Background Independence

0:00 We liberalize that, so it should be experiments not only perform within that context, but ones that are based on perhaps very different ways of theorizing and conceptualizing about the world. this thing would involve the possibility of experiments in such an enormously and completely unfounded space that it would seem that that idea would have to remain very much as a completely ideal object and not something that could bear any relation to an actual periodization of the I think no one really believes that we will be able to do any such experiments. But the hope is, at least coming from the stream theory side, that in terms of experimental verification, that there are sort of indirect things that we may be able to test. Like, for example, I showed you this picture, the background fluctuations, and I said that this most likely has nothing to do with quantum gravity, but nevertheless, the question is, still, there's a question as, what is this scalar field that gave rise to these fluctuations? Somehow the whole mechanism of inflation should be explained in a natural, intrinsically natural way from such a theory. But the only way I can see how that might really work is that we should be able to pin down the correct theory, more or less uniquely as to what it is, and we're still very far from that. That's the problem. and theories for quantum gravity, it will be very difficult to discriminate between them. I mean, some of them are maybe in obvious contradiction with the experiment anyway, but even if we can eliminate some of them, it will be very difficult to discriminate between different concepts if we're not able to narrow this number of choices we seem to have so far. Last question. Well, a comment for us to bring up the telecomological problem in my experience.

2:30 But you said that in your point of your screen slides that the string theory is . Well, it doesn't, but I mean, isn't there always an additional class of scale the field? No, no, no, yes, there's lots of extra stuff in strings here, in particular there's a dilaton, that's a field that always naturally appears, something that, this is for example something where one might get some experimental evidence, if you find deviations from the principle of equivalence, there are such experiments on the way. Then you would say, well, that's already first indication that, say, something like strings here is on the right track. In fact, this dilaton also appears in Calus-Lat-Line type, like series. You don't need to go to string theory for that. But indeed, there's lots of extra stuff. But what I wanted to say is that among this extra stuff, there's also classical space-time, but we will have a dilaton, we will have a standard model, we may have extra particles. And this is where loop-quonon gravity actually fails. They're not able to produce, out of the equations, a classical That's not equation. You know, something that looks like... It's better to say you have the classical thing as a hybrid representation. No, even that is... I didn't quite understand that question. The question is... Did you say those dilettons are going to lead to a violation of the control program? Yes. So that, theoretically speaking, is inconsistent with classical GR, Yes, because what you will find, I mean, we know that the principle of equivalence holds to very, very high accuracy, and therefore it would be, if you definitely find a deviation from it, that would be definitely evidence for something beyond Einstein's theory. And that, I think, is something that string theories would be very happy about, because that's precisely the kind of thing that string theory... But that's no problem, because... It's not a problem. in chapter 12, except Mr. Parsons. Yeah, yeah, yeah. Okay, thank you. Thank you.

5:00 Thank you. Okay, so we continue this morning's session with Oliver Poulet from the University of Okay, thank you very much. Okay, well let me start by saying that I don't, I mean in this talk I'm going to be saying some rather pedestrian things about background independence in classical theory. and suggesting that there are reasons to be sceptical that various ways people have tried to pinpoint what background independence is are on the right track. Having said that, I don't think for a second that there's something absolutely right about the loop approach in being background independent. So the question is, what should we say about the classical theories and what properties do the classical theories have, which are the analogues of the sort of properties that the loop quantum gravity theory has in comparison to string theory. Okay, so let me just motivate my reason for worrying about what people have been saying. It seems to me that there's a prima facie tension between two sorts of things you're likely to come across.

7:30 namely between the textbook wisdom about general relativity, and in particular the general covariance of general relativity, and what various people who work on loop quantum gravity say. So the textbook wisdom has a negative part and a positive part. The negative part is that despite Einstein's hopes, general covariance as a requirement on a theory is physically vacuous. that general covariance was a way of implementing a generalized principle of relativity, that it would abolish, you know, sort of something like Newtonian space-time as something that acts on matter but isn't acted back upon it. But then he pretty much immediately dropped that view in response to Kretschmann's objections. Okay, so that's the negative part. general covariance doesn't pinpoint what's special about general relativity. The positive part says, well, this is, you know, very roughly speaking, what's special about general relativity. And this, in a sense, is the only thing that's special about it. In general relativity, the space-time background is dynamical rather than fixed. And then there are various ways in which people have tried to pinpoint exactly what this notion of dynamical rather than being fixed is. Absolute Objects program initiated by Anderson that Michael Friedman had something to say about. And there are alternative approaches. I'll get on to that. So one thing I'd quite like to stress is this solely. And I think we need to come back to that claim in trying to pinpoint what's special about general relativity and in particular what background independence might be. So my thought is this, is that if you look at a special relativistic theory of a certain type, say special relativistic electromagnetism, and you compare it to a general relativistic version of electromagnetism, the way in which the matter fields, the dynamics of the matter fields, the kind of quantities that are observable, the role that the space-time background, oh, sorry, not background, the space-time is playing in determining the dynamics of the matter fields, all of that is basically the same in the two theories. You can either look locally at the local dynamics of the matter, in which case there's a sense in which the GR metric is just as much a background structure as in special relativity, or you can do the special relativistic theory in a background-independent way and treat this fixed field as a pseudo-dynamical field.

10:00 There's something artificial about that, but one thing I think is right about it. It highlights that in terms of the observable content, there's no reason to think that in special relativity... Well, there's no reason to think that the type of things that are observable in special relativity are radically different from the type of things that are observable in general relativity. So all of that, I think, is sort of just under the surface of the textbook wisdom, and I think it's basically on the right track. Okay, so the quantum gravity wisdom is the following. General relativity differs from special relativity in being background independent. Now, of course, there may be no tension here if background independent turns out to mean, you know, doesn't have any fixed fields. And perhaps that means doesn't have any absolute objects. But people go on to say other things. In particular, you'll find background independence is standardly linked to some substantive notion of general covariance. The claim that, well, the claim that I've just said I'm doubtful about, that there is this difference in the nature of what's observable. Okay. So, what is general covariance? Well, here are two sort of clearly non-sumstantive notions. general covariance 1 the equations of the theory transform covariantly under arbitrary coordinate transformations what I take to be pretty much the same thing actually there's a way of writing your equations which aren't in terms of a particular coordinate representation at all you can think of the equations as relating particular kinds of geometrical objects the standard wisdom is Any sensible theory, you know, our initial formulation of it may not conform to these constraints, but any sensible theory can be put into this form. Right, what about the following? If a particular mathematical object, so M is the space-time manifold and O are the various fields we find on it.

12:30 If something of that sort is a model of a particular theory, T, then this thing is also going to be a model of T. So if that's true of T, you might say that it's a generally covariate theory. Now, I think it used to be the case that people thought this was as trivial. But I think that would be moving too quickly. It seems to me that the recent things people have said about background independence suggest that that's moving too quickly. So, suppose that this particular object is a model of our theory team, where we're dividing the fields in this model into two sorts. The fixed fields, where these are just ones that intuitively then vary from solution to solution, and the ones which are the real dynamics. the real dynamical fields. So here are two questions. If such a thing is a model of the theory, is this also a model of the theory? And according, at least, to the way of thinking about it that I'm just about to describe, it isn't. What about this, right? If this is a model, is this a theory? Well, I mean, that really is a theory-dependent thing. It'll depend on the nature of the theory. So, let's... Okay, so let me give you this quote from Rivelli. Rivelli, in this passage, says three things. I'm just first going to look at the first two. So, he says, A field theory is formulated in a manner invariant under passive diffeomorphisms or changes of coordinates. If we can change the coordinates of the manifold, re-express all the geometrical quantities, dynamical and non-dynamical, in the new coordinates, and the form of the equations of motion does not change. So the sort of standard wisdom is any theory can be put in that. Any sensible theory can be given a formulation for which that's true. The more substantive thing is meant to be following a theory. invariant under active diffeomorphisms when a smooth displacement of the dynamical fields and the dynamical fields alone over the manifold sends solutions of the equations into solutions of equations. So Rubella's version of general covariance is the following. If this is a

15:00 model of theory, then so is this where we're only acting on the dynamical fields. Okay. So here's a non-diffomorphism invariant formulation of a theory. You think of the theory as involving a map. So this is just basically Klein-Gordon theory. The only dynamical field is a scalar field 5. But the way you think of the theory is involving a space K of kinematically possible models, where each point of K takes this thing with a fixed metric as a starting point and involves a scale of field. And then you've got some equations. I've written on generally per variant Klein-Gordon equation, picking out the subspace, which is your space of solutions. And if a particular object is a point here, then diffeomorphism R is going to be a symmetry of the theory in two senses. First of all, if you consider trying to do something like this, then this object isn't even going to live in K. And then if you consider doing something just like this, this is going to be a well-defined map on K, but it's not going to fix X. I write non-generally covariant equation here actually there's no reason to think that so you can think of the theory as involving this sort of apparatus but write your equations generally covariantly and that would be the sort of revalued way of thinking about it so thought of right the theory would be invariant on passive diffeomorphisms you'd be allowed to re-describe you'd be allowed to do a diffeomorphism in the sense of you know re-describe a point in here in terms of a completely new arbitrary coordinate system, but you wouldn't be able to do an active dipomorphism in the sense of use the dipomorphism to drag the fields around for them, because that would, well, if you did this, you wouldn't still have a solution. If you did this, you would be sort of throwing away the apparatus of your theory. Okay, so that's clearly what Rovelli has in mind. But what about thinking of the following Klein-Gorlin theory? Now we think of the base manifold as just the Venn manifold,

17:30 and the dynamic of fields as both the scalar field and the metric. But the equation of motion for metric is just the curvature vanishes. So now we still have what we had before, that doing this sort of transformation didn't fix the solution space. But that doesn't matter, because that's true for general relativity as well. It turns out that this does live in S, which should be S double prime, if it's listed in the solution space. Now, of course, Rubelli's intuition would be, well, look, the fact that this doesn't live in S double prime for GR doesn't matter, because for GR, G isn't a dynamical field. And the point is that you should act on all the dynamical fields with a different moment. But, you know, what's at issue is with what Wright will just barely say that in a case, when we formulate it like this, the metric for this version of special relativistic kind of theory isn't a dynamical field. Very interesting. Okay, well, so the question is, is G a dynamical field? So, let's just go on to the last thing that he said in this quote. He says, distinguishing a truly dynamical field may be a field with independent degrees of freedom from a non-dynamical field. disguises as a dynamic field might require a detailed analysis of, for instance, the Hamiltonian of theory. I'm not entirely sure how he thinks we're going to get this result from a detailed analysis of, for example, the Hamiltonian form of the theory. I think it's clear what he means by saying, I mean, I think it's clear that in our example, there's a sense in which this field, G, doesn't have independent dynamical degrees of freedom. But let me use what Rebelli says as an excuse to move on to something John Erman has claimed.

20:00 okay so John Ehrman is also someone who thinks that you can make out this interesting difference between substantive general covariance and non-substantive general covariance according to Ehrman the substantive requirement lies in the demand that diffeomorphism invariance is a gauge symmetry where that just means for Ehrman that you're thinking of two models related by diffeomorphism as different ways of representing exactly the same possibility. And he says it's substantive because this requirement is not automatically satisfied by a theory which is formally generally covariant. Okay, so let's go back to our example here. Right, what Ehrman seems to be saying is that, okay, the way you've set up the theory, it is true that if this is a model, this is also a model. But it isn't the case that automatically when you've got two things like this, you have to regard this object and this representing the same state of affairs, i.e. transformation from this one to this one as a gauge transformation. And Ehrman thinks that there is, in physics literature, a generally accepted apparatus that applies to a very broad range of theories that serve to identify gauge freedom. Okay, so we've got a theory which is diffeomorphism invariant in the sense that we've got a solution space on which the diffeomorphism group acts, and, well, a kinematic space on which the diffeomorphism group acts and it fixes the solution space but we want to know when those diffeomorphisms are gauged Ehrman claims that when you apply this apparatus to GR it tells you that diffeomorphisms are gauged transformations whereas formerly gently co-variant forms of special relativistic theorists need not satisfy. So why does he say this? the apparatus, right? So he talks about both the Lagrangian formalism and the Hamiltonian formalism. Okay, so I'm just going to talk about the Lagrangian case. So suppose that our equations, theories, are derivable from an action. We're going to say that a particular Okay, so a particular group, G, is a variational symmetry group,

22:30 just in case the infinitesimal generators of G leave this object, L, invariant up to a divergence term. And then we know where, if this group, G, if the parameters of G are arbitrary functions of the independent variables, we know that there are going to be S-independent equations relating the Euler expressions. And Ehrman's proposal is that in such cases, variational symmetries connect equivalent descriptions of the same situation, i.e. a gauge transformation. And it's this apparatus which is meant to tell us that for GR, diffeomorphisms are gauge transformations, but for the Klein-Gordon theory, they're not. And this is his, right, what I'm giving you now is from his paper. Okay, so we've got our non-generally covariant form of the Klein-Gordon equation, but, right, we could have written it generally covariantly and also got that equation from an action principle, okay? But this action principle is one where we only subject phi to Hamilton's principle. So this metric is not to be varied. And Ehrman's claim, so having pointed all this out, Ehrman goes on to claim, the action admits the Planck-Curray group as a variation, which is the apparatus sketched above rents the verdict that there is no non-trivial gauge freedom in the offing. Well, I think this is problematic. First of all, the diffeomorphism group is the symmetry of this action. we're not varying the metric just is irrelevant to whether this particular mathematical object is varying under diffeomorphisms. Secondly, Ehrman's criterion isn't really what he needs. So he says, if this group, he says, look at what physicists do. If the group is a variational symmetry in the sense of Nernst's second theorem, then it's a gauge group. Okay, so that tells us when to see gauge, right? But what he actually needs is, it's a gauge group only if, right? It's a second theorem, variational symmetry. And he's already said that his apparatus just doesn't tell us one way or the other.

25:00 The theories which, you know, aren't given in a, don't have a Lagrangian form. and finally I mean it seems to me that physicists see this went to see gauge apparatus as motivated by the desire to avoid indeterminism right it's not there's no other reason to think that these symmetries are gauge symmetries and this provides just as good grounds for regarding the diffeomorphism group as a gauge group in our going to flick back to our client okay so all of that gives us just as good reasons to regard diffeomorphisms in this case as gauge symmetries as in this case okay so here we've got gr and here's our earlier klein-gordon theory and here was our very the one we talked about first and basically here you've got exactly the same kinematic space you've just got different sets of generally covariant equations picking out different subspaces in the space of solutions Okay. So that's all to say, there's nothing that you're going to get from some sort of formal apparatus, which is going to tell you when, which is going to give you a difference between special relativity and general relativity in terms of the status of diffuborphisms. okay so let's go back to this idea that gr has no fixed fields but sr has fixed fields okay so this is how anderson motivated his absolute objects program right he says the dynamical quantities depend on the absolute elements and not vice versa okay there are various senses that depend, might, should, might be taken here, right? Does their very definition depend on the existence of this absolute element, or do their dynamics depend on the absolute element and not vice versa? In this quote, it seems clear that it's the second in mind, and absolute objects affects the behavior of other objects, but it's not affected by them. Okay, I like this. I mean, this does seem to me right in pointing to something that marks out gr from sr and also it does appear to connect to the problem that we had right at the start of the previous talk when you have einstein's equations and this bit's quantum

27:30 and this bit's non-quantum and you know that's not a tenable state of affairs right it's the fact that the geometry is acted on by matter that means you can't just you know leave it as it is so it seems to me that this way of thinking about it does connect to well perhaps not the problem of quantum gravity but why we have to why we're compelled to look for it in the first place ok, so Anderson's view how long have I got? might have to start going through some other slides Anderson's view is that basically non-dynamical fields are absolute objects, where absolute objects are fields that are the same in every model. One thing might strike you about this, you know, what's the connection between this notion of action and reaction and fields that are the same in every model? I mean, here's an obvious thought. If the field's the same in every model, then it's clearly not being acted upon by the other fields. Right? So it looks as if this gives us a sufficient condition for when an object is non-dynamical in its intuitive sense. Right? But it's not clear that it should be a necessary condition. Okay. Well, I mean, this definition was modified in one important way by Michael Friedman the requirement wasn't that you have the very same field in every model, but it's sort of the same locally. For any point, there's got to be a neighborhood at that point where the field in that neighborhood is locally diffeomorphic across models. Okay. Brian Pitts has written an interesting article recently on problems with Anderson's approach, and he certainly endorses this freedom of modification. But one thing he doesn't endorse, which was a modification that was sort of more implicit in the way Friedman discussed it, is that Friedman effectively restricts the subject matter to a fairly limited class of geometrical objects. It's basically, you know, the familiar tensors, connections, that sort of thing.

30:00 Whereas Pitts thinks you need to consider at least tensor densities to deal with the counterexamples we're about to consider. He also notes that whereas for Anderson the covariance group is a more abstract notion and the diffeomorphism group might be a subgroup of the covariance group of the theory if you've got internal signatures, for example. Friedman only talks about, well, sets things up in terms of the diffeomorphism group. I mean, that's fine for us, because that's all that's relevant to background independence. Okay, so let's consider some of the counter-examples. This one, which I think was due to Garrosh, and this one due to Toretti, they put in different directions. I mean, this one basically points out that Anderson's definition seems to be, well, sorry, Anderson's, it's really Friedman's modification of Anderson's definition, but it's the locality modification, and that's good. Ends up classifying objects as background, sorry, elapsic fields, which, when they're Tourette's example pushes in the other direction. It says, look, here's a background field, but your definition, sorry, here's an absolute object, but your definition doesn't class it as such. Okay, so in the Garrosh case, the point is that if you've got, well, any time-like vector fields are always going to be, any two time-like vector fields are always going to be locally dipheomorphic to each other. So, the velocity field for dust in a general relativistic space-time, for example, comes out as an absolute object according to the definition. Okay, a standard move is, well, rather than taking the velocity field as the object, sort of the basic object in theory, we should be taking the momentum current as the primitive thing. And Pitts endorses this and basically says, actually, if you looked at what Anderson said originally, Anderson has this prohibition on formulating your theory in a way which involves irrelevant variables. So, you know, that gives us a way of choosing this way of formulating it, even in the necessarily dust building. So, one reason for saying, actually,

32:30 thinking in terms of this is that U is just sort of arbitrary in regions where there's no dust, right? So it looks as if the formalism is in some sense suspicious, but then you can say, well, considering universes which are necessarily filled with dust, right? Writing things in terms of row U looks as good, but Pitt says, no, there's something in Anderson which can favour fight in that case. Tourette's examples involve theories where the theory says space-time has got a fixed you know, space-time has fixed spatial curvature and then maybe I put sort of standard special relativistic matter equations into that space-time but actually the value of the fixed curvature can vary from model to model and one way of getting this kind of thing using field equations is if to the metric that the Barthel tensor vanishes and the Einstein tensor vanishes, then that forces the metric to be a decision metric, but, you know, it doesn't fix the value of the curvature, the radius of curvature. Okay, so one response to this is, well, if we split the metric into a conformal metric density and a scalar density, right, then this turns out to be an absolute objective and sense. And just thinking about geometry, suppose I mean, there are different ways of thinking about it where you might think that actually this is a natural quantity anyway. You know, if you think that the length scales are basically you know, lengths all to do with, scales all to do with, you know, comparisons of congruence, then you might think things should be, well, actually, let me not say anything more about that. Here's what I find unsatisfying about these sorts of responses to the counter-example. Okay, it's, you know, even if we were forced to treat this as the correct formalism,

35:00 right? Intuitively it shouldn't be classed as an absolute object. So the response is, well I'm going to stick with the Anderson machinery and I'm just going to say for this particular theory I'm going to change the variables in a way which is going to mean that the theory no longer has any absolute objects in Anderson's sense in it. But that's the wrong way to go. Even if we were forced to think of you as an element of the theory, intuitively it's not a background field. The answer shouldn't be, well, let's reformulate the theory without U. The answer should be to modify your definitions to explain why U isn't intuitively a background field. And a similar thing can be said here. Even if I was forced to take the metric as the basic variable, the fact that there is this slight variation from model to model intuitively doesn't stop the metric as being a sort of background or absolute objective theory. I mean, just to mention in this paper by Pitts, having tried to salvage Anderson's approach in these various ways, it turns out that this very move causes a problem in GR because you can split up the metric in GR and it then turns out that the volume element is an absolute object in Anderson's sense. Okay, so maybe Anderson's approach to consider fields that were at least locally the same from model to model is the wrong way to go. So one proposal was made by Huskies in, I think, the mid-80s. Maybe non-variation fields are those fields that we don't vary in the action. So let me go right back to, you know, this stuff from Anderson. Sorry, from Ehrman. I mean, the point about our... I mean, this is the action principle version of our... Okay. So, right. Here, we want to say that despite difamorphisms being a symmetry of this theory, the metric is intuitively a background in this theory. And maybe what's going on is that, well, when we consider the actual principle version that corresponds to that theory, we don't vary this.

37:30 So that was Hiske's idea. Well, the problem is that you can come up with new formulations of the theory where you still have the same fields, but you do vary them in the action. Okay, so I didn't need to go back. This is, I mean, this is, okay, remember what I was saying at the start. There's a sense in which GR and SR, if you focus on particular things, are really not very different at all. So you might think of the models as being the same kind of thing. You've got these generally covariant equations. The equations are derivable from an action, but, yeah. When you say special relativity, it means you take the tree, here's a background metric, but it cannot be the big option. well in this case we're treating it as flat but so in terms of global topology it need not be so I should think of it as a flat but possibly although we can get these equations from action principles there's something odd about this one because we're not allowed to subject g to Hamilton's principle if we do we get triviality ok but then Sorkin set things up like this, right? Now we have the following action, right? We have the standard matter Lagrangian, but we've replaced the gravitational Lagrangian with this Lagrangian density, where this is a new field. And when you vary this, you just get the flatness condition popping out of the field equation. And then, of course, you get another set of field equations, Other dependent variables in this action are subject Hamilton's principle, and when you vary matter, so when you vary the metric, you get the standard matter fields that are common to all the theories we've been talking about, the SR theory, GR itself, and this theory. But when you vary the matter fields, you're going to get a new set of equations for this new field. And one thing you're going to get from that is the stress-energy tensor of the matter field's batches. Excuse me, but I simply have to worry about, I mean, imposing, say, vanishing the Greenland tensor by Lagrange multiply.

40:00 I always have to worry about LaGrange, multiply becoming dynamical in some sense, which here do you think of the theta as the probability of the real theta? It really isn't LaGrange multiplier. Well, you think of it, so I'm thinking of it as, in a sense, a physical field. I mean, our models for theory have this field sitting in there, but there's a sense in which, you know whatever its dynamics are it doesn't matter because you know we're thinking of the observable stuff as all being in the theta dynamics of the metric and the theta are fixed by the other field equations and so it's almost as if the matter provides a fixed background you know acts as a background field for the theta I mean that's what I'm going to go on to say that's the word that we should look at so it's as if matter acts on but the theta doesn't act on energy at all. Did you say it's the stress energy? I think, well, certainly, if you... Yeah. Sorry, it's... No, it doesn't... I meant the... Yeah, I did say that. I shouldn't have done it. It's the invariant divergence. Of the mass... Sorry, the stress energy. Yeah, but I mean there's a route to it from, so I mean there's a route to it from the equations you get when you vary the matter. okay so i mean one response to this is well look you've changed the theory now right this isn't just a reformulation um you've got this new theory this new field here um but i mean again that seems to be the wrong thing to say because it's not enough to say oh well this is a different theory in which are, in which the metric is a dynamical field, right? Intuitively, it's not a dynamical field. It is a background field, even in this theory. So, you know, there seems, whether you vary the field in the action principle doesn't seem to have anything to do with whether it's a background field. Here's another example, right? So,

42:30 So, in the old way, we could have models of exactly the same kind as GR. But now I feel the equations are the standard Klein-Gorn equation, but it's the vacuum Einstein equation. So, again, we've got an action. well, we can think of getting these equations from actions but only if we don't vary G in when we're requiring S1 to be stationary or we can do this Sorkin thing and have a Lagrange multiplier to get the field equation that the Einstein tensor vanishes I mean, notice that amount of arbitrariness here because we could have also had the Ricci tensor And then we would have had two different theories with different field equations for the Lagrange multiplier fields, which really is sort of highlighting one defect of this. I mean, the Lagrange multiplier field is in no sense, you know, an observable physical field. But, you know, if we're to take this formula seriously, that's how we should think of it in these theories. So, what should we say about metric in these theories? it seems to be dynamical in every sense that you could want it varies in non-trivial ways from model to model and you get it to field equations from an action principle where all the fields are varied is it a background field well it seems to me that there's one sense in which it obviously is even though it's a dynamical field and it's in the sense that Anderson originally gave us in this program, in this theory the dynamics of the metric are just what they are independent of what matter is doing whereas the dynamics of matter is tied up to what the metric is. Okay. basically finished. So the moral is that the machinery of diffeomorphism and variance It's a bad guide to what fields are absolute, fixed, background, whatever.

45:00 And also, the moral is that whether a theory is a sort of non-dynamical, sorry, whether a field is a non-dynamical field, whether a theory has absolute objects or non-dynamical objects, I mean, clearly that is one way in which general relativity differs from the special relativistic theories we're familiar with. But it's not the only novelty about GR. because this is a theory which in some ways is quite unlike GR it's got a field which is dynamical but in one sense a background is a background field it acts but it's not acted upon there are things that I haven't discussed one thing that people say but doesn't really get explored in all of these formal proposals about pinpointing what it is that makes them feel the background field. It's the idea that background fields help to define kinematics and the dynamics of other fields. And this is certainly something that, for example, Lee Smolin has focused on in his sort of intuitive gloss on the difference between SR and GR. Well, I mean, go back to this way of thinking about it. It seems to me that when you set the theory up in this way, it is the case that the metric in this theory is sort of playing that role, right? Where, you know, we've got this sort of fixed background with respect to which we're defining the dynamics of a particular field, right? but it's not clear that the metric in this theory which we want to say is very unlike GR has any more of a sort of kinematic defining the kinematics or the dynamics of the other fields than it has in GR I mean it's certainly true that in this theory the metric defines the dynamics of the matter fields right at the beginning that's true in GR as well. When you look at the matter, the field equations you get when you vary the matter degrees of freedom. Right? The standard special relativistic equations, basically. The role of the metric in those equations is exactly

47:30 the same in these two theories. Okay. So there is this thing that no one's really focused on yet, which is the idea that background fields help define the kinematics of fields. And, you know, it seems to me that's not going to give us an interesting difference between SR and GR. Okay, also the relationship between the gauge states of dipheomorphisms and also the nature of observables in the presence of background fields. Again, it seems to me that you're not going to find an interesting difference between general relativity and special relativity here either. If you formulate a special relativistic theory, generally then you shouldn't think of diphenomorphism as gauge transformations in that theory. And if you think about the observable content of special relativistic theory, artificially thought of as a cosmological theory and general relativity, it seems to me that there's no difference. I mean, one common thing people say is something like the following. In special relativity, the value of a field at a point, right, is observable. And where they might think of that point is being defined by some lab frame. Well, it seems to me that's the wrong way to think about it. I mean, first of all, you should be comparing the sort of cosmological interpretation of the special relativistic theory with general relativity. So you shouldn't think that the only way of understanding special relativity in terms of... And the sort of coordinate systems of a special, the special coordinate systems in a specially relativistic theory, I mean, you shouldn't think of them as defined in terms of, you know, sort of lab frame outside the system. That's not how you should think of inertial frames in, you know, Newtonian gravitation. Okay, so that's the first thing. Okay, so how should you think about it? Well, these coordinate systems are encoding, you know, metrical relationships between field values. So, the fact that you can, you know, in the theory you treat, you know, phi of x as if that was an observable, means that you set up phi, and then you later measure phi at x, which means you measure it at a certain metrical distance from, you know, your initial phi. And that's what's observable. But that looks just like a GR observable, right?

50:00 is a value of phi at one point, right, the theory predicts that if the value of phi is such and such at this point, then in a certain spatio-temporal distance, the value of phi will be such and such. And that's a completely diphthomorphic invariant fact. Okay, so final thought. Well, it's all very well to say all of these definitions aren't really getting at this intuitive idea of action-reaction. there anything um to be said about making that idea precise and just a sort of throwaway first thought um here's one way of putting the difference between the sort of toy theory we had here and gr right if you consider it in terms of initial data right you consider a space like hypersurface you have the you know metric data and the matter data on that surface and then you ask, what do I need to take account of to work out what the metric is going to be to the future? And the answer is, I only need to worry about the metric data defined on that hyperservice. If you ask, what data do I need to take account of to, you know, find out what the matter is going to be doing to the future, I need to take account of both the matter and the metric data. So that's one way of sort of trying to pinpoint the So that's just the first thought about it. Thanks. Thank you for the very fun-grained analysis. General commandments, background dependence. And Alexander? I'm probably sort of jumping ahead of the train, but I want to go one step further right away and see what these extensions become in one grade there. So, when you add one component to this. And it seems to me that the whole issue of whether there are backgrounds or dependent fields now shifts to something which is in a way conceptually different. of whether, because you know you cannot talk about objects at all when they are built or not, so the question is now whether you know the sense of which we talk about the background is the presence of the manifold is linked to the very possibility to talk about geometry, to make distinction, to see where geometry sits in the motor.

52:30 So, this kind of distinction is something which is still very much couched in the possibility to call something an object, whether it's a real object. which possibility may not be available in quantum gravity, in air quantum gravity. So, is your thought that to the extent that the quantum gravity, the theory of quantum of gravity is background-independence, is background-independent. You don't have this ability to talk very well. Well, yes, I'm saying the problem, the way usually the word background is used in the way, let's say, at least Molly talks about gravity. It's not so much about whether there are background-independent fields, because we don't know yet whether there are any in terms of both like such things. But it's whether we can tell where a mozo in a background manacal structure can all. There is a manacal thing on it. There is a special temporal thing. So whether one can actually, you know, your analysis starts by saying, here is a new gene, here is the metric. way general relativity works now i'm saying the problem seems to become something different when you can actually not separate the existence of geometry from from the question of what lives in this geometry the very question of the existence of the background becomes a problem Here, we're talking about objects as being background-independent or not. I guess, I think that thinking about whether certain objects are background-independent,

55:00 backgrounds. So it's not whether the objects are background-independent, it's whether certain objects are backgrounds. So not, I mean, I think there's, you know, I think you're going to gravitate the manifold, it's just going to be thrown away. But that's sort of by the by. I mean, what's interesting is that you're not treating the various fields. So if you're looking at the classical theories for some illumination about what's going on. It's that you're not going to be treating the fields differently. But for instance, I think you would agree that the space-time of general relativity is not the Riemannian manifolds, but is the class of all Riemannian manifolds related by the form of it. So this is an object, this is a background, this is not a manifold, this is a class of manifolds, but it exists, it has a perfect meaning. And this is exactly covariance. And another point is that you mentioned covariance or chromatism invariance like gauge transformation, which is perfectly right. And you use coordinate transformations as representative of the different ones. because this makes the formalism of general activity like the formalism of gauge theory. But you have another choice, which is never use the components of the tensor or the vectors in the language, but instead use the tensor formulation. So, for instance, instead of writing g nu nu and nabla nu 5, you will write that as the action

57:30 And here you have never coordinates, you have never frames, you have never small letters in pieces. So this is combined. And by definition, because the sensors are in your own business environment. So you don't have to think about the gage transformations. So I think it's true that the concept of gage transformation can be very important for some but if you want to discuss I think the best thing to do is just to use the gage transformation But I think the, okay, so just dealing with the last claim first, I mean, okay, so I did in fact set it up using, writing it down as if I was worrying about coordinate components. But that, you know, I could have written the equations to make it more manifest that, you know, even in the case of this theory, or even in this one, what's going on is, Do you actually have no indices out there? Well, yeah, that's true. But I fear, you're just too small. No, I fear nothing. Yeah. So, you know, it seems to me that there is important differences between this way of thinking about it, and this way, and this way, and this way, right? And in all cases, I can write the equations and talk about what I'm doing without saying anything about components. in the case of thinking about this theory in terms of an actual principle whether I submit a certain field whether I require that a certain field base is just irrelevant whether I write things in a kind of freeing way I also did want to reply to the first first claim I probably don't agree that the basic object of GR is an equivalence class of Ruhlian metrics. It seems to me that the way to think about the way models of GR represent the world is that, you know, I take out of the equivalence class, I take a particular one. And when I do that, it's the manifold equipped with that particular metric that represents space-time. But, you know, I could have picked another one, and it would have been equally good.

1:00:00 and there's nothing in this model that isn't in this one which is of any physical significance but the thing that's apt to represent space time is the manifold equipped with a metric, if I then think of the mathematical object which is the equivalent class of manifolds equipped with the diphenomorphic equivalent class of certain manifolds equipped metrics, I mean that's a mathematical object which is not suited to representing space time You can choose the gauge and work in this gauge, or you can work with the gauge invariant projects. You always have the choice. Even in spatial relativity, even in geography. This is the same in geography. What is the surface of earth? This is the class of equivalence of some manifolds which are formed. So, general relativity, spatial relativity, geography are covariance. So, but I just wanted to make the point that, you know, you've got to bear in mind that it's not a sense in which, oh, it's, you know, really the points in this, well, it's not really the equivalence classes that are the things that represent. I mean, you've got to be careful about how you understand that thing. Right, so that's all I was saying. But even if you go to the equivalence class, you've still got the notion, you don't, it doesn't pop out that that's then the background object. because, you know, a particular equivalence class is one solution and a different equivalence class is another solution. And it's this difference from solution to solution that, you know, suggests... It's the fact that you've got nothing common from solution to solution apart from the base manifold that suggests there's no absolute object or background. Okay. Could you go back to that applying border, the dynamic background? Yeah. So what is the sense in which the metric acts but it's not acted upon? Well, it acts in exactly the same way as we think of the metric as acting in. I understand how it acts. How is it not? Why is it not acted upon? Well, because the field equations are just the field equations of vacuum geom. Oh, because there's nothing on the right-hand side except zero. But, you know, I wouldn't feel really uneasy with this, apart from the philosophical aspects of this, but I wouldn't feel uneasy if somebody gives me a Lagrange and tells me, well, there's some set of fields with which, with respect to which you're not supposed to vary, keep them as a background, the other fields vary.

1:02:30 You must always watch out that we do not run into some kind of unhabilitation.