Markus Hanke

The MTW method has one limitation though - it assumes that all dimensions are of equal size. If that’s not the case, then the result obtained by this procedure may become scale-dependent. We all know about compactified dimensions (ref String Theory). I’m wondering though - is the opposite possible? What I mean is - could one configure a spacetime manifold such that one of its dimensions becomes detectable only at large scales, but is hidden at smaller scales? I can’t think of a way to do that, but would like to hear others’ opinions on this.

I don’t quite understand what you mean here - vectors, forms and scalars are themselves just tensors, so all of these objects are already on equal footing from the beginning. The slots (indices) simply tell you what mappings are possible, and what the rank of the resulting tensor will be.

I’m with MTW - I look at it as a map that maps vectors and forms into real numbers (or other maps).

Another way to look at this is that in curved spacetimes (of which FLRW is a specific example), energy-momentum is not - in general - a globally conserved quantity, even though it remains conserved everywhere locally. Thus it is not surprising that light does not retain its original frequency when traversing large regions of non-flat spacetime. I personally think this is a better way to view this, since, after all, these galaxies remain in free fall and do not undergo proper acceleration at any time, despite the velocity-distance correlation.

There’s also the matter of the star’s shape to consider - in the frame of the particle, the star isn’t spherical, but a flattened disc along the direction u. Thus both the time and the space parts of the gradient must be considered, and treated in the same way (for covariance) - therefore (2.37) is indeed reasonable so far as its general form is concerned.

I would have answered the same. Does anyone see a reason why this would be incorrect?

Are we not perhaps overthinking this a little? After all, the quote given is from MTW, which is a book about spacetime. My feeling is that what the authors had in mind was a physical space, as well as physical procedures to determine its dimensionality. The more abstract notions mentioned here are all good and well in pure maths, but they don’t necessarily relate to GR.

Very nicely put +1

I’m sorry, maybe I’m being a bit thick here, but I’m still not 100% sure what ‘structure’ were are talking about exactly. The definition says it locally resembles a Euclidean space - so can we assume the presence of a connection and a metric, or just a connection, or neither?

On manifolds with curvature, covariant derivatives do not, in general, commute; thus, again in general, they wouldn’t end up at the same place. But answering this question requires that your manifold is endowed with a connection.

Yes, it would appear so…but I’ll defer to what the real mathematicians here have to say on this.

Good and interesting point, I hadn’t remembered this remark from MTW. I remember from topology that proving that the boundary of some n-dim region on a manifold to be a (n-1)-dim region, requires the notion of homeomorphisms. But I don’t know what exactly needs to be in place for this to work. Nice one!

You don’t need a metric for this, but I think you need to know at least what kind of connection your manifold is endowed with, for this to be possible at all (someone please correct me if I’m wrong). Given a connection, you can use geodesic deviation - you analyse all the possible ways that geodesics can deviate in the most general case (no Killing fields) on your manifold. From this you can deduce the number of functionally independent components of the Riemann tensor and torsion tensor, which straightforwardly gives you the dimensionality of the manifold. This is independent of any metric; it even works if there’s no metric at all on the manifold. Practically doing it may, however, be pretty cumbersome. Anyone know of an easier way?

Can you make precise what exactly you mean by “volume” here? Is it a 3-volume of space, as in a geodesic ball for example? Or a 4-volume of spacetime? Both of these depend on the metric, as do all measurements of lengths, angles, areas, and n-volumes in general on a differentiable manifold. For 3-volumes this is immediately obvious without any further ado, since FLRW spacetime is not a vacuum solution, and thus the Ricci tensor does not vanish. Therefore 3-volumes are not, in general, conserved as they age into the future, irrespective of your choice of coordinates. The only way to get them to be remain constant would be to choose a cosmological constant of just the right value so that you end up with a constant scale factor. But that is not compatible with observations. For 4-volumes of spacetimes, the determinant of the metric enters as part of the volume element, so when you perform the integration to find the total volume of a given spacetime region, the result will explicitly depend on the times you use as integration limits, since the scale factor will be in there. So again, expansion has an effect here. All of these are general mathematical results in differential geometry, and not specific to just GR as a theory. I’m happy to show you the mathematical expressions if you need to see them (but I’m sure you’ve seen enough of me around here to know that I’m not just making this up). Otherwise Misner/Thorne/Wheeler has a good overview on how to construct general volumes on differentiable manifolds, or you can take a quick look on Wiki as well.

Funny coincidence…just as this thread gets posted, do I get back online, after a fashion anyway Yes all is well with me, I was just without Internet connectivity for a while. I live in Norway now, and helping to build up a monastery here. My Internet connection is still basic, so I mightn’t be quite as active as before. But it’s good to be back And thanks for the book recommendation @studiot, I’ve had this in my library for a while, but haven’t gotten around to reading it. Hopefully soon!

Hi all, just wanted to let you all know that I’m temporarily away from the forum, since I’m currently staying at a place without any Internet access. Will return once circumstances permit 👍

Well, that’s part of it. But basically what it means is that spacetime does not exist in anything else - it’s not embedded in some kind of higher-dimensional space (=background), and the curvature that forms its geometry isn’t of the extrinsic kind, in the same way that a cylinder is extrinsically curved. IOW, nowhere in the theory is there any reference to anything that isn’t spacetime. This is in contrast to the Standard Model, which assumes a background spacetime - and a specific one at that - on which its fields live. You simply can’t have a quantum field without a background on which it lives. As Genady pointed out above, there are no distinct parts - a specific aspect of energy-momentum simply equals a specific aspect of spacetime geometry, up to a proportionality constant. Of course, when you are dealing with a specific scenario you can run things both ways - ordinarily you start with a distribution of energy-momentum, and calculate curvature from that. But because these are equivalent, you can do it the other way around as well - if you are given a specific geometry, you can calculate in what general way energy-momentum has to be distributed in such a spacetime. What’s important to remember is that the field equations are only a local constraint - they do not themselves uniquely determine geometry/energy-momentum, but merely constrain what forms these can take. To fix a unique spacetime, you need to also supply the right number and kind of initial and boundary conditions; physically these conditions usually describe the distribution of distant sources (as opposed to local ones given by the energy-momentum tensor), as well as overall symmetries of the spacetime. Another important point is that even if the local geometry of a spacetime is uniquely determined, the Einstein equations place no constraint onto its global topology, so many solutions are topologically ambiguous.

Gravitons, if they exist, would be massless spin-2 particles - and as such, they would be perfectly stable and can have no decay modes.

Right, I see. So, what we are really talking about when we speak of the “rest frame” of a quantum system is a region of space V where: \[\langle p\rangle =-i\hbar \int _{V} \psi ^{*}\frac{\partial }{\partial x} \psi \ d^{3} x=0\] wherein x is to be understood as the appropriate collection of coordinates. Ordinarily the region V would need to be all of space, or at least large enough to ensure proper normalisation of the wave function. So in essence this is a statistical statement about a region of space, given by the volume integral of some (probability) density. This is remarkably different from the notion of “rest frame” as used in (eg) GR. Just trying to develop a more intuitive grasp on the relationship between classical and quantum frames here.

I’ve also asked it a lot of technical questions about GR, just to see how capable ChatGPT is in that area. Unfortunately I must say that a sizeable chunk of the answers it gave were either misleading or straight up wrong - which only goes to show that this is not an acceptable source of scientific data, at least not without fact-checking the answers you get. I have, however, noticed that you can point it out to the system if an answer is wrong, and it seems to be learning from such corrections. Yes, that’s a good point. And this may be the reason why (eg) ChatGPT sounds a litte “hollow” and “woody” in its answers, if you know what I mean. But now imagine if we did add these extra information channels to the system - for example by giving the AI access to a camera and letting it observe human non-verbal behaviour that goes along with spoken material, and correlate the two. Or letting it analyse videos with spoken audio in it. And so on. This would allow the neural net to not only learn the mechanics of language, but also context and usage. I still think that given enough data of the right kind, it should be possible to end up with something that communicates verbally in a way that is indistinguishable from a conscious human - or at least indistinguishable without special tools and analyses.

Why not? What else would it require but a model of natural language, and a sufficiently large set of precedents? I don’t see why verbal interactivity cannot be simulated by a machine, to such a degree that it becomes indistinguishable from a conscious agent - at least in principle. You can never be sure, based purely on verbal interaction. In fact, if you were to interact with me directly in the real world, you’d probably be left wondering; I am autistic, so my face-to-face communication style is - shall we say - unconventional and not quite what you would expect from an ‘ordinary’ person, so you’d be forgiven for mistaking me for an AI

But isn't this just conjecture? How can we be sure that a simulated brain is functionally indistinguishable from a biological one? So far no one has succeeded in accurately simulating even a single neuron (except as rough approximations), since, when you look at it more closely, it's actually an incredibly complex system. How can we be sure that it isn't the case that some part of the biological hardware is actually necessary for a brain to function like a brain? I'm not claiming it can't be done (I don't know, and I'm not an expert in this either), I'm only urging caution with this assumption. I think it needs to be questioned. I don't understand this inference - why does being able to verbally articulate something imply that there are necessarily inner states? And why does the presence of inner states imply awareness? Any and all human languages have a finite number of elements, and a finite number of ways these elements can be combined, due to the presence of rules of grammar etc. It is conceivable to me that one can write software that simply trawls through the entirety of all written and spoken material that has ever been digitally captured, and, based on this, will be able to verbally respond to any question you pose to it in seemingly meaningful ways, based on precedents and probabilities within already existing media. In fact, if my understanding is correct, this is roughly what ChatGPT does. However, this is purely a mechanical and computational process, and I wouldn't agree that there are any kind of 'inner states' or 'awareness' involved in this. Of course there could be, but how can we be sure? My feeling is that any sufficiently complex language model will eventually become externally indistinguishable from a conscious agent (based on verbal interactions), even though it is entirely mechanical in nature. I guess this is just the classical 'philosophical zombie' thing.

Yes, so this is the crux of the matter. Since Bell's theorem implies an absence of local realism, then in what sense is the concept of a 'rest frame' even meaningful for quantum systems such as electrons? At a minimum, being at rest wrt to something would imply that both position and momentum of that object are simultaneously known precisely, or else the notion of 'relative rest' is meaningless. But a vanishing commutator like this implies we are in a classical situation, so there's a contradiction. So are we to understand that quantum systems do not possess any rest frame in the usual sense, except perhaps as a helpful approximation?

QGD is conceptually similar to LQG (one might argue that the latter is a specific example of the more general former), but I would not consider it to be a quantum field theory in the ordinary sense since it does not involve quantised operator-valued fields on a fixed spacetime background. Nonetheless, I agree that this general approach to things - ie some form of quantisation of geometry itself - is probably the most promising when it comes to quantum gravity candidates.

It's not that GR modifies the background, it's that it does not have any background. It's a fully background-independent theory - in contrast to the other interactions. This may well be the case still, it's just that in all likelihood this would not be described by a quantum field theory. I think such a unification of forces will require us to let go of the notion of a smooth and continuous spacetime, which renders the concept of a QFT meaningless.