The Physical Content of General Covariance

0:00 The problem that I want to talk about is one that historians of general relativity feel to varying levels of acuteness. On the one side, when you look at what Einstein says about general covariance, it's very clear that it's quite central to his understanding of the general theory of relativity. And I've got some typical remarks that one finds. So shortly after his completeness of the completing of relativity in a general covariant form, he writes to Besso, my wildest dreams have been fulfilled, general covariance, where the stress is Einstein's, Einstein's perihelion motion of Mercury, and so on. And then later, a typical remark from his writing, for example, from The Meaning of Relativity, I won't read the passage, but he stresses repeatedly that there's a very close connection between general covariance and the realisation of a generalised principle of relativity. Now, that's what the historian finds, such as me, when you work through Einstein's papers. When, however, you start looking at modern formulations, by modern, I mean over the last 20 years, say, of space-time theories, you discover that every space-time theory that you run into just about has some kind of generally covariant formulation. I'm thinking of Newtonian space-time theories, special relativity, general relativity, and, well, this could be extended to include just about anything you'd like to put in there. As a result, it's hard for someone now working to see that general covariance is anything special about general relativity. And since these theories variously satisfy principles of relativity or not, it's especially difficult to see any kind of connection between general covariance and a relativity principle. Well, what I want to try and do in my talk is to provide a way of thinking about this problem. I've got a... The starting point for the story that I want to tell is actually a piece of historical trivia, or at least it starts out that way. I want to point to a difference in the mathematical methods used by Einstein and the methods that he uses now.

2:30 And it's summarized best by looking at the sorts of differentiable manifolds that Einstein used to construct his theories. In brief, what I'd want to say is the following. Nowadays, we use what you could call a general differentiable manifold. That's one whose point set is a set of unspecified elements. In exactly the same place, if we try and make Einstein sound as much like us, you will find Einstein using R4, or open sets of R4, where by R4, of course, I mean literally the manifold whose point sets are the quadruples of real numbers. okay and that'll be the starting point and i'll tell a bit of a historical story about that and then that will lead on to the claim that when einstein talks about a coordinate system what he's talking about is the coordination of a physically possible space-time with this number manifold and that it's not the same thing as far as physical principles are concerned it's not the same thing as what we would now call a coordinate chart and i'll go into greater detail I'll end up with a claim that the covariance principles, when thought of in the light of this result, actually turn out to be principles that do have physical content. They do make physical claims. I'll even say that the claims look like relativity principles, although I'll immediately back off and say that I don't think that the relativity principle-like character is something that can be transferred through to modern formulations. Part of the story is going to involve looking at the modern formulation of space-time theories, and I'll talk about so-called active covariance principles and make a few remarks about them to help me along the way. And then finally, fairly breathlessly as I run out of time, I'll try and get to an episode that I only came across quite recently and is fascinating. The earliest, and I think still one of the best-known objections the notion that there is physical content in general covariance comes from a paper that Kretschmann wrote in 1917. When you actually sit down and read Kretschmann's paper, you discover that what everyone thinks he said is not what he said, and what he really said is far more interesting and has connections to something that many of us have been worrying about, the so-called point-coincidence argument and many other things as well. So that's a kind of tease in case I never get there

5:00 to make you want to go out and read the write-up of the paper. Let me start out now by talking about Einstein's spacetime manifolds. This now repeats the claim that I made at the beginning. Where we now have a so-called general differentiable manifold, Einstein literally used R for its open subsets when he needed to provide a manifold-like structure to build the spacetime models. I simply see this as a natural part of the tradition that had been doing this since the latter part of the 19th century. Going back, at least, to the work of Riemann, one finds the notion of manifold introduced in Riemann's inaugural dissertation. Riemann is actually quite vague about what we would call manifold, about what it actually is. He talks about a, quote, n-fold extended magnitude, and gives a few vague physical examples. I turned to Klein for this is Felix Klein for an explication of what Riemann really in square quotes meant and Klein says at the foundations of his researchers Riemann laid n variables x1, x2, dot dot dot xn, each of which can take all real values Riemann denoted the totality of the value system as manifold of n I take that to be saying that a manifold of n dimensions is in effect Rn. I mean, what is the totality of the value system of n real value variables? I mean, that's what it has to be. And then he continues, by a fixed value system, x1 prime, x2 prime, up to xn prime, he meant a point in that manifold. So a point is simply one n-tuple of reals. And he noted in 1873, this is actually noted in 1873, that this definition of manifold is in agreement with the usual terminology. Now, let's proceed forward. What was Minkowski's world, the four-dimensional world? Well, if you read it literally, his world simply was R4. And this is the passage where it's introduced in his 1908 lecture on space and time. Oops, that has a first line. There it is. We will try to visualize the state of things

7:30 by the graphic method. Let x, y, and z be rectangular coordinates for space, and let t denote time. A point of space and a point of time, that is a system of values, x, y, z, t. I will call a wall point, so just read it literally, a world point is an n couple of numbers perhaps coordinated in some way and the manifold of all thinkable x, y, z, t systems of values we will christen the world. Same thing again. And if you start reading around, you will find quite a well established tradition that identifies manifold literally as rn. And Limit Chavita certainly falls into that tradition and others as well. It's interesting to pause for a moment and wonder why this might be the case. I think one of the principal attractions of this is something that was remarked on quite often in introducing this particular style of geometry. And that is that if you use number manifolds as your manifold in geometry, then you promote a very free commerce of results between real analysis and complex analysis as well, and geometry. and this was quite an attraction as they say our literature comments on this for example in the introduction of this book that's now called Tensor Analysis or Tensor Calculus rather and also recall the influence of Klein's Alumin program which told you not to think so much about the structures that you use to build your geometry but think about the invariance of the groups and that tends to direct attention away from what your manifold might literally be and of course I must mention that a new tradition that did not use non-manipodes to do their geometry. This was emerging in the 1910s at the time of Einstein's work, although I don't think Einstein actually used that tradition. Names like Weill and Hausdorff come to mind there. Okay, so I claim that Einstein simply kept in with the Minkowski tradition, and it's interesting when you look at Einstein's own definition of the space-time manifold, I've transcribed it and written up for you. It's in between the quote marks. Yes, well, that's right. That's the problem. He doesn't actually define what his manifolds are. And for a long time, this puzzled me greatly

10:00 as to why he didn't do this. His method is the one that I think anyone who's read Einstein's early papers on relativity is one that we're very familiar with. The primitive notion is taken to be the coordinate system. And it's assumed you just know what a coordinate system is at the very beginning. off defining vectors, tensors, by the transformation properties under coordinate transformation. Now, what's going on here? Well, I've done a little work in the mathematical background to Einstein's work, and nowadays we're always told it was the Ritchie-Levice Vita paper of 1900 that was the great mathematical fountain from which poured all the results that Einstein was going to put into his theory. Well, in fact, that's not exactly right. There were branches of mathematics that Einstein drew on in 1912 and 1913. On the one side, there was that Ritchie-Levice Vita tradition, differential geometry and variants of quadratic differential forms, and there the names of Riemann, Christoffel, Ritchie-Levice Vita certainly come to mind. But before Einstein turned to this branch of mathematics, with the help of Marcel Grossman, of course Grossman introduced him to that, to certainly the latter part of that tradition, before that, he had immersed himself in the methods of vector analysis. These, of course, go back to people like Grossman and Hamilton, and then after that Gibbs and Heaviside. And they were then applied, through the work of Minkowski, to special relativity. And so what Einstein was doing around 1912 was immersing himself in that new four-dimensional vector treatment of things through such sources as Zollefeld's resource article in an Arlander Physique and Lauer's text relativity. And there, one finds the sort of thing happening as in vector analysis. Now, in vector analysis, I think just a standard exposition in vector analysis, you don't start out defining manifolds, usually. You start out defining vectors and scalars and transformation properties. And my guess is that Einstein simply falls in with this tradition. Okay, now, so there's the claim so far. Oh, and a second little piece of trivia. We now know of the Leverettia Vita paper as the great paper in the 1900, Ritchie Leverettia Vita paper, as the paper that is the turning point in tensor analysis.

12:30 Curiously, the word tensor doesn't appear in the paper. Where does the word tensor come from? Well, it was used for a quantity of lesser interest in vector analysis, and you, of course, know exactly is appearing in vector analysis in the Maxwell stress tensor or in looking at mechanics of solids. And it was known as early as 1901 that one could think of tensors in terms of the Ritchie by Lichovita analysis. But it was actually Einstein, as far as I can tell, who made that usage the standard. And so if you look much before 1913, you simply will not find this branch of mathematics known as tensor calculus or tensor analysis. which rather surprised me I first read this in Bevelin's book in 1920 on invariance of quadratic differential forms provided to me You didn't read it I'm sorry? You didn't read it I didn't, it was written in the 1920s and it attributes the use of the term tensor to Einstein's 1916 paper which I think is three years too late the term tensor in the context of invariance of quadratic differential forms Okay, now I have to apologize because I'm going to stray a little bit now from history of science to do a little bit of foundation of the space-time theories. This is all work that I'm going to come back and apply. What's the principal burden of a space-time theory nowadays? The principal burden is to present certain mathematical structures that somehow represent what can happen physically. Okay, so this is a picture that I'm going to repeat over and over again. What's basically going on in a space-time theory is that you represent what can happen, okay, or a physically possible or actual space-time, a PPS, you represent that by some kind of mathematical object. And the mathematical object I'm going to use is going to be an n plus 1 tuple. The first element's just a differentiable manifold, and the remaining things in the tuple are geometric objects. And then I'll think of a space-time theory as simply being a set of models. I mean, this is not you, I guess. We've heard of this before. I'll simply think of the space-time theory as a set of models, each of which represents a physically possible space-time. For example, general relativity typically looks like this.

15:00 You have three spots, you have a triple, you've got manifold. You've got manifold. Oh, well, I can see it. You've got manifold, Lorentz's signature, and then, say, a symmetric cancer represented matter fields, although, in principle, you could have more in there, but that's not relevant to the story. Okay, now, I want to define the so-called active general covariance and Leibniz equivalence principle. The term Leibniz equivalence is a recent one, so I'll define it. It's, in fact, due to the fevered imaginations of myself and John Irman, so I apologise. People have pointed out that the name might not be exactly right, It's sort of stuck now. Here we have physically possible space times, and I've tried to draw them, at least on the previous slide, as realistically as possible. I sort of mean the thing that we're actually in, or something like it if the theory turns out to be false. And then that's going to be represented by the sequence of models. Now, it's possible to cluster these models together under the condition that any two models can be inter-transformed. So start with a particular model, call it T, then take any diffeomorphism that you like that maps the manifold of T onto perhaps the same manifold, perhaps the different manifolds. It doesn't really matter. And then the operation of the carry-along will give you a new triple. And it's customary for both of those models, the model and its diffeomorphic copy, to be in the theory. If that's the case, then I say that the theory satisfies an accurate principle of general covariance. Now, does the model and its diffeomorphic copy represent the same physically possible space-time? If it does, then I say that the theory satisfies the requirement of Leibniz equivalence. Now, loose talk, which is, I think, the common kind of talk, will grip those two together. satisfied, you would just call that a generally pervariant theory, although I think it's actually profitable to try and distinguish the two requirements. Now, the issue at hand is just what is a physical principle? Okay, since I'm going to work, I'm sorry, it's very deceptive,

17:30 I can see this here, like a little child, you know, if you can't see me, I can't see Okay, just what are physical principles in a space-time theory? And I offer this criterion. A principle whose truth depends at least in part on properties of the physically possible space-times represented. Okay, such a principle I take to be a physical principle. If its truth is independent of whatever's going on in the physically possible space-times, then I don't see how it could be a physical principle. And let me give you an example of a physically vacuous principle. let's represent the physically possible spacetime by a model T you know that the manifold M has many coordinate charts and you can have kind of coordinate representations of the model in many different coordinate charts and it will look let us say like a triple with R4 or some open set in the first slot where the manifold was you'll have a matrix of components for the remaining geometric objects Now, it seems to me to be just a matter of mathematical definition, in fact, part of the definition of this whole story, that if T is a model of the theory, then that model is represented by every such component structure that's produced by the coordinate charts. Now, that is simply going to be true no matter what's going on over here with physically possible space-time. So that's going to be a physically vacuous principle, and I would label that incidentally as passive general covariance. And in brief, what I'm urging is that we commonly tend to read Einstein's covariance principles as being principles of this kind and therefore wonder how there can be anything physical going on here. What I'm going to urge is that we ought to think of them rather as another type. And I wonder if that's on the next slide. No, it's not. We'll get there. Okay. Now... Hello. Thanks. What I want to urge now is that the active, not that passive version of covariance principles, but the active version of covariance principles are in fact physical principles. I immediately add the rider, in order to stop so many cries of outrage, I immediately add the rider that in modern space-time theories, the physical content is so small

20:00 that it doesn't really do much harm if you ignore that. And that's what people usually do, the fact that there is some physical content there. It's usually ignored completely, I don't think it does any harm at all. Perhaps in the Cauchy problem there are some considerations there, but apart from that, you really need to worry about it. Now, what's the basis of the claim that these covariance-Leibniz-equivalence principles are physical principles? Well, it's simply the following observation. Diffiomorphic models are just mathematically distinct things, okay? Unless you have a special case. No, no, for me. No, okay. Good, okay. Okay, unless you have the following special case, and this is where the diffeomorphism is either an identity or a symmetry. But aside from that, you simply have two different objects, and to say that they are not is to then say the diffeomorphism must be a symmetry or an identity. I'll give you one example. What happens if you have two diffeomorphic models, and the point sets of the manifolds are disjoint? Geometrically the same, but I would urge... Okay, but if you have disjoint point sets, then you can distinguish the two by mathematical property of set membership. I give you P as the point set of one manifold, is it a member of the other, and I think the answer is no. Otherwise, you're simply being incoherent when you say that the point sets are disjoint. And we seem to think we can say that they're disjoint. Okay, well... I'm now talking about global with global distance or global when you go on by defining your... I usually mean global here. Okay. But I'm not sure that much hangs on with that. It makes a mathematical difference as well. I agree with you. Okay. So I think we'll have an interesting question time now. It's necessarily global. I'd be sure if it's in the same. Okay, now let me now just tease through the idea. We start with a model T of a theory that represents a physically possible space-time. You make a diffeomorphic copy of an HT. General covariance tells you that that, if the theory is actively generally covariant, that tells you that that is also a model of the theory. It is a, as I claim at least, a distinct mathematical entity. And so I take the default assumption that the properties under which it differs with

22:30 the original model represent something in the physically possible space-times it represents, a different physically possible space-time. You then add in a lot of its equivalents, and you then discover that the two models represent the same physically possible space-time. You think you might be back where you started, but in fact you're not. You've come a long way now. What you now have is the following picture. You have a model and all of its dipheomorphic copies representing one physically possible space-time. This then tells you that only those features upon which the models agree, significant. In other words, only the invariance of the transformations connecting the models can be physically significant. And so now you have a nice recipe for picking out those aspects of your models that are physically significant. So, for example, the fact that your point sets are disjoint is something that under this principle you would then deny any physical significance to, although it is a mathematical difference. now let me turn to turn back to Einstein and see how this affects Einstein's work or Einstein's way of formulating a space-time theory you recall the basic picture is you have a physically possible space-time which is represented by a model and I'll take the simplest case general relativity and I'll ignore the question of the stress-energy tensor so one simply has two slots and a metrical object. In the manifold slot, we put in R4 for its open subset, since I don't want to make claims about global topology, but I'm as limited to R4 as a global topology. And in the metrical object slot, I urge we simply put literally the matrix of reals GIK. I don't want to put in there the tensor, whatever that is, because the tensor is, perhaps at the simplest level, the equivalence class of all inter-transformable matrices or something more abstract derived from that equivalence class. So the picture that we now have looks something like this. One has one physically possible space-time, and where we nowadays would have manifold plus Lorentz-Signature metric in the second slot there, I urge that if we try and cast Einstein in as close a language as we can to our own now, we literally have R4, or open subsets, and a matrix GIK.

25:00 And then one physically possible space-time under Einstein's general covariance requirement is going to be represented by every one of the equivalence class of these pairs such that one can be transformed into the other. And there the transformation rule is exactly the one that we know from just introductory tensor analysis. And I point out what's going on here. One is coordinating the physically possible space-time with the manifold. That coordination is, in Einstein's work, labelled coordinate system, and that's what he takes to be his primitive. But there is between decimal transformations now, to local, to local, to local, to local to local? Well, no, or open sets of local, I mean, I just, I tried to write in R4 or open set. Locale? I mean, I really don't want to get into the local, global stuff. Local in the sense of finite, but not in the decimal world. If this, for example, is a Minkowski space-time, then one can certainly have a global transformation. You can change from Schwarzschild to the isotropic coordinates, that sort of thing. But that's probable. Okay, think about the way that Einstein developed his general relativity. his coordinate systems never covered all of the physically possible space-time. And so what he had to do was actually piece together the story from different what I would call different pairs like this. It's not the normal viewpoint on the surface, I speak. Isn't this one of the awkwardnesses of reading Einstein's early work? the space-time out of the bits. And that's what these are. These become bits that might well overlap or not. But, you know, I'm not going to finish it. If we worry about that, it's really a mopping up operation. Or if it can't be mopped up, then I'm sorry. It's a problem Einstein had. I'll take that out of the discussion. I'm sorry? I'll take that time out of the discussion. I'll give it back to you.

27:30 Thank you. Now, I want to urge that the general covariance construed in that way for Einstein's models, that that is physically significant, a physically significant principle, for exactly the same reason. If you take two inter-transformable models, where, remember, GIK and GI'K' are matrices of reals. Okay, they're inter-transformable. In one, I mean, you could even take a Mikalski space-time here. This GIK might just simply be the diagonal matrix for all ones, matrix with all sorts of numbers in it. Now, I urge that those two structures are distinct mathematical structures. And if you're going to say that they represent the same physically possible space-time, you then have to say that there are certain parts of the models that can't be physically significant because they disagree on certain things mathematically. And only things that they agree on can be the things that have physical significance. And that's the basic assumption that makes Einstein's covariance principle as construed here, into physical principles. And to drive the point home, what I urge is the following, and this, I guess, is the basic point of the paper. Einstein represents physically possible space-times by pairs that look like this, with R4 and GIK, or an open set and GIK. Of course, he has to tile them together sometimes. It's a little awkward. I urge that we think of that as analogous to the representation of a physically possible space-time by a pair, manifold plus metric. nowadays the practice has been to think of this relation as being analogous to this one here the representation of a model by its components in some other coordinate chart now mathematically they're very similar and so usually one does no harm but in terms of the physical principles I'm talking about one does great harm because one then ends up concluding that the covariance principles that Einstein much about, end up as being mathematically, as being physically vacuous. Now, this connection is actually fairly strong. If you look at a metric g and it's carrying along h star of g, and then look at the, and then simply write out the equations that describe that relation in some particular coordinate chart, you end up recovering almost exactly the coordinate transformation law that we know for tensor components. But it's interpreted, obviously, quite differently. But formally, it looks

30:00 Okay, now what I urged was that... Can you just put the last bit of pink? No, orange. Can I cut it off? Just the last bit of it. Thank you. Now, what I urged is that the combination of general covariance and Leibniz equivalence principles, or let us just say general covariance, or covariance principles, what they do is enable you to sort through the mathematical structures and decide which ones are physically significant and which ones are not. And the brief answer is the invariance of the physically significant entities. Now, in the modern formulation of space-time theories, we use manifolds that are extremely impoverished of structure. So there's actually little urgency to try and sort through a space-time manifold and figure out which bits of this might be physically significant and which bits might not be. Einstein in the 1910s using R4 or its open subsets as your manifold, then you have quite another problem, because R4, as you know, has considerable further structure. I mean, to begin with, every point in R4 is distinct from every other point in R4. And of course you don't mean anything physically significant by that. And so it goes on. Now, if you just look at the way Einstein talks about his coordinate systems, what he does is he ascribes a kind of default physical interpretation to the various structures of R4. So, for example, in principle, the fact that every point of R4 is distinct from every other point, in principle, could represent inhomogeneity of space-time. Or, he talks about the X4 coordinate as though it were a time coordinate. And so there's a default interpretation that you that hypersurfaces of constant X4 are then simultaneity hypersurfaces. I mean, you understand where I'm going here. All of these interpretations are going to have to be deprived of physical significance, and you're going to do that with the covariance-leibnance-equivalence principles, and you're going to show that these are not invariants of the transformations of the covariance principle. That's why I'm heading. I'm just giving you the list here of default interpretations. Correspondingly, if you take the congruence of X4

32:30 curves, you have an absolute rest frame. Or if you take the set of congruences of straights in R4, then you have a set of inertial frames. And then, I mean, there are many, many more than this, but I'm just listing it enough to give us the flavor of the story. Then, if you look at the values of coordinate differences, the default interpretation is that if it's an x4 coordinate, you're talking about a time difference. If it's an x1 or x2 coordinate, you're talking about a difference, a metrical difference in space. distance, okay, which of course you want to get rid of. Now before I go on and show how those principles are applied, let me recall exactly what the various theories look like. To begin with, special relativity, as Einstein routinely formulates it, certainly in the earlier days, actually had models simply of this form here. One has R4, one has eta IK, which is the components of the metric cancer. Now, the crucial thing is that this eta IK was only ever this diagonal matrix. Okay? And you see that the way he always writes the line element for special relativity as in the simple diagonal form. And so the covariance group of special relativity, if it has models of this form, will simply be the symmetries of this particular matrix, which is the extended Lorentz group. And then, of course, you move up to general relativity, where the models have this form here that we're very familiar with, satisfying the field equation, and that is, of course, generally covariant. Now Einstein has a halfway house, which is the principle of equivalence. And the principle of equivalence works in the following way. You simply take special relativity formulated with models of this type, and then you say, but wait a minute, what happens if I expand the covariance group so that transformations to accelerating frames are included as well? And we make sense of transformations to accelerating frames by looking at the default interpretation of R4 on the previous slide. What happens if you do that? Then you get a model of special relativity which looks like this. You have open sets of R4. You must have open sets now because your map no longer takes you back onto all of R4. You have a matrix GIK whose components are no longer all constant. And, of course, a stress-energy cancer, which I put in there for completeness.

35:00 And the group of transformations that's involved here is larger than the Lorentz group, but still smaller than general covariance. Now, and of course there's the default physical interpretation of the varying GIKs representing a gravitational field. So pictorially, what happens with the principle of equivalence as Einstein sets it up is this. You start out with special relativity, which is in the green here, where one only ever has the diagonal matrix, and then you expand the set of models by allowing transformations to what we call uniformly accelerating frames, and this gives you orange models of that form here. Now, in principle, you're transformed to a new physical circumstance, one that Einstein interprets as having a gravitational field. But he then says, no, folks, you haven't gone anywhere at all. In fact, this orange model and the green model you started with both represent the same physically possible space-time. And what I claim now is that under that reading, we can make sense of what Einstein says literally about the principle of equivalence, which, at least in recent times, has proven more and more intractable to make sense. I urge you, for example, to look at Friedman's book on Foundations of Space-Time Theories, where Einstein's multiple confusions are paraded for you, as you might guess I disagree that he was confused on most of what Friedman says he was. So, for example, the coordinate system K and K''. Einstein in 1922 says, of the complete physical equivalence of the system of coordinates, k prime, we call the principle of equivalence. I simply read that as the in effect the Leibniz equivalence principle that I've been talking about. And then elsewhere in 1916 he writes, the requirement of general covariance of equations embraces the principle of equivalence as a quite special case. Now I say that follows immediately from what I'm saying. If you now continue on and expand this even further to allow general transformations you then have a general covariance. Now, why should Einstein think that his covariance principles in particular, the progression from Lorentz covariance the extended covariance of the principle of equivalence and then finally general covariance why should he think that these are relativity principles when nowadays people tend not to think so? Well, it's simply a direct application

37:30 of what I said earlier. You use covariance principles to pick out those aspects of your mathematical models which are physically significant. This is a simple table here, and I've listed down the left hand the default physical interpretations of R4, and then across the top I have the sequence of theories. In order to make the table pretty, I've started with a default theory that has no covariance principle at all, and then I go through special relativity, augmented by the principle of equivalence, and then finally general relativity. Now to begin with if you have no covariance principle, then everything is an invariant, so it's all physically significant. If you then have the Lorentz group as your covariance principle, as your covariance group then the distinction between points ceases to be an invariant the absolute simultaneity you'll recall was the X4 hypersurfaces, they're no longer invariant, and the absolute rest frame their end. That is no longer mapped back into itself. But inertial frames and metrical significance of the coordinates do remain. Yeah. And then as you expand the group, you gradually lose physical significance for the remainder of the properties. For example, you knock out the physical significance of inertial frames, which is exactly what you want. I mean, that's what a relativity principle does to begin with. If you extend it You somehow knock out inertial frames having physical significance. So I'm not misunderstood. I'm not making the claim, on the basis of what we have here, that general relativity does satisfy a generalized principle of relativity. Please understand that that's not my purpose. I simply think that the setup is different now, the way we set up general relativity. Now we start with manifolds that just don't have all of this extra structure need to be denied physical significance. So we don't carry this over. But what I am urging is this. If you want to understand what Einstein was saying, you can think of it this way, then it begins to make sense. And I don't want to condemn him in 1910 for not using the formulation that we use now. What I also urge, incidentally, is that one can use the type of talk that I've given here as a kind of translation manual. When Einstein talks about coordinate systems, about here, as opposed to the way that people tend to think of it now as a coordinate chart.

40:00 And I learned that some things will make sense that didn't perform. Good. Now I can tell you a bit about Kretschmann. Kretschmann's 1917 paper is usually remembered as making the following remark. there's no physical content in general covariance because you can always reformulate a space-time theory so that its equations are generally covariant. And for a long time I thought this was just a splendid remark and really hit the nail on the head. I no longer know why I thought that. Because with just a little thought you begin to see there are big problems here. Granted, granted that you can always take a space-time theory, sit down, fiddle its equations, perhaps add in an extra quantity here or there, and eventually you'll end up with something that is generally covariant. What I want to know is what makes you believe that it's the same theory that you started with. I mean, you've changed... Typically, you've added extra quantities. So, for example, if you take special relativity formulated with just that diagonal metric and expand it so it becomes generally covariant, you add extra structures. You now add matrices that have non-constant components as representing ventricle structure. Now, I say you probably have good reason to think that it's the same theory, but that reason is based on a physical assumption, and that is the extra bits of mathematical structure that you've stuffed in there. These extra bits are purely auxiliary quantities, and they have no physical significance. But that is a physical assumption, and if you make that physical assumption, you can then no longer support the original claim, that if you reformulate the theory into a generally covariate form, then there's no physical content in general covariance. you had to add physical content to get there. Or another way of thinking about it, is there a general principle that's underpinning that? Are we saying something like the following? Pick P, any formal property of the theory. I want to show that P is physically vacuous. Is it adequate if I then go and show you that I can take some other theories, many other theories, and reformulate that in such a way that P is always present? I can give you an example that seems to say to me principle is not right, take the presence of the string of letters E equals MC squared

42:30 in a theory. Now you give me any dynamics you care to give me, and I'll have some tricky clever way of changing its equation so that somewhere the formal property of having E equals MC squared is going to appear. Now does that mean that you now think that E equals MC squared in special relativity is physically vacuous? Well I hope not, although maybe we this at question time. Okay, now, I don't think that's actually what Kretschmann said. What Kretschmann really said was the following, and this is my rather mediocre translation. The claim that any space-time theory can be made generally covariant was actually dependent on an assumption, which I'll read to you since the wording is quite crucial. I've actually missed out a word. It says, imagine that all physical observations consist in the of pure topological relations coincidence in brackets between space-time objects of perception and hence immediately that no coordinate system is privileged above any other by them. Okay, now that's the condition from which he draws the conclusion that any space-time theory can be made generally covariant. Now, what's interesting is what he says and also the footnote. The footnote is to what we now, is to in an islander physique, to the page that contains what we now call the point coincidence argument. Very briefly, the point coincidence argument says, at least in modern language, that if one has two inter-transformable models, they must represent the same state of affairs, because they share all point coincidences. And what's an example of a point coincidence? For example, the collisions of particles. That's the argument. argument for general covariance, okay? Now, the interesting thing, then, is that Kretschmann seems to be advancing exactly the same argument that Einstein advanced. He's saying, given the point-coincidence argument, general covariance then follows. So, I actually don't think that they're disagreeing on this, and I want to stress in particular that that assumption underlying the point coincidence argument, is a really quite profound physical assumption. It's based on the following idea, and that is that the physical content of a space-time theory is fully exhausted

45:00 by the catalogue of its space-time coincidences. That is, if you capture all those coincidences, you've got all there is, and anything else is going to be supported for this. Now, what's interesting about this episode is not just that they present the same article, but it's the second footnote. The first footnote is this asterisk, and the second footnote is this double asterisk. And that is a footnote to a paper that Fretzman submitted to the Arlander Physique in October 1915, which teases out what is, to my eyes, almost exactly the Poin-Coin-Coin-Coin in very great detail. And the reason he actually uses this locution, pure topological relations, is that that was the one that he himself had used. Now, October 1915 is prior to any record that I know that once I'm thinking about the point coincidence argument. I mean, I don't know what this means, but I find it interesting. Notice that October 1915 was the date of submission of the papers. In what context did the paper present? ...the discussion of the relativity principles and the physical significance of coordinate systems was the theme, which is exactly what he says here. You see, it's that no coordinate system is privileged because that was the theme of the paper. Okay, so this actually was just the beginning of Frenchman's paper. What happened after that is now even more interesting. if we're not to think of relativity principles in terms of principles that deny physical significance to bits in the manifold how then are we to think of relativity principles well I think Kretschmann hits it right on the head, the way that we now think in effect Kretschmann's major point in the paper is the following

47:30 he agrees that general covariance is something I'm sorry he derives general covariance from something like the point coincidence argument point is to insist that the covariance principles that Einstein talks about are in fact not relativity principles after all. And his argument runs in modern language as follows. One must define relativity principles in terms of symmetries of structures defined on the manifold. That's the basic point that he makes. Now, it's a little hard to see that in a paper because he doesn't use that kind of language. And at least I'm not a very good German reader and I find the German very convoluted. So my guess is that other people have as well. and this might have contributed to a slowness in reading what's really in the paper. What he does is he actually doesn't talk about the structures defined on the manifold. He's a little more phenomenalistic about the story. He takes the structures that interest him to be the trajectories of light particles and the trajectories of free particles, so roughly conformal an aton structure. And he looks at their symmetries, and he defines his relativity principles in terms of those. that I certainly see floating around now a lot, and that is that special relativity satisfies the relativity principle with the largest group, that is the Lorentz group, and that general relativity is, in his words, a completely absolute theory and satisfies no relativity principle at all by simply grinding through the results that we know about, about symmetries of structures in semi-Roumanian spacetimes. The interesting thing is that there's, in fact, a lot more in that paper, and I'll mention two of the results that are interesting. We normally think of Bile as deriving the result that conformal structure determines the metric up to a conformal factor. Well, this result is already clearly in Kretschmann's paper as a part of this analysis. And he also has an interesting method for defining a preferred coordinate system in terms of invariance of the curvature tensor. Okay. One more slide, folks. Jorgen has been ridding me for doing nothing but pre-publicity for this paper, and I'd hate to disappoint him. I'd like to give you some publicity for the next paper, which I just happened to have brought it up and off.

50:00 When I realised, folks, when I realised that Einstein was using number manifolds, the scales fell from my eyes and all was clear, or at least that's what I say in this paper. I'm going to word that if you keep pursuing this idea that Einstein's space-time manifolds are number manifolds, you solve two other problems that at least were troubling me in historical work. One was, there seemed to be great trouble in figuring out how Einstein mashed together coordinate charts, frames of reference, relative spaces. Well, I think that entire issue becomes solved by this story. Then there's a second one that has simply been a bother for people who have been worrying about Einstein's work on the point coincidence argument and the so-called Hall argument. I mention that for those who've been following that literature. We are repeatedly troubled by having to ask be read as active transformations or passive transformations. I urge that if you adopt the viewpoint presented here that you will no longer have that problem. There is simply one type of transformation that has some of the properties of active transformations and some of the properties of passive transformations, and everything works just full-endedly. Thank you very much. Thank you for a very provocative talk. Some people have been understanding this. Some of the great patients, I agree with your conclusion, perhaps no with the first part. Two remarks first. Lagrange and the distinct state Marks-Reller consider dynamical systems in terms of general covariance and the notion of defunctable manifold, which a long time after that is present implicitly or explicitly, explicitly in some part of Thompson's in 1965, to the book of Article 900 of Renzi and Levi-Civita, the Mathematician Alan, it's not purely mathematics. We have both embarked in classical dynamical adaptation, in arbitrary coordinates, and

52:30 and continuous media and I know that Einstein used directly this paper much more and larger and so on. Oh, please don't misunderstand me. He did. He certainly did. The blame was that Well, from Elika Arthorn first, from 22, and let's speak from Elika Arthorn, and after, from Wittling, in 36, which is the global definition of differentiable metaphor, the first, 36, the modern art is genetic. So, I agree with your presentation, because poor Einstein has no good in global orientation at the time of the creation of the general relativity. But now, I agree with your conclusion, that is to say, the so-called covariance principle is Elie Carton, I, in the, not his certificate, in 31, exactly the following, I'll say perhaps in French. La théorie de la relativité générale de M. Einstein n'est pas, comme beaucoup de gens le croient, basé sur un principe de covariance générale ce qui serait une simple pathologie mais il est basé sur le fait que en chaque point, la géométrie de Newcastle est large the first theory general relativity of Mr. Einstein It's not based on the fact, as many people think, even Einstein, on the so-called Gawairo's general principle. But based on the fact that in each point, the geometry of Winkowski is very clear. So, you different of this, but for the people now, the geometrical model, in the geometric sense,

55:00 from the 36th, is exactly equivalent to the dynamical system given by, you know, but the remaining, the remaining, generally. And the components are locally used for calculus, but no more. And there is no general, according to your conclusion. But the earth is clear, that you can see the point of view of Einstein at this time, and it's not the point of view from its sense of physics. Yes, that's exactly what I was trying to say. I didn't want to be... If you have two parts, I think that I know the paper of Frenchman in Sydney, and I am finishing with just a remark. It's not so known that even the classical mechanics can be, in different forms, completely collided with respect to a whole change. is in coordinates together in an Arangian form in an Amitian form in their museum non-privileged coordinates and it's completely equivalent to the classical mechanics thank you very much can I ask you the presentation is from you

57:30 Also, if you may be right in general, that Einstein considered nameless man, he told. Nevertheless, in this, I think it's called answer through a paper by the Sitter, he remarked that this so-called mass horizon arises in one of the forms of the Sitter method. Maybe not a real mass horizon, but maybe a coordinate thing about it. But unless it is proven that this horizon cannot be removed by a coordinate transformation, it has to be opened up there, it's a mass transformation. And in this sense, I think Einstein has the concept of coordinate singularity in the sense of space-time. I can see the argument change that by the data. Okay, and then meeting Arbiter theorists in general, there's this discussion, I think it's in Anderson's book, there's a distinction between idea and special relativity in terms of general coordinates and interviews with its matrix Gij of X. And now the difference of them, with the vacuum general relativity and special relativity, lies in the difference of the structure of B equation. Ritchie tensor equals to zero, and total curvature tensor equals to zero. So it had a very different system of differential equations. So I think after making a theory generally covariant, one must look at the archipi, the differential archipi structure of the equation. And the theory behind that is probably one of the cut-down versions of the theory of differential systems. And so the discussion then has to use this theory, which is difficult. Robert Herman uses it. Yeah. I understand your point, though. You're saying that one can still have a generalized principle of relativity and general relativity

1:00:00 if I look to this tradition that you talk about. Is that the claim? No, I'm not claiming that how words are being used, but the way of analyzing the theory of art that has been made generally covalent in the sense of Frenchman. discussion has to be in terms of this theory of differential equations which is independent of the independent and the independent and that's the cut-downs program which he formulated at the beginning of the century I think he has given a description of his work of life at some time and there he says one of the basic ideas was to give the theory of differential equations which is independent of the kind of dependent and independent It's a very valuable mediterium, which is almost surely because everything can cut down, you know what I'm talking about. And the last one, Mark, is that in Kretschmann, he gives arguments in both directions. For instance, also, he gives an analysis of the curvature and the look of the eigent feedback doesn't seem to really make the new Kretschmann's work. And so this is less relevant because there are these specified directions at each point. Well, let's pause. Let me put a question to everyone here. Who knows what happened to Crutchland? Serious question. It's arisen in my work with my colleague John Herman. We found a paper he wrote in the early 20s and we don't know if there's anything further. I think I've seen about superconduction but I can check it I can check it I think I'm going to make the same point I think some disagreement on what is meant by General Covary I think we can all agree that Newtonian theory A if we want to use polar coordinates it's still the same Newtonian theory I think we can also agree that we are using Newtonian theory using a coordinate system rigidly connected with the Earth even though it's an accelerated frame of reference it's still the same Newtonian theory now in the same sense one can do it with special relativity

1:02:30 one can do special relativity in an inertial frame and eta no longer has a diagonal form like it is minus one, minus one, minus one and one can introduce accelerated frames of reference that means coordinate transformations involving both four coordinates and it's still the same theory the principle of equivalence has nothing to do with that because that makes a statement about gravitation we are simply talking about rewriting special relativity in different forms because in particular Falk who advocated that and it hasn't been elaborated by many others since then, you can use special relativity which is still the same physical theory, but you can use any accelerated train of reference you wanted. The thing which distinguishes it from general relativity as such is that you're always talking about a flat space time. I just want to disagree about the remark on the principle of equivalence. Since I've written a paper about it, I won't say it all again. Three things. First, I was very interested in your comment about Einstein using the word tensor and so forth. That was very interesting. I looked up the English dictionary and the first use actually of the word tensor was by Hamilton, if I remember. I don't think that's a disagreement in the context of differential geometry that was all I just wanted to confirm that you agreed to that, it wasn't a disagreement next thing was all the business about the coincidence argument and so forth, it appears explicitly in 1915, my belief is that a lot of that is already implicit in the 1913 paper with Grossman if you read that I think it is already and I would guess that Kretschmann was reacting to the 1913 paper if I've seen it a little bit more all I can say is I don't think so if you can show me where the other point is that the coincidence argument applies to the whole world as it does evolve the thing I think that really casts light on this

1:05:00 trying to say the previous thing is that I think you have to actually look at the specific structure of the theory which makes it possible from information given back here to say what will be happening up there that is something to do with very fundamental physical problems and I do believe that if you make this split into 3 plus 1 you see that general covariance in 4 dimensions as it was used by Einstein is actually incredibly important and incredibly valuable and incredibly powerful and that when Kretschmann makes Newtonian mechanics generally covariant. This is rather a superficial thing on the theory when it already exists, so to speak. But if you actually look at it in terms of how initial data is propagated there, you see that there is this very characteristic difference between Newtonian theory in the general case and general relativity. And I think that is where one really wants to look at it to see the real physical content of four-dimensional general covariance is that it's actually really in how it applies to the initial data problem. to go into that as much as I'd like to, but I think we could discuss that, I think that... There are two remarks. The first that Einstein learned the lecture, the lecture dynamics from Tertull, and Tertull has several chapters called Introduction in Factor Entrepreneurship, so he must have continued this, or he was educated, self-educated in this tradition. And the stapler, if I remember well, the first time he mentioned that the only similarities of importance in physics was in a letter to Besso in February of 1915, if I remember well. So perhaps it was before Kershman's paper, I have no idea about it. I can look after it in the paper, which I have to read myself, and I can check. May I just... May I just... May I just add a demurre there? And basically, we discovered letters of Einstein to his then-fiancé in a marriage. one can document, Einstein's reading of all of everybody that's later mentioned about the documented reading of Marx, also about the documented reading of Helmholtz, the one

1:07:30 name that does not occur is Föppel. However, the name that does occur is Derude. We know he was reading Derude's Ethere of Ethere, which was essentially the textbook on Maxwell's. That was the textbook which he learned from. When or where he later read Föppel is a matter of speculation, but as a matter of fact, there is no mention of Popple in those letters, but there is mention in Brutus' textbook. And if he had studied, one could have a counter-argument as well, although it's not decisive obviously. If he really learned from Popple and was so familiar with vector analysis, why does he always write things in component form, even as late in 1905? So I would not take it as axiomatic as he read Popple. Obviously, the argument is not decisive. It could always be that the letter where he mentioned Popple is missing, you know, and he could turn up 10 years from now. But there is no evidence for Popple. There is definitely evidence, however, that he read Brutus theory of the ether and studied as a student at A.T. I mean, not in class, but wow, wow. Well, is that on this point? Yes, on this point. The purpose of imperialism point is that Einstein's discussion of electromagnetic induction is exactly parallel to the purpose. Sure, but that's 1905, where the evidence that he was reading Brutus from 1897, 1898. about what he had read before 1904. Yeah, well, the stem was made, Einstein studied Maxwell's theory from Drude, from Poppel. I'd say there's no evidence for that. He could have read Poppel later. He could have read it in 1904, 1904 and a half. Well, it depends on Maxwell's theory. You know, he'd read Lawrence of 1895. Sure. But the evidence that he read this is earlier. He mentions all the Lawrence papers. He mentions in a letter in 1899, I think, or 1900. He's got to study, he's got to bubble down and study Lawrence's work seriously. He's already studied the Buddha, did I say, from the evidence of 1897, 1898. I think the evidence is the strongest thing that he read Buddha first. Not using vector form, not using vector form in the special relativity paper was actually beneficial for him, so he could display transformation property. Using vector forms of special relativity is a little bit, it can't be complicated. It can be, but it's not just done properly, for example, in what's-his-name's textbook. It can be done. Seminal places where things are usually not done. What I'm saying is there is no direct evidence that he went off.

1:10:00 I'm telling you, isn't there? I do. Probably you'll disagree with much of what I'm saying now. To me, the co-branded principle is, well, let's say this. The theories, the co-branded principles are theories whose basic objects are representation group. And so the physical interpretation is nil, except if you identify coordinate systems with reference systems of the physical observed. And you can't do this. However, as soon as this general group of difeomorphism is broken to a lead group then you get some preferred Gordon systems but this is done by the dynamics you didn't mention at all and there I come back to what Peter Harbaugh said because we just add that the curvature potential vanishes and that's the dynamics and that miraculously although it's a tensor equation does not restrict the Gordon systems but it selects of isometries, the extended Lawrence group or the Blunker group. So you get a set of preferred reference systems which you call inertial systems and so on and so forth. And Einstein theory in vacuum has the dynamics that the rigid tensor vanishes, but this does not select an equal. So you have no preferred systems except. Yeah, but the Lorentz group comes in with what you said, Merovitz, with the formulation of Carton, that the tangent bundle, or the manifold, has a specific characteristic, and that's where the Einstein equation comes. But as far as this identification of reference system and column system is made, this is sort of a thing I can't understand in Einstein's setting down theory. So you kind of say why Einstein would do it or don't believe that he did it? And when you say every theory can be formulated covariantly, okay, you take the heat conduction equation, you can do it. But if you look at it, you see that the general covariance group of Differomorphisms is broken down because an absolute object appears.

1:12:30 And I call an absolute object an object that is represented by a representation with regard to a D group, and not a group of Differomorphisms. So there is a distinction between the various covariance principles and the various theories. In Einstein's theory, you have nothing that breaks the group of the thermo-physicians. Whereas in all other examples we gave, you have objects, absolute objects that break this group down to a legal, And I claim that it's the dynamics of the theories that makes this project. I find myself agreeing with you. I mean, I mean, I mean, I mean, I mean, I mean, I mean, I mean, I mean, I mean, I mean, I mean, I mean, so the only thing I, I think the talk about certain objects has been something that's been pursued quite recently in philosophy of space and time circles to very little effect, I'm afraid. The basic idea is one that I think is, I mean, the most promising way, I think, of cashing out relativity principles is in terms of absolute and dynamic instructions, such as in the readers of Anderson's book. And recent work in philosophy of space and time simply found it impossible to have a precise criteria for what an absolute object is. You, I think, suggested one just through there. And I would urge you to write that up and send it off to a philosophy of science journal in order to inform the philosophers of science that there is a good definition there. The current work has tossed around many definitions and failed to provide any. I haven't heard that one, incidentally. Please share it with us. I'm interested to introduce two remarks. One is that there is an interesting discussion on the covariance by Wilson at the same time as the freshman. Yes, yes, I did. Where is that? I think it's in the proceedings of the U.S. National Academy of Sciences. Yes, yes, yes, yes. And the second one is all the confusion about the covariance and the general covariance and other things is just maybe explained that at the time when Einstein has formulated his theory, he didn't have clearly two ideas.

1:15:00 of the space-time structure, the geometrical structure of the space-time, and he didn't see exactly the difference between Scobaridge's general and invariance of the space-time structure. And he didn't use Nielke's law at that time, and so on, and if we introduce, according to Carlton, then all the things I think will become very clear, and the discussion over general covariance has a historical interest for it. If I could just say a word in summary, what has been emphasized for me from this discussion, is the need to keep very distinct historical context, right, to investigate the questions of what Einstein knew and when he knew it, and the context of what the modern approach to these questions is. And I could just remind you that it's not God-given that you use manifolds without additional structures as the basic mathematical tools for doing space-time theories. If, for example, the Einstein... There's no good theory on this side. But if, for example, the Einstein-Grossman theory, which is non-generally good theory, had been found to be the most useful gravitational theory, we would have to use manifolds with additional so-called absolute objects, as the mathematical object, and the pure manifold theory would be much less interesting to us. So Einstein, of course, didn't have, this is all Einstein, Einstein didn't know that he had to feel his way towards, and his work promoted the development of general abstract manifold theory, and you have to keep these two things very, very separate. By the way, I have a paper also on Einstein's search of general covariance, which provides a somewhat different point of view than Gordon's economics, One last remark. I was just wondering what you said, that Einstein was promoting this use of money, but what parallel concurrent developments are taking place in mathematics, which promotes this goal? So what hospitalizations do you have up this time? I think kind of might probably be the one who could answer that question. I don't really know.

1:17:30 You know what about it. In conclusion, we don't understand Einstein's theory better than he likes when he likes it. On that note, we're now ordered to have lunch, ordered to have a swim, and ordered to have a nap. Thank you.