perform and obviously what we've now shown is that's the contraction of these two four vectors in the primed frame is equal to the contraction of the four vectors in the unprimed frame here I have a sum over lambda and here I have a sum over mu but that doesn't matter that's just a dummy index over which I'm summing the important point is that the abdi the result of this summation is the same in the tube frames and here I actually made no special reference or use of the particular four vector in question I only utilized its transformation properties I did not say anything about the specific Lorentz transformation I simply encoded it in these very general matrices so there's a very general property we see that these kinds of contractions over to four vectors will be sum over their components one upper and one lower and these things are the same in all reference frames and obviously that's very important because if the principle of relativity is to be upheld then we know that the physics in all reference frames should be the same therefore we know for example that the Lagrangian of the system must contain only Lorentz scalars so more on that later so let me just summarize the result here for any four vector if I form this inner product in the primed frame it's equal to the analogous in a product of the same four vectors in the unprimed frame and this of course means that they're the same in all reference frames okay so with that background technical information in mind let's now explore the implications of all of this for electromagnetism which is of course the topic of this course and let's first consider the famous continuity equation describing local charge conservation in the previous lectures we derived the continuity equation from first principles we didn't have to use anything from electromagnetism we just used the simple fact that charge is indestructible and moves from one place to another in a continuous fashion and we were able to derive the following very important equation which relates the time derivative of the charge density Rho to the current density J through its divergence and we see that's the sum of these two terms D Rho by DT and the divergence of J must always be equal to zero to two charge conservation so a decrease in the charge in a particular region is due to the flow of the charge or a current through the walls of the region and this equation must be true in all reference frames since encodes a physical conservation law the concentration law cannot change depending on whether or not I am moving relative to the physical phenomena in question so with this equation we would argue because we derived it from first principles and it embodies a conservation law must be something that is relativistically invariant so the important consequence is that if this equation holds in all reference frames we must be able to cast it in the form of a for vector equation in which we have a contraction to produce a Lorentz scalar because we know Lorentz scalars other things that are invariant to changes in reference frame so let me write down the four vector equation version of the continuity equation this equation is precisely equivalent to this equation where we have a few things to note here first of all this object is manifestly a Lorentz scalar because we have two four vectors one of them a contravariant one of them covariant with lower and upper indices and we have the same index here which is being summed over due to the Einstein summation convention and we just where it lends to prove that those things do not depend on the reference frame here in this expression we have the covariant for gradient which we've just been talking about so this object is 1 over C D by DT in this in the 0 or timelike components of the four vector and then the gradient operator D by DX D by D wine D by DZ in the spatial components what is this J object this is a new four vector which I have introduced here and it must be equal to C times Rho the charge density in the zero or a time line component of the four vector and then it must just be equal to the current the standard current in the space like components if the two equations that I've written down here the continuity equation in its usual form and this four vector equation are to be equivalent so the seed the one over C here and the C cancels there we have D by DT of Rho that's the first term and then when we take the dot product of the gradient operator and J of course that's just the divergence so this equation with this definition is exactly to the continuity equation and furthermore because we have the Einstein summation convention over here and one of them is lower index and one of them is an upper index we know that this object must be relativistically invariant and that actually is sufficient to prove that this objects J that I've written down here is a bona fide genuine for vector and that object is referred to as the for current and in this way we've been able to obtain a new type of for vector we know that this J which comprises one component of the charge density and three components of the current density now when you put those together in this particular way noting also this dimension here with this multiplication by C this is a four vector and we call it the four current because we know that J is a four vector and we know what a four vector is the definition of a four vector is something that transforms under Lorentz transformation we now know how the four current will transform when we change reference frame in particular we'll be able to say what happens to the charge density and current density when we change reference frame so this is a big step forward which we essentially get for free using this machinery so what is the Lorentz transformation of the four currents we know that J mu in a primed reference frame will just be equal to this expression where on the right hand side here we have the contravariance four currents in the unprimed reference frame and as usual we do is contraction over the repeated index nu so let's do a quick example of this in use let's consider it for example the let's say the X component of the currents in the primed reference rain how is that related to the charge density and currents of the unprimed reference frame well we know that JX primes is equal to the one component J one of the of the four currents in the primed reference frame and that in turn is equal to this object which involves the Sun over knew from zero to three of one the one new component of the Lorentz transformation tensor lambda times J nu and we go back to our definition of the lambda which can basically thought of as a four by four matrix and we just plug it in and work it out and we find this where on the right hand side we have the X component of the currents in the unprimed reference frame but we also see the charge density of the unprimed reference frame appearing on the right hand side gamma as usual is the Lorentz factor and V here is the relative velocity between our two reference frames so this is not there's no physical particle moving with velocity V this is us moving with a part with a velocity V it's the difference in the velocity the relative velocity of the reference frames and we see that the current gets modified as we change reference frame and that's actually natural because as we move along relative to our system it appears that a static charge distribution is moving and that constant constitutes a current and indeed what you'd expect perhaps from a Galilean transformation would simply be JX minus V Rho but of course being relativity we pick up this extra factor of gamma the Lorentz factor so we see something non-trivial emerging in these transformation equations and we get all of this basically for free just using the machinery of special relativity applied in the context of electromagnetism the conclusion of this is that the components of the four currents get scrambled up and mixed together when we change our reference frame and that's not at all intuitive so now I want to introduce a different type of four vector the so called four potential so what I will do here is simply introduce as like an assumption or in Anne's adds a particular form for this four potential in terms of the scalar and vector potentials that we've been working with earlier in this course I will then just take this this assumption and then explore the consequences of that and then in the end that will justify the choice of this four potential error right down so without further ado what is this four potential let's introduce a contravariant four potential a mu where in the time like a component we have 1 over C of the scalar potential and then in the space like components will have the three components of the usual three component vector potential and at the moment let's just posit this as an assumption okay so let's explore the consequences of this first of all let's remind ourselves how the magnetic and electric fields are related to the usual components of the potential so from the last lecture we know that the magnetic field B is given by the curl of the usual vector potential whereas the electric field E is given by mate minus the gradient of the scalar potential minus the time derivative of the vector potential that's what we had from the last lecture and we also had the concept of a gauge transformation and again transformation is something where you can alter or change in some way the scalar potential and vector potential in a very specific way such that it leaves the electric and magnetic fields alone as we discussed in the last lecture this corresponds to a kind of local symmetry of the problem so let's remind ourselves of the gauge transformations we can write down let's say a new vector potential in terms of the old vector potential plus the gradient of some scalar field sigh and for the scalar potential I can write the new scalar potential in terms of the old scalar potential minus the time derivative of the same scalar field sign where just to emphasize this this sigh is a proper field that exists everywhere in space and at all times and we can choose anything we like for this scalar field sigh as long as it's twice differentiable with respect to the spatial coordinates and the time we then chose a specific gauge that so-called Lorentz gauge and in the Lorentz gauge by definition of the Lorentz gauge we choose that the divergence of the vector potential is exactly canceled by 1 over c-squared of the time derivative of the scalar potential that's a choice that we make it's a gauge fixing procedure but whatever gauge we choose we know that it's not going to affect the physical observables that's the whole point of this gauge transformation okay so why am I talking about these gauge transformations why is that important well if we look in the Lorentz gauge we know from the last lecture that's Maxwell's equations reduced to a pair of in homogeneous wave equations in terms of the potentials we have one equation for the vector potential in terms of the regular current and we have one equation for the scalar potential in terms of the regular charge density let me just rewrite that second equation for the charge density a little bit differently let me just divide through by C through the whole equation and then I will use the fact that mu naught epsilon naught is 1 over C squared and I'll get the following ok so just a very small modification there to the way in which I write the equation the equation itself is the same I just divided through by C now these equations can actually now be rewritten in an even simpler form by using this box squared notation this down version so let me convert these equations into that form where just as a reminder the Box squared means though the fashion minus 1 over c squared d squared by DT squared so it's just the operators that appear in our in homogeneous wave equations and using this box squared notation it just simplifies the equations a little bit so the nice thing is that in this representation I have these two in homogeneous wave equations for the potentials which basically embody all of Maxwell's equations in terms of the potentials in the Lorentz gauge and in here we have a on the right-hand side we have a component of the we have a components of the current J we also have seeds times row and this is reminiscent of course of our for current let me just write the four currents upstairs as a reminder the four currents J nu is equal to C Rho in the time line component and then the usual three current J in the space flight components and so we can actually write these two equations in a four vector form if we assume that the for potential takes this form so let's just write it down and see what it looks like we have a very simple equation in our four-vector language which actually embodies all of Maxwell's equations it tells us that box squared of the four potential is equal to minus mu naught times the four current this is a four vector equation meaning that it holds in all reference frames and this is really embodying all of Maxwell's equations in this four vector form okay but now we have to go back and re-examine the assumptions somewhat first of all if this expression is to be a four vector expression then anything that isn't manifestly a four vector for example the four potential here or a four current here must be a Lorentz scalar so mu naught for example must be the rent scalar well that is because it's just a number it's a universal constant of nature but what about this this is box squared this is something involving these differential operators and they involve space and time components and of course they will change individually when you do a Lorentz transformation from one reference frame to the other but miraculously of course this box squared is precisely this thing involving the four gradient and so this is a Lorentz scalar it is something that is the same in all reference frames so if I were to do a Lorentz transformation of Maxwell's equations here um this box squared operator would not change when we go to different reference frames so that's exactly what we want this object is indeed a Lorentz scalar that's one aspect the other aspect is I assumed in all of this that this a mu object which we called the four potential I assumed that this was a proper four vector a bona fide e4 vector now we can actually see that it must be a four vector precisely because of this equation in this equation we have Maxwell's equations reduced into this one equation and we know that the phenomena of electromagnetism described by Allah Maxwell's equations is relativistically invariant we know that when we change two different reference frames the the physical phenomena described by Maxwell's equations it doesn't change so the very effect that this expression describes Maxwell's equations and Maxwell's equations do hold in all reference frames actually implies that this object that features in this equation is itself a four vector why because everything else in the equation is either a four vector or a Lorentz scalar there's only one remaining hole in our argument which is that the derivation of this expression which is Maxwell's equations in full vector form appears to require the specific form of Maxwell's equations in described as in homogeneous wave equations over here and that required Lorentz gauge so one more thing we need is to show that Lorentz gauge written here actually is something that holds in all reference frames but of course we see that it does and that's because this Lorentz gauge condition can actually written itself as a four-vector equation it can be written in this way which is manifestly Lorentz scalar we can write it as the for gradient of the four potential if we write this out we simply recover this expression and of course that means that the concepts of Lu the rents gauge holds in all reference frames if I apply Lorentz gauge in one reference frame it will automatically apply in all other reference frames so what that means is that these Maxwell's equations in Lorentz gauge can apply in all reference frames and that's then everything we need so we did several things here first of all we introduced the four potential and we showed that indeed it is a four vector given that we know what Maxwell's equations are Lorentz invariant and we actually derived Maxwell's equations in for vector form so here it is again this is the four potential and we've argued that this is indeed a genuine four vector and what does it mean to be a four vector it means that one can do a Lorentz transformation it means it transforms under the symmetry operations of the Poincare group which include Lorentz transformations and so what that means is that we can take for example the four potential in a primed reference frame and related to the four potential in an unprepped referring where those two reference frames are related by a constant relative velocity so those are inertial reference frames as an example let's consider what happens to the scalar potential Phi in a primed reference frame and it's equal to C times a zero in the primed reference frame and so we can relate that now to the four potential in the unprimed reference frame and with the explicit expression for our Lorentz transformation tensor lambda and using the Einstein summation convention over this repeated Greek index new here one apart and one lower we obtain the following where in this expression the V here is the relative velocity between the two reference frames and so we see that the scalar potential in one reference frame actually against mixed up with the scalar and the vector potentials in a different reference frame again this is a highly non-trivial result and it also tells you that in a dynamical world where the dynamics of the system could even be provided by a moving observer we see that electricity and magnetism are not distinct things would get mixed in together so we have a beautiful description of the theory in terms of the potentials here generalizing to the four potential but of course the potentials are not observables the observables that we are interested in are the electric and magnetic fields so how do we incorporate these effects of special relativity and this formulation in terms of four vectors how do we write that in terms of the fields themselves before we discuss that very important question let's just pause a moment and consider this aside let's revisit the idea of the cross product this might seem totally irrelevant to the current discussion but actually we'll see that having a deeper mathematical understanding of the cross product will actually help us to understand what's going on in electrodynamics and at heart that's because the magnetic field is derived from the vector potential through the curl of the vector potential and the curl is of course nothing other than the cross product with the gradient operator consider for example angular momentum which is defined as R cross P R is the position vector P is the momentum vector and we'll ask ourselves is the angular momentum L actually a vector a question arises where did this idea of the cross product being related to funny determinants come from anyway when you first heard about the cross products back in school you probably thought thought that's a bit weird and arbitrary but there it is and then probably never gave it a second thought well maybe now it's time to give it a second thought it actually turns out that angular momentum has some funny properties that you might have not have considered before and that's actually related to the facts of this definition through the cross product let's consider the following example imagine we have a car that's driving towards us as pictured here on the left because it's driving towards us the wheels are rotating in a particular sense and by the right-hand rule we can associate with that rotation of the wheels a certain angular momentum and that's vector on this car here is this vector pointing to the right so I'm just going to draw this vector on suggestively on this car whereas in on the right hand side we have a car that's driving away from us now we get from the left hand car to the right hand car by doing a PI rotation we rotate the car around the axis and it's driving now in the opposite direction the wheels are still rotating in such a way that the cars driving forwards but now the angular momentum vectors are of course pointing in the opposite direction and that just arises because the wheels are rotating in the opposite direction as far as the observer is concerned so how does the angular momentum vector l which have drawn here as the red arrows transform under the rotation well we have this usual equation where this arm here is a rotation matrix and in this particular case we're considering a PI rotation in the XY plane let's say and this transforms the angular momentum vector L from pointing to the right - L primed where it's pointing to the left what I've considered here is a particular symmetry operation which is a simple rotation it's a so-called proper rotation let's consider a different transformation an improper rotation the so-called rotation reflection transformation in this example of improper rotation the so-called rotation reflection on the left hand side I see a car driving away which of course therefore has an angular momentum vector for the wheels pointing to the left hand side but on the right hand side I also have a car that's driving away but it's the mirror image of that car so we see for example this pattern on the car and these lights on the top these are the mirror image of each other but the car is still driving away so the rig wheels are still rotating in the same sense and therefore the angular momentum vector is still pointing to the left so even though we do a reflection of the image of the car itself the angular momentum vector does not get reflected and so we get a transformation equation like this where L Prime's is minus R of L where here I'm using a different transformation matrix R tilde which encapsulate this improper rotation and the defining feature actually of an improper rotation is one where the determinant of the transformation matrix is equal to minus one so the fact that the magnitude is either plus or minus one means that there's no scaling and we're just doing rotations and reflections but if it's minus one of the determinant then it means that we're doing a reflection whereas the determinants of a proper rotation is equal to plus 1 and therefore we can combine these two expressions into the following which holds for any kind of transformation let's consider a general transformation matrix M and to get the correct angular momentum we need not only to transform the vector M times L but we also need to multiply by the determinant of that matrix and this is what defines a so-called pseudo vector and so what we have in general is that a true vector a so called polar vector is one that has the usual transformation property that we'd expect whereas pseudo vectors or axial vectors carry with them this additional minus sign for improper rotations which is encoded in the determinant of the transformation matrix M so these are the two different kinds of vectors that we might not have considered before the fact that the angular momentum is a pseudo vector can be traced back to the definition in terms of the cross product R cross P and in fact it's totally general results that any vector that is arising from the cross products is actually a pseudo vector these differences between vectors and pseudo vectors and the symmetry properties of pseudo vectors can be understood in terms of a more rigorous and mathematical description of the cross product so let's just sketch that mathematical structure now let's introduce a tensor object with two indices I and J and I and J are elements of x y&z their components and on the right-hand side of this definition we see the position vector or components of the position vector RI or RJ and components of the momentum vector P I and P J and this object is defined to be antisymmetric which means that l IJ by definition is minus l j i and that also tells us right away of course that L III for any I XY and Z is equal to 0 so we have a three dimensional vector for the position R and we have a three dimensional vector for the momentum P and when we form the product of our MP there are many different things that we can do and one of them is simply to consider all of the the different combinations of RI and P J but what we're doing here is considering a specific way of doing that which is anti-symmetric so this combination here is anti symmetry seing the result and we're left over with a 3 by 3 tensor or matrix basically but because it's anti symmetric and because of these properties we actually only have three independent entries to that matrix expressing this tensor L as a 3 by 3 matrix we have something that looks like this in the component form we only have three independent elements which are this one this one and this one because these other elements are just - those things and the diagonal are equal to 0 because it's antisymmetric now one of the amazing things is that the components of this matrix is three independent components those components actually transform like components of a regular vector what I mean is that they transform under rotations and translations of our coordinate system in particular it can be shown that the 1/2 element of this matrix transforms like the Z component of a vector let's call that L said the 1/3 component of the matrix transforms rather strangely like - the Y component of a standard and the to three component transforms like the X component of a standard vector and of course we know that the diagonal entries are equal to zero and by the anti symmetry property this must be minus L said this must be plus ly and this term must be minus LX so here I'm just writing these components out in this particular way I'm just labeling those things as LX ly and LZ just because they transform like the x y&z components of a vector and so what we do is we just use take those three components and we just assemble them into a vector that we call L which has components LX ly and LZ but this is not a true vector we just put them into it of there are just three components we just put them into this three component vector form because they transform like components of vector however this thing is in truth a pseudo vector and we were able to show that a moment ago just pictorially by looking at this example of the cars and the rotations and the reflections of these things so this thing that I've denoted here as L is not actually a vector it's really a pseudo vector but when we write down the cross product of two vectors and we write it down LX ly and LZ in this way really they're just the independent entries of this L tensor matrix and one final thing if LX here is given by this combination of the components of the R and P vectors if minus L Y is given by this combination and if LZ is given by this combination and if we assemble LX ly Nell's ed into a vector that whole thing can be expressed in a very concise way as the determinant which is the thing that we know and love and that's just like a shorthand simplified way of getting the right answer but it should be understood that really what we're doing here is forming this tensor object and then just looking at the three components of the three independent components of this l tensor and just shoving them into a vector form but really it's a pseudo vector so you might be wondering why on earth we're talking about this the reason is that's in electromagnetism we have a definition of the magnetic field in terms of the cross product of the gradient operator with the vector potential a so from this we might conclude that the magnetic field is not actually a vector field but a pseudo vector field maybe it would be better to introduce something I will call B IJ which would follow as d by d R I of the component a J of the vector potential minus d by dr j of the component are a i of the vector potential i could then write out a three by three matrix let's say it's really a tensor but let's just write it in matrix form and it's not be too pedantic about it and in exactly the same way as in our previous work on the definition of the angular momentum we'd see that this could be written in terms of components of this form and it is those components that we usually assemble into this thing called B which is a vector or we now should probably recognize this as a pseudo vector as I now want to show you it turns out that's the generalization of this concept to four-dimensional space-time well then just the three dimensions that we have here gives rise to something called the electromagnetic field tensor which involves both the magnetic and the electric fields not just the x y&z but exe Y and E's and that is this total thing this electromagnetic field tensor which is the fundamental objects which features in the theory of electrodynamics and of course if we were to have a four by four matrix or a four by four tensor and it is anti-symmetric tensor then we know it before diagonal elements will be equal to zero and the off diagonal elements will be related by the anti symmetry property and that tells us that we have six independent entries in our four by four matrix and magically those six entries are the three components of the magnetic field and the three components of the electric field so let's see how all this works the central objects that were interested in in electromagnetism when we're considering the relativistic formulation is a similar kind of object it's called the electromagnetic field tensor and it's denoted F mu nu so it has two indices mu and nu their Greek indices so they run over there four components of space-time and it's defined in a similar kind of way to the discussion we had a moment ago except instead of having different combinations of three vectors we now have different combinations of four vectors we have the four gradient and before potential and it's an anti-symmetric tensor and so we just include these things with the indices the other way around and this guarantees the anti symmetry property but otherwise you see that it's kind of pretty similar to the definition of the magnetic field in terms of the curl of the vector potential so let's have a look at this interesting object and discuss some of the properties of it the first important property I want to discuss is the fact that this electromagnetic field tensor is so defined is gauge invariant it means that I can change the gauge for the electromagnetic potentials the scalar potential and the vector potential um and this object F mu nu will not change that means that this electromagnetic field tensor isn't observable because it does not depend on the gauge choice and we'll see actually that it's related to the electric and magnetic fields so first of all let's establish this fact that this object is gauge invariance first let's just remind ourselves at the four gradients written here D mu is 1 upon C D by DT in the time light component and then the usual gradient operator for DX dy DZ in the space light components whilst the covariance for potential is equal to minus the scalar potential over C in the time light component and then the usual x y&z components of the vector potential in the space light component of this four-vector so with these definitions in mind let's prove that the electromagnetic field tensor is gauge invariant to do that we'll perform a gauge transformation of our potentials let's consider a new vector potential which is related to the old vector potential through the gradient of some scalar field Tsai and some new scalar potential related to the old scalar potential with minus D by DT of sign being via the extra factor so we're going to just use these definitions to transform the electro magnetic field tensor so this is what we have where here we have the definition of the gauge transformed for potential which can also be expressed directly in terms of the old before potential and indeed this second term here which accounts for the gauge transformation can actually be written in terms of our four gradients so what happens when we substitute in this gauge transformation of the four potential into our definition of the electromagnetic field tensor well then obviously we obtain the following we get the contributions from the old four potentials and then we get an extra term and these two terms can be considered separately the two terms in these brackets this one is clearly the definition of our electromagnetic field tensor in the old system before we did the gauge transformation and this term is equal to zero and that's because in here we have these second mixed derivatives of the scalar field sign we have things like d squared Tsai by D X mu DX noon and in this second term which enters with the minus sign we have the same kind of things but the order of the derivatives is opposite so as long as we have smooth differentiable functions for our scalar fields I then of course those second mixed derivatives are equal and this term will exactly cancel so what we've managed to prove explicitly is under our gauge transformation the electromagnetic field tensor is invariant that means that we can change our gauge from whatever gauge to whatever we like and the electric magnetic field tensor will be unchanged thereby telling us that's the electromagnetic field tensor is an observable quantity that doesn't depend on the gauge so if it's an observable what observable is it so here we have the electric magnetic field tensor written out again of course what we have in this expression is time and space derivatives of the potentials and we know that the electric and magnetic fields are actually related to time and space derivatives of the potentials through these important equations and so what we can do is just go through element by element in the electromagnetic field tensor and compute these derivatives and then see what we get in terms of the electric and magnetic fields and when you do that I won't go through all of the steps because that's a bit laborious you can try that for yourself but when you do that you get the following result so this might look a little bit complicated but it actually has an understandable structure so if you imagine dividing this into blocks containing a magnetic field and electric field then we see this lower 3x3 block is exactly what we had previously for the magnetic field when we were discussing its in terms of being a pseudo vector so here this is the generalization of that treatment except now expanded to four dimensional space-time and using this definition of the four gradient and the four potential this is all just rolled up into one consistent picture so the real thing that we're dealing with in electromagnetism is this electromagnetic field tensor so we see that electricity and magnetism are all just dimensions of the same thing they were all just different different dimensions of this single object and because this is this tensor here is related to four vectors we can of course change our reference frame by doing a Lorentz transformation of those four vectors and we will see the electric and magnetic fields will get all scrambled up together so even just changing our inertial frame of reference will change our perspective of what it means to have electric and magnetic fields those things will get mixed up together and the precise way in which they do that is contained within the mathematical structure depicted here so let me just emphasize that this is an anti-symmetric tensor and I'll also mention that typically if we're going to depict it in this kind of matrix form here then the first in is referring to the Rose and the second index for columns so this is not a symmetric matrix is an anti-symmetric matrix the difference between swapping the rows and columns of course is just a minus sign so it's not that important in the end so let's have a look now it's a Lorentz transformation of the electromagnetic field tensor this is important because if I change my reference frame using the Lorentz transformation then I want to be able to say what happens to the electric and magnetic fields and that will be contained in the electromagnetic field tensor so because the electromagnetic field tensor consists of four vectors if we want to find out the Lorentz transformed electromagnetic field tensor we can just consider it from the definition in terms of four vectors and transform each of those individually what I mean from that concretely is the following let's say I want to know a particular component of the electromagnetic field tensor in the primed reference frame I just used the definition of the field tensor in terms of the four vectors the four vectors in question of course being the four gradient and the four potential arranged in this anti-symmetric way and each of those things have primes and now we just use the usual Lorentz transformation to convert these four vectors in the primed reference frame to the unprimed reference frame and then we'll get the following where as usual here the Einstein summation convention over the repeated Greek index Sigma one upper one lower lambda lambda and Sigma is implied we can tidy the algebra up a bit here and we get this just by factorizing out those Lorentz transformation tensors there and then we can recognize of course that this objects in the brackets here is nothing other than the electromagnetic field tensor in the unprimed frame and so finally we have the transformation expression for the field tensor and so we have this very important equation note also on the right-hand side we have a double summation over lambda and also over Sigma so when we do a normal Lorentz transformation of a four vector we have one factor of the Lorentz transformation tensor there when we have the transformation of a tensor itself we have to have as many factors of the Lorentz transformation tensor as we have indices so in this case the electromagnetic field tensor has two indices mu and nu so we have two factors of this thing and as with the Lorentz transformation of a simple four vector we can actually view this as a matrix product so we imagine that's the electromagnetic field tensor is a four by four matrix and these Lorentz transformation tensors are 4 by 4 matrices and then we'd get something like this where here the all of these objects are matrices and in particular we have the F electromagnetic field tensor in the primed frame on the left hand side whereas we have the electromagnetic field tensor F in the unprimed frame on the right hand side but just taking into account the proper summation over these various indices the matrix products we form is actually lambda f lambda so we've got to get the order of the matrix products right so if we write it all out we get something like this so this is a triple matrix product we see here on the left hand side of the equation the field tensor in the prime frame sandwiched between the two lambda matrices here is the field tensor in the unprimed frame and this is the full expression if you were to do this fold matrix products you would find a relation between all of the elements of F Prime's - all of the elements of F of course you don't have to do it as a big matrix products you can just use this expression directly to find any given particular element you're interested in so let's do an example of that so as an example let's have a look at the y component of the magnetic field in the primed frame how does that relate electric and magnetic fields in the unprimed frame the primed and unprimed frames are of course just related by a relative velocity along the X direction which we'll call V so what we find is that this V Y Prime's of course that is equal to the 3 one element of the electromagnetic field tensor in the prime frame remember that the first index in the field tensor is the row of this matrix and the second index is the column so if you remember the expression for the field tensor then the 3 1 elements is indeed the Y component of the magnetic field and using the Einstein summation convention and just going through all of that we can show that this is equal to gamma of 3 1 + beta gamma of f30 and they're the only finite terms so I'll leave it to you to go through the matrix products of those things and if we now convert or translate back the meaning of F 3 1 and F 3 0 we'll find that this is actually equal to gamma lots of be y plus V upon C squared of easy so of course this is rather interesting what it tells us is that the magnetic field along the y direction in the primed frame is related to the magnetic field along the Y direction in the unprimed frame plus there's a piece coming from the electric field as well so this is indeed quite interesting note also that if our velocities v were very small compared to the speed of light then this would be a very small term that we wouldn't maybe notice also the Lorentz factor gamma would be approaching 1 so in the non relativistic limit of the velocity much less than the speed of light note here that the velocity is in particular the relative velocity between the reference frames nothing we're not talking about anything moving specifically at this frame this speed other than the observer in the non relativistic limit we see that B Y is approximately equal to B Y in the primed reference frame but when we're moving quickly and we take into account the effects of special relativity we see that the electric and magnetic fields get scrambled up when we change reference frame according to this expression let's have a look at the electric field let's have a look at the electric field along the X direction in the primed reference frame looking at our electromagnetic field tensor that's equal to C times the f1 0 component of the field tensor in the prime frame and if we go through the calculation again and we expand out that's triple matrix product and we tidy up a bit of algebra we find this is equal to C times F 1 0 in the unprimed frame which is of course just the electric field along the X direction so this is again some more interesting it tells us that if we have a relative velocity between our two inertia reference frames along the X direction then the electric field along the X direction is not changed so what else can we learn from this transformation so let's take a look that's the electric field along the z direction in the primed frame which is just see lots of F three zero in the primed frame so we're looking at the electric field along the Z direction in a reference frame that's moving relative to ours with a relative velocity along the X direction so what does the Lorentz transformation give us for this well we get one piece that comes from basically the electric field along the Zed direction in the unprimed frame and then one piece that's interestingly comes from the Y component of the magnetic field in the unprimed frame and there's an overall factor of gamma in here which is Lorentz factor so this is also a rather interesting result and we're gonna explore some of the consequences of this now let's consider a charged particle Q moving along the x axis with a velocity V subjects to a uniform magnetic field that's acting along the Y direction and let's assume here that there's no electric field present what is the force on the particle well we use the Lorentz force law that tells us the force on the particle is equal to the charge of the particle times the electric field but there's no electric field here plus the velocity of the particle cross the magnetic field so this is a cross-product between V and B here and the magnetic field is only acting along the Y direction and what this tells you is that the force only acts along the z direction so the force here is equal to a force along Z times the unit vector along Zed and specifically the force along said is Q times V times the magnetic field along the Y Direction B Y let's consider now changing the reference frame and seeing what happens to the force in particular let's choose a reference frame which is Co moving with the particle so our reference frame will jog along with the particle along the x axis at velocity V so in our reference frame the particle is not moving so the force in the primed reference frame is the Q the charge on the particle which is conserved that's the same times the electric field but here the velocity is now equal to zero in the primed reference frame and so that's it we just have that the force in the primed reference frame must be the charge on the particle times the electric field in the primed reference frame now if the principle of relativity is to hold that the physics is the same in different reference frames then we have a bit of a puzzle because in the unprimed frame and prime frame we have a force acting along the Z direction which has this magnitude and in the primed reference frame we have Q times the electric field and there's no electric field in the original system but of course now we understand what's going on because if we were to work out the electric field in the primed reference frame we would see it's actually related as we see from this expression to the magnetic field in the unprimed frame and in particular if you to work out all the components of the electric field you'd find that it was only the electric field along the z direction that was finite due to the magnetic field acting in the original frame the unprimed frame along the Y direction there's no electric field acting in the original primed system and so we can actually write down what the force is because we know what the electric field in the primed frame is and the answer is it's gamma V well we still have the charge Q in here sorry times V Y and so this is very interesting because it tells us that the force in the unprimed frame is almost exactly the same as the force in the primed frame but the mechanism is it very very different in the unprimed frame it's due to the magnetic part of the Lorentz force whereas in the primed frame it's due to the electric part of the Lorentz force but the magnitude q vb y q vb y is exactly the same in the in both cases except for this thorny extra factor of the Lorentz factor here so this requires a little bit of further explanation in the nonrelativistic limit these forces would be exactly the same because the Lorentz factor would be close to 1 and of course that's what we observe in everyday life we would observe that if I jog along with a particle of course we see the same force and doesn't matter that I'm moving along the particle sees the same force so in the nonrelativistic limit that we used to this makes eminent good sense however if things are moving very quickly if the relative velocity between the two reference frames is very large or if the speed of the particle itself feed is very large then we seem to run into a problem because the two forces are not the same and we would expect from the principle of relativity that the physics of the of the particle should be the same in to inertial reference frames these forces don't appear to be the same the resolution here is just to go back to our classical mechanics and remember that of course it is the four force that should be invariance is the fourth force that it should be the relativistic quantity we can and these are just the regular three forces so the solution to the quandary is to basically to say okay so let's what would happen to this particle it feels a force it's therefore accelerates according to Newton's second law and we'd see a certain trajectory it's the trajectory really that is the thing that is the observable it's what happens to the particle as a consequence of the force that we observe and in relativity if we go to a different reference frame then apparently we have a slightly different force but the trajectory will remain exactly the same and the reason for that is that the force of course is really the by Newton's second law the rate of change of the momentum and it's the momentum the change in the momentum that we actually see in the trajectory but the rate of change of momentum depends on of course taking a time derivative and we have time dilation as we go from one reference frame to the other so there's a little bit of a subtlety here it turns out when calculating the trajectory in the two reference frames but when you work through it it turns out the trajectory is identical to you to these two different forces in the primed and unprimed reference frame um as I said it's easiest to see in the nonrelativistic limit where even in that case you would see that the forces are basically the same even though the mechanism for the forces is very different okay so let's now take a step back and revisit Maxwell's equations but we'll do it now in terms of the for vector formulation and we already saw how Maxwell's equations could be written in full vector form in terms of the potentials but now let's see how it looks in terms of the electromagnetic fields themselves which were of course contains in this electromagnetic field tensor F so the result is that we can write Maxwell's equations in this nice four vector form and this equation being a four-vector equation of course is something that applies in all reference frames we can do a Lorentz transformation of the individual four vectors that feature in this expression and we've just explored how we can do a Lorentz transformation of the electromagnetic field tensor so this is basically Maxwell's equations actually it contains all the information in Maxwell's equations when coupled with the Bianchi identity which we considered in one of the assignments so the first thing to do of course is to verify that this equation that I've pulled out of the Hat here is in fact equivalent to Maxwell's equations in the for potential formulation that we derived rigorously earlier on in this lecture so what we'll do is we'll just plug into this a left-hand side here the definition of the electromagnetic field tensor and we obtain the following in terms of the four potential a new and we can play the usual trick of reordering these derivatives so let's just unpack this a little bit and write it in the following way so in the second of these terms here we see a contraction in the usual way of this gradient operator and this of course gives us the down version box squared what about this term well this is actually the Lorentz gauge condition so in the Lorentz gauge this is equal to zero and previously we worked out Maxwell's equations in terms of the four potential in the Lorentz gauge so in the Lorentz gauge which we'll stick to for now of course then this first term is equal to zero so in the end the left hand side is just equal to minus box squared of a and so we can summarize by writing box squared of a mu is equal to minus you know what of the four current J mu and this is exactly what we had before in terms of the potentials so this is indeed exactly equivalent to our previous result but now we're writing Maxwell's equations in terms of the electromagnetic field tensor which might be nicer because this is directly now in terms of the electric and magnetic fields which in the end are the things that we want so these equations really describing the physics contained in Maxwell's equations actually I should be more specific the information contained in this full vector equation is the information that's contained in Gauss's law and as the Maxwell ampere lon I mentioned earlier somewhat obliquely that Maxwell's equations were fully described this for vector equation plus the Bianchi identity so let me just remind you of their Bianchi identity actually it turns out the Bianchi identity contains the information to the other two of Maxwell's equations so the Bianchi identity is as follows it's these various derivatives of the electromagnetic field tensor which involve these cyclic permutations and the Bianchi identity says that this object is equal to zero this is very easy to prove you can just go through it element by element and show that because the second mixed derivatives are equal for these smooth well behaved functions that this object goes to zero so that's easily proved by just putting in this definition of the electromagnetic field tensor so this Bianca identity just as an aside tells us about the information contains in the other two of Maxwell's equations the the one that's the divergence of the magnetic field is equal to 0 and Faraday's law one final thing I want to show you with regards to the electromagnetic field tensor is what happens if we contract the indices what I mean specifically is let's consider this in a product this is a Lorentz scalar object because we have the contraction of thee of the indices in the Einstein summation convention over this repeated index mu and over the repeated index nu so we sum over both of those things and using the same logic as before this must be something that is invariant dependent no matter what our reference frame so if we change our reference frame we still have the same value of this object so this is a Lorentz scalar it's a relativistic in Merrion if you work through the maths of this and just work out all of those components summed over what do we get and leave this for you to confirm you get B squared minus 1 upon C squared of the electric field strength squared so this object is the same in all reference frames it's a Lorentz scalar and we know that because it was obtained by this usual process of contracting these indices here so this is rather interesting because it tells us that the the magnetic field strength squared minus 1 upon C squared the electric field strength squared is something that doesn't depend on our reference frame and actually this reminds us a little bit of the expression for the energy density in the fields it's not quite the same but it's related so this week all's you sub am for electromagnetic internal energy and this was one half of epsilon naught E squared plus 1 upon mu naught of B squared and there is obviously some kind of connection between these two expressions which you can see a little better if I multiply this expression by mu naught and then remember that epsilon naught mu naught is 1 upon C squared and now you can see that these expressions are almost exactly the same except that there's a plus sign in the internal energy whereas this is a minus sign so we'll return to this in a moment when discussing the lagrangian for electromagnetism but for now suffice to say that when we contract the electromagnetic field tensor with itself we get something by construction that is a Lorentz scalar it's the same in all reference frames and it's the difference between the electric and magnetic fields this internal energy is not that object because it differs with from the minus signs the plus sign so we can infer that this actually does change with our reference frame as we go from one reference frames on the other it looks like the energy stored in the fields will change and that's correct the energy is not itself a Lorentz scalar it's a component of a four vector that changes with reference frame so what is the four vector for which the energy is a component well of course that's the four momentum that we discussed in the context of classical mechanics and relativity for now let's simply leave it at that the fact that the energy stored in the electromagnetic fields is frame dependent but that this object is frame independent and they only differ from each other by this minus sign the last topic that I want to address in this lecture and indeed this will be the last topic in the entire course is a Lagrangian field theory for electromagnetism we talked about many things in this course Maxwell's equations and latterly the reformulation of this in terms of relativistic for vectors we talked about Lorentz invariants we talked about gauge invariance we talked about ideas of causality and locality so once you draw this all together finally in this final topic and discuss a Lagrangian formulation of this problem and will write down allogram for electrodynamics this Lagrangian can then be used with the principle of least action to finds the equations of motion for the system they tell us how they feels interact with each other and how they interact with charged particles and current will finds that when we plug in our electromagnetic Lagrangian into the Euler Lagrange equations of motion we will actually generate Maxwell's equations but before we do that I want to discuss the general principles for constructing a field theory these kind of principles that I will talk about really apply to any kind of theory within physics all known theories that we have satisfy these principles and so these are really black like guiding lights for us to construct our field theory in terms of Lagrangian for electromagnetism in the end when we write down our Lagrangian we'll be able to check that it's correct by plugging it in to the Euler Lagrange equations of motion and generating back Maxwell's equations that's the ultimate check here because we believe that Maxwell's accrete equations are correct and described nature so what are these principles well there's three important principles that I want to talk about they are locality Lorentz invariants in gauge invariance so let's take each of those one by one so the first of these is the important concept of locality which Einstein encapsulated very nicely when he said there should be no spooky action at a distance so the idea of locality is that dynamical information about how a system evolves does not travel through space instantaneously there is a minimum time it takes for the information to travel from a distant point to us and that minimum time depends on the distance between us and that phenomena that minimum time is related to the distance and of course the speed and that minimum time is simply Delta X over C that's the minimum possible time because the maximum possible speed is the speed of light so in in an infinitesimal time step DT only information about the neighboring points DX can possibly affect the physics and this is basically connected to the idea of causality if we have a causal relationship between things then we require that the information cannot just travel instantaneously so this concept of locality implies that the equations governing the dynamics of a system are differential equations the dynamical evolution at a certain point in space depends on what's going on at that point and in the neighboring points DX away from that point in a time period DT so the equations therefore local equations that apply point wise in space and these are differential equations involving the values of the fields at those points and the gradients of the fields at those points in the rates of change of the fields at that point so these are local equations and their differential equations in the end this results in our formulation in terms of the action principle and the principle of least action of course yields the Euler Lagrange equations of motion for the dynamics of the system in terms of a Lagrangian and those equations are differential equations the second very important and very general principle is that of Lorentz invariants the equations governing the dynamics of a system should not depend on the inertial reference frame of the observer and this is the principle of relativity we have the same physics in all inertial reference frames so and we switch to a different reference frame the equation equations governing our system and the physics of our system should not change and we know how to change the inertial reference frame it's by doing Lorentz transformation and therefore the equations governing the dynamics of our system should be Lorentz in Varian's they should be Lorentz scalar quantities which do not depend on the reference frame if it were otherwise and different observers would calculate different physics for the same situation and of course that violates the principle of relativity this means that in terms of our Lagrangian we must use only Lorentz scalars these should be obtained by suitable contraction of the indices of various four vectors and tensors and so on for example our Lagrangian should only contain terms like this or terms like this because we know that these are Lorentz scalars and on the last slide we also looked at a contraction of indices for tensor objects such as this so all of these things are Lorentz scalars and these are things that don't depends on the reference frame so are good candidates for things from which we can construct our electromagnetic Lagrangian so I'm not saying that any of these things in particular have to feature but the kinds of things that go into our electromagnetic Lagrangian can only be these kinds of objects actually as we'll see this kind of thing here does feature that's coming up so if the Lagrangian is a Lorentz scalar what that means is that when we plug it into the principle of least action then all observers in different inertial reference frames will calculate the same minimum action for the same physical situation and that will have the same physics and therefore using the principle of least action with Lorentz invariance Lagrangian tells us automatically that the principle of relativity will be respected the final guiding principle that I want to talk about is gauge invariance this is a little bit more of a subtlety so the gauge symmetry of the underlying fundamental potentials or fields involved in the theory which for electromagnetism there are the scalar and vector potentials so these fundamental potentials have a certain local internal symmetry which we call a gauge symmetry and the facts of this symmetry implies that our Lagrangian must be gauge invariant in particular we know that physical properties observables cannot depend on the gauge choice and so the Lagrangian is basically a physical observable because through it we determine our equations of motion so the Lagrangian itself must be gauge invariant that will then guarantee that the equations of motion our gauge invariant they don't depend on our gauge choice at this therefore respects this local gauge symmetry of the fundamental potentials so the idea is that there's a wider or broader kind of symmetry in these problems they are not simply global symmetries where we can rotate or translate our coordinate system and of course we talked about Lorentz symmetry as well doing Lorentz boosts along different directions there's actually an even bigger symmetry which is a local symmetry an internal symmetry between the components of the potentials which can be applied from point to point in space and also changing in time and this wider symmetry is the gauge symmetry it turns out that all known theories are gauge theories in physics there are ones that have these hidden local symmetries and we want to incorporate that into our definition of the theory and our formulation of the Lagrangian we insist that it must be that the Lagrangian itself must be gauge invariant and again that restricts the kind of things that can appear in our Lagrangian so to satisfy these three principles of locality Lorentz invariants and gauge invariants we actually have quite a few restrictions on the kind of things can write down so what is the Lagrangian of classical electrodynamics well it starts with the action principle let's write down the action the classical action the action denoted s is equal to the integral of the lagrangian DT but here we're going to be talking about field theories electro dynamics is of course a field theory which is a continuum theory it exists everywhere in space so we're not just interested in the total lagrangian it makes more sense to discuss the Lagrange density which is denoted with this curly now where we integrate over space and also time here so this curly L here is the ground which density or the grandjean density if you like and we see that this is an integral over the whole of space-time and we can denote that in a slightly more compact notation by writing the the integral over D to the 4x and this is supposed to mean the four components of our four-dimensional space-time now we use one of the most powerful concepts in all the physics the principle of least action it tells us that the correct classical physics is obtained when the action is minimized and this condition of the action being minimized for the correct classical physics can be written mathematically simply is d s equals zero so it's a stationary point of course you could also have a maximum action that's satisfied this but in most situations there is no maximum action the action just can continue to increase increase and increase as you get further away from the truth physics but the true physics will be the minimum in the action which is unique so some lengthy algebra is then required to convert this D s equals zero condition into the equations of motion for the system in terms of the lagrangian itself that is straightforward but a bit laborious we did that already in the classical mechanics and relativity module however it's a little bit more complicated when you have a full field theory with this integral here over the difference spatial dimensions as well as time and also when you have different fields and different components of the fields so from the relativistic Lagrangian density as this condition we obtain the Euler Lagrange equations of motion when we do that so let me just write down that expression in terms of the fundamental potentials that feature in the theory of electrodynamics and this is what we obtain it's straightforward but a little laborious so I won't prove this it's in all of the textbooks and I've written this specifically in the index notation of for vectors where in here we see X mu of course being the usual four vector for the event the curly L is the Lagrangian density and then also in here we have the four potential a new so I'll just put here as a reminder that's X mu the event is C T and the position R whereas the four potential a mu is the scalar potential Phi upon C and the vector potential for future reference will also just write down the for current which is of course C Rho the charge density and J the usual three current density and as usual in this expression the Einstein summation convention is assumed so our general philosophy is that we're going to come up with some kind of Lagrangian that when we plug into this equation we generates as our equations of motion Maxwell's equations of course the Maxwell's equations that we come up with will be something in relativistic for vector form and we've just worked that out and the equation that we're seeking it when all said and done is going to be this one this is Maxwell's equations written in terms of the electromagnetic fields tensor and the four currents J but of course when we have the Lagrangian we can do far more with it so this will give us greater power to do problems in classical physics in terms of electrodynamics so what is the Lagrangian that we seek well first of all we're going to decompose it into three pieces the first piece is the Lagrangian for the free sources and by sources here I mean charged particles or currents so these are the free particles and currents not in any potential not in any fields just isolated and of course that's basically just the kinetic energy of the sources secondly we have the Lagrange density for the fields themselves again in isolation without any sources present just the lagrangian for the fields in isolation and finally we have an important term which is of course the coupling between the sources and the fields so the branch density here will describe the coupling between the charge density and the current density and the electric and magnetic fields so we have the two things in isolation and the coupling between them so if we were to consider a single particle queue of some mass M moving with some velocity V then of course we know what Lagrangian for that system is for the free particle which just contains the kinetic energy and we know that Lagrangian is minus MC squared multiplied by the square root of 1 minus V squared over C squared this is the relativistic Lagrangian for a free particle and of course we can generalize that to an arbitrary distribution of moving particles or currents and I actually don't want to say anything more about the Lagrangian for the sources at this point they are just the usual relativistic lagrangian's for those particles what I want to focus on is Lagrangian for the fields and the coupling okay so let's now focus on Lagrange density for the fields well in the previous slides I was discussing how we are guided by certain principles in constructing our Lagrangian in particular our Lagrangian should be something that is Lorentz invariant and gauge invariant and that actually restricts the kind of things we can write down let me write down one such object and then I'll show you why this is the right thing to write down imagine that our Lagrange density is given by the contraction of the electromagnetic field tensor with itself this has the two nice properties that we're looking for its Lorentz invariants and it's gauge invariants and we showed that in the earlier part of this lecture its Lorentz invariant because we're doing the contraction of these tensors using the usual Einstein summation over repeated index mu here and repeated index nu and by construction this gives something as we proved that is Lorentz invariant but also this object is gauge invariant and we also proved that so this object has the right properties that we're looking for but in fact we can also put in here any constant let's call it alpha in for any constant there and that Lagrangian of course would still be Lorentz invariant and gauge invariant so it's we have something that's a plausible Lagrangian but we've not defined yet what this what this constant alpha is and we can express this Lagrange density in terms of the electric and magnetic fields and this is what we obtained we found this in the previous part of this lecture good now one of the motivating reasons for using this this object in our Lagrangian apart from it having the nice properties of Lorentz and gauge invariance is the relation between the Lagrangian the Hamiltonian and the energy and so if we take this Lagrangian as written let's perform a Legendre transformation to convert from the Lagrange density to the Hamiltonian density I won't go through this it's a little bit technical something you can try for yourselves if you want but converting from this Lagrangian we obtain the following Hamiltonian and where we have this overall minus sign here but basically it's the sum of those two terms rather than in the Lagrangian where we have the difference of those two terms and this might of course remind you or be reminiscent in in usual classical mechanics when the Lagrangian is the kinetic energy minus the potential energy and Hamiltonian is the kinetic energy plus the potential energy so so we have some things that sir looks somewhat familiar here especially more so because in the previous parts of this next lecture course we actually worked out what's the energy density of the fields was and we had this result and from this expression you can see that this is this energy density of the electric and magnetic fields is very close to this Hamiltonian density and we just need to fix this value of the constant alpha sitting in there and of course we know that the Hamiltonian density should be the energy density and here we're looking at the part of the Hamiltonian density that relates to the fields so we fix the value of alpha so that these things become equal and we know that alpha therefore must be minus 1 upon 4 mu naught and that guarantees that the Hamiltonian density is indeed the energy density of electromagnetic fields so that all works out very nicely and of course this pins down what's our lagrangian for the fields should be it has to be minus 1 upon 4 mu naught of the contraction of the electromagnetic field tensor in this way this has all the right properties its Lorentz invariant gauge invariants and then when we convert it into the Hamiltonian it has it corresponds to the energy of the fields in the right way so we can be pretty confident this is the right thing and it turns out it is the right thing ok so let's try to work out the final part of our Lagrangian the coupling between the electromagnetic fields and the sources which are the charge distributions and the current distributions contains in the four current so we have to have some kind of coupling between the four current and the four potential so far we've just considered the fields in isolation and their sources in isolation of course Maxwell's equations describe the coupling between those things how they feed back into each other and that's what we want to write down here what I'm going to do here is kind of make an educated guess about what this is going to be and in so doing we'll write something that has all the right properties down and then we can plug it in to the Euler Lagrange equations of motion and see what we get under the find that it does indeed reproduce Maxwell's equations so I'm going to motivate this with the following very simple example is a simple limit that we know must be true it's that of a single static charge particle Q let's call it in an electrostatic field so I'm not talking about any currents and not talk about any magnetic fields I'm not talking about anything moving this is a totally static situation in that case I know what the Lagrangian has to be so in the electrostatic limit I know the Lagrangian takes this very specific form this first term here in the nonrelativistic limit would simply be the kinetic energy of a particle and in the relativistic limit we know what the lagrangian for a free particle is with no potential so by the word free here I mean it's the gran Gian without the potential and in special relativity we know the results for this for a single point particle and there it is this second term here is the contribution from the potential and the potential in electrostatics is the potential energy here is minus Q times the electrostatic scalar potential which I'll call Phi here evaluated that's the position of the point particle so this is the potential and what we can see from the Lagrangian is usually you have kinetic energy minus potential energy and so therefore appearing in Lagrangian we get the free part which is basically the kinetic plus Q times the scalar potential so we can kind of think about this term as being like kinetic and this term being the potential and we have L is t minus V essentially so this is the result that we expect and we know is correct in the electrostatic limit and for a single point charge so what is the action for this system the action s is of course the integral DT of the Lagrangian not writing down the Lagrange density here just a Lagrangian and let's just call this first piece here s free it's the free part of the action and then we have this extra piece which is coming from the interaction between the charge and the fields which is done here through the scalar potential and Q of course is just the charge which is a constant so let's call all of this term the coupling part of the action okay so that's definitely the correct results in the electrostatic limit how are we going to obtain the action for the full electro dynamical situation well we have our guiding principles we have our guiding principles of locality of Lorentz invariants and of gauge invariants and this action that I've written down here for especially this term involving the coupling is manifestly not Lorentz invariant there's no contraction here of four vectors if I change my reference frame this is definitely not going to be constant and the reason as we already explored in this very lecture is that if I change my reference frame we know that the components of the four potential get mixed up so here I'm just involving one component of the four potential namely the scalar potential Phi and if I change my reference frame this is going to change and I'm gonna get a different value of the action so this is manifestly not Lorentz invariant and we need to fix that if we're going to write down a full electric dynamical action another thing that I want to address is the fact that here we're considering just a single point charge whereas in the end we want a continuum field theory and so I will also want to replace the charge by a charge density over some infinitesimal volume element detail so the amount of charge in a region D tau is ro D tau where Rho is the usual charge density and so if we make this substitution we can actually look at a Lagrange density so I will just write down that the coupling part of the action will be the integral of Rho times the scalar potential fine and then integrated over the volume T tau which I'll write DX dy DZ and then integrated over time as before in the action expression there DT and of course what we see here is this integral over the four-dimensional space-time as before and therefore we can sort of extract the Lagrange density this is Lagrange density now in the electrostatic limit has just Rho times the scalar potential Phi so what we now need to do in the dynamical theory is to come up with something that is Lorentz invariant this objects we've written down here is valid in the static limit but it's not Lorentz invariant so when we're going to the dynamical case we need to think of something that's reduces to this in the static limit but that has full Lorentz invariants so in electrodynamics we required Lagrange density corresponding to the coupling to be little Lorentz invariant and this thing that we've written down so far in the static limit is manifestly not Lorentz invariant so let me just write down a suggestion for this Lagrange density corresponding to the coupling I will suppose that's so the grams in is minus the contraction of the four potential with the four current so this is definitely a Lorentz invariant object because we have two four vectors and we're contracting the indices here so this is definitely Lorentz invariant and if I just write out what you get from this we have the following and in the first term here you see magically that we get five times row as we want but now for this to be properly Lorentz invariant we have to subtract off a dot J where a is the regular three vector potential and J is the regular three current so we've dreamt up here something that magically does the trick it is Lorentz invariant and also in the static limit we know that it has the right form it has the scalar potential multiplied by the charge density Rho and so finally we've arrived at the Lagrange density for classical electrodynamics it involves three pieces the first is just the the free source Lagrangian so this is basically just the kinetic energies of the source is something trivial that we know from classical mechanics the second piece tells us about the Lagrangian of the electromagnetic fields themselves and the third term tells us about the coupling between the sources and the electromagnetic fields which takes this incredibly simple form it's just the contraction of the four potential with the four current so this actually turns out to be the correct Lagrangian and we sort of guessed the answer in various parts here using analogies between different things um however we can now demonstrate that this is the correct result by simply plugging it into the oil lagrange equations of motion and seeing what we get and the answer is that we get Maxwell's equations back so we have this wonderfully consistent picture in the end and this Lagrangian that I've written down here embodies the covariant formulation of electrodynamics which involves these fundamental relativistic objects the electromagnetic field tensor F here the four potential a and the four current J okay so that's actually basically everything that I wanted to say about the relativistic formulation of classical electrodynamics and the Lagrangian formulation this also brings to an end to this course on electromagnetism I hope you found it interesting and insightful and of course there's many more things that we could say on this topic it's a huge rich and very beautiful topic but hopefully this has been a good introduction

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Length: 162min 11sec (9731 seconds)

Published: Fri Apr 24 2020