Markus Hanke

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.

The geodesic equations used to obtain free fall orbits are a system of differential equations - meaning any exact solution depends explicitly on initial and boundary conditions. The planets are (roughly) all in the ecliptic plane, because they all formed from the same protoplanetary accretion disk, meaning they all share one common boundary condition. So yes, GR handles this just fine, and so does Newton. The maths of how to obtain orbits in GR for simple Schwarzschild metrics are found in any undergrad GR text; I’ve worked through these calculations myself, and, while cumbersome and tedious, they demonstrably yield the correct results. It’s curvature of spacetime, not just space. In fact, for situations such as planets and stars, gravitation is mostly due to the time part of the metric. Spatial curvature gives you tidal forces, but the ‘downward force’ of gravity is overwhelmingly (by a factor of c^2) due to time dilation, in conjunction with the principle of extremal ageing. Of course, in the geodesic equation you don’t really separate these effects; there’s just a trajectory in spacetime. If you were to take the emotion out of your posts, then people would take you more seriously. Some of the points you make and questions you ask are valid and worthy of discussion, but your approach to argumentation is off-putting and unscientific. You’re sabotaging yourself here.

We do know how light interacts with all the particles of the Standard Model. Furthermore, postulating unknown particles (which would need to have some very strange properties) to explain cosmological redshift is a far bigger step than simply accepting metric expansion, which is a natural consequence of the laws of gravity, and requires no new physics. So you don’t gain anything. Sure, but such scattering processes are wavelength-dependent, so spectral lines are not shifted uniformly. That’s precisely the point - cosmological redshift affects the entire spectrum uniformly, which is how we know it isn’t due to scattering. This has been pointed out multiple times already. There is no physical medium that ‘stretches’ - metric expansion just means that the outcome of length measurements depends on when they are performed, in a specific way. They are not invariant under time translations. Expansion is accumulative, so it depends on total distance. Thus you simply look at what is behind the empty region, and that will tell you the expansion of the entirety of space between you and the observed object. There are no experiments, and there never will be, since metric expansion only becomes apparent on scales of ~MPc. All evidence is observational and large-scale. But of course you can locally test the laws of gravity, of which metric expansion is a direct consequence. This has been done extensively, as I’m sure you know.

I see. My immediate reaction to this would be that the two cases are interchangeable only in the classical domain; once you take into account the quantum fields that make up matter and vacuum (which GR of course does not do), then you will find that neither the strong nor the weak interaction are invariant under scaling of this type, so it is difficult to make sense of ‘rubbery’ measuring intervals. As such, the curved spacetime view is the more general one, as it applies to a wider domain within the real world.

I don’t understand what this is in reference to? The layout of your posts is a mess, and I made no mention of the dark sector anywhere. As I pointed out to you, the concept was abandoned because it is in direct contradiction to observational evidence.

What exactly is meant by “effects on all clocks and rulers”, as opposed to geometry? Do you have a reference to what Kip Thorne actually stated? This doesn’t seem to make a lot of sense to me.

In the absence of a self-consistent model of quantum gravity, we do not yet know the underlying mechanism that makes gravity work the way it does. General Relativity is an effective description of the large-scale dynamics of gravity, and as such it is very successful; but it has nothing to say about the underlying nature of spacetime. That is outside its domain. For the same reason we also don’t know yet what exactly went on prior to about 10^-35s after the BB. This, however, does not cast doubt on the fact that the BB happened, because this is an inference based on extrapolation from current observational data.

You made numerous references to a class of models collectively called tired light, as an alternative to explain cosmological redshift. The trouble with these models is not so much the exact mechanism, but the fact that none of them actually corresponds to available observational evidence. For example, Zwicky’s original scattering model is immediately falsified by the fact that we...well...don’t observe any scattering (which would visually blur distant images). Others are falsified because they are wavelength-dependent, and thus can’t account for cosmological redshift. I don’t know of a single tired light model that is actually consistent with available data. This is why this line of thinking has been pretty much abandoned by the scientific community.

Let me try to explain this a bit more. In a local gauge symmetry, you apply a smooth and continuous transformation to your fields at every point in spacetime, and the parameters of the transformation can vary from point to point (in global gauge symmetries, the parameters are taken to be the same everywhere). You do this by replacing the ordinary derivatives in your Lagrangian by appropriately defined gauge-covariant derivatives. For example, in QED the gauge group is U(1), so the transformation is essentially a rotation by some angle, and the corresponding gauge-covariant derivative introduces a new rank-1 object in the Lagrangian - which is just the electromagnetic vector potential, and thus the photon field. So what does this mean? Local gauge symmetry means that fields have redundant degrees of freedom that allow a very specific type of change to happen in the field configuration at each point. Since the symmetries are continuous, by Noether’s theorem this corresponds to the existence of a conserved quantity (a charge of some kind). Consistent changes in field configuration plus conserved current equals an interaction between fields. So this is the central idea - redundant degrees of freedom (local gauge symmetries), consistently defined across all fields at each point on spacetime, allows for interactions between fields, and the interaction mechanism is itself a new field, as is apparent by the formalism of gauge-covariant derivatives. Without gauge symmetry, fields wouldn’t have any way to interact in this self-consistent manner. Note that in a global sense (accounting for all fields at all points in spacetime) nothing really changes at all, because all interaction currents are made up of conserved quantities - you still have the same Lagrangian after an interaction happens. All you do is ‘shift things around’, so to speak. This is the great beauty of it. This can be very elegantly described as connection forms (called gauge potential) on fiber bundles, so a knowledgable of differential geometry is very helpful here.