Markus Hanke

No problem. It is far more understandable though if one has access to the textbook, where all of this is described in much more detail.

Actually, what I am hoping to find is precisely a connection between ER=EPR and the issue of locality/separability/realism. What if the violation of Bell's inequalities precisely implies that spacetime is in fact multiply connected? Shouldn't it be possible, at least in principle, to retain both locality and realism, while still violating Bell's inequalities, if the underlying spacetime is multiply connected in just the right ways?

That would probably help things - it's in chapter 15 of the book. The essential train of thought is this - suppose you have an elementary 4-cube of spacetime \(\Omega\). We know that energy-momentum within that cube is conserved, so (in differential forms language): \[\int _{\partial \Omega } \star T=0\] If we want to obtain a metric theory of gravity, the question becomes - what kind of object can we couple to energy-momentum, that obeys the same conservation laws, in order to obtain the field equations? For this, consider that the boundary of our 4-cube consists of 8 identical 3-cubes, each of which is in turn bounded by 6 faces. We can now associate a moment of rotation with each of the 3-cubes; to provide the link to energy-momentum, we then associate that moment of rotation with the source density current in the interior of each 3-cube. Let \(\bigstar \) define a duality operation that acts only on contravariant vectors, but not on differential forms (the Cartan dual). The moment of rotation is then \[\bigstar ( dP\ \land R) \] wherein R denotes the curvature operator. We also find that this expression is just the dual of the Einstein form: \[\bigstar ( dP\ \land R) = \star G\] So let's put this all together. First, we create a moment of rotation in our 4-cube of spacetime: \[\int _{\Omega } d\star G\] Apply Stoke's theorem: \[\int _{\Omega } d\star G=\int _{\partial \Omega } \star G \] Rewrite in terms of the curvature operator: \[\int _{\partial \Omega } \star G=\bigstar \int _{\partial \Omega } ( dP\ \land R) \] To associate this with total net energy-momentum (at the moment it's associated with source current density), we must sum not just over the boundary of the 4-cube (which are 3-cubes), but also over the faces of each 3-cube; so we must apply Stoke's theorem again, to get \[ \bigstar \int _{\partial \Omega }( dP\ \land R) =\bigstar \int _{\partial \partial \Omega }( P\ \land R) \] But because \(\partial \partial =0 \), this automatically yields \[\bigstar \int _{\partial \partial \Omega }( P\ \land R) =0 \] But because the bracketed expression is just the dual of the Einstein form, this implies \[\int _{\partial \Omega } \star G=0 \] This is the exact same as the conservation law for energy-momentum given above. So we can associate the two: \[G=T\] So, using the concept of a moment of rotation, some elementary geometric considerations, and the "boundary of a boundary is zero" principle, the form of the Einstein equations is uniquely fixed up to a proportionality constant. MTW even obtain this constant somehow, though I don't quite follow their thoughts on this minor detail. Anyway, that's the general idea. If you can get access to the text, it is all described in much more detail there.

That is essentially what I was hinting at, actually...I think our current concept of what is fundamental (quantum fields and their interactions and excitations) just doesn't really cut it, from a philosophical point of view (not saying those things don't work!!). I think there is a whole lot more going on than we currently realise, and it will need a major paradigm shift to reveal it. Something else just occurred to me earlier today - your mathematical definition of locality makes a tacit assumption: that the underlying manifold on which the field 'lives' has a trivial topology. But what happens if that is not the case? For example, what happens if the manifold is multiply connected, or has closed loops, or whatever else may be the case? Properly defining 'locality' becomes more difficult, then. P.S. I'm aware of course of ER=EPR, but haven't really arrived at a conclusion about what this really implies.

It would certainly be non-trivial. However, what I was trying to point out is that, if someone develops QFT first, then they would very quickly realise that a standard QFT for spin-2 bosons that couple to energy-momentum (a reasonable ansatz if one wants to find a model for gravity) yields something that is physically meaningless. So they would begin to wonder if perhaps gravity can't be described via a QFT at all. This would eventually bring them to consider metric theories instead - and GR is the simplest example of that. Yes, this is the strong field regime I mentioned in the last post.

The only textbook I know of that explicitly mentions this is Misner/Thorne/Wheeler "Gravitation". It's a very beautiful connection between GR (and also electromagnetism!) and that topological principle. If you don't have access to this text, I can try and summarise the derivation here, if I have time over the next few days. Interesting observation, but...isn't the AdS/CFT duality the exact opposite of this? Only spacetime within the bulk would have a time dimension, whereas the boundary on which the CFT lives is purely spatial. However, I may have this wrong, so please correct me if necessary.

Have we not answered this already earlier on this thread? Any scenario where the non-linearity of GR cannot be neglected - so essentially the strong field regime - will display effects that differ from the weak field approximation. Even the article on GEM which you shared mentioned some of these effects. I just see it as the simplest possible metric theory that fulfils all relevant self-consistency criteria. This is an issue only so long as one tacitly assumes that GR must be the classical limit of some quantum field theory - which implies the assumption that gravity works in the same way as the weak, strong, and EM interactions. But actually, there is nothing in physics that indicates that this must necessarily be the case. Also, it is actually trivially easy to write down a QFT for a spin-2 boson interaction, but it is just as easy to see that such an attempt yields something that is physically meaningless. So clearly, gravity doesn't work in the same way as the other forces, hence it is not a surprise that it isn't renormalizable. Personally I think gravity isn't a fundamental interaction at all, so attempting to apply the usual quantisation schemes to it is simply an error on our part; a category mistake, if you so will.

Quite possible, I am not sure what the convention for the notation is, on this one. I thought I have seen people use circles on both integrals...? Indeed. Did you know that the form of the GR field equations follows from this seemingly simple topological principle? Both this principle, and the generalised Stoke's theorem above, are IMHO among the most beautiful results in all of mathematics The name kind of rings a bell somewhere, but I wouldn't be intimately familiar with what he did (even though I live in Ireland). This is probably a good time to reiterate that all my maths are entirely self-taught, so there are large holes in my mathematical knowledge. I really only ever looked at those areas that are directly relevant to the areas of physics I am interested in. Gladly I'm somewhat out of my depths on this one, since I've never really studied QFT in any detail. That's a shortcoming I am intending to rectify when I have the time and inclination.

Lol...I given no guarantees, this is second hand information. Coming from what I would consider a reliable source, I simply assumed it's true

I don't think that follows logically They have 20 degrees of separation on average from the Philosophy page, but not necessarily from each other.

FTC = Fundamental theorem of calculus AdS = Anti-deSitter Space CFT = Conformal field theory So essentially the AdS/CFT correspondence relates two formally completely different theories - a geometric theory of gravity (such as M-Theory, or Loop Quantum Gravity) in the bulk, and a conformal field theory (a special type of quantum field theory) on the boundary of that bulk. The point here is that two very different theories over different domains can describe the same physics. No, I haven't heard of that - I'm not sure if that is really the same thing.

It's not so much the equivalence principle, but the principle of least action. Well, you can also go into a stable orbit (no acceleration once there), and the universe will look just the same. This is not due to Unruh effect. By making the metric time-dependent, i.e. by reducing the distance between all points on the manifold.

I don't think it is that simple. The duality we are referring to here is a duality between distinct types of physical theories - geometric theories of spacetime (on the bulk) on the one hand, and conformal field theories (on the boundary) on the other side. How does this relate to the FTC? Try this: \[\oint _{M} d\omega =\oint _{\partial M} \omega \] Even for this, I am struggling to make a connection to the AdS/CFT correspondence.

Lol...I haven't actually tried this myself, it was in a presentation on information theory and data science by the mathematician Dr Hannah Fry. I don't think so, but it is interesting in the sense that it shows that encyclopaedic information has a tree-like structure, and the concept of 'philosophy' sits pretty far down, basically at the roots of that tree of knowledge.

This may be so, but I'm somewhat troubled by the implications. What does it even mean for the fundamental elements of the world to be inconsistent with realism? My immediate impulse would be to say that it means they really aren't fundamental at all - but I will have to ponder this for a bit.

Well, GR is a purely classical theory, so it doesn't support the idea of 'multiple realities'. There are of course lots of observable quantities which do depend on the observer (they are not invariant/covariant), but observer-dependence does not imply multiple realities. Indeed - but it isn't so much the finer details of how exactly the observer moves, but rather the very fact itself that the observed vacuum depends on the motion of the observer at all. You travel in an accelerating rocket and see a sea of particles around you; then you stop, and pop! - they are all gone, though you are still in the same region of spacetime. If you really think about this, it poses very serious questions about what is really fundamental, and what is not. The result has already been replicated for deSitter space (dS/CFT correspondence), and even Kerr spacetimes (Kerr/CFT correspondence). It suggests that the duality itself is an expression of some deeper connection between bulk and boundary - I bet there is some form of underlying duality that relates the two for any kind of spacetime, independent of its specific geometry (or perhaps for some physically significant subset of geometries). Finding this would be a major breakthrough. Good point +1 Never looked at it this way.

Another useless - but actually interesting - fact: start at any random Wikipedia page, and click on the first link that appears on the page. This will bring up another Wiki page - do the same here. And again. And again... For over 95% of all Wiki pages that you start with, within about 20 link-clicks on average, you will arrive at the same page - Philosophy.

Ok, that's a good point, and I think I get what you mean. I looked at the situation only spatially, but not along the time line. So in that sense, as you explained it, it is indeed local - later measurement outcomes do not depend on distant parts of the system, since the statistical correlation has already been there from the beginning, and thus remains local at each branch of the experiment. So the situation does not in fact fulfil the non-locality definition you gave. That's a pretty self-consistent view on this, as it avoids any clashes with SR. This would seem to imply then that we have to let go of realism; it also implies that the spatiotemporal embedding of the underlying wavefunction that describes the system is non-trivial - it cannot be located anywhere in spacetime in any self-consistent way. This is consistent with my own view which I keep exploring - that the underlying structure of reality is not in any way spatiotemporal in nature.

Yes. Essentially yes. No, that isn't really right. I think what you are referring to here is what is called the equivalence principle - it states that in a small enough local frame, uniform acceleration is equivalent to the presence of a uniform gravitational field. The emphasis here though is on the terms 'uniform' and 'local' - essentially, it means that if you are locked in a small windowless box, and you measure a constant acceleration in some direction, then there is no local experiment you can perform that will tell you whether this is due to the box being accelerated, or due to the presence of a uniform background gravitational field. The problem though is that the gravitational field of a real-world source (such as a planet, a star, or whatever) is not uniform - it is tidal in nature. This is why the equivalence principle holds only on small local scales, where tidal gravity is negligible. But it doesn't hold globally, so gravity is not the same as acceleration, in a general sense. If the box you are locked in is large enough, you will eventually be able to detect tidal effects, which allow you to distinguish between the presence of a gravitational source, and mere acceleration. These two cases are not physically equivalent. Geodesics are world lines that a test particle in free fall traces out; an accelerometer carried along with that test particle will read exactly zero everywhere along that world line. Anything that is not in free fall - i.e. anything where a co-moving accelerometer reads something other than zero - are not geodesics. The world line of a meteorite freely falling towards earth (outside the atmosphere of course) traces out a geodesic. A rocket that fires its thrusters somewhere far away from any planets etc will trace out a world line that is not a geodesic. In both cases, if you project the world line onto a standard 'flat' coordinate system, you will get a curved trajectory. The source of gravity isn't just mass, it's a mathematical object called the stress-energy-momentum tensor. A tensor has the important property that it is covariant under coordinate transformations, which physically means that all observers agree on it, regardless of where they are and how they move. In other words, if a body exudes 'x amount of gravity' in one frame, it does so in all frames. The only thing that changes is the coordinate values the observer assigns to each event in spacetime.

No, this isn't correct. Please refer to my comments on the other thread about this.

They are almost Maxwell, but not quite - they are invariant under hyperbolic rotations and translations, but not under boosts, because two of the quantities in the equations don't transform in the necessary way. Actually no, they can't. The GEM field is a rank-3 tensor, which arises from a rank-2 potential, giving the Lagrangian density \[\mathcal{L} =-\frac{1}{16\pi } F_{\mu \nu \alpha } F^{\mu \nu \alpha } -\frac{G}{c} J^{\nu \alpha } A_{\nu \alpha }\] This is different than would be the case for electromagnetism (which arises from a vector potential, and has a rank-2 field tensor). The above is a good approximation for the linear part of the Einstein equations, so it is a weak field approximation to GR. Hm, I don't think this is obvious at all. For one thing, since it is not Lorentz invariant, it's also not CPT invariant, so adding this into the Standard Model is not trivial. Renormalizability also cannot just be assumed, this will have to be specifically checked; I don't know if anyone has done these (pretty complex) maths for the case of GEM.

I should point out here that not all relevant quantities are observer-dependent. Locally, all proper quantities - such as proper acceleration, proper length etc - are invariant, and all observers agree on them. Furthermore, all those quantities that characterise the geometry of spacetime and the distribution of energy-momentum therein are tensors, so all observers agree on them, too. So there is only one reality, which is characterised by suitable invariant and covariant quantities (which may not always be obvious to us). What I do find fascinating though is that the vacuum ground state of quantum field theories is not one of those quantities - it is explicitly observer-dependent. Hence, where one observer sees a vacuum, another observer may see a thermal bath of particles. This raises some interesting questions about the ontology of what we usually consider to be the fundamental building blocks of our universe (particles). It also makes it obvious that there is a deep link there somewhere between GR and QFT.

Yes, I agree.

I am not sure which definition you use...when I speak of 'functions', I always use the standard textbook definition as a relationship between sets, or more accurately between elements in sets.

Assuming their minds, bodies, and technologies are roughly similar to our own, they would certainly get to GR sooner or later - but they might formulate the model differently. GEM is not Lorentz-invariant, unlike Maxwell's EM. It works only in the low-velocity, low energy, weak gravity domain; it's only an approximation. It gives fairly good predictions for slow-moving observers in our solar system (so it can be very useful, since the maths are much simpler than full GR), but it fails pretty spectacularly for relativistic scenarios, similar to Newtonian gravity. Any observation that involves motion at relativistic speeds, and/or strong gravitational fields will quickly show discrepancies with GEM. The Wiki article you quoted mentions one such case, have a look under the "Pulsar" section. Also, the polarisation modes (and general dynamics, really) for gravitational radiation are wrong in GEM, which would have been noticed with the advent of gravitational wave observatories. To get back to the original question, it is quite likely that without Einstein we might have used GEM for a while, but eventually people would have started to notice that some observational data deviates from the model, which would eventually have led to GR in some form.