David J Baker: Spacetime substantivalism & Einstein's cosmological constant / Cartesian ontology of GR

0:00 Speaker is David Baker from Princeton University. He's going to talk about substantivalism and the ontogenity of all of the concepts. Okay, so the basic idea here is to take some new physics or, you know, more accurately speaking, some old physics that's newly come back into people's headlights. And try to apply it to this old problem in philosophy of physics about the ontology of space and time and see whether it can rule out one of the two sort of babbling positions in substantivalism and relationism. So let me just give you a little road map of where I'm going. I'm going to start out giving an overview of the ontological problem in GR, the controversy between substantivalists and relationists in different possible positions. Next I'm going to show how the cosmological constant lambda appears in the field equations of GR and what sort of effects it has. And then finally I'm going to show how lambda can strengthen the causal argument for substantivalism if in fact it takes the form that Einstein originally supposed it to. Here we have some candidate ontologies for space, time, and GR. The first one is the one that I want to defend here, which is sophisticated substantivalism. Sometimes it's called metric field substantivalism. There's a related view of metrical essentialism. It's very similar. The idea is basically that space-time consists of the metric field, as well as the set of the manifold of space-time, and that this should be viewed as a substantial particular. And the alternatives are, first of all, manifold substantivalism and associated relationism, which is just the opposite of this, which is the view that spacetime is properly understood only as the manifold of points, only as the differential and topological manifold, and that the metric is, instead of being part of spacetime, a physical field in spacetime. And then the final position is liberalized relationism, which is, you know, the view that it's the same understanding of space-time that takes place in the sophisticated substantivalism view, but it's saying that this is just a set of, or it's just a set of relational spatio-temporal properties of objects, and so being, you know, just a conjunction of the properties of objects, it's not itself a substance.

2:30 The next step is to reject manifold substantivalism, I think, because I don't think it's very philosophically worthwhile. So I have a few arguments against it. The first is that I think it's a very impoverished ontology in the sense that it takes a lot of properties that we normally think of as spatio-temporal and that really define the way that we talk about space and time, like distance, space-time interval, past, future, things like this, volume. And it says that, in fact, these are properties of a field of force or a physical field or, you know, a geometric field is a way that they sometimes describe the metric. And I think that if you're going to take that position, you need a very strong positive argument for it to chuck those intuitions out the window. And I don't see how the promoters of that view could put forward such an argument. The second thing is that I don't want to talk much about the whole argument here, but in light of the whole argument, the manifold substantivalist response to that is to say, well, space times related by diffeomorphism represent different physically possible situations. And recently, Oliver Pooley has written, you know, this involves basically the attribution of primitive identity to the spacetime points. And hexeity or primitive identity, I want to say, is kind of bad metaphysics. So that's a metaphysical reason to reject this. The third argument is that, you know, Supporters of manifold substantivalism will talk about Leibniz equivalence between space times as being equivalence under diffeomorphism, but I think that's sort of a distortion of the original debate between Leibniz and Clarke, you know, involving Newton, which is, you know, which was more about... The fact that, from Leibniz's perspective, objects can be translated in space by a certain distance with no empirical effects or a boost of object's velocity without empirical effects.

5:00 And specifically under this view, Newton's bucket argument doesn't have anything to do with substantivalism versus relationism because it's about inertial structure, and inertial structure is a property of the metric. So those are some reasons, I think, to get manifold substantivalism out there, or out of there, rather. So we're left with sophisticated substantivalism and liberalized relationism. And now I want to give this argument that's been suggested in a couple different places for substantivalism about the space-time metric. So it starts out with what I think is a plausible assumption, although maybe people will want to argue against it. That acceleration, or, you know, equivalently a change in kinetic energy, is something that requires a causal explanation. If we see something accelerate, we want to know what the cause of that acceleration was. And, you know, it's very clear from looking at GR that gravitational waves, that gravitational radiation is capable of accelerating objects and creating kinetic energy in detectors and things like that. If you deny the existence of the spacetime metric, it's hard to see what a continuous causal explanation could be for these sorts of accelerations that we see. So the upshot is... Here we have a relationist rejoinder that I think works pretty well, which is, you know, in any physically realistic case, of course, a gravitational wave is going to have a source and a... Subsequent titles might say to that as well, where did the energy go between emission and reception? The response to that, which is due to Colonel Heffer, his paper on conservation of energy in GTR, is that, you know, GR doesn't really have a strict conservation law. You know, total energy is sort of conserved in asymptotically flat spacetimes, but our spacetime isn't like that. And additionally, gravitational energy takes the form of this pseudo-tensor, which, among other things, isn't localizable.

7:30 So we have good reason to reject the idea that gravitational energy is really energy. And so, what we can say then is just, well, there's some energy that disappears at the point of emission, and some energy appears at the point of reception, and that's all that happens. So, you know, I think that argument basically is a way out for the relationist, but I think it runs into problems if you introduce a cosmological constant. So now I have the obligatory equation, you know, which is the only one that will appear here, where you just step land into the field equations like normal. What's important about it, of course, is that lambda leads to a constant positive scalar curvature or an average curvature of spacetime. And this is going to lead to accelerating the expansion of the universe. So free-falling objects are going to accelerate apart in a universe with lambda greater than zero. And one of the phenomena, so the acceleration is proportional to lambda times the distance between the two. One of the things that this leads to is that when two points are far enough apart, you know, they're going to be moved apart fast enough by lambda that, you know, this is sort of a heuristic way of looking at it. Actually, there's a more technical explanation, but you can look at it this way. They're going to be moved apart faster than light, and so then you have a de Sitter horizon, which is a light-like surface or a causal barrier. That's going to be important to my argument. Some things to note about the physics here. The first is that physicists will often use the term cosmological constant to talk about a contribution to gravitation from the vacuum energy density in quantum theory. And I just want to set that aside for now sort of beyond the scope of this study. I'm talking about lambda as Einstein originally envisioned it as a constant in the field equations that's not derivable from material sources. And it's also interesting to note that when Einstein introduced the cosmological constant, he was motivated in large part by relationist ideas, you know, mocking relationist ideas where, you know, originally it was thought that with lambda, the field equations had no matter-free solution, which he saw as being necessary for a mocking interpretation of GR. And in fact, when the decidere universe was discovered, that was later disproven.

10:00 And another thing of interest is, you know, these recent observations that you may have heard about in cosmology, that there is an accelerating expansion to the universe. Now, a lambda of the sort that Einstein envisioned isn't the first place that people are turning for an explanation, but it is one of the possible explanations of this sort of behavior. So lambda does seem to be, again, something of a live possibility these days. So I'd like to give some ideas now for what an ontology would be for lambda. And I think that a good first pass is that we might consider it as being kind of a ground state for the gravitational field, sort of a basic amount of curvature that space-time has by itself, and then when you introduce matter into it, that sort of perturbs it and adds additional curvature that, and of course, you have to qualify that because, of course, the perturbation is nonlinear, as usual, since this is general relativity. But I think still it's a useful way of looking at it. And an equivalent way of looking at this is that you can say that space itself is a source for the gravitational field in the universe with non-zero lambda. And I'll go more into that in a second. But first, here's the actual argument. So what I'm considering here is a test universe. It's a de Sitter universe, so it's empty of matter, and it has lambda greater than zero. So every observer has a decided horizon that no causal signals can cross. Here we have two objects, O1 and O2, just test particles, and they're outside of each other's horizons. And you'll see that, of course, by the effect we described before, they're going to accelerate apart. And the question is, what is the causal explanation for this acceleration? Because, in fact, it will be observable if their horizons overlap. There will be a few observers around in here who can see the two objects moving apart. So what is the explanation for that behavior? And of course, since by stipulation they're out of causal contact, the exclamation for O1's acceleration can't be O2. And it seems like the only thing that's in this test universe that we're talking about is the spacetime and the scalar curvature that I talked about before.

12:30 So that seems to be where we need to look for our causal explanation. So that would give us reason, I think, to say that in this universe, spacetime has causal effects, and so we should say that it exists. And as far as energy is concerned, you might wonder whether this sort of argument I don't think it can because lambda's energy is localizable. It's basically described by something like a stress-energy tensor of the normal sort. It's just lambda times the metric. And so, you know, this is localizable, unlike the gravitational pseudo-tensor. So I don't think the same arguments can be used to say that this isn't real energy. So here's just another way of putting the argument. And the idea is that lambda, you know, or at least the energy that it describes, you know, it seems to be a causally efficacious property of something out in the world. You know, you vary lambda, you see different behavior in the objects. So in a relationist universe, a relationist would want to say that all properties are properties of material objects or material fields. But lambda's effects, it seems very counterintuitive, very difficult to describe them as properties of objects because they don't really depend upon the matter configuration of the universe. It has the same gravitational effect no matter what the arrangement of objects is. So to call it a property of the object seems pretty mistaken. And as sort of a parting shot, I want to say also, it seems kind of strange to say from a relationist point of view that you could explain the energy density of lambda because, you know, on a liberalized relationist view. You know, you want to say that empty space is just a set of possible relationships that objects compare to each other, a sort of spatial temporal possibility, and to say that these facts about what possible spatial relations are determined, you know, are, you know, so for instance, in a closed universe, the volume of empty space is going to determine, in part, what the total energy in that universe is, and...

15:00 So the idea that these sort of possibilities could determine something like that seems very counter to relationist ideas and conclusions that I hope that I've drawn and that I hope people have drawn, which is first, as I began with, I think the only two viable ontologies for space-time are sophisticated substantivalism and liberalized relations. In a universe with non-zero lambda, it seems to me like the physics seems to rule out the relationist interpretation. So, it seems to me that relationists, you know, had better hope that experimentally the cosmological constant turns out to be zero. We've got plenty of time for questions. What's the set of constraints between zero plus the boundary term when you integrate that with the same boundary? Okay. No, that's for the revision. Yeah? What if we just forget general relativity for a second and think about particles obeying the law of inertia that live on? In that case, say you have a compact manifold with constant negative curvature, two particles will move away from each other, sort of exponentially, locally speaking. I guess the same kind of behavior you'd have to consider in space-time physics, other than space-time constant negative curvature. Do you think there's any, if we were talking about, sort of... Would you be tempted to say all the same stuff, or would some things be different? I think I would be tempted to say the same thing. But there's no... there's nothing sort of playing the lambda role there. The physics is... I mean, it seemed to me like you were happy to concede to the relations, but in a sense to speak of the geometry of a close space-time. There's no problem with that, and the problem had to do with... Well, so, looking for a cause for the acceleration, but, I mean, if you can sort of say straightforwardly, these things are just obeying the law of inertia, they're just moving along the straight trajectories of the space they live in. Right. Yeah, I mean, so, even in cases, though, where objects are moving along what we think of as locally straight as trajectories, we do look for explanation for that, you know, in practicality when we're doing gravity, you know, when we see something...

17:30 And so, following a geodesic path that happens to accelerate it, you know, we look for what the mass is that's distorting the space time. So, I don't know, I suppose that in just a space that was sort of curved by stipulation, it might be a slightly different situation because you're not talking about a space that's a dynamical medium. So, that's true. I would have to, I think I might have to revise some of my views. If we had, for example, a black hole, which is a vacuum solution sometimes, then there's net time, and yet... The object's magnetic field to be holding the relational cause of the motion unless we've got a substantival. Isn't that an argument for substantivalism? And then I guess the real question is, why isn't that argument just as strong as the argument that you want to give to Landler? I mean, I think one of the reasons that it's not as strong is the energy issue. You know, so yeah, I do think that Kerr-Beck argument is our sort of prima facie argument for substantivalism. The problem for that, as has been pointed out in the literature, is that The gravitational field only carries a pseudo-tensor of energy, so among other things, you know, if you look at the value of that at a point, any coordinate transformation, you know, for any point you can have a coordinate transformation taken to zero. Right, so, but in this case, I think what you have is, you know, you have a term that's on the left side of the field equation, so the sort of first pass idea that is describing a property of spacetime, and it does have... In the original causal argument, the way it ended up there, I think, well... So what kind of liberalized relations do you have? Yeah, I guess in a way this slide is pretty sloppy in the way that I wrote down the argument here.

20:00 Really, I would say that the strongest way of putting it is that you want to say that you have... Subjects include mathematics, geometry, algebra, mathematics, quantum mechanics, physics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, Yeah, I mean, so I think the answer is that the slide here is sloppy a bit, but I do think that there's reason from this to believe that the metric is a substance because, you know, otherwise there's nothing local that can be said to cause the motion. So, yeah. I guess, so I'm interested in how you can go from that very simple world, where maybe your argument holds, to say something about our world, right? Because objects in our world, in fact, aren't sort of causally isolated from every other object, right? So, here's one way you might try and do it, and then I'm going to try and block it off by beating around the drum, I guess. So you might say, well, what this shows is any world, you know, you might try and run this argument, well, look, I've given you a world in which GR is the laws of that world, and so I can run the argument there, and so I should be able to run the argument in any world where GR, general relativity, is the law of that world. Okay, but suppose you advocate a sort of Lewis-style best-fit account of laws. Then it's just not true that general relativity represents the laws of that simple world. So one might say, by stipulating, I mean you're assuming general, according to the relationists, that general relativity wouldn't hold in that world. So your example is sort of stipulating general relativity holding in that world, which is already adding more in than the relationists would concede. I mean the simplest law is just going to be there's two bodies moving apart at such and such a rate or something. That's been actually a pretty good argument. Yeah, right, so now I do think that even if you reject the argument about the sort of causal situation in the toy model, you're going to have to say something about the energy density as well.

22:30 So, I think in that way, you know, it might lead you to something like, for instance, a substantivalism about space slices, sort of like what Cooley has advocated recently, because what possesses the energy isn't space-time, it's going to be a space, you know, it's going to be the spatial extent. Yeah, I think that's a pretty good response about the twi model. It'll be the case in our universe, though, if our universe is enough of a deciduous universe, that, you know, you'll surely have some situations out there where you have two objects that are pretty isolated from each other outside of each other's horizons, and you still get this sort of behavior. Like, you could have very far apart pieces of debris out in an intergalactic space that are going to behave roughly like what I described. So... Which one of them will have some other stuff in it that is inside its horizon? Yeah. I mean, the idea is that the relations can always stay the same thing as the... Substantialists would say that it's not an independent entity. Right. So there's a couple of things that could be meant by that. One of them would be that it would be like a matter field that is a source for... So that's sort of the... that's what these sort of quantum vacuum pictures lead us to in a way. And, I mean, for one thing, it would be kind of strange if you had a field like that whose only interaction with other matter was gravitational, which is what the, so I would be, I'd be hesitant to say that there was a matter field there that doesn't have any interaction with other matter through the other forces. You might also look at it as being sort of a field of force that, you know, operates separately from gravitation.

25:00 So, you know, you might have, you might say that, well, there's this force field that acts on things to accelerate them apart, and then in addition to that we have general relativity, which is our force law for gravity. I mean, since GR is non-linear, I think that would be a very difficult position to hold as well, because you're not just summing lambda's contribution with other material sources. You have to... I have a question. Oh, yeah. Well, okay, let's just go. Oh, okay, yeah, yeah, go ahead. Some of the remarks you made when you were answering my earlier question here seemed to me to be in tension with your sort of... More or less acceptance of the reputational wave energy just disappearing here and coming in there, because you said you needed some sort of continuous... Yeah, that's, uh, yeah, so I guess I did sort of assume a denial of some of the stuff that you argued for. Um, yeah, um, I mean, so, in that case, I guess that... I guess that I would cast the relationship response as being sort of a setting aside of some of these locality ideas that I talked about and that that I think, well, it's not a move that I would want to make. It's something that I might accept as a good argument against a position that I'm setting forward. And I think the same can be done in the case of the... I'll get my head around where the debate is now and sort of follow up on Nick's question in a way. So if you go the Barber or Brown-Pooley way, then they're going to take the set of solutions and they're going to throw away all the ones without... And so on and so forth. In this model, so do we say general relativity is a substantivalist theory, or do we say this world it is, this one it's not, and this one it isn't?

27:30 I sort of leave that as an exercise to the reader, right? You could draw either conclusion depending on what you feel motivated towards. I myself kind of feel motivated towards the idea that it is. It tells that GR is sort of a substantivalist theory just because, you know, there's this constant that's not determined by any first principle. And, you know, so the fact that the field equations take that form, you know, and in a particular universe lambda equals zero, well... It seems to me that the theory could probably still be different from the substantivalist theory, but I understand that that's not the way that everybody's going to react to that. There's going to be a shifting between what's the laws and what's the sort of boundary conditions, right? Right, right. So the rotation thing is going to be seen as an initial condition for one and not for the other. There's a couple ways of looking at it. You know, you can look at it as sort of a boundary condition or a constant of integration, or you can look at it as, you know... A constant of nature. And I think John Ehrman has a paper, I don't know if it's published, where he talks about this distinction a little bit. I'm trying not to take a stand on that issue, I guess. But yeah, it is a definite question about what morals you want to draw from this. Alright, so Ed's going next. Ed Slowik from Winona State University is going to tell us about the Cartesian ontology of general relativity. This paper may be sort of a light-hearted interlude, I think, between the papers that are coming and going. My interest in this actually came from reading Carl's paper, along with Rob Reinerscheffert's paper, on how to think about general relativity. Is it closer in spirit to Newton? So, in the 15th edition, one of the later editions of Einstein's work, Relativity, especially in general, confirms Descartes' conception of space in a roundabout way. And this led to the 96th article by Ryan Ashabit saying that if that's sort of the case, What relevance, really, since Descartes' theory is supposedly a relational theory, what relevance does then maybe the substantial relational debate have for modern GTR? So the first question I think to ask is what is the relationship then between Descartes' theory of space and general?

30:00 So as I was just mentioning, that point was raised in the 96 article, and then Heffer's response was that he thinks that GTR is closer to the absolute substantial conception. So that's the first question. The second question, which is really the more important one, is actually, this is actually the answer I'm going to be giving to it. I think that GTR has more in common maybe with Descartes' conception than with Newton's, although that's sort of a... This whole paper is sort of, I guess you could say, an eccentric experiment in theory comparison. But the deeper lesson, I think, is this one here. Question number two. Are there any necessary and sufficient criteria for what a relational theory is as opposed to an absolute theory? And this actually follows up, I think, on the conversation we just had with David in his paper. My answer to that is I think the answer is no. But I think there's a lot of conventionality in what we stick to as a relational or an absolute paper. I mean, space-time. And so there's a sense in which how you decide to divvy up these space-times according to what you consider a substance to be or what you consider a relation to be is going to determine that outcome, and that there's a lot of room for that. Starting off then, my sort of claim is that if Descartes' physics is good enough for Einstein, why shouldn't it be good enough for us? If Einstein thought that his theory was closer in relation to Descartes, why is that? Essentially, one of the reasons he gives in that extra material. By the way, too, it's kind of strange that at this point in Einstein's career that he actually made this argument because this additional material came in the probably late 40s, early 50s, long after he'd become disenchanted with his early Machian views about what generalism meant. So it is sort of strange. Why is he sort of espousing this sort of conception of GTR? Given that he had come to discover that it really wasn't Machian, and so here's sort of his argument that Descartes helped to usher in, as he said, the concept of field and its final claim to replace in principle the idea of particle material points. Essentially then he goes on to argue that Newton's physics seemed to have a sort of a different setup where the ontology is essentially material points against an independent background space, and that independent background space can exist separate from the material points.

32:30 Whereas he thinks that Descartes sort of introduced the notion of the physical field. In that last line there, physical fields, on the other hand, only occur within a ponderable mass. They serve only to describe the state of this matter. Going on then, Einstein links GTR to confusion physics by conceiving the metric tensor G as a physical field, a rotational field, therefore sort of exists inside a ponderable mass. So I mean this sort of actually follows up really nicely on the conversation we're just having. How should we consider, how should we think of the metric field or any other field essentially in modern theories? Are they sort of physical types of entities or are they fields in some kind of background, again some kind of backdrop substance? And if the argument is, well, it depends on how you think of substances, and substances are sort of a metaphysical issue, then maybe deep down there really is not much of a difference between what you might want to call sophisticated substantialism and sophisticated relationalism. They're more or less the same view. Now here's two of the big important descriptions that come up in Einstein's later additions to the book, and so it's worth actually reading things out in full. He goes on, if we imagine the gravitational field, the functions, GIK to be removed, there does not remain a space of the Flat-Mankowski type, but absolutely nothing, there is no such thing as an empty space, i.e. a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field. Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space. The notion indeed appears absurd, as long as physical reality is seen exclusively in primable bodies. It requires the idea of the field as the representative of reality in combination with the general principle. That's sort of the, I think, the important line there. There exists no space empty of field, right? And of course, and then this last line, which comes from the preface, which he also added on later on. I wish to show that space-time is not necessarily something to which one can ascribe a separate existence into the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way, the concept of empty space loses its meaning. I should also note, too, that I think this sort of metric field relationism, as you might want to call it, is becoming popular again. Rovelli, Deeks, and I think a few others have actually put forward views which are very similar to this, just conceived of metric field, or maybe any other field, or energy source, whatever you have in the space-time, as a physical thing. And essentially, then, you have relationism by default.

35:00 So, moving on, then. This is sort of an old debate, and there's a lot that can be said here, and so I'm going to kind of give a condensed version of it, and we could really do a whole paper on this alone. How should we think of Descartes' physics? So let's look a little closer at Descartes' physics and Einstein's metric field of relationism. Is it R1 to R2 using Ehrman's classification? R1... Our one is the more strict requirement that all motion is relative motion of bodies. So you can sort of interpret this as what a structure of the space-time you're assuming is a Leibnizian, which Ehrman and others believe is real in velocity differences. Or is it the metaphysical claim R2, which essentially just rejects space as an independent background substance. So R2 is just more or less the minimal metaphysical claim that rejects space existing separately from any kind of field or physical object. Descartes' physics clearly is not R1, since R1 only allows relative differences in speed and velocity, acceleration versus severe. Now, why is that the case? There's sort of two reasons, and this, once again, is sort of a long story. Descartes' physics and its collision rules really do depend head-bottomly on motion. To make sense of his physics, you really have to sort of describe individual states of motion, which seems to be against that sort of minimum Leibnizian space-time background where you only have relative differences. John Ehrman actually in the last PSA gave a nice explanation with slides, which I don't have, of how this worked. I mean, in the fourth collision rule and the fifth collision rule, you have this sort of scenario. You have a small body approaching a large resting body. It hits it and then rebounds back. And so that seems to be contrary to your R1 Leibnizian spacetime. And that's well known. I mean, this has been discussed a lot. One of the other interesting facets, though, that's not discussed as much with Descartes' physics is some of his descriptions, actually, of rust in motion. Descartes conceives of rust in motion really as opposite states of bodies. This comes up in the Principles in Part 2. And I have a quote there. He even says,

37:30 Maybe the R1 view of motion, because they really are opposite states. So that's a very sort of a last vestige, sort of a scholastic conception of bodies. That's problematic. So overall, and actually in the longer version of this paper I talk a little bit more about some other views. If you actually look at most of the theories in the early modern period going on that claim to be relationist, they often make, I guess you could say, you know, or say discrepancies like this, which seem to violate a strong view of relationism, and I really think that it isn't really until the 20th century that you get consistent attempts to develop relationism that really do stick to R1. Most of the theories, I think, before the 20th century tend to flip-flop back and forth between statements which sound very strict, but then they move on towards... Either adopting some kind of specialized framework, a reference frame from which to analyze the motions of bodies. Now you can get into a big debate on this, and I know Nick and myself have been probably putting off a long debate on this ourselves about how to think about privileged reference frames. But it's not so clear that you, you know, I could talk about this more in a question and answer, but it's not necessarily clear that that really does uphold a strict relations view. Moving on then, how about R2? Is that, could that be used to distinguish relations theories from substantial theories? I think what I have here, while it may be necessary, it may not be sufficient. And this is something that you guys have been working on Newton are probably familiar with. In the Principia, you really don't, this is actually sort of the Stein-Basali argument, you really don't have any espousal of space-time as an entity there. Their argument is that in some sense Newton is giving a definition. And so on and so forth. So, I mean, if R2 is just more or less the denial or not wanting to espouse space as a substance, then maybe the scolium is sort of, and the principia is also R2.

40:00 And then the other claim here, and this is actually maybe the question I was asking earlier, it's often been claimed that there's only a conventional or just an arbitrary distinction between supersubstances, really the only substance, there is nothing else, spaces sort of, all of us are just sort of like maybe little thick portions of space. And Descartes' brand of quantum relationism, Descartes of course viewed that space and matter were identical, and even used the notion that there's only a conceptual distinction between the two. The essence of... So what really is the distinction between the two, between supersubstantialism and Descartes' Relationism? They seem to be almost the exact same view, but would we want to say that supersubstantialism is R2? That seems kind of problematic. So I mean, this is sort of the problems that come into play once you start moving towards these more sophisticated versions of the debate. And I think, like I said, it trades on how do we conceive of substances. Anyway, let's go to the fun part, which is actually thinking about how GTR and Descartes' theory are related. Let me give you a little brief on the vortex theory here. In the vortex theory, of course, you have all these little particles of matter moving around a central located sun or any other object, because there's a sense that there are little vortices going around the Earth, which explains gravitational attraction around the Earth. And then you have all these little, essentially in Descartes' universe, you had a homogenous space, which then God sort of divvied up into small segmented areas. ...gave it a certain, a conserved amount of motion, and then after a long period of time, the collisions of these equally sized parts broke matter down into sort of a three-part structure. Very fine, subtle matter, which is sort of debris that's left over from a lot of the collisions, small globule-sized atomized little pieces, and then large tertiary matter, which is just much larger pieces of volume. And of course, there really is no difference between these three types of particles. Because all space is just extension, it's just essentially their volume which is the difference. Carl's argument then, and I'm going to have to go through this a little faster, is that the metric field, in contrast around the Shabbos, he wants to argue that the metric field of GTR explains planetary motions, like what Descartes' vortices tried to do, though obviously not in the same way. In important ways, though, the metric does much more than Descartes' subtle fields, so Carl wants to argue that there's a sense in which the metric explains, as Shabbos points out, those fluids do not fill all of space.

42:30 And so it could not play the role that the metric plays of defining the B. The metric also serves to define absolute acceleration and rotation, a function that Descartes relations could not allow any material thing to do. And that's going down to the conclusion there. The relationship between GTR and Descartes are interesting, but by no means do they outweigh the... A and B are my little additions there. So the first argument then against sort of a comparison between GTR, and here's a little follow-up on the first one, the fluids of the vortices whose job was to explain the orbits of the planets were not like 19th century ethers in that they did not fill space. They were in direct contact with the planets and perhaps even filled the fine gaps in all ordinary bodies, but that is very different from filling all space uniformly. So Carl's argument sort of is that And the metric obviously fills all of space, whereas, and that of course is used to explain planetary motions, the metric doesn't play a role in all that, but in Descartes' theory you really don't have that because you just have those small particles pushing around larger bodies, and so there's sort of a disanalogy there. I mean, that's sort of true, but essentially... The longer story of Cartesian gravity really includes all of matter, because in Descartes' theory, how he explains it, it's kind of a convoluted view, is that he thinks that the smaller elements, especially the primary elements of matter, have a larger centrifugal tendency to leave the sun along a radial line, and he argues that an equal volume of primary and secondary elements has a greater centrifugal tendency than an equal volume of, say, tertiary matter. And so what happens then is that, given that the tertiary matter has less centrifugal force, the primary elements will rise above it and push it down, and that's gravity for Descartes. There's a great example, by the way, when you're explaining to your students about a theory that has simplicity. In Descartes' theory, there really is only one law of motion, and it also explains gravity. It's really a beautifully simple view. All of these terms get sort of put on a balance, I guess you could say, with the tertiary. So really all of matter is involved in explaining gravity. It's not just the primary. All of it is involved in gravity. So I think that if you look at it that way, then really since all matter is involved in gravity and matter is space, of course, then you have that sort of explanation.

45:00 So, why is it that Descartes' theory doesn't explain acceleration? Well, actually, there's a sense in which it does because Descartes' account of motion, and this comes up actually in Ron Shevitz's paper in 2000 on relative motion, is that Descartes' account of motion does allow an absolute distinction because motion is change of place, and place is defined in sort of Aristotelian fashion as the little particles that surround it. It's sort of the common boundary between the body and the outer sort of containing surface. And so, acceleration... In fact, understanding motion in general, you have to think about the matter that's surrounding a body and that body's sort of boundary. So there is really a close connection there, too, between work and understanding motion. Going on, then. There are some interesting other things to say about here, and here's one of the more important points I was going to make. Recent physics juggles GTR in that there is a very strong reciprocal interrelationship between the geometric content of the theory and the material content. And in this respect, I think, there can be a claim to be made that Gay-Kurtz theory is closer in spirit to GTR than its use in substantial space. And Descartes cosmology, of course, actually I did this explanation so I can go on, right, all of matter is filled with space. The, you know, the primary essence of matter is extension. And so, almost given that definition, they're going to obviously have a very strong correlation. Now here's maybe where you have a little bit of a contentious sort of debate. The Cartesian, this is what I want to argue, the Cartesian correlate of the metric of GTR would presumably consist of the surfaces, shapes, and volumes of all material bodies. Given the identification of spatial extension with matter, so I'm saying if you're trying to look for a correlate of the metric in Descartes' theory, you could think of it as just being the surfaces and volumes and stuff in the space, whereas the Cartesian version of the stress-energy concept of the space could be represented by the same thing, right, the space and matter are identical, or you could think of it as just the conserved quantity of motion and sort of as a universal force or energy. So therefore, there's really a sense in which the interrelationship is real in Descartes' definition of both body and place is really a radical sort of interrelationship there. In fact, if you look at Descartes' definition of motion, you see how this comes out.

47:30 Motion, as he defines it, is the transference of one part of matter or of one body from the vicinity of those bodies, and either that contributes to it and considered at rest in the vicinity of some others. So he defines motion as change of sort of place. And then he just, in the very next line he says, by one body or part of matter, I hereby understand everything which is transported. So motion depends on body, because bodies define place. But then how do you define body? Well, it's everything that moves together. It's a beautifully circular definition, and this has caused a lot of confusion and worry amongst Cartesian scholars. But one of the ways you can sort of look on this is that Really, body and motion are sort of ontologically on a par. There's a sense in which they both sort of exist, and you can't see one of them as more basic than the other. And so that is, like I said, it's sort of a radical sort of notion of an interrelation. So my little upshot here is consequently, much like in GTR, direct interrelationship is manifest between the content of the space-time, either viewed as matter or quantity of motion, and the metric, if you think of it as the three-dimensional extension in volumes. For the sake of time, one of the problems, and we were just talking about this in the last talk, is what do you do with the vacuum solutions to GTR? I mean, this is a big problem. One of the things you could say, does this sort of undermine the analogy between GTR and Cartesianism? Because, uh, consistently about the metric and the absence of matter. And so, is that a big problem? Well, it is, it is a problem, I should say. Um, one of the things you could say, following Ehrman and Norton, is just to identify sort of the gravity waves as a material concept of the space-time. Here we're thinking about vacuum solutions with gravity waves. In principle, the waves' energy could be collected and converted to other types of energy, such as heat or light. In his really nice paper on the lack of conservation, there's that line to claim that it's not tensorial, so it's only preserved from particular reference frames, other reference frames you might get zero, right? Now, the interesting thing about this, though, is this is exactly like in Descartes' theory, because in Descartes' theory, quantity of motion is dependent on particular inertial reference frames. Quantity of motion, in fact, isn't even preserved by all inertial frames, some that we'll view maybe as...

50:00 So there really is a strange sort of connection between the pseudo-tensorial nature of the vacuum solutions without gravity waves. Where can you find a reference to this? Well, one of the things you could do if you're a Cartesian trying to find a relationship between GTR and Einstein, how about that case in the beginning of the universe before God imparted a quantity of motion to the universe? Descartes said that God was able to divvy up space into equal spaces, and this is before quantity of motion is entered. If you think of quantity of motion as sort of the energy, stress-energy content of the space-time, then there you have sort of a space that has metric distinctions but doesn't actually have an energy yet, doesn't have the force imparted to it. So there is a sort of strange relationship there between Descartes' theory and Einstein's theory. And so then, the moral of the story, then, what conclusions can you reach from this, from this sort of eccentric interpretation or analysis of theories? My argument, and of course this is a lot more looking at other cases, is that to draw a continuous line of development from past theories dubbed substantively or relational to contemporary theories introduces a lot of unwarranted choices. Is a relational theory our one? When these theories keep together, it really depends on that choice. And what I guess you can say constructively or on the positive side is that at most what the examination of these physical theories was is that certain aspects of non-modern theories resemble certain aspects of contemporary theories. And so the vacuum solutions do seem to be closer, of GTR, do seem to be closer to, I think the interrelationship between matter and space seems to be more closely resembling to Descartes' theory. I think you could say that in the modern sort of context there are certain aspects of contemporary theories that can be related to earlier theories, but there is, I don't think there's any one clear line of development. And so my last line is that on a Kuhnian reading relationism does not constitute a full-fledged paradigm since there doesn't seem to be any agreement on the extent of relationism, except possibly R2, although once again that may not be sufficient. Yet, given the lack of consensus on what theoretical context classifies as space-time ontology and the Kuhnian route, what you could say is that the whole ontological dispute is sort of pre-paradigmatic. There doesn't seem to be any general agreement on how this could ever be settled. I should ask you another question.

52:30 In this kind of Cartesian reading of general relativity that you want to offer, do you want to go the supersubstantivalist route and think more? Yeah, I mean, I guess that's sort of the route. I mean, really, if you think of Descartes' theory, it really is sort of supersubstantialism, because essentially there really is only one thing, however you want to describe it, and you do seem to have that, I think it's the same thing as supersubstantialism. I mean, it really seems like, if you try to put a causal reading on Einstein's equations, it goes from left to right. I guess if you read supersubstantialism as the metric side. I mean it really is difficult because it's in which well I mean motion seems to be so involved with understanding the distinctions amongst bodies of course that all plays out against against his plenum and so there's a sense in which there's never going to be any sort of any change Thank you for your time, and I look forward to hearing from you in the future. Why do we have to look at the general relativity to make this case? I think it was the second point about there being no sort of one doctrine that characterizes relationism. It seems to me, surely, just comparing Descartes and Leibniz, he's already going to make the point, right? I mean, Descartes is a substantivalist. He thinks space exists. He thinks it's matter. Whereas Leibniz thinks space is some kind of construct, right? It's ideal. That seems like a big enough difference. Oh yeah, I already said it's already broken down at that point. And then maybe the Leibnizian idea is the one you want to trace forwards and see if that's a... Yeah, I mean, that's a possible route to go.

55:00 One of the claims that could be made, though, is that Leibniz seems to be so influenced, at least in part, by reading Descartes, and so, I mean, even Newton, to some extent, and that they make these claims, I mean, and this is sort of that bipolar tendency, and they want to make statements which are strictly in relation to R1 in terms of motion can be assigned to other bodies. But then, of course, they seem to want to invoke physics or make statements that, no, but there really is a sense when, you know, this is Leibniz saying this point, there really is a sense in which the motion is in this body as opposed to the other. And what I sort of see is that that sort of problematic duality, or whatever you could say, seems to be in Descartes as well as in Leibniz. Although Leibniz is much more clear, at least in parts of, say, the Leibniz court correspondence, where he's trying to think of space as an abstraction. But then again, so is Descartes, I mean, in parts of the Principia. He does say that, and I want to think of it as an anomalous way when he talks about the difference between space, the vulgar conception, versus sort of the philosophical conception. So, I just, I sort of see the similarity between the two, but I mean, you're absolutely right, I mean, if you just look at the two of them and sort of, and look at how they approach it, there's sort of a, there's a difference, but... I mean, yeah. I mean, there's a sense of what you're right, but I mean, here's another way to get at the difference that I'm pointing at is, I mean, Descartes, for Descartes, it's considered the arguments against the existence of the vacuum. So for Descartes, it's just logical impossibility. Space is matter. Leibniz has to give this argument that, you know, God is going to fill everything so that there's, you know, his kingdom is as large as possible. Yeah, no, that is a big difference. I think that's one of the fundamental differences, is that he's suggesting that Descartes really defines the vacuum away, whereas Leibniz really has to give sort of a separate argument for it, so that there is a big distinction. Often differ quite radically from Descartes. Huygens allowed for the possibility of being a small vacua on sort of the scale of particles. And so, I mean, one of the big problems, and this is a great example for students about what's wrong with Kuhn's theory, all these guys supposedly are in the Cartesian paradigm, but there's a sense in which, is there any single one doctrine which runs through all of them? I think the same holds true for substantialism versus relationism, in that there's a sense in which, as you go on, that theories seem to pick and choose a little bit from earlier views, and whether you want to call it substantial or relational often depends on which main components you find to be consistent or to be the true view of, say, substantialism or relationism. But again, how are we going to make that decision? Yeah, but that is a question.

57:30 Do you use the word potential or potential difference in any of your work? And if so, how? I don't know if I understand what you're saying. Well, I understand motion to be work. Work is a force times distance. But on the other hand, my handbook of physics says force is undefinable. So, I think some of the engineers use the expression mu-dm, in other words, potential difference times some derivative of mass equals work. So, we look at potential differences as causing motion. Now, I don't know much about Einstein, but... It's interesting you raise that, because I know that in Westphal's argument, one of the things he wanted to claim was the problem of centrifugal tendencies. But, I mean, clearly in relationism there's a sense in which it's all, at least one view of relationism, and David actually sort of suggested this, that you sort of see a space as, you analyze space or behavior of bodies in terms of potentials, and so it's just the possibility, and space is just all the possible relationships that can exist between two objects. So, I mean, there's a sense in which relationism supposedly is predicated on the notion of potentiality, I guess, from the very beginning. But, you know, one of the other conceptions, of course, is, let's say, like with Descartes, is that, no, it's all filled with matter, so there's a sense in which you always have, there really is sort of a certain set number of bodies between any other. And there's the potentiality there, I guess, is just what other bodies can fill in the distance between two objects. I don't know if I really answered your question. I'm hoping you might be able to help me out a little bit because I've never been able to make sense of Descartes' ontology. The most troubling part is that I don't have any sense of how he could deal with just basic cohesion. What happens if something is one large body or several... Or, you know, plurality of small bodies moving together. And, you know, I take it the only conceptual resources he has is what's the actual motion that's imparting to this matter, right?

1:00:00 Or does the whole thing, you know, go and push this one away, right? I mean, this is... A, do you think Descartes has the conceptual resources to make, and then just sort of an invitation to, you know... What key point do I need to know to make sense of Descartes' ontology in this, if you think it can be? Are they conceptually coherent? That is one of the big problems that they've got, I mean, it's sort of like you postpone the problem by defining bodies as sort of a... Large bodies, whether it's tertiary matter or even bigger bodies, their sort of unity is defined by the... the relative rest of the particles. And so, fine, but then exactly how do you explain their ability to stand up to impact and hold together? And so by that definition leads into that problem. He really doesn't have one. He sort of gives an analysis of bodies in terms of sort of a common method that was used by the other mechanical philosophers in terms of pores being in the body and that when they're in impact, these pores are collapsed and they eject matter and then after impact, the pores open up and take it back in and it swells up again. But that doesn't explain how all the rest of the parts that stay there collapse and then swell back up. Subtitles by the Amara.org community The cohesion problem is a problem that, strangely enough, a lot of people didn't actually seem to write to Descartes about. I mean, Moore later on did write about the relativity problem, saying, if motion's going through a tower, should we also see it as the tower going through resting air? And Descartes really didn't answer that in a straightforward way. He sort of switched to an answer that talked about two guys pushing off a shore and where it really seems to be... Some people have argued about conflating of the kinematic and the dynamic sort of aspects of it, and maybe showed that he was unhappy with it as well. But yeah, I mean, it's interesting that that wasn't a problem that was raised at the time, but it seems... Next talk will be Bill, who will talk about the methodological value of coincidences and arguments for general relativity and dark matter. We'll hand out here a mic.

1:02:30 Well, this paper is a follow-up to one that I published in Philosophy of Science last year. In that paper, I explored the evidential warrant for accepting that general relativity is the correct theory of gravity to apply at galactic and greater scales. I described the dynamical discrepancy for galaxies, clusters of galaxies, and other large-scale systems. That's a problem popularly known as the dark matter problem. And I showed that in light of this, the warrant for GR at galactic and greater scales is relatively weak. In this paper, I address some additional considerations that I did not have space to address in that earlier one. In particular, I want to talk about whether the apparent agreement between four different kinds of measurements of the masses of galaxies and larger structures constitutes a good evidential, methodological, or other basis for preferring general relativity over a particular class of rival gravitation theories. The overall conclusion here is that these coincident measurements do provide some grounds, although weak and defeasible grounds, for thinking that GR is better than its rivals at galactic and greater scales. So general relativity makes predictions that agree to high precision with all of the available evidence regarding interactions that take place over scales corresponding to roughly the size of our solar system. I'll call these stellar system scales. But in fact, GR also has a much stronger kind of confirmation at stellar system scales. These are the parameters of the theory. These measurements from phenomena are the basis for epistemically robust theory comparisons via the parameterized post-Newtonian, or PPN, framework. The result of applying the PPN framework is that, at stellar system scales at least, GR has a very high degree of empirical support. It's clearly better than every rival gravitation theory so far articulated. On the basis of these stellar system scale successes, GR is inductively extended to cover all phenomena, all non-quantum phenomena anyway. This is analogous to the pattern of reasoning used by Newton to establish universal gravitation, wherein diverse phenomena, including pendulums, the orbit of the moon, the orbits of the planets, the orbits of the Jovian moons, and so on, are each used to make independent and agreeing measurements that the power law of the force of gravity is inverse square, and then that inverse square law is inductively extended to cover all phenomena whatsoever. For interactions taking place at stellar system scales, then, GR is highly confirmed. For interactions taking place over galactic and greater scales, however, the situation is quite different. The existence of the dynamical discrepancy, the dark matter problem, shows that the observed motions within galaxies, clusters of galaxies, and other large-scale dynamical systems are actually inconsistent with the predictions of general relativity, given the amount and distribution of mass that's observed to be present in those systems.

1:05:00 Now the dynamical discrepancy for galaxies and other structures is analogous to the discrepancy that we knew about for Uranus' orbit beginning in the 1800s. The motions of Uranus at that time were inconsistent with the predictions of universal gravity given the then known distribution of mass in the solar system. Two possible solutions were available. Either modify universal gravity or posit the existence of previously unknown mass. The latter, of course, was the kind of solution pursued independently by Adams and by Le Verrier. In 1846, Le Verrier's prediction of the geocentric position of an unknown mass was good enough to lead to the telescopic discovery of Neptune. The case of Mercury's excess perihelial precession is similarly analogous to the dynamical discrepancy for galaxies. For Mercury, however, the solution turned out to be a new theory of gravity, namely GR. Note that both types of solutions, matter solutions and gravity solutions, were tried for both of these problems. Uranus and Mercury. Just as for Uranus and Mercury, there are two classes of possible solutions for the dynamical discrepancy in astrophysics. The members of the first class, called the matter solutions, postulate the existence of about 100 times more mass than is visible in those systems. The distribution of this mass on the assumption that it exists is fairly easy to determine from the observed motions. It's much more difficult to say what this stuff is. It's called dark matter because it neither emits nor absorbs electromagnetic radiation at any wavelength. Dark matter has so far eluded direct detection despite at least 30 years of active searching. The discrepancy itself, by the way, was first discovered in the late 1920s. It wasn't taken seriously by the astronomical community until the mid-1970s. A plethora of dark matter candidates have been proposed over the years, ranging from otherwise unknown fundamental particles to black holes. Many of the candidates have been ruled out on empirical and theoretical grounds, or they've been shown to be unable to resolve the entire discrepancy. The matter candidates that do remain viable have little to no positive empirical support. Just about the strongest claim that can be made is that it's not impossible, as far as we can tell at present, that the remaining candidates could resolve the dynamical discrepancies. Members of the second class of possible solutions, the gravity solutions, postulate no unseen matter and instead modify the action of gravity at large scales. There's no empirical reason to think that a matter solution is more probable than a gravity solution or vice versa, just as in the Uranus and Mercury cases it was impossible to tell in advance which type of solution would ultimately succeed.

1:07:30 The empirical constraints on theories of gravity offered as solutions to the dynamical discrepancy are surprisingly weak. Obviously, because of the epistemically robust solar system tests of GR, gravity solutions must be empirically equivalent to GR at stellar system scales. But at larger scales, their predictions may diverge, even radically, from those of G.R. One possibility is that G.R. will turn out to be the stellar system scale limit of some successor relativistic gravitation theory, in the same way that Newton's universal gravity turned out to be the low velocity weak field limit of G.R. Well, how can we decide, then, between G.R. and the potential gravity solutions to the dynamical discrepancy? It would be ideal to construct a theory testing and comparison framework analogous to that Unfortunately, the following difficulty, which I call the dark matter double bind, seems to preclude the possibility of constructing such a testing framework. In order to evaluate the empirical adequacy of any gravitation theory at galactic and greater scales, the mass distribution in those systems has to be known. But because of the astrophysical dynamical discrepancy, that mass distribution is not known. In contrast, in order to infer the mass distribution from the observed motions, a gravitational law must be assumed, but such a law cannot be legitimately assumed, since that's the very thing which is at issue here. Okay, so to talk a bit about the coincident measures that I want to focus on here. I agree with Harper and DeSalle that Newtonian methodological ideals inform current theory testing in gravitational physics. An important part of Newton's use of reasoning from phenomena in the argument for universal gravitation is that diverse phenomena yield precisely agreeing measurements of parameters of the theory. The coincidence of these measurements lends strength both to the unification of these apparently diverse phenomena under a single gravitational law, and also lends strength to the inductive extension of that law to all possible cases. The methodological value of such coincident measures, then, is that they are the foundation for stronger arguments in favor of a theory than would otherwise be possible. There's a set of measurements of the dynamical masses of galaxies and larger structures that appear to provide independent, coincident results. The question then arises what the coincidence of these measurements does for GR at galactic and greater scales, what the coincident measures at stellar system scales do in the PPN formulas. Dynamical mass is calculated by one of four methods. We've got rotation curves, velocity distributions, X-ray temperatures, and gravitational lensing.

1:10:00 No matter which technique is used, the discrepancy is always found between the dynamical mass and the mass that's visible in the system whose mass you're trying to determine. One of the four techniques yield apparently agreeing measurements of the masses of these systems, and since the techniques seem to be independent of one another, the coincidence of their results is sometimes taken to provide grounds for thinking that the dynamical discrepancy will have a matter solution rather than a gravity solution. However, the apparent coincidence of the measures is not as evidentially or as methodologically significant as it's sometimes thought to be. Now, that said, I still think that these measurements taken together provide the best available reason for preferring matter solutions over gravity solutions, and thus for thinking that GR, rather than some rival theory, correctly describes the action of gravity of galaxies in greater scales. But, as I want to show you, that is not a very strong preference. So first, for spiral galaxies, which have a well-defined plane and sense of rotation, a rotation curve can be taken. And from the rotation curve, an overall mass distribution for the system can be inferred. So by the well-known relationship between an object's velocity along the line of sight and the Doppler shifting of its spectrum, the observed redshifts of stars and clouds of gas in a galaxy give the component of rotation along the line of sight at a given radius from the center of the galaxy. So a graph of redshift against distance from the galactic center yields a rotation curve. Now, the mass interior to any given radius can be calculated from the rotational velocity of objects orbiting at that radius. The principle is the same as measuring the mass of the sun from the radius and speed of the orbit of a planet around it. The observed rotation curves for spiral galaxies show that the absolute value of the rotation at any given radius is much higher than is predicted, given the visible mass and the Newtonian limit of gr. Even more importantly, however, instead of falling off asymptotically to zero, as you would expect, observations of gas clouds at extreme radii from galactic centers show that the rotation velocity actually remains flat or even rises as you go up to several times the radius of the visible disk. On the issue that the Newtonian limit of GR correctly describes the action of gravity in galactic and greater scales, the only way to account for those observed rotation curves is to hypothesize the existence of a spherical halo of dark matter There are a number of elements surrounding every spiral galaxy, and that halo has to extend to several times the disc radius, and it has to contain about 100 times the mass that we can see.

1:12:30 An alternative to this extravagant excess of invisible mass of unknown type is to propose a new account of gravity at galactic and greater scales. The well results are found for elliptical galaxies and for clusters, and in these systems the internal motions are essentially random, so we don't have a uniform sense and plane of rotation, so instead of doing the rotation curve thing that I just described, you do velocity dispersions. So you obtain the velocity dispersions spectrographically as well. This collection of relative redshifts of the stars in elliptical galaxies or of galaxies in clusters gives information about the motions within those systems. The Virial Theorem, originally developed in thermodynamics from principles of Newtonian mechanics, can then be applied to determine the gravitational potential needed to produce those observed velocities, and this yields a value for the total mass of the system. So the Virial Theorem is on the handout there. It's m equals r times the average of the squares of the velocities of a given radius divided by alpha and g. g is the gravitational constant. Alpha is a constant having to do with oblateness, but it's usually of order 1. That's the velocity dispersion method. The third method is x-ray emissions. We have observations that show that there are x-ray emissions from clouds of diffuse gas enveloping many galaxies and clusters of galaxies, and this gives us our third measurement of those systems. This technique depends on the assumption that the gravitational potential of the system is the only plausible mechanism for continually heating this gas, and you have to continually heat it because it's continually x-rayed. So the intensity and the spectrum of the X-ray emissions determines the amount of heating needed, and this in turn is converted into a value for the gravitational potential, and hence the mass of the system in question. As in the case of the rotation curves and velocity dispersions, the masses found by the X-ray method are roughly 100 times greater than the luminous masses of the systems in question. Previous measurement techniques are in practice performed using only Newtonian physics. The systems satisfy the weak field low velocity limit, and thus no relativistic contributions are expected. In contrast, this fourth mass measurement technique that I'm turning to now, gravitational lensing, depends on the specifically relativistic parts of geo. So what's going on in gravitational lensing? Well, in rare cases, we have a foreground galaxy or a cluster of galaxies that lies

1:15:00 along the line of sight to a background galaxy or cluster or quasar, as the case may be. And the gravitational field of the foreground object deflects the light of the background object. From the appearance of the image of the background object, the mass of the foreground object Gravitational lensing is well understood theoretically, and given that GR is correct, the image patterns that will be produced by different magnitudes and configurations of mass in a lens, the foreground object, given different distances and alignments between the observer, the lens, and the background object, these images can be predicted. Conversely, from the observed image pattern, the mass of the lens can be inferred. And again, when we do this, we find a very large discrepancy between the gravitational lensing mass and the mass visible in these systems. With each of these methods, a general trend has been found wherein larger systems tend to have a higher ratio of dynamical mass to luminous mass. This is true both within and across system types. Larger spirals have a larger dynamical discrepancy than smaller spirals, and clusters have a larger discrepancy than individual galaxies do. The methods, moreover, seem to agree with each other about the masses of systems of a given type and given parameter. So for similar spirals, spirals that have similar diameters and similar luminosities, you find the same degree of discrepancy across different instances of those spirals. Advocates of gravity solutions to the dynamical discrepancy might try to discount the agreement between these four different measurement techniques by claiming, as I once actually heard one of them claim, that they must all be wrong by the same amount. It's hard to see how this could plausibly be the case, however. A better way to challenge the evidential or methodological value of these measurements is to point out that the agreement between them is actually much less close than it at first appears to be. There turns out to be plenty of room for gravity solutions to the dynamical discrepancy. So here's my critique now of this apparent agreement. The first point is that it's often impossible to apply more than one method to any given system. Rotation curves, for example, are possible for spirals but not for ellipticals, and the velocity dispersion technique applies to ellipticals but not to spirals. There are relatively few cases of gravitational lensing known, and the available gravitational lensing results do find that the masses of spirals that act as lenses are roughly the same order of magnitude as is typically found by rotation curves for other spirals with similar diameters and luminosities. Similarly, where the lens is an elliptical galaxy, the mass derived from the lensed image agrees to within an order of magnitude with the typical masses found by velocity dispersions for ellipticals whose parameters are similar to those in the lens.

1:17:30 But, so far as I'm aware at least, no single galaxy has had its mass determined both through gravitational lensing and through either rotation curves or velocity dispersions. Part of the reason for that is that systems that act as lenses are normally extremely distant, and hence too dim for detailed spectroscopic work that would be necessary to do the rotation curve. This means, then, that what we really have is an agreement that's based on an analogical argument. All there is, really, is an order of magnitude agreement for analogous types of systems, rather than for individual cases. This is suggestive, but it's not definitive. It's unlikely that the agreement can be made much better, however, even with further observations and improved techniques. Fairly large margins of error are present in lensing calculations because of the need to make assumptions that can't be definitively checked observationally. We have to make assumptions about the diameter of the lens, about the overall shape of its mass distribution, about its distance from us, and the distance to the background object. And all of these are subject to various kinds of error. It doesn't seem like we can eliminate it and get it very tightly constrained. Well, it is true that many galaxies and clusters have their masses estimated both from velocity dispersions and from X-ray temperatures, and that those results essentially agree. But the margins of error there, too, are not small in either case, in either kind of measurement. Even if the errors there can be reduced, two other points are important to mention. The first point is that the X-ray temperatures and velocity dispersions are based on the same Newtonian principles, and hence it could be argued that they're not really independent methods. The second and more telling point is that if an alternative gravitational law were assumed in these techniques, then agreeing measurements of a different amount of mass might well be found. To my knowledge, no one's tried this yet, but it's, I think, something that could be done. This is to say that the mass results from velocity dispersions and X-ray temperatures are fairly sensitively model dependent. Plus, the fact that their results agree in a given case could indicate that the correct value for the dynamical mass has been found. Or it could be construed as an agreeing measurement of the parameters of an alternative theory of gravitation, where by itself the agreement between velocity dispersions and x-ray mass measures only confirms the overall size and character of the dynamical discrepancy itself. It doesn't give preferential support to the hypothesis that hidden mass is the cause of the discrepancy.

1:20:00 As a final note, let me mention an observational study that was published about two years ago by Vogt et al. They reported that in a few galaxies, the orientation of the cloud of X-ray gas is different from the orientation of the luminous matter, so the two ellipsoids are not aligned. This is an interesting result because it implies that there must really be a halo of dark mass in these cases. On theoretical grounds, it's clear that the shape of the gravitational potential that heats the gas has to be the same as the shape of the overall mass distribution, so the fact that the X-ray emitting gas cloud is oriented differently than the luminous mass indicates that there must exist a distribution of hidden mass in the galaxy, and that it dominates the luminous mass and then is oriented in the same pattern as the X-ray cloud. If robust, this result shows that there really is dark matter. Note, however, that this result can hold even if the correct theory of gravity for galaxies and larger structures is not GR. That is, even the Booth observation is perfectly consistent with alternative theories of gravity. So that result doesn't prove that the dark matter that is needed to heat the X-ray emitting gas is the complete solution to the dynamical discrepancy. Instead, it could indicate a complex solution involving both dark matter and a new law of gravitation. That's an ugly solution that no one wants, but the universe has defeated our expectations more than once. The upshot of all this is that the four methods for measuring dynamical mass at galactic and greater scales do not really yield closely agreeing measurements of the masses of large-scale astrophysical systems. The hint that we get from these techniques is somewhat loose and could be merely coincidental. Whatever agreement is present, it certainly does not have the same epistemically robust character as the multiple agreeing and precise measurements of the parameters of GR that are obtained from solar system phenomena via the PPN formulas. It can be denied, however, that the four methods discussed here do give roughly agreeing results. The roughness of agreement must be taken into account, as should the fact that what is being measured is the mass of those systems on the assumption that the Newtonian limit of GR applies to those systems. Strictly speaking, these techniques give us agreeing measures, insofar as they are agreeing, of the value of the total forces produced by the combination of the mass distribution plus a gravitational law. The techniques don't tell us the relative contributions of those two components.

1:22:30 The measured values could result from just the visible matter plus a new law of gravity at large scales, or from the Newtonian limit of GR plus dark matter, or from some combination of new matter and new gravity. That said, my intuitions are, with those of most physicists, the available evidence seems more likely to yield a matter solution than a gravity solution, which is to say that on balance it seems more likely that GR is the correct theory of gravitational interactions at galactic and greater scales. That position is not strongly warranted, however, and a gravity solution to the dynamical discrepancy remains open. Thank you. What was the name of it again? Dama. Dama. Is that the one out of Italy? Yeah. Yeah. You have heard of that? I have, yeah. Didn't they report a zero result recently? Oh, they did. I think so, yeah. I go to conferences every once in a while. I hear one of theirs. Right. They were convinced, but I guess they're now saying they don't see any longer. Well, my understanding was that they hadn't actually detected a signal for dark matter, but maybe I'm thinking of a different thing. Well, I mean, if you look at the popular press, the dark matter solution's been solved about six times a year for the last ten years. So it always turns out to be a false signal, or once you reanalyze the data, there's nothing there. So then my other question is about whether folding in mass-energy distribution will not affect this. My understanding of all of that is not very strong, but what I think I understand is that they have to build in a lot of assumptions about how much mass is there. In fact, they actually have to include a much larger mass component than would even be required to take care of the dynamical discrepancy in galaxies. So you have to assume the existence of a lot of dark matter to make even GR fit the cosmological scale result. So there seems to be plenty of wiggle room there, too. They give 30% of the total dark matter. Well, it depends on whose model you're talking about, but yeah, somewhere between 10 and 100% extra dark matter. And it depends on what scale you're talking about, too. But you say for these studies it's 100 times? For spirals and ellipticals, it tends to be between 10 and 100. I'm looking for some enlightenment here, and that is, it sounds like most of the...

1:25:00 There are treatments that count for the missing matter if we add more mass in the galaxy than we see. And so one just sort of naive question is why wouldn't a very massive black hole at the center of matter that is missing also needs to be in between the different galaxies for some of these things? The galaxy just isn't massive enough. Right. So you could potentially throw in a really big supermassive black hole and make up all the extra mass. You'd have a really high peak and then it would drop off, but then we actually have the inverse of that for rotation curves for spiral galaxies, so it's the only way to get the observed rotation curve is if you embed the spiral in a massive halo. I can follow that. What are such definitions like Thorne, Lee, Lightman, and Meek? Are you assuming that this is clear? The way this usually comes up in the debates over dark matter is that there are some people who like... There are a lot of people who want to use GR and the Newtonian limit of GR and want to use it at galactic and greater scales, and so they're going to opt for solutions that involve new kinds of matter. And then on the other side are people who think that, well, having all this extra stuff that you can't see is too much like the ether, so let's pick another solution and maybe try to modify gravity to do the work. So that's usually how it works out in the literature. Right, but how do I know whether, say, a branch of a scalar field is a new kind of matter? I haven't thought about that. I think that all of the stuff that I would call matter solutions are going to be things where people are postulating the existence of new particles. So, for example, some of the supersymmetric particles are candidates for dark matter. Those kinds of things are going to be found in matter solutions. So, new particles, you could usually say new fields for the gravitational field. Well, in a way it's hard to say, because there have really been only two theories that have been offered until now, and only one of them, well, no, actually that's not true, but until just recently, there was only one that was relativistic, but one theory, Modification and Atomic Dynamics, put forward by Milgram, there was an activistic version of that published this year by Bekenstein. What goes on in mod, essentially, is the modification of Newtonian dynamics is that you change the action of gravity when accelerations are very small. So it's not just an inverse-square law, it's inverse-square plus something else. And then one is the, they call it vile gravity, it's Mannheim of Casanas.

1:27:30 And what we do is add a linear term to the field equations. Essentially, that linear term is so small that you can't see it at small scales like at a stellar system scale, but as you get something as big as a galaxy, its contribution adds up and changes the total acceleration. I think that Mannheim's is. Virtual numbers of what the agreements are actually. You could just sort of say order of magnitude, but that's very... Well, it is really loose, and that's really almost as much as you can say. The trouble is that you're comparing... Different systems usually. Say you're talking about the masses of spirals. Calculate the mass of one spiral from its rotation curve. You get a good number for that. But then you add against, say, a gravitational lensing case, and you have to build in so many agents into the model that it's hard to say exactly what the margins of error are. And then to rid them is difficult. And in fact, if you want to talk about a general claim about all spirals, well, you can't really, because the... The size of the dynamical discrepancy depends in part on the parameters of the spirals themselves, on their diameters and on their luminosities, and so there isn't really a general number that you can use for different experiments and shows, you know, the dynamical discrepancy, the matter discrepancy that we've measured with our technique for spirals as a function of the spiral size. I've seen one, but the literature is huge, so it could be out there. That's the kind of thing that physicists think to me. That's a good point. Thanks again. Our last speaker is Peter Bukulich. We talked about black hole complementarity. Does the black hole complementarity answer Hawking's information? So the answer to my title question is, well, kind of, sort of. And what I'm going to be doing in here is trying to...

1:30:00 Sort the wheat from the chaff as far as the responses to the paradox, which I'll quickly explain, and in particular, as I'll say more about, I'll be looking at black hole complementarity, which is now, it might be too much to say that it's the accepted solution to the information loss paradox, but at least among people working in quantum gravity, it's them, well, you know, what do you think the right response to Hawking is? To the extent that they're comfortable giving an answer, most of them, the majority, are now going to say one thing like black hole complementarity in sort of vague terms. So I'll talk briefly about what that is and then I'll step through the three most prominent versions of black hole complementarity and I'll say a little bit about, mostly about Where the chaff in those arguments lie and things that are parts of the argument that we should really discard, but to start with Let's step through the information loss paradox. I assume not everyone is familiar with it Hawking in the 1970s They told us that black holes are not completely black, but they give off heat. They are thermal objects that give off very, well, for large black holes, they give off a very small amount of heat. As they get smaller and smaller, they'll give off more and more. What does that mean? Well, if the black hole is giving off radiation, that means it's giving off mass energy, that means it will get smaller and smaller and hotter and hotter, and it seems like it's going to disappear completely. And this is just encoded in our citation value of the stress-energy tensor, plug it into Einstein field equations as a sort of first approximation of what we expect from quantum gravity, then we will know that the black hole is going to get smaller and smaller and presumably eventually disappear. Now into the space-time from containing and evaporating black holes. The black hole is right here. This is a region from which nothing can escape off to infinity because it would have to go faster than light, and that's not possible. So why is this a problem? This is a problem because if we consider this space-like surface right here, in general it's going to be the case that there will be quantum correlations between the inside and the outside of the black hole.

1:32:30 Quite simply, what we can usually, the way we can picture this is just suppose we've got Alice and Bob, and each of them grab a particle, the two of which are in a singlet state, so they're, you know, nice L-type correlations between the two. Alice sacrifices herself and goes and jumps into the black hole. While Bob is out here, well, there are going to be correlations across this space-like surface. That means that the state of the exterior surface here is going to be a mixed state, but then if we go and evolve that state forward, the correlations from inside the black hole can't escape, so this final, the state on this final time-like hyperspace-like hyperspace, evolution from a pure state at an early time, we're assuming, to a pure state here, but then future evolution is going to be a mixed state. Why is that a problem? Well, we have evolution from a pure state to a mixed state. That cannot be unitary evolution, and so we have some real problems with quantum mechanics, or so a lot of physicists argue. There's debate about this point, and we can talk about it a little bit, what all is packed into this unitarity, but high-energy physicists especially get very upset when you go violating unitary or suggest that we will. To save the unitary evolution of black holes, this is what's known as Hawking's information loss paradox. I guess I should mention in passing that just this last summer, Hawking actually capitulated and said that he now no longer thinks that his information, his little paradox actually shows that there is non-unitary evolution. He thinks that information does escape from black holes. Probably shouldn't take too much time in this presentation to talk about that. Okay, when Hawking suggested his paradox, there wasn't a whole lot of response to it at the beginning, but in the early 1990s, people started developing these two-dimensional toy models of Quantum gravity, semi-classical quantum gravity in particular, and people started getting interested in this problem and tried to figure out, okay, well, how could we get a unitary picture of what's going on in the evolution of a black hole?

1:35:00 And a number of different solutions were proposed. A nice survey of those was presented by Blot, Ehrman, and Ritchie in a nice paper in 1999. Pretty much everyone has given up. There are no more remnant theorists out there or people buying into thunderpops or baby universes or all these other possible solutions have just sort of fallen beside them. It seems like the most crazy, contradictory, absurd solution to Hawking's information loss paradox is the one that pretty much everyone now buys into, with a few exceptions. But anyhow, what is black hole complementarity? To begin with, it's a suggestion that an external observer can treat a black hole, someone who isn't falling into the black hole, can treat the black hole as a heated membrane at Now, at just that level of description, This is just, can be seen as a nice calculational device that can be used in classical general relativity. Thorne and others have a nice book called The Membrane Paradigm of Black Holes. That's what we're appealing to. The suggestion, then, that Susskind, Thurlesius, and Ugloom, who are one of the first proponents of Greece about black hole complementarity and one of its most vocal proponents, they suggest that, let's say we've got Bob here outside the black hole. He sees this heated membrane. Alice goes and falls in. She has her particle that is entangled with Bob's particle. Those quantum correlations are going to get passed off to the membrane and then eventually radiated back out into the external universe. And if Bob were able to perform all these measurements on the outgoing Hawking radiation, he would be able to reconstruct the original state and no information is actually lost.

1:37:30 Now that's all well and good, you might think, except that there's something that we call the quantum no-cloning theorem, or what is that if? The correlations are in the outgoing radiation or in the membrane. They can't also be inside the black hole here. So what does that mean? Well, that means that if the information, loosely speaking here, about Alice is going to be in the outgoing Hawking radiation, Alice herself must have been destroyed passing through the event horizon. The problem with that is that the event horizon of a black hole is a global feature of the space-time. We don't expect Alice to notice anything unusual as she's passing through the event horizon, and so she should just pass peacefully through until the tidal force is near the central singularity, tear her apart, and then tear apart her molecules and atoms, etc., etc. But until that point, she should be a perfectly happy individual. So this looks like a big contradiction. So what do these people then suggest? Well, what Susskind, Thurlesius, and Ugglund suggest is that, well, look, once Alice goes into the black hole, she can't talk to Bob anymore. She can't tell Bob, look, I made it through, or anything else. So maybe there are just these two different perspectives. Maybe there's Bob's perspective, where Alice is destroyed, and then there's Alice's perspective, where she doesn't notice anything unusual, and we'll suggest that these two perspectives are complementary to one another. And this is going to be our solution to the black hole information loss paradox. Well, that's all well and good, but you can't just take a contradiction and say that I'm going to ignore it. What is it that is really motivating this, is driving this particular claim? Well, the real work comes in. In the claim that the global state, the pure state that we saw, that has the interior and the exterior of the black hole, and there's these correlations on it, Susskind's Relation Snuggling suggests that this is unphysical, they say, because it encodes correlations that can only be measured by a super-observer. Now, the picture here is that Bob's outside the black hole. He can perform his Bell measurement.

1:40:00 Alice is inside the black hole. She can perform her measurement, but they can never get together over drinks to compare their measurements and find out whether or not they got their correlations or anything else. So, you know, the state, the supposed state, is supposed to be encoding all of these correlations, but there's no way of testing them. And so then it seems, then we've got the physicists claim that, well, it's unphysical, maybe we don't really need to worry about this. Now, Balot et al., in their nice paper, rightly worry about this particular paper. They say, look, we're looking, this is some sort of operationalist, verificationist philosophy that doesn't seem very convincing, especially if it's supposed to be some real principled answer rather than just some heuristic or theory device or theory construction or something like that. If we're going to be operationalists, why should we care about the interior of black holes at all? And we're not going to be going in there and bringing back results and talking about them at seminars and anything else. Why should we even consider this an area of interesting physical research? To go further and ask, why should we be committed to this particular account of verificationism, this operationalist account of meaning that the black hole complementarians are appealing to, when People who advocate black hole complementarity tend to be string theorists, or at least highly sympathetic with string theory. So string theorists are going to reject the framework of Hawking's information loss argument because that's based on local quantum field theory. Local quantum field theory is something that string theorists don't think is a fundamental story about quantum gravity. So, why should we go through all this trouble and be worried about Hawking's argument? Well, that's one of the things I want to give an answer to today. The reason that complementarians, that is, advocates of black hole complementarity, are worried about Hawking's argument It's because even though they don't think that quantum field theory is a fundamental theory, even though they don't think it's going to be the right theory of quantum gravity, they do think that local quantum field theory is going to be a highly accurate, effective theory, and it should hold good as long as the curvature isn't terribly high or we don't have high Planck scale energies or anything else that pushes us into the true quantum gravitational realm.

1:42:30 The worry is that the merely effective applicability of local quantum field theory is going to be enough for us to secure Hawking's argument here. And the way we can show that is we can foliate the spacetime in such a way that all of the local energies that we're considering are nice, low, well-behaved, etc. Stay away from the central singularity. ...for long enough that the black hole is going to be shrinking down, shrinking down, and eventually the black hole is going to get small enough that there's no way we can avoid quantum gravitational effects. So at that point, you know, microcausality could be violated. It could be the case that information escapes. But there's no longer enough mass energy left in the black hole to carry away all the information that would have been packed into it. Thank you for watching. Remnants of a black hole could return enough energy to the outside universe to restore the purity of the outside state. So this is the worry, that just the mere affective applicability of quantum field theory is going to be enough to be able to run the information loss paradox. Susskind and Thurlacius in 1994 run what I think is a promising line of argument. They suggest here that in this paper what they want to do is challenge the commonly held view that as there is no strong curvature in these knife slices, for example, an information loss paradox can be posed without detailed knowledge of the underlying short distance physics.

1:45:00 So their suggestion is that, well, look, you think that you're able to run Hawking's information loss paradox just appealing to local quantum field theory, but you need to make some assumptions about the underlying quantum gravity, and those are some assumptions that I might not be willing to grant. So I think this is the right argumentative strategy, but I don't think they actually succeed in making, establishing their argument. And why is that? They once again appeal to a verificationist philosophy. That is, their argument here is that, well, look, you can't pose this in an effective local quantum field theory, Hawking's argument, that is. Well, why not? Well, because there's no way to verify the sort of claims that are made by Hawking's, by the local quantum field theory. Just to give you some examples. Well, first of all, we said, look, Alice performs her measurement inside, Bob performs his measurement outside. They can't get together over drinks, right? This was in 93. One thing that people pointed out to him right after they published their papers, well, maybe there are some ways, even though Alice can't signal out. Alice could leave a signal inside the black hole, and then Bob could jump into the black hole, and as Bob's on his way to central singularity and about to die, he could be sitting there and saying, I have found a contradiction in nature and, you know, and all. And they say, well, gee, if Alice is going to leave a message inside the black hole, let's take a look at what sort of energies would be required for this particular, for this to happen. Quick calculation reveals, not surprisingly perhaps, that Alice is going to have to be sending messages with Planck scale energies for it to be able to stay inside the black hole long enough for Bob to be able to do his measurements on the outside, Hawking radiation, and then jump into the black hole himself. Or another example that people suggested to him, and they said, look, you're not actually talking about a membrane at the horizon. Instead, you're talking about a stretched horizon, membranes slightly above the horizon. So what if maybe Bob goes down and says, you know, just sort of in free fall, says there is no horizon there, and then escapes and goes back out to infinity. Won't this pose a contradiction for your account?

1:47:30 It's easy to see that if you're going to get within a Planck length of the actual horizon and then turn around and get yourself back out, you're going to have to undergo Planck scale accelerations, which again is going to bring you into the realm of quantum gravity, and we don't know what's going on in quantum gravity there, which is supposed to be the basis for their claim that you can't even pose the paradox without appeal to Planck scale physics. This is actually from the paper. I think that's supposed to be a rendition of Lenny Susskind there. I thought it was so nice that I should share it with you all. The problem is that this is, even though this is being brought in in a slightly different argumentative format, this is still just another appeal to verificationism. And so we don't have any particular reason to get into this because the argument... For the paradox is based on the nice slice argument and not on our ability to verify the correlations that are encoded in the quantum field theoretic description. All right, so that was Susskind, Tuft, or rather Susskind, Thurlesius, and Uglum's proposed account to try to listen from Hawking's argument. He proposes a much more direct argument for the claim that these two representations, Alice's representation and Bob's representation, that these are complementary and incompatible. And this is based on his claim and comes out and says, well, look, if we just look at the operators that we would assign to Alice and the operators that we would assign to Bob, we're in the Heisenberg picture here, of course. We're going to find, he says, that these two operators don't commute, even though they're space-like separated. You'll be shocked by that, and I hope you are. But what is his argument for this? Well, he repeats it several times, but here's one particular version of the argument. I'm just going to read it out loud here.

1:50:00 His argument is, any measurement made by Bob Implies the introduction of states by acting on it with operators that create or remove particles seen by Bob, which for Alice would be outrageously energetic. These particles would cause gravitational shifts that seriously affect the ingoing objects, including the fragile detectors used by A. Thus these observations, he says, cannot be independent. So the claim here is that Bob's out here, he's measuring the airplane Hawking radiation, he measures, you get some particle, it's got some energy, omega, and then that particle climbed out of the gravitational well of the black hole, so then if you ask, well, what energy would this thing have had back when it was right next to the horizon, passing by Alice going through, well, it would have been exponentially larger, and even just the gravitational influence of this thing would have seriously disturbed or destroyed Alice. And so that's going to be a big, you know, Alice is going to notice that, and so Toft says that these two are actually incompatible complementary observables. What are the problems facing Toft's account? First of all, it's an appeal to measurement collapse. He's claiming that Bob performs this measurement at this late time that collapses the wave function of things, and we all know that measurement collapse is a kind of problematic concept, so we might worry loudly about that. But in this context, an even greater worry is that measurement collapse is non-unitary. We've got a violation now of unitary Schrodinger evolution, so if we're invoking non-unitary evolution, why should we care about Hawking's paradox when the whole worry about that was that we had non-unitary evolution? But perhaps even more of a problem, well definitely more of a problem for a Tost argument, is that it's just fallacious. It just doesn't follow. The easiest way to see this is to recast this scenario in the Schrodinger picture. Toff's argument is based on the claim that there's a large commutator between the longitudinal component of the stress-energy tensor right outside of the horizon and some measurement performed on an outgoing Hawking particle at a late time that this commutator is going to grow exponentially with time.

1:52:30 Well, okay, we can accept that. What does that mean if we're looking at things in the Schrodinger picture? Measurement, right outside the horizon, we're going to get a wide dispersion in the results that we get in Bob's measurement. Okay, well we can accept that too. But does this pose any problem for the health of Alice falling into the black hole? The answer is clearly it doesn't. Either the stress-energy-tensure measurement was performed or it wasn't. If you didn't do the measurement, well, Bob's results aren't changed at all and Alice is perfectly healthy. On the other hand, if you do perform the measurement, well, Bob will be able to tell you perform the measurement, if he's able to make a measurement on an ensemble of such particles and stuff like that, but that doesn't cause any particular problem for Alex either, and there's no problem with microcausality in this particular context. Toph's argument is just confused, as far as I can see. For Linda, offer what I think is a better direction to go. Their argument, roughly speaking, is the following. They say, look, we've got an effective semi-classical field theory here. That is, we think we can, we're considering quantum states operators on a given background space time, right, a classical space time. Well, if the energies are too high in certain states or interactions, then we're no longer in the spacetime that we're trying to consider. That is, we're going to be in some different spacetime, so we need to impose some sort of cutoff. Okay. Well, we put in our cutoff, and if we have an energy that's larger than our cutoff here, then we're going to be taken to some different spacetime. But cutoffs are frame-dependent. If you, for example, are just staying here, just say, you know, consider we're in some Minkowski space-time, whereas I'm some Rindler observer and I'm accelerating, you and I are going to have different accounts of what high Planckian energies are, so we're going to use different cut-offs.

1:55:00 We have some maximum, we have some, we set some cutoff based on the fluctuations that it induces in our spacetime via the stress-energy tensor, and then your quantum description, the complete set of operators that you have using your cutoff, and the complete set that I have using my cutoff, are going to provide complete complementary descriptions of a single Hilbert space. Now, in this case then, this means that they're not going to be the same operators, but then what we're suggesting is that these operators might fail to commute with one another, and so we're really going to have a case of good old-fashioned complementarity in these cases. And then black hole complementarity is going to be a special example of this space-time complementarity. It would also apply, for example, to the Minkowski and Rindler observer. Well, the nice things about this is that there's no verificationism here. That's a good thing. There's no appeal to measurement collapse to call for some implausible destruction of observers. It has what I want to argue are the correct conceptual framework. That is, what we're looking for is the limitations of the theoretical descriptions that we're using. We're saying, well, what are the theories that we have and when does that theory give out? The problems with this is that there's a somewhat explicit violation of microcausality here without a very good account of how it is that this microcausality isn't going to come back to haunt us. But even more of a problem is that it's a very, very speculative account. You wouldn't expect general space-time complementarity to hold. You wouldn't expect to end up with the same Hilbert space if you're using different cut-offs. It seems plausible, at least it's got the right conceptual framework, but physically it's not very well motivated yet. Just some concluding remarks. First of all, and this is something I haven't really argued here, but complementarity is often mistaken for some sort of verificationism. That's, I think, what Susskind, Farlashius, and Uglim have in mind here. I think that's a particular mistake. It doesn't require the impossibility of performing certain measurements that account for the procedures that require the impossibility of sharing information, so we don't need to keep these two... We don't need to prevent Alice and Bob from communicating. One of the nice things about Kim Ferlinda's suggestion is that it is clear that this sort of complementarity would hold even when people can.

1:57:30 I'm a Minkowski observer and you're a Rindler observer, we can both perform our measurements, then we can get together after drinks and actually compare our measurements and everything else, even though, according to Kuhn, Verlinde and Verlinde, these are complementary, these are supposed to be incompatible observables of some sort. Well, in what sense then are these going to be incompatible? Well, they're going to be compatible, according to Kimberlin and Berlinda, because we're not able to apply a semi-classical theory that allows us to interpret both of these measurements. So we can go through the procedure, but then when we try to get together and ask what it means, that's the point at which we're going to get lost. And that seems to me to make a lot more sense than trying to argue that people get blown up as they fall into blackboard. Let's say the wheat, what it has to offer us as a proposal, I suggest that the three key points here are first of all a claim that the evolution of black holes is unitary, that we're able to secure that unitarity by having the correlations cleared off by Hawking-like radiation, and that really what allows us to escape Hawking's paradox are violations of micropausality. If people have questions, I can talk a little bit about modern support for black hole complementarity that goes beyond these particular arguments that I think are a little bit weak. Final interpretive point that I just want to end on is I think what's really important for philosophers of physics is, in our interpretive work, is to be looking for the boundaries of various theories. Every theory we have, we know is wrong, is at best an effective theory. So one of the things that we want to do if we want to know what these theories are telling us about the world is tell us where do these theories apply and where do they not apply.

2:00:00 And that's a very important interpretive task and one that I think we haven't taken seriously enough yet. Thank you. When this paradox is set up, there's something that is, to me, paradoxical about the paradox, namely that it depends upon a high-energy theorist's sensibilities favoring unitarity and a geometrical general relativist's sensibilities about interpreting gravity, rather than, say, a high-energy theorist's treatment of gravity as a field or spin-to sort of phenomenon. In order to get this paradox going, you have to introduce premises which probably no one actually, I don't know why anyone would believe both of these premises. I guess you believe in one or the other. These premises are, you know, estimatrices are really what's fundamental and what we should be after coming from the particle physicists. Something about quantum mechanics being fundamental or something along those lines, and someone from general relativity arguing that, look, we need to take these space-time pictures seriously when we've got singularities and black holes, etc., etc. As opposed to something more like a universal course. It seems like it ought to be a natural view for a product of physics. I could explain more where the geometrical premise enters in, if that would be helpful, but do people just not notice this, or do the string theorists actually take the geometrical viewpoint about gravity themselves? One thing to keep in mind here, they do take the geometrical viewpoint, they do think that these are the space times that we need to deal with, at least at an effective level, right? What they're interested in doing is figuring out, okay, what is a good semi-classical approximation going to look like in this case, in the case where we've got black holes. Now, what actually drove the work on this, Hawking presented this paradox in the 70s. There was a little bit of stir about it, but really, no one took it seriously until the early 90s when people started dealing, playing around with these two-dimensional toy models. We're actually working on getting these two-dimensional models and trying to model.

2:02:30 What they're actually trying to do is get the semi-classical equation and get solutions to it. And we can just throw up the full semi-classical. And so what they're trying to do is get nice simple two-dimensional models where we can actually get some solutions to these equations. And they actually did this. And they actually did it to try to see, well, in this simple case, what happens when the black hole radiates down, right? And, well, you can get solutions, but you've got to put in some sort of boundary conditions. You put boundary condition in, you have to, you know, you know by the time you get to singularity, you've got to put in some sort of boundary conditions. So they start that, and things, you know, the wires, they didn't have a decent picture of what was going on. He got negative energies out and all sorts of, and negative singularities and all sorts of nasty stuff like that, so they said, okay, well, let's move the boundary conditions to the event horizon instead and see what we get out here in our toy model once again. And once again, they've got this little, it looks better, but you've still got these negative energies coming out in a thunder pop, and that's why they actually moved to the stretched horizon. Once you go outside the event horizon, then they argue everything's well-behaved, and so what they're trying to do here is get some sort of nice semi-plastical picture of what, you know, rough and ready, what's going to happen when you've got the evolution of a black hole. Now that is assuming a background space-time that then is going to be modified. ...semi-classically, right, that you can actually feed back into the equation. So that's what they're trying to do at this point, is just work at a semi-classical level, and then the worry is, well, can we even make sense of this at this point? And I think what they're now starting to come, the commitment that they're now starting to have is that, well, the violations of the semi-classical picture are far more subtle than we thought. That is, before we thought, well, as long as you stay away from the singularity, it should be a pretty good description. Now they think, well, there are violations of this sort of picture that are much more subtle, that are over astronomical distances. And one way of cashing that out is that it's violations of microposality.

2:05:00 Anyone else? I think we are... Yeah, Karl. Yeah, just a quick question. You draw these...