Background independence / Backward light-cone simultaneity & the twin paradox (& others)

0:00 So this is basically the outline of what's to come. The beginning of the story, I'm sure, is familiar to most of you. So, Weinstein originally thought when he was searching for a very specific theory of gravitation that general covariance would be one way of implementing a general principle of gravitation, and that's what he believed after he had found the field equation. And then, of course, Krachman comes back to him and says, no, almost any theory can be put in a generally covariant form. So, in particular, general covariance can't have a physical significance. A response by some people, so I think the orthodox position is yes, that's why general covariance is not a physically interesting concept. But more and more recently, people have been responding to that kind of question in the following way. They sought to distinguish between a formal notion of general covariance and a substantive notion of general covariance. And the idea is that generally covariant versions of pre-general relativistic theories satisfy formal general covariance, but they don't satisfy substantive ones. Getting clear on what this substantial principle is, is one way of getting clear on what the conceptual logic is for general mathematics and C.R. Well my response to this has been, that sounds ridiculous, Surely generally covariant versions of pre-generally relativistic theories aren't nearly formally generally covariant, and in particular, surely they satisfy active as well as passive versions of generally covariance. I put those words in the scare quotes because it seems to me that that kind of terminology is actually very ill-chosen. I'll go on later to do my best to make sense of those. Anyway, so that was what I used to think, and part of the point of giving this talk is that I've actually come to regard that as a mistake by myself, and yes, after all, there is an interesting distinction you can draw between formal and substantive general covariance, and yes indeed, it is the case that some pre-general relativistic theories satisfy only the formal notion. However, before I go on to say what that is, I'll consider Erman's attempt to...

2:30 I think that the term fails, but it sort of points in the direction of where there is a genuine interest. However, it is not automatically satisfied by a theory that is meaningful.

7:30 So what are Ehrman's reasons for denying that the Drew Robinson group is a gauge group? Well, he looks to the practice of physics.

10:00 So he says that physics literally contains a generic set of data. The verdict of this is that it's all to do with... Now, the way he phrases it, the idea is that this adroits with our theory, so Bumman seems to say that we can change the coordinates of a static notion, a theory's invariant under active dimorphism, smooth displacement of the dynamical fields, and the dynamical fields alone.

15:00 On the one hand, we've got this kind of active-passive difference, whether we're just doing a coordinate, and then we've got the coordinate chart. So here's a proposal. We haven't moved fields, but when we use D, then we've got an active dimorphism. So if you choose D and 1...

17:30 The 1D here and the inverse here, we're going to end up with a change from our original coordinate to the very same coordinate, but in the first case it's just a re-coordinate, but now we're thinking of this as a genuinely distinct state of affairs because it respects the old coordinate. The question is, is, so with this language, our old principle, if Mg buys the model, so is M d star e d star i, that just becomes T's invariant under F. So that language hasn't yet given us a new principle, but here's a new principle. Okay, GC5. This is basically what we're getting from Reveille. You need to draw this distinction between two types of physical field in theory, the non-dynamic ones. It's only this thing. It's two very different ways. There are two different ways of thinking. It's only GC5 again because it went to the first and I want to get to the other thing. We've got this distinction between...

25:00 And you're acquiring that this is where we're only acting on the dynamics of. Now the GR just, you have just that, so GC5 is equivalent to GC5. And then the question is, okay, I want to say, if I say the metric is a field of this kind, then pre-relativist theorists don't satisfy this. But then what's your story about what kind of objects these are? Now Ehrman, I mean this is very like Ehrman's definition of the dynamical symmetry, and in his argument that if something is a space-time symmetry then it should also be a dynamical symmetry, that appeals to GC3, because it says, you know, I've got by GC3 m d star a d star g, but then d star a just is a, because it's a space-time symmetry, and so this must also be a model. But of course, the interesting case is where we have a failure of this because, in general, something like this is something where I've got d star a and d star d isn't even a model of the theory. We definitely have a failure of GC5 when we've got a theory like this where this thing isn't even in k. So there's something, I mean, there's an interesting difference. There are well-known difficulties in exactly defining those so-called absolute objects.

27:30 I wondered how is that reflected in this story? Is there a connection between the dynamic object and the non-dynamic object? Yeah, absolutely. So, the second part of this talk then seems to address this question... I draw a distinction between a dynamic and a non-dynamic field, so you can reformulate GC5 as, you know, only act on the non-absolute objects, right, and then... We would get a violation if you're saying that you make costume metrics as an absolute object because perhaps you've got the treatment as the criteria for what's an absolute object. And then I think GC work just becomes equivalent to Adan's principle that the diffeomorphisms are... The things generally covariate in that sense just becomes that the diffeomorphism is invariant in Adan's sense because Adan had something that was... We can look at the clock at this time for instance.