Markus Hanke

No he wouldn’t. The distance from horizon to horizon would explicitly depend on where it is measured, when it is measured, and also how it is measured (since the horizons are not spherical). The spacetime in between these black holes is not stationery and admits very few spatial symmetries, so figuring out what any one observer would measure here is highly non-trivial. But he certainly wouldn’t see any perfectly circular orbits. You cannot use Newtonian forces to describe or visualise this, since the radiation field here couples to a rank-2 tensor as its source. This really does require the full machinery of GR.

How do you know this? The metric in close proximity to a BH merger is complicated enough so that it cannot be written in closed analytical form - it can only be modelled numerically. Given that such a spacetime would admit at most one spatial Killing field (and even that only approximately), and is not stationary, I think it is highly unlikely that any observer there would measure a perfectly circular orbit for the black holes. Maybe it is possible to find some coordinate system where this might be approximately the case over a short span of time only, but even that I’m not convinced of.

Two black holes in close proximity cannot be of the Schwarzschild kind - so their event horizons would be nowhere near spherical. Also for a single Schwarzschild BH, no stable orbits exist at the horizon.

For a BH merger, there is only one wave field, not two, so I’m not sure what you mean by ‘phase difference’. Also, gravitational waves and the energy-momentum they ‘carry’ do not add linearly (GR isn’t a linear theory), so the argument wouldn’t hold anyway.

I would disagree with this, since - as mentioned - several “toy models” exist that do just precisely this. Genady has mentioned ‘t Hooft, and another example is the model by Donadi/Hossenfelder: https://arxiv.org/abs/2010.01327 I think the key here is the distinction between correlation and causation - the necessary correlations exist from the beginning, but the measurement outcome is still caused purely locally by the detector settings only, just like in ordinary entanglement. The very fact that such models can be written self-consistently while demonstrably reproducing all predictions of standard QM makes it difficult to outright dismiss the concept as nonsense. Experimentally, superdeterminism would show up as small deviations from Born’s rule under specific circumstances, so it might be possible to experimentally distinguish it from QM.

The thing here is that no experimenter has the freedom to choose the parameters of his experiments - in some sense, all experiments and their outcomes are preordained from the very beginning, so it isn’t possible for these constituents you mentioned to not be exactly correlated in the right way. One might say that it is the correlation that reveals the experiment, not the other way around. There is no freedom of choice in superdeterminism. Unfortunately I do not understand all of it, since this is not an area of physics I have reliable expertise in. So I’m not in a good position to evaluate its scientific merit. It should be mentioned though that this isn’t the only such proposal - there are several superdeterminism candidate models in existence, all of which are consistent with QM and the known Bell experiments. Personally I dislike the idea greatly; it even scares me a little, since a superdeterministic universe would be one in which no one possesses any degree of free agency. But I struggle to find a decisive argument against it, since the maths appear to be consistent, and I must acknowledge that it would eliminate some difficult issues, not least of which the measurement problem.

I wouldn’t look at it this way. What it means is that there are in fact correlations between observer and system that are not accounted for in standard QM, which fundamentally assumes statistical independence. I don’t like this idea much either, but the fact of the matter is that the universe started off very small and very dense, and has a finite age - so the concept perhaps isn’t so absurd after all. But of course, this would have serious implications for the philosophy of science, since one could no longer cleanly separate an experimenter from his experiment. How much could we reliably know about a universe where this is the case? Here’s a very recent and quite interesting paper on this - see under ‘conclusions’ at the end for a quick summary. https://arxiv.org/pdf/2308.11262.pdf

Bell’s inequality can fail to hold in another way, too - namely, by virtue of one of its underlying assumptions to be false. In particular, it assumes statistical independence of observer and system, meaning the experimenter actually has completely free choice in how he sets up his experiments. If this is violated, you get some version of superdeterminism, which can preserve local realism even if Bell’s inequalities don’t hold. I personally don’t like this idea, but the more you think about it, the less easy it becomes to dismiss it outright, especially since it also provides a solution to the measurement problem.

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.