Markus Hanke

You’re right of course, my terminology was sloppy. Let’s, for simplicity’s sake, say it was a homogenous ‘soup of particles’ (though this is problematic too, but you get my drift). Ah, I get you now. I hadn’t looked at it quite in this way...something to think about. Great reference +1 I was not aware of this field of maths. Goes on my reading list!

I think it is not particularly helpful to think about the situation in this way, because if you look at the Lagrangian that describes the electromagnetic field, you will find that it does not contain any mass terms - neither in classical field theory nor in QED. Saying that “the field has mass” is thus misleading at best.

Perhaps I have simply overthought all this - ultimately it comes down to gravity. You start off with an early universe that is basically a homogenous soup of energy. If you just naively scale this up, then it is difficult to understand how gradients of entropy could evolve. The problem though is that a state like this is in an unstable equilibrium - even the tiniest fluctuation is enough, and gravity will kick in, drawing the ever so slightly denser regions together in themselves, creating in homogeneities. Over time this leads to what we see now - clusters, galaxies, and stars. And once you get gradients of energy like this, from which work can be extracted, it is no surprise that entropy increases at different rates in different places. It is not a big step from here to local complex systems. I had made an attempt to connect this to the stationary action principle, but I now think this might have been ill-conceived. The trouble is that the action principle is not well defined for all types of systems; in particular, in most cases it does not apply to dissipative systems, which is what a biosphere would be. So this is problematic. Another issue is that, even if an action principle exists, the action itself might not be unique. So to make a long story short, I no longer think there is necessarily an issue with entropy and action principles. Whether or not this is sufficient to explain the sheer degree of local complexity we see is another question again. Could you elucidate on this a bit? It’s the variation of a line integral, in the form it’s usually given. But I think I may just be missing the point you are attempting to make. This may be related to my above point about not all systems admitting well-defined action principles. Also, the action is a global property of an entire region in phase space (ie an integral over time), whereas the second law is a local statement (in time). If the region has no clear boundaries, then it will be difficult to make sense of the concept.

Yes, certainly. This is one of the standard solutions that can be written in closed analytic form. I don’t know what you mean by “quantum charge”. GR is a purely classical model, and the Q parameter is not quantised. Either way, charge (along with spin and mass) is just a parameter within the metric, it’s not a function of coordinates. So it can’t be localised anywhere - it’s a global property. The paper you referenced introduces an additional charged scalar field, so we are no longer dealing with a vacuum solution. The original RN metric (the topic of this thread) contains no such extra fields, so I’m not sure what ref 7 has to do with this thread.

To be honest, I’m not sure what the main point of this discussion actually is - in Schrödinger’s scenario, the radioisotope that sets of the mechanism is never in any superposition relative to the cat; at any given time it has either decayed or it has not. Likewise, the cat is not in any superposition, it’s just that its state is unknown until the box is opened. This was not intended as a real-world example of quantum superposition, but merely as an analogy to demonstrate the basic idea - the uncertainty here is merely epistemic, but not ontological. I still maintain that for all intents and purposes, in the real world, decoherence prevents a system the size of a cat to be in a superposition for any reasonable amount of time (I’d guesstimate no longer than perhaps 10^-30s or so). And that’s not even considering how you establish such a superposition in the first place, before decoherence.

Because a system the size of a cat cannot be isolated from its environment by any practical means. When you scale up a quantum system, you also scale up the degree by which it will undergo decoherence due to interactions with its environment. For a system the size of a cat, decoherence happens so quickly that any quantum effects become entirely negligible almost instantaneously (not that you could even establish such a superposition in the first place!). Colloquially put, the system’s quantumness - if there is any to begin with - bleeds out into the environment, and this happens the more quickly the bigger the system is. In principle it is possible for such a superposition to exist, if you could somehow find a way to completely prevent the cat from interacting with its environment; in practice this is not possible by any conceivable means. Even superpositions in very small systems - like on atomic scales - are difficult to maintain for any length of time for this same reason; this is eg one of the fundamental issues in quantum computing.

Two quick comments on this: 1. Mass, charge and spin are global properties of the entire spacetime - you cannot localise these quantities at any particular place 2. None of the metrics in the Kerr-Newman family will realistically appear in the real world in an exact way, because all four of them require asymptotic flatness - meaning these require an otherwise completely empty universe. A more realistic - yet still idealised - family of solutions would be the Vaidya spacetimes.

I think it’s more complicated than even this - because in my opinion language is more than a simple mapping into the outside world. It is strongly contextual, and meaning isn’t inherent (as it would be in a mapping), but given only through its actual use by people. Thus, language is more than an abstract set of rules and maps - it’s a cultural and social convention, and as such it is fluid and permanently evolving. You cannot separate language from the context of its users. I’m pretty firmly with Wittgenstein’s philosophy of language in this regard. I’m not saying that comparative philology isn’t a worthwhile endeavour (it’s quite interesting!); only that there are inherent limitations to such a project. I don’t believe this is true. Consider the example I gave earlier of เกรงใจ in Thai - this is a very subtle social concept that is quite specific to Thai culture. It is a real ‘thing’ in the outside world (an aspect of culture), but there exists no adequate translation for this in English or any other European language. Even trying to explain this concept in all its subtleties requires an entire paragraph of text at least, and even then it isn’t guaranteed that the reader will understand. Whole guide books have been written about it! Another example of such a thing is the word fa’alavelave in Samoan, which roughly refers to a social obligation created by something that has happened in the extended family, and for which material resources need to be raised so as not to loose face in the community (it also means simply ‘trouble’ or ‘problem’). You can verbally understand the explanation, but you won’t understand what fa’alavelave truly means to a Samoan person, unless you have lived in Samoa (it took me a long time to fully understand all implications of this when I lived there). The concept simply does not exist outside its cultural context, so no other language has any way to adequately express it in all its subtlety.

A language consists of much more than object-nouns, though. Furthermore, not all languages cleanly distinguish between word types. For example, I live in Thailand at the moment, and the Thai language does not, in many cases, make a distinction between noun, verb, and adjective (there are ways of marking a word to be of a specific type, though, if necessary). Furthermore, there are many cultural concepts here that have no equivalent in any other language - eg เกรงใจ, which means something like not wanting to cause an unnecessary burden for someone else. There is also a large number of personal pronouns that denote subtle differences in social status between speaker and listener, and have again no equivalent in any other language. Also, you have language registers - meaning you use different vocabulary for some things depending on who you are talking to (monks, people with status, members of royal family etc). How would you handle such things? Basically what I’m trying to say is that languages are not generally a 1-1 mapping into each other, unless they are very closely related already. Some things will be like this in many languages, but you will always have words that cannot be mapped like this, because language always reflects local culture.

Well, there simply won’t be a B field if the situation is a static one; the energy then is just the integral given by joigus. You can’t increase this without adding more electric charge, which in the real world implies moving charges into the spatial region in question (and thus the temporary existence of B). Sure. Nonetheless, it is often helpful to go to the full covariant formalism simply to illuminate the underlying physics. In this case, the point is that splitting the EM field into E and B fields is an arbitrary (mostly historical) choice; in reality though there is just one electromagnetic field that permeates all of spacetime, and the energy stored in it, as captured by the energy-momentum tensor, does not in any way depend on which observer defines it. Thus, whether the source distribution is static or not relative to any given observer is irrelevant for the underlying physics. The only way to increase total field energy is to move extra charges into it, ie increase its source density - which requires work to be done. Relative motion alone doesn’t qualify. I think while it may be formally possible to attribute some notion of ‘mass’ to an EM field (as an equivalent to its total energy), in practice this would be a fairly useless quantity, since it would be a global property of the entire field - which stretches into infinity. No observer could ever measure this mass. This is why in relativistic EM theory (the OP did reference the energy-momentum relation) we use the energy-momentum tensor of the EM field instead - which is a local quantity.

Indeed, I didn’t. Thanks for clarifying. And I had to make an edit to my post, as I was typing it in haste, and it got all muddled up and imprecise. Yes, it is interesting that the path integral formalism in QFT gives the same results as the action principle; but I’m not sure if this can be considered a derivation. I rather think these are different formulations of the same principle (but I’m open to correction on this)?

Could you further specify precisely which quantity you wish to obtain? Is it the total energy stored in an EM field? If so, then you can work this out using Poynting’s theorem - this should help: https://www2.ph.ed.ac.uk/~mevans/em/lec14.pdf This formalism can be straightforwardly generalised to the relativistic case, where it is written in terms of invariants of the EM energy-momentum tensor - meaning it applies no matter what reference frame you work from. I don’t think looking at this in terms of mass and momentum is very helpful. But maybe that’s just me

The principle of least action is a general principle of nature, which applies both in the classical and the quantum domain. It says that a given system will evolve such that the variation of the ‘action’ - a quantity which equals the time integral of the Lagrangian of the system (being the difference between kinetic and potential energy) - vanishes, ie it is stationary. This is equivalent to the Euler-Lagrange equation. Hence, to find the evolution equation of a system, you can first work out its Lagrangian, and then make the variation of the action vanish. For example, the Einstein equations emerge in this way from the Hilbert action. This is amongst the most fundamental and most powerful known principles in physics.

That’s true. But so far as GR is concerned, it doesn’t exist in a vacuum - one has to also look at the wider context. For example, if, in addition to GR, we also take into account QFT, then we know that event horizons of Schwarzschild black holes carry entropy (as a function of horizon area). This implies that the bulk enclosed by the surface has a finite, well defined number of non-trivial degrees of freedom associated with it - which physically means that the bulk must have some kind of ‘structure’, or is granular, or is topologically non-trivial. This of course runs counter to a naïve application of GR alone, under which spacetime in the Schwarzschild bulk is everywhere smooth and continuous, and singly connected outside the singularity. More recent results (ref Netta Engelhardt et al) support this. This is one of the indications that make me think that GR isn’t fundamental. In fact, I’ll eat my monk’s robes should it turn out to be - without salt and pepper, but perhaps with a bit of chilli 🌶 sauce.

I would tend to agree. Good point!

My interpretation of Newton’s words is that ultimately this is what he is asking for, hence my response. They’re measurements of space and time, but that doesn’t mean they don’t have physical consequences, so in that sense they are also a physical object. I don’t think one can cleanly distinguish between these. Energy is simply the conserved quantity associated with time-translation symmetry under Noether’s theorem, so it is itself a measurement of space and time. I would say that the fundamental dynamical quantity of GR - the metric tensor - is in fact a tensor field; would this not make it a field theory? Furthermore, one can write GR as a gauge theory, with the Lanczos tensor being the gauge potential. This should make it a field theory for sure, no?

Yes, but spacetime and its degrees of freedom is - at least in my opinion - only an effective description of gravity. I do not think it is fundamental at all, and the underlying mechanisms that give rise to the appearance of smooth spacetime aren’t yet known (though there are of course some interesting hypotheses out there). To put it slightly differently - the domain of applicability of GR is limited, at least in a ‘downwards’ direction.

I think it is still very much open. GR is an accurate and very valid description of gravity (within its domain of applicability), but it isn’t an explanation, because it has nothing to say about the underlying mechanism. We simply don’t know yet how and why macroscopic spacetime with its observed degrees of freedom comes about; we can only describe its dynamics. This is why research into quantum gravity is so important.

Interesting and very valid thoughts. Thank you! I think all you could prove is that the dog isn’t infinitely divisible - it must be made of constituents that interact in certain ways. However, I don’t think you can determine the nature of these constituents via a top- down approach; if I was to somehow replace every single atom of the dog with a nano-machine that interacts with its environment in the same way an atom would, then nothing should change - you’d still have the same dog. I know this is controversial, but I don’t see what could possibly be different about the dog as a whole. The question that arises then is why simple systems should organise themselves into vastly more complex ones. It is interesting to note that the universe at large is a sea of increasing entropy interspersed with islands of low entropy - which is what living ecosystems fundamentally are. Left alone, these islands of low entropy grow and spread about, and it is not immediately obvious why that should be so, given the principle of least action.

Well, kind of. But I haven’t, in my own mind, arrived at a rigorous definition of the concept just yet. I don’t know the answer to this - personally I prefer to think of the lower levels as kind of a boundary condition to the ‘higher’ laws. Consider, for example, the laws of evolution - clearly, they are closely connected to lower levels, but are they determined by them? Can you start off with - say - statistical mechanics, and eventually arrive at the laws of evolution, perhaps through a simulation? Or how about the laws of psychology, sociology, or macroeconomics? Are they derivable from, say, the Standard Model? I think these are important questions to ponder.

Well, this is what we assume (of course with good reason) would happen - but how can we show this? Good point. Ok, but who is to say that “fundamental” necessarily has to be absolute and global? My thoughts were that ‘fundamental’ might be scale-dependent, so that laws can form a hierarchy, wherein each set of laws is fundamental on that level. So basically fundamental to me would mean irreducible. In that picture, nature would come about as a set of strata, ie a multi-level hierarchy of laws, each one of which being irreducible. Each lower level would then form a boundary condition of the next higher level, but does not uniquely determine it. This is not a claim - I don’t necessarily believe that nature is like this. I am simply speculating out loud, to see what implications emergence might have.

I’d like to throw in another question - given a system consisting of a very large number of constituents which interact via known laws, is there a mathematical prescription that allows us (at least in principle) to determine all global degrees of freedom of said system from its local degrees of freedom? For example, could one derive Navier-Stokes equations from electromagnetism (ie H2O molecules -> water) in a purely mathematical manner? How exactly are these dynamics mapped into each other? What happens if the interaction mechanism is changed - can we predict what global effects this will have? On a more philosophical level - are all large-scale and global degrees of freedom of the universe uniquely determined by its fundamental constituents, or is it the case that nature is actually made up of a hierarchy of laws, with each one applying to a specific length scale only, and each hierarchy being irreducible? For example, is the particle zoo of the SM the only possible choice to obtain a universe that looks like ours on large scales? This is of practical importance, because it would mean that applying certain laws to the wrong level would be problematic. This is obvious in the down-scale direction, but we implicitly assume it’s ok to go up-scale, for example by applying GR to systems with very large numbers of constituents, and expecting the same degrees of freedom as on smaller scales.

Yes, GR simply doesn’t have anything to say about the earliest moments - that’s outside its domain of applicability.

The FLRW metric is a solution to a classical field equation, using classical fluid dynamics as boundary conditions. The problem is that beyond a certain point in the past, quantum effects (both within the primordial plasma, and within gravity itself) become too large to be neglected - hence, taking the FLRW metric beyond that point would mean you are extending GR beyond its domain of applicability. The alternative approach is to treat the universe in its entirety as a quantum wave-function, even if you don’t know the precise laws of quantum gravity; it then satisfies an evolution equation not dissimilar to the Schroedinger equation, which is called the Wheeler-deWitt equation. Finding solutions to this is, for technical reasons, very difficult - however, one solution we do know of is the Hartle-Hawking state. In this state, neither time nor space have a boundary at the beginning, but instead have ‘poles’. These poles do not coincide, so there would have been an initial region that was just 3D space. The poles themselves are not boundaries, in the sense that you cannot extend geodesics ‘beyond’ them, even though geodesic completeness is maintained. What this means for time is that, if you were to extend a geodesic into the past, there eventually comes a turning point past which you can go back no further; instead, whatever direction you choose to go to will be the future again. It’s like the North Pole on Earth, from which all directions are south, without it being a boundary of any kind. In the same way, a pole in time is a point at which all temporal directions are necessarily the future. Something similar would be true for space as well. Hence, in the Hartle-Hawking state, space and time would be unbounded, yet still finite (in the past). Note though that this is only one specific solution to the Wheeler-deWitt equation; others are possible, which lead to different scenarios.

Evidently not. Nonetheless, many of the answers given are genuine, and thus of value to casual readers, if not to the OP No. I already explained this on another thread - metric expansion means that measurements of distances outside gravitationally bound systems are time-dependent, so the outcome of such measurements depend on when they are undertaken. Metric expansion is just a specific example of the absence of time-translation symmetry in a system. There’s nothing that is being physically accelerated by any forces - you could attach accelerometers to each galaxy in the universe (including our own), and they would all read exactly zero. Yet distance between them is measured to be increasing as the universe ages into the future, irrespective of where you are performing the measurement from. Spacetime is not a “thing”, substance or fabric subject to mechanics of any kind. I don’t think comments such as this will help your argument.