David Albert on the (non)-time reversal invariance of classical electromagnetic theory

0:00 I thought it was all I saw, and I'll leave you there soon, to see the whole thing. Yeah, it's from your house that goes. On the very day, that's probably the whole town. He was a brilliant student who was supposed to give a talk at Rob's conference. He became the father of Soren, an eight-pound and I believe 15-ounce suit. And mother of a great mother. Well, now that Jeff has announced the joyful news about the new baby coming into the world, I have a sad community to announce that David Malamud will usher out of the world. David Malamud will usher out of the world. David Alberts, I should not find the next character of Collective Hand. I, too, want to thank Jeff and Robin for putting together this wonderful opportunity for a great good use to Robin. So many collaborators and friends and students and admirers have come to such a reason. Can I be heard in the back? I prefer not to use the microphone. I want to talk about a certain claim that David Albert makes in the first chapter of his book called Time and Chance that was published three years ago. Most of the book deals with issues and the foundations of statistical mechanics and connections or possible connections between statistical mechanics and questions involving the direction of thought. But before all that, in the first chapter, he discusses a topic

2:30 that's in large degree separate from the others in terms of time reversal invariance or what one should mean by the time reversal invariance of physical theory. discusses a few examples. He agrees that Newtonian particle mechanics is time reversal invariant, but he then surprisingly announces that classical electromagnetic theory and many other canonical examples are actually not time reversal invariant. He's certainly aware that in saying that he's going against the received view. In fact, as he puts it, all the books say that classical universal invariants, but he thinks they're mistaken, because they trade on an incorrect and unmotivated account of how the time reversal operator properly understood axon magnetic fields. It's a wildly theoretical claim. John Norton said it was like getting warmed up by announcing that the sky is green. And what I'm going to do today is tell you why all things considered, on balance, I think it's blue. I feel a little sheepish about the talk, because there's no chance in the world that I'll convince David Albert, and I'm not sure there's anybody else in the world who needs to be. So I don't know if the talk has an audience, but I will try to say a few things along the way dependently of David Albert. This is the first time I'm talking about this material. I'll stick with the plan I have, but maybe in the future I will de-emphasize David Albert and put the emphasis more on just the discussion of what one can mean by time reversal invariant. I'll be making a few remarks. I guess the most important one I can describe in advance. I'll discuss a way of thinking about time-reversal invariance that works well within the framework of general relativity. The usual way one finds it discussed, even in mathematically sophisticated books, takes for granted that the background space-time structure is Minkowski's space-time, and in particular that the space-time structure exhibits certain symmetries or isometries.

5:00 particular temporal reflections of a certain kind, or if one's talking about parity reversal, spatial reflections, but certainly in general relativity, in general, one doesn't find those symmetries, those isometries present, and yet it seems to me it makes perfectly good sense still to say of electromagnetic theory when posed against the background of curved space-time, that it's time reversal invariant, so it would look like a way of thinking about to that generalization. But I'll discuss a way, one that I think is simple and natural, which is fully equivalent with the usual way in the context of Minkowski's space-time, but naturally allows itself to be lifted to the context of current space-time. So here's the plan. I'm going to first begin with a brief account of the standard account of time reversal, a brief description of the standard account of time reversal invariance and the time reversal operation. Then I'll discuss David Albert's objections, and then I'll shift gears. I'm going to be talking about both classical E and M and time reversal invariance, invariance within the spirit of space-time geometry. So I'll give you a brief description of the invariant four-dimensional formulation of classical electromagnetic theory and suggest that it has a very simple natural geometrical interpretation from a certain standpoint it's much simpler to think about E and M from that in that formulation than a textbook formulation in terms of the components electric and magnetic fields. And then I'll say something, I'll discuss this way of thinking about that does not presuppose that certain background symmetries are present. Roughly speaking, it will go like this. I'll characterize at least a fragment of classical electromagnetic theory in terms of two structures on space-time, a tensor F, which represents the electromagnetic field, and a vector field J, which represents the charge current source of the electric magnetic field, and I'll stress the fact that both of those objects are only defined relative to a choice of background temporal orientation,

7:30 and suggest that one can think of the time reversal operation as one which one moves from the pair as determined relative to one temporal orientation to the pair corresponding pair as determined relative to the other temporal orientation. Then I'll briefly talk about the parity-reversal operation within the same spirit, and only after all of that will I come back to talk about magnetic fields and, rather more directly, confront what David Albert has to say. Let me begin with some acknowledgments and mention some other papers. The book that I've referred to, in case you don't know about it, is Time and Chance. It was published by Harvard University Press in 2000. I can mention and recommend two other papers on the same topic. One by John Ehrman has already appeared in International Studies in the Philosophy of Science. Both of these are critical discussions of what David Allard has to say about the time reversal invariants of classical UNM, but the other is by Frank Arzinias, and it's going to appear in studies in history and philosophy of modern physics. There's some differences in our point of view and emphasis, but I think those differences are formed by much more significant agreement. Have there been any papers defending these? Not that I know of. and I don't think they're going to be doing that. I also want to thank a few people for helpful correspondence. Bob Garrosh, John Norton, and Howard Stun. They give me some very helpful comments in an overdraft of the piece. Okay, so let me begin by talking about the standard account variance. And for the moment, I want to put aside the complications of relativity theory and a few other complications. I want to allow myself to talk without qualifications about space at a given time or about the state of a system at a given time. I want to take for granted that I have a background time coordinate and eventually some space coordinates. To

10:00 To avoid complications, I'll imagine that the time coordinate runs from minus infinity to plus infinity. None of that will very much matter. I'll start with a standard characterization that's schematic in the sense that there are two placeholders which we'll have to fill in by considering some examples. I'll make use of the notion of a state of affairs or an instantaneous state of the system, however that's characterized, and also a time-reversal operator. Let's imagine that we think of a history for a system as a map which assigns to every T some state of the system, S of T. So we can say this, a physical theory is said to be time-reversal invariant if for any history of that kind, which is allowed by the theory, which is compatible with the dynamical constraints of the theory, a corresponding time-reversed history is allowed as well. Now I have to characterize, say what I mean by this time-reversed history. There are two different operators here. This notation might be confusing. Let me distinguish them. R is supposed to be an operator that acts on instantaneous states. It takes one instantaneous state to another. T is an operator that's induced, which transforms histories to histories. The time-reversed history is generated by considering in effect two operations that can be performed in either order. If we forget about the R for the moment, think of the initial states as being rerun in opposite temporal order. We start with the times that were at plus infinity and work our way to the times that the states are at states that were at minus infinity. But we also characteristically transform each of those states along the way. The idea in the background, of course, is usually explained in terms of watching movies and the movies run backwards. I can imagine, for example, watching a movie of a point particle in Newtonian physics moving

12:30 from left to right, and then seeing the movie run backwards to the projector, starting with the final frame and ending up with the first frame. When we run it backwards, instead of seeing the particle moving from left to right, we see it running from right to left. So let me help explain this in terms of the simplest possible example. A single Newtonian particle. And there its state is characterized by a pair. And I want to emphasize, David Albert hasn't yet made an appearance. This is just a standard way of thinking about things. A single Newtonian particle, we characterize it with both the position and the velocity. And the time reversal operator transforms it into a pair in which the position is left intact, but the velocity is reversed. The induced operator then starts with this state, which assigns to every t a position and a velocity, and it assigns a time-reversed history. parts to it, I can think first of what happens if I just replace the state of t with the state of minus t. There I just have these two entries. And then I apply the operator r that leaves x of minus t intact, but it puts this minus sign in front of the v. I claim that makes very good sense. It corresponds perfectly with our intuition about the particle from left to right and then moving it then when it's run backwards, moving from right to left. It occupies the same position as before in the rerun, but occupies those positions in inverted order. But insofar as it occupies them in inverted order, this velocity is no longer oriented from left to right, but now from right to left. That's why we've built the velocity. In general, if one element of the state description can be characterized as a time derivative of some other. If the first is kept intact, then the second will have to be fixed, because a minus sign enters when we take the time derivative. Okay, well that's the first example. There'll be only two. The second example is classical electromagnetic theory,

15:00 and I'll start with a standard textbook description. I'll characterize the state of the electromagnetic system in terms of four objects, an electric field and a magnetic field. The vector fields, we can think of them as signing a vector to every point in space at a given time, and then two objects that represent the source for the electromagnetic field, a scalar charge field and a vector, a charge density and a current density vector field. And the time reversal operator, as standardly understood, leaves the electric field intact and the charge density, but it reverses the magnetic field and the current density field. Actually, David Adler doesn't discuss sources at all in his account, term of my preference for including them here. If there's any chance that you're worried here, you might well just consider the special case for the source advantage. Do you mean he considers instead the forces? When he only talks about the E and B field. Nothing more? When he formulas Maxwell's equation, he takes rho and j to be 0. Then it would seem he's leaving out something actually essential. You're not going to talk about something more than the EMB field, the definition of R is pretty much arbitrary. I don't think so. I think the same points can be made even if R and J are 0. But let me go on and we can decide if it makes a difference. Okay, well, anyway, as soon as I tell you what the state description is and how R is supposed to act, everything else is determined that T is reduced. initial state is just of this form I have these fields at a given time it makes assignments at all points then we have a time reversed history now I'll just write it in this form for purposes of later comparison where I have four fields corresponding time reversed fields and they have this form so in In the case, I've replaced a T by a minus T, and then I've also introduced minus signs

17:30 according to the prescription of the time reversal operator R. The field at Tx is now taken to be what the electric field was at minus Tx. And similarly, in the case of rho, nothing is done except for this rerun of the order. but in the case of B and J, I introduce them on. Okay, and now I can make the simple claim that the original history satisfies Maxwell's equations if and only if the time-reversed history does so as well. Let's just take one example, I'm sure this is clear, let's take this example here. Curl of e is equal to minus the partial derivative of e with respect to p. So the claim is that if the initial fields satisfy these equations, then the transformed fields do as well. So let's just look at the second one as an example and consider what happens if we think of E and D as now replaced by their counterparts in the second line. If I look at the left-hand side, if I now consider E of minus Tx, nothing is really going to change because the curl involves only partial derivatives with respect to spatial variables and those haven't changed. On the right hand side, I'm taking a time derivative of b, but because I'm taking the time derivative of b of minus t, a minus one is introduced by that partial derivative. On the other hand, there's now a minus sign out in front of b, and those two cancel each other out. And so the right hand side also is left intact. And if we had equality before, we're still of quality, and the same discussion of products that the others. So that's what you'd find in almost any textbook on possibly seeing if this issue arises as the subject of time reversal invariance arises at all. Now let me begin to talk about David Albert.

20:00 I have a question. Please. The way you've defined time reversal invariance without any strong constraint on what kind of, what far too many systems would be time reversal invariant. For example most dynamical systems studying ergodic theory or most dynamical systems studying ergodic theory, the system is isomorphic to its time reversal. Just take t to minus t there's some mapping from the state space to itself. There's a truth true for example for all Bernoulli systems you run them backwards, they're isomorphic themselves so there's always some R which does the job. I'm not suggesting that by virtue of there being some R, it's time-reversal invariant. Whether it's time-reversal invariant depends upon the choice of R. For the moment, I only want to make the, I think, uncontroversial claim that if one takes these characterizations of states and accepts these accounts of how the time-reversal operator applies, then those two theories, at least those fragments that are under discussion, do qualify as time-reversal and variation. I agree. I think those R's you've written there, but for reasons you haven't directly stated, are eminently good time-reversal operators. But once we give examples where everything comes out perfectly fine, but the R is not a legitimate R for the purpose of discussion of time-reversal. I think that's the kind of issue that will be discussed. Specifically, David Albert objects to that R in the case of electromagnetism. He thinks it's a cheat, in a way. It's been introduced so as to guarantee time reversal of variance, but it's not been properly noticed. But I better move on now. If you're still concerned, later we can discuss. So I want to discuss two objections that David Albert raises. There are quite different types. I'll mention it first, but it isn't going to concern me. The first involves the very characterization of instantaneous state for the kind of systems you're interested in. And the second is the action of the time reversal operator in the case of electromagnetism. The first, David says that on any proper account, instantaneous state descriptions at different times should be logically,

22:30 conceptually, and metaphysically independent of each other. It has the view that the only appropriate entries in the characterization of an instantaneous state are qualities or magnitudes that in an appropriate sense are intrinsic to the time slots. And this requirement fails to be sure if we for example characterize instantaneous particle states in terms of positions and velocities because if a particle has a velocity at a given time it's certainly going to be determined by considering what its positions are at all times after that particular time its positions at all times before just by taking a limit limiting He claims that the things that get to be called instantaneous states in physics books should actually properly be called dynamical conditions. Well, I don't want to discuss this really. I'm not convinced that there's anything objectionable about this notion of instantaneous state. I'm not going to pursue it because I don't think it makes any difference. If you're interested, there are several papers which discuss specifically this issue. The one that John Ehrman paper does devote some time to this. There's also a paper by Sheldon Smith that's recently appeared in Studies in the History and Philosophy of Modern Physics with a short reply by Frank Arzinas. And one published paper that John Ehrman showed by a student at St. Paul, which I also thought was very nice and in a very good way of directed attention to what was important in this debate. Anyway, David Albert thinks that what are usually called instantaneous states should rather be called dynamical conditions. But if you take the characterization I gave before, the schematic characterization, and systematically substitute the expression dynamical condition for incontainment state, you arrive at a formulation of what it means for a theory to be time-reversal invariant that I think he accepts. So let me just go through it. Now we can read it this way. The physical theory is time-reversal invariant,

25:00 yet for any history of dynamical conditions allowed by the theory, the time-reversed history of dynamical conditions is about as well. The real issue, I think, is just how one understands the action of the time-reversal operator on the states are. So let me turn to them. So first, David agrees that if we have two vector quantities in basic dynamical condition, I'll try to remember and use that expression. And the second is the time derivative of the first. then if the reversal operator keeps the first intact, it must reverse the second, just by considering what happens to these time derivatives. So he certainly agrees that in the case of Newtonian particle mechanics, R is appropriately understood to take the dynamical condition x together with v to x together with minus v. So we agree that the theory is time reversal in variance, despite the disagreement about whether a particle's velocity should count as particle's instantaneous state. But where he takes the stand in effect is by saying that only if a certain quantity is at the time derivative of some other that's kept in fact, is it appropriate to think of the time reversal operator it's a vector property as flipping it and multiplying it by minus one. He puts it this way, magnetic fields are not that so specifically he objects to thinking of the time reversal operator as sending b to minus b. Magnetic fields are not the sort of things that a time transformation can possibly come around. Magnetic fields are not ideologically or conceptually the rates are changing. And one can think of them as proposing an alternate transformation.

27:30 On top I have the standard transformation that I had before This is the alternate set, and they're the same except in the second line. Again, David doesn't discuss currents, but because the current density is a velocity-like object, I'm pretty sure he would want to say that, yes, that that one is reverse. I think he would agree to that much. So the only place where one has disagreement is in the respect of second law. He thinks that just as the electric field is kept intact, so the magnetic field should be kept intact. There shouldn't be a minus sign out in front as there is a thing. And if you consider the second set of outer transformations, or how I call them, and then look back at Maxwell's equations, one sees that though the first and third equations will be invariant under those new transformations as well, neither the second nor the fourth will be invariant, and they won't be invariant even if jj is 0. That is crucial. As we saw before, one minus sign enters when we take a time derivative with the minus t there, but now we won't have a minus sign outside. The minus sign in front can cancel, so the left-hand side will be sense left intact, but a minus sign will be introduced on the right-hand side, destroying these areas. Okay, I've proceeded kind of formally now. Let me suggest what I think is the picture that's on the back of David's mind, why he presents things this way. Let's just strip things down and consider two times a slight, let's say initial and final, earlier and later. And the world line of a particular particle which you might as well imagine is moving from left to right. And the E and B fields are vector fields within each slice.

30:00 Let's just consider the value of those fields at two points where the world line intersects and let's imagine that at both those points, neither of the vectors vanish. So then we have a well-defined acute angle between those two vectors. What happens now if we consider the time reversal operation? Well, the picture, I think, is that we interchange the position of those two time slices. Now we have f first, or below, and i above. But if we take that line which is moving from left to right and flip it, or invert it, it's now seen to be moving from right to left, we still have the points of intersection. And I've drawn this in such a way that the vectors are just where they were. I have that configuration on slice f and the same configuration here, and similarly on this the e and b just lie there picture these are fields which lie within the time slices and they aren't the time derivative of something they just live in these slices and are kept there when we shuffle the order of the decks and the cards and if you do think about them in that way then And it certainly seems appropriate that we have B left intact. So this corresponds to a pictorial representation of those Albert transformations. If we worked with the standard transformations, when we get to the time-reverse version, we'd be replacing B by minus B. So here we have the vector which looks like this, and instead of having the initial angle we have pi minus the initial angle. Let me give you an initial response which will set up what comes later. Just make a few remarks right away. The first is the answer, the response, which I think you find coming from most

32:30 physicists who think about this for a moment. The V is what's usually called in classical textbooks an axial vector field in contrast to a polar vector field. The claim is it can no more just fly there than an angular velocity vector field can. And the inversion of the end of time reversal is no more mysterious or unmotivated than the change from clockwise to counterclockwise rotation induced by that operation. And there is a very nice strict geometrical analogy between the magnetic field when it's thought about geometrically and an angular velocity. So I think that's in some sense a complete and adequate answer, but I'm going to come at it a bit differently and make a few points. First, I want to say that because of the trade-off in relativity theory between electric and magnetic fields, one can fully characterize the operation of time reversal invariants without making reference, characterize the operation of time reversal without making reference to magnetic fields at all. So the asymmetry between magnetic and electric fields, the probable value, I think just cannot be essential to the issue of time in which invariants. But finally, and most importantly, I think we get some considerable insight into all this. If one works with the invariant formulation, rather than the formulation in which one has components of electric magnetic fields. That version seems to me, in some respects, much more complicated. It's much more to keep track of than invariant formulation. So, let me move on now. Say a little bit about space-time geometry and temporal orientation. I'm going to use some language from the framework of general relativity, but if you're not familiar with it, I suggest that you track what I'm saying in terms of flat space-time and custom space-time. Well, suppose we have a relativistic space-time, a four-manifold connected and a Laurentian

35:00 manifolds. Most important, the Sorensen manifolds determines a little cone in a tangent space at each point, and so determines the partition of vectors. You can think of them as time-like, null, and space-like. On the sine convention I'll use, plus, minus, minus, vectors will be time-like if the inner product of themselves, which if the inner product of the vector of itself is positive. The only assumption I really want to make is that it's temporarily orientable, which means that it admits a continuous time-like vector field in the background. At each point, that vector in the field lives in one of the two cones of the light cone. You can think of it as marking what we'll count as the future world. Once we have this marker, we can talk about future vector vector, time-lock vectors. So somewhat more formally, this vector field tau determines the temporal orientation given a time-lock under the tackle here. Time-lock vectors C at a point is future-directed or past directed relative to the marker tau if the inner product of the vector we're thinking about with tau is positive. That will be the case if they're in the same mode, and if they're an opposite mode, the inner product will be negative. It can't be zero because no two time-like vectors can be orthotic. And the whole game, as I want to play, is to consider a number of geometrical definitions and constructions, and to keep track in each case what role, if any, is played by a choice of a background, temporal, or equation. Suppose we consider, first of all, just the representation of a particle moving through space-time, a particle with non-zero rest mass. There are two things we can attribute to it right off that do not depend upon temporal orientation. I can characterize a curve as time law, first of all, just by virtue of talking in terms of the tangent line. I'm not going to worry about collisions, so I'll imagine that the curve is smooth, say,

37:30 or at least differentiable to second order. So at every point there will be a well defined tangent line. I can say that the curve is time-wise if it's tangent-line of everything, it's time-wise. I can also talk about the curvature vector at a point to the curve. I can use the same characterization as it would for a curve in Euclidean geometry, but here it bears a natural physical interpretation. this curvature vector can also be thought of as a four acceleration. And I claim that too has a characterization that's independent of the choice of temporal orientation. But usually when one represents particles, one works with a parametrization and one has a notion of a four-galoxy. At each point, one works with the unit vector, which is aligned with the tangent line and represents the line by that unit vector. But there are two choices at this point we might work with. There are two candidates for the four-velocity. There's this vector and there's that vector. We don't get to decide which one of those is the four-velocity until we have a temporal orientation. If we wanted to think about the characterization of four acceleration in terms of four velocity, we have two choices. If we orient the curve in the up direction, if we take the unit tangent vector that's pointing upward in each case, we use the left-hand side to characterize the acceleration, the rate of change of the four velocity and the direction of the four velocity. This delta is the derivative operator, the covariant derivative operator, uniquely determined by the metric. But if we work with a different temporal orientation, in each case when we add c, we'd have minus c. If we plug that tangent vector field, four-velocity field, into this expression, the minus sign would appear twice, and they'd cancel. So I have two different characterizations of this magnet. Could I ask for a point of clarification in talking about full velocity?

40:00 We understand in physics that the time component of full velocity is 1 over the square, 1 minus v square over c square. The spatial component is the ordinary velocity times that same gamma factor. Now, if you reverse the time, the ordinary velocity reverses. but the gamma part goes into itself so it isn't clear that the time reversal of four velocity is necessarily designated by minus the four vector C sub A it could very well be plus or minus they both satisfy the requirement I believe but can you clarify that? I urge you to think about this in terms of invariant geometry and not with respect to components relative to some background frame of reference I want to think of the formula I think very simply as just a tangent vector to the curve. And whatever else is the case, if I reverse the tangent vector, what I mean by that is replacing this by minus vector. But still, you are talking about reversing time. It's only time that's determined this foliation. Well, maybe I just want to stick with what I have so far. I've introduced the notion of temporal orientation. And for present purposes, that just means a choice of one lobe of electron at a point rather than the other. And all I care about is which one of these two vectors is the one that gets to be described as pointing in the future direction. Everything that one ordinarily finds in standard presentations of relativity theory can certainly be recovered in the language you describe when you take projections and consider coefficients and components. But I'm going to place some weight on thinking about it in varying terms. I actually think that one cuts through a number of issues that otherwise would be confusing. Now just one thing, the four acceleration of the curvature is always orthogonal to the tangent line. Not by virtue of any dynamical restriction, just by virtue of the way you want to find the curvature of that.

42:30 If we're now thinking about how equations of motion constrain the motion of particles, we're going to look to constraints between the possible tangent lines of four velocities and induce, well now I guess I'll say forces acting on the particles which would be orthogonal to them. And we can think of the, wow, I'm going to have to make some choices as to what to tell. I think of the F field, which represents the electromagnetic field, this way. It's a second-order tensor field on the background manifold, which is anti-symmetric. I'm going to give you a little story, a standard story as to why one wants two indices and why it should be . We can think of it as doing nothing more and nothing less than coding at every point in space-time the net electromagnetic force that would be experienced by a charged test particle at that point, depending upon its charge . Let's for the moment forget about the charge and just imagine that we're dealing with a fixed particle with unit-positive charge. The claim is that all possible relations between four-velocity and impressed force, we have one that's incredibly simple. If the particle has four velocities c, that's its direction of motion at a given point, then the force it experiences there is given by contracting the vector into this object f. Think of that with one index raised as just a black box. You give me a vector, a contravariant vector with one index up, and I'll give you back to you another vector, which is this one, which is one true index up. And to say that the mapping can be expressed in this form is to say it's a linear map. Just for the second of the tensor, a linear map. One could imagine all kinds of complicated possibilities, but electromagnetic force on a particle could depend upon lots of things other than the four-velocity and its dependence on a four-velocity did not have to be linear perhaps, but that's in fact the way electromagnetic theory plus electromagnetic theory works.

45:00 There's a single object which encodes the forces that's experienced by particles it opens, and it has this simple memory of time. Okay. Well, I've told you what the force on a particle is, so now we have all the ingredients for determining the equation of motion for a, By a test particle, by the way, I mean one whose charge is such that for the purposes at hand, its contribution to the electromagnetic field can be ignored. If one's troubled by that, there are various ways of trying to get around it. What's really important in determining the motion is the charge to mass ratio. So one could think in terms of complicated limiting schemes in which the charge to mass ratio constant and q goes to zero, so the perturbation of effect goes to zero. But I'll just follow the track of the standard textbooks and allow myself to talk about self-qualification. So the equation of motion is just m equal ma. That's the equation of motion in long-divian theory. On the right-hand side, we have the four acceleration and the m out of from, which is mass, it's rest mass. On the left-hand side, we have the force that's determined on this 4-velocity, on the particle with that 4-velocity, with the proportionality back to QM in front. So we have FB both in there. By the way, this does give a full characterization of that because if I had two different maps which have the same action on all four velocities at a point, because the four velocities span the tangent space, and those two x would have to agree insofar with the union of action and the vectors that span the dimensions. So I have a characterization of this field F. I'm not saying that's how it's defined,

47:30 and I have no commitment to a claim of that kind, but it is a convenient characterization to use. And again, it's going to standard assessments. So just two comments briefly. If you look at this equation, it can't make sense unless F is anti-symmetric. The right-hand side is an acceleration, which is automatically orthogonal to the four-velocity. So the left-hand side better be orthogonal to the four-velocity, which means if one contracts on A with that same vector C, one has to have zero. That's to say that for every four velocity of a point, if you contract twice with that four velocity, you're going to get zero. But that's just to say, if that's true for all unit time like this, if there's four of them, then it has to be the case that that's going to be symmetric. So f is nothing more than an object which makes an assignment. You give me the four velocities, I'll give you an orthogonal vector. That's an orthogonal vector which represents the force of the target. The second thing is that this prescription I've given for determining f can be thought of as involving two elements. There's one hand in the varying mapping from the tangent line for the force vector. I could characterize things that way, but I don't know of any natural mathematical formalism in which you represent mappings from lines to vectors, and when you're algebra, the usual way is by going from vectors to vectors. So what we do is introduce a temporal orientation in a sense as an auxiliary factor, so that one represents the line in terms of one of these two choices for what the four velocity is going to be once one has that in addition then I can use the prescription before and I get out an edge you just said you don't know of any natural mathematical formulation but actually what you're doing is giving just that this would be the natural mathematical way to describe

50:00 from tangent lines to forfeits to force things. Right. I meant without first introducing this auxiliary structure. This is what a mathematician would do. It's just the right thing to do, just what the doctor ordered. Good. So anyway, what's really important for my purpose is that there is a dependence on a temporal orientation. And I shouldn't seem surprising that I'm going to place a lot of weight on this. And when I think about time reversal orientation, I'll consider what would happen if we considered the same mapping from lines to vectors, but represented in terms of the other possible temporal orientation. It's important that it's the same mapping in a sense, but I'm just representing in terms of two different temporal orientations. And I'm going to suggest that that's a way of thinking about temporal orientation, which in no way depends upon background space-time flight. Okay, I haven't said anything about J and its characterization. A similar story can be told. You can think of it as coding at every point in space time to charge density there as determined by different observers. The rule is this. An observer with four velocities see at a point. It takes the charge density there to be a certain number that you get by taking the inner product of those two vectors, J, and four velocities. But once again, this characterization essentially depends upon the choice of temporal orientation. Because I haven't gone directly from the tangent line to the charge density. I've gone from the four velocity to the charge density. And now I can characterize Maxwell's equations in this framework. I've told you how to think about f and j, and I can formulate Maxwell's equations this way. In a different notation, this might first one just come out and said df is zero. It says that this anti-symmetric tensor field is closed. It's more to the exact vector potential, and one also has this second equation here.

52:30 Okay, at this point what I was going to do is show you how to decompose the electromagnetic field relative to an additional utility structure and orientation to recover E and B. I think And we'll just show you a transparency, which is something of that later. If we want to talk about E and D, we have to introduce auxiliary structures. We have to in particular introduce, at least in the simplest case, Let me stay in Minkowski's space time for the moment, where I have what I'll call a reference frame, which for my purposes is just the future-directed unit timeline vector field, which is constant, which means if I start with the vector one point, I get the vector any other point by parallel transfer. And now I can use that frame of reference for decomposition of the field in a standard way. F in a way that's standard for any anti-symmetric second-order cancer. So I'll kick off the component of eta as this black, brown frame. I can decompose it into components that are tangent to and orthogonal to eta. And I can decompose F this way. I can get E by contracting with data on one index. But to get B, I have to introduce this anti-symmetric tensor epsilon, space-time volume . It's a completely anti-symmetric object. I would have told the story about how to think about this and the fact that it's an additional constraint that space-time emits such an object, but if it does, it emits exactly two. And the point is that though all these other objects are defined simply relative to eta, here the magnetic field is only defined relative to the choice of the background-cranial reference and the volume. Here we get the decomposition of the two.

55:00 I'll just flash this up, if one uses that decomposition a straight forward computation allows one to take the two invariant equations and break each one up into two parts, essentially by projection tangent to and orthogonal to atom. And all I've done is recover the standard formulation in this notation which I'm using. It shouldn't seem too surprising, for example, to correspond to this one. So that's one way of thinking about the recovery of the standard formulation without talking about foolishness. Okay, let me get to the most important, at least I want to do this. I want to suggest two ways of thinking about time reversal in variance. Let's say, let's imagine we're in the context of Minkowski's space time. Let's first think about just the first and second panel. This isn't a great drawing. I wanted to represent a particle which is at rest relative to some direction for a certain part of its history and then begins accelerating to the rest. It has a four, I'm giving a temporal orientation, so it has that four velocity and this four acceleration. What happens if I consider the time reverse history? The way one usually thinks about it, one just as it were, flips the curve. And this is supposed to look like what we'd get if we took this curve and flipped it over. It ends up being pretty vertical. And now rather than moving off to the right, it's coming in from the right. It's moving from right to left, it's still accelerating towards the right. And this is, imagine that we have a horizontal line here and reflect with respect to it. This acceleration vector gets mapped onto that one. And as a result of the reflection, a four velocity would have been pointing this way.

57:30 But now because we're keeping the orientation intact, they'll have it pointing in that way. And that, I claim, is what you get in a standard description of how the time of rehearsal operator works, the one you get if you look at those coordinate transformations that I had before. What's important is that here we're leaving the temporal orientation in fact and reversing the history. We're considering the image of the history, as it were, under an isometry of the space time onto itself. its image under any mapping whatsoever because I want to preserve all the elements of the relations. I want to preserve what its curvature is. Okay, well another way of thinking about the same transformation was to go not from one to two, but from one to three. Here, I have not changed the configuration of the one there the acceleration vector is exactly the way it was before what I have done though is to change the temporal orientation whereas before the four velocity pointed of that way now it's pointing down this way I've gone from C to minus C but that doesn't change the acceleration because that can vary now it's important that two is a is a distinct history from one, and three is a distinct history from one. Those cases have gone from one history to a time-inverted history. But I want to claim that two and three are two different descriptions of the same history. They are in the appropriate sense, I'm not sure what to say, perfectly isomorphic. Every element of structure that's been introduced is preserved if I take a mapping, which, if I take that flipping matter, everything either goes over into this counterclockwise. That wasn't true in the transition from one to two, from one to three. We have a trade-off. Rather than keeping the time orientation intact and flipping the history, I've kept the history intact and reversed the time orientation. And insofar as these are perfectly equivalent descriptions, I want to say

1:00:00 If you would agree that this can be thought of as an appropriate representation of the time of Earth history, so should this one. It doesn't matter if you have this arrow pointing down the narrative somehow. An essential feature of our convention is in the space-time diagram of the arrow after the wall. What's important is the internal relations among the elements Anyway, which gives us a very simple way of thinking about time reversal variance. Consider tensor fields of a certain kind, which have the property, this isn't true of all, but which have the property that they're only determined relative to the temporal orientation. and consider time reversal as an operation which takes the first field as determined relative to some initial temporal orientation to the corresponding field as determined relative to the other temporal orientation. We have one example here. If we're considering the four-velocity to the curve, it certainly is defined relative to a temporal orientation. So when we consider its counterparts under the time reversal operation, we have to go from that vector to this vector. I want to claim that simple way of thinking about things induces the transformation of fields like the F view that represents the electron view. And let's just see what happens in a particular case of electromagnetic field. I had a prescription before. The particle has charged Q and 4 velocities C, then the force of experience is given by contracting the 4 velocities to F. as Shelly suggested, as a way of talking about a mapping which in itself makes no reference to temporal orientation. That was the mapping from the lines of the vector. Well, let me just work with the other possible observatory structure, the other possible temporal orientation. That way, our four velocity is not c, but minus c.

1:02:30 But if I use the same prescription, I get this result, that here I compute it with the first temporal orientation, I have the original f and the original four velocities. On the right-hand side, I'm thinking about what the time-reverse field would be, whatever it turns out to be I'm using this as a characterization to determine what it should be well here in the appropriate position I use the four-velocity but now I have to find a reverse four-velocity so this characterization of t on the is such that this expression must equal that one because we have an invariant force or of the variance of acceleration. Well, this equation can be re-expressed this way, that t of f plus f kills of four velocities. But that's to be true for all four velocities at a given point. And if it even just does so for four linear independent vectors, there, and so on, is left with this conclusion that the time-adverse field is monetized. That's a discussion which does not presuppose anything about the space-time background except the temporal orientation, but it's perfectly equivalent with the standard one in Eccleston's was covered in that context. And then similarly, I get a similar characterization for the time reverse version of j. The prescription there was an observed with four velocities c at a point takes the charge density there to be this inner product. Well, I just once, as before, used the same characterization, working with the initial temporal orientation and now the time-reversed orientation. It's the same argument. I'm using this prescription to determine what t of j should be, and now I work with the time-reversed four-velocity, which is minus c.

1:05:00 So I'm led to this equation, and once again, if that's true for all four velocity vectors at a point, I have to have j going to minus j. And so here's Maxwell's equation, the invariant formulation, and here are the time transformations that one ends up with, and it's perfectly clear that the equations are invariant under that transformation. The first one doesn't depend at all, I believe it happens. The second one requires that though F and J both depend upon a temporal orientation, I guess that's what's really going on here. They both are only defined relative to the temporal orientation, but the character of those dependencies match. If that's the way I put it here. Both f and j depend on are only defined relative to choice of dependent orientation, but their dependencies match in the right way. It might be that someone's worried at this stage that with all the, with the geometrical presentation that I'm doing, it somehow had to come out that the equations, the field equations for this Maxwell field had to be pragmobersal invariant. I don't think that's true. So let's just consider a ridiculous example. an equation which involves f and j. It's totally unmotivated, but it's one that's not time-reversal and invariant. Take f, contract it with itself, and then apply the derivative operator, so there's one free index in the left and one free index in the right. It's a well-formed tensor equation, but it's not invariant in this transformation. If I replace minus F by minus F, the left side is left intact, but the right side is not. When the dust clears, this ends up being something really very simple.

1:07:30 These fields just undergo those simple transformations, but that really is what I think is worth. Now, if I had another half an hour or so, I could trace through in detail the decomposition of those fields what happens when one operates the time reversal operator, consider what happens to the electric field and the magnetic field. And I guess there just isn't time to do that. Thank you. There is time for questions. Yeah, David, I'm puzzled about why it is internally inconsistent for Albert to argue, on one hand, that you should flip a vector that's defined as the time derivative of another vector, and yet to argue that you should leave both E and B unflipped and assert violation of time reversal invariance because what independently of Maxwell's equations does he have access to as constituting mutually reinforcing definitions of E and B and if the divergence of E proportional to rho convinces you not to flip E and then the equation that relates the curl of B to the time derivative of E and the current density, which is essentially a time derivative of moving charges, why can't he see that as an argument for at least flipping the curl of B? And then you can express vast arrays of magnetic fields as integral transforms of their curls. I completely agree. I think the fact that in the chapter, at least, I've never discussed this with him, he doesn't think about the characterization of the fields. He just works with this picture in which somehow or other, we get to the fields, and they're just arrays of vectors on these time slices. And captivated by that picture, he thinks that we have pictures on the tap.

1:10:00 If I had had more time, I was going to say that when one thinks about a change in temporal orientation, there are two possibilities. One can have a change in temporal orientation that leaves the spatial orientation intact, or one can also change the spatial orientation. And David Albert's got a kind of halfway house in which it doesn't really correspond to either one of those possibilities. If one does reverse the spatial orientation, one does reverse that as well, then the curl term, of course, goes to the minus curl term. Sorry. I'm just wondering, how much does your image work for solving this problem? I think I don't know enough about it entirely Entirely. Yeah. It completely doesn't work. Wendy? Oh, um, does anyone here talk to David about this sort of stuff? I read John, okay, and I prefer to Frank. Exactly, exactly. I read John, I earned a paper, and I thought, you know, this is right and and clearly David Allen was conceited with Frank. And what David Allen was conceited with Frank? And I think Frank's paper is the same. So what David Allen was now saying? Okay, so I'll have to be, do you really believe that? So I think the answer is, so let me try and say what David would say in response to your paper, and what I know he would say as a sentence comes to my paper, and actually it also relates to that. So to what Gordon said. So first thing to note is, it's not true that David thinks that the invariance of the electric field

1:12:30 It's got anything to do with the invariance over the charge density on the flip. Now, you shouldn't think, on his view, as being characterized in terms of forces. In particular, he thinks you're begging the question when you say that the force law is going to be invariant, which you essentially say is going to be invariant. No, that's begging the question. These are separate on the multiple things, just like position is not defined. It's separate on the multiple things. And the only way you can argue for a non-trivial time reversal transformation is if you have some argument in terms of multiple things, some other multiple things from which we already know about the time reverse. So he accepts that, okay, can you find voltage in terms of positions? Then, of course, given the way that positions we all agree must go on the time reverse of voltage and voltage. But electric fields and magnetic fields are not defined in terms of anything else, are not defined in terms of the force or the force or anything like that. And I guess the default assumption is, well, when you look at the entity that's not defined in terms of something else, nothing happens in the time. So that's the first thing to say. Then as to the more general arguments, the basic answer to all of that is, and this is where it starts to equivocate, he really ought to say if he was consistent, look, it's not a tensor, it's not invariable to Lorenzbuss. these invariances. Now, at that point, you know, one has, you know, still out of cryptids, and I just threw out my hands. But I think the coherent answer would be, yes, it's not invariant on the power, it's not invariant on the risk boost, it's not a tensor. All of that's thinking of equations. It's not a multiple thing that doesn't change on any space-time transformation. Remember, all the equations that equated to something that does change on their space-time transformation, all those equations are not invariant on Now, when you start to point out that, and so Lorentz invariant also goes, all of that goes, you start to go to like, oh, gee, maybe there's a central difference between time reversal and boost. But then I think, essentially, David said, you can't do that. You can't say, oh, there is a difference between time reversal and lorenticals because you assume that, essentially, the time reversal operation and the lorentical loop operation can use because they're inherent to say that.

1:15:00 That's really, I think what he would say. Can I say something about the first point? Yeah. I'm sure that in the end I wouldn't get very far, but I would try to say something like the following. I don't know that I'm assuming the invariance of the force law let me try and do this in very operational terms and talk about the movie the force on a particle can in some crude idealized sense be represented as a displacement vector I mean I could imagine carrying around a certain Well, I won't try to describe the device. The proposal is that one can come up with sort of devices such that a charged body large enough to hold this device would hold it in its arms and there'd be a vector displacement. It could even be a computer, a screen being stretched in some way. Now, suppose that we watched a movie and we have some particular moment that we're interested in. We tag that moment in the movie about a light going around. And at that moment, when we see the movie run in a straight direction, we see that the spring has been displaced this much. And now we run the movie backwards, and at a certain moment we see the light, and we look at the displacement, and by God, the displacement is just the same. Maybe I should say that I can characterize the displacement in terms of position. And he agrees that the positions are position. So what he's going to say is in a backwards run movie, that the electric field is not, and that's because it shows that the same was on top of the backward run movie. It's wrong to identify the electric field with the displacement that it causes. According to the backward run movie, forget the electric field. Or claim that there is a relation between a tangent one and .