Markus Hanke

Take that a step further - as thought experiment, imagine that you are carrying some kind of recording equipment 24/7 for the entirety of your life which records what you hear and see, and then save that data on some hypothetical large-capacity DVD-like disc. Now it is immediately obvious that information requires no specific spatiotemporal embedding - your direct experience of seeing and hearing took place in what appeared like (3+1) dimensions; the same information is stored in your brain in a very complex network that can probably be considered a Hausdorff space with some non-trivial number of Hausdorff dimensions; and also on the DVD, which is in itself (2+0)-dimensional. Interesting to note that there is no real distinction between spatial and temporal dimensions in a Hausdorff space, and no notion of time on the DVD at all...even in its real world embedding, all the information about your life exists on the DVD simultaneously (as it is small enough to be considered local). How we construct a play-back device for the DVD is an arbitrary choice. In all cases, the exact same information (roughly at least) is referenced, so clearly information is quite independent from its spatiotemporal embedding. These embeddings are just arbitrary choices on how the same data set is represented - in the same way that a computer’s graphical user interface is an arbitrary choice of how to represented the information (which are just bits) in its memory banks. There’s nothing inside the memory banks that resembles a little “file” icon, or the letter “A”, or even any notion of a two-dimensional screen. None of this is intrinsic to the actual information contained in the computer’s memory; they are just arbitrary conventions imposed from the ‘outside’ in addition to the raw data. This is quite necessary for our mind though, because we would not be able to comprehend the data set in its raw form. So again - could the imposition of a particular spatiotemporal embedding (such as (3+1) of how we experience the world) be something the mind does to structure and order information, and assemble it into a linear and coherent model of the world?

Ok thanks, I expected that. My understanding is that CPT invariance always implies Lorentz invariance and vice versa (a duality?), so it can only ever be a local symmetry. That has interesting implications though, because what it really means - at least in my reasoning - is that one can only meaningfully speak of ‘future-oriented time’ for larger ensembles of particles. For individual particle interactions, I don’t see how the notion of ‘past-to-future ordering’ could possibly be intrinsic. So another technical question to you @joigus: when calculating the Dyson series in order to get to a S-matrix, how does one know the correct order of the terms? It looks like this is a series of nested integrals, so the order of the terms is not in general arbitrary. I know there is such a thing as a time-ordering operator, but my question is - does the ordering somehow follow from the theory itself, or is it imposed ad-hoc? Hopefully this makes sense. I think this would be problematic, because it leaves electric charge unchanged. There will certainly be a large number of scenarios that are TP invariant, but some systems are not, notably anything that explicitly relies on electroweak interactions. Trouble is, unlike T and P, C is not spatiotemporal, in the sense that it doesn’t directly arise from coordinate symmetries. But perhaps C/P/T could be connected to topological invariants somehow? This way one might be able to treat them under a common framework that is independent of particular coordinates or metrics. It would mean though that vacuum spacetime would have to have a very non-trivial topology, even on scales of the Standard Model, which is potentially difficult to reconcile with GR.

Ordinary spacetimes for simple ensembles of particles don’t have a topology of this nature, so I’m not sure how this is relevant...?

What I mean is that CPT symmetry implies and requires Lorentz invariance, and vice versa. We know that Lorentz invariance is a purely local symmetry - it holds across small enough patches of spacetime that can be taken as Minkowskian. However, an ensemble of many such patches of spacetime taken together is generally not guaranteed to remain Minkowskian. At the very least, the same must hence be true for CPT symmetry as well. Just exactly what “local” encompasses will depend on the circumstances - every time you add another particle to the ensemble, the failure of spacetime to be Minkowskian becomes a little less negligible by degrees, and eventually there comes a point where one can’t ignore it anymore, at least not on the characteristic length scales of the interactions in question. Determining where that point is would be quite a difficult undertaking, since the gravitational effects of each particle do not add linearly. One might argue that a very large number of particles is needed, but I am not so sure, because it is also a matter of scale - what is negligible on our scales may already be very significant on QCD and electroweak scales. Perhaps - and I am just speculating here - the second law emerges precisely due to the failure of background spacetime in that region to be exactly Minkowskian. It would seem too strong a correlation to be a mere coincidence. Purely technically, if one was to be really strict, there is no such thing as a pure CPT symmetry in the real world, because the presence of even a single particle already makes the spacetime region non-Minkowskian - we just choose to ignore this, because the deviation is vanishingly small by all practical standards. So maybe it would be better to say that what is CPT symmetric isn’t the physical system as such, but rather its dynamics, i.e. its Lagrangian. Question to you, since you know much more about QFT than I do - is there some version of the CPT theorem that holds in curved spacetime QFTs as well? I don’t see how it could be possible, since in general curved spacetimes there won’t be any time-translation symmetry.

Great, thanks It would appear that his thoughts were very roughly along the same lines as my own thinking on the subject. I shall definitely add this to my reading list

I think the issue with this is that CPT symmetry is a local symmetry, just like is the case with Lorentz invariance. From what I remember (and I am by no means an expert on QFT, it is the one area of physics I am least comfortable with), to calculate the scattering matrix of an interaction, you perturbatively sum all possible Feynman diagrams of all intermediate states into what is called a Dyson series, which then allows you to obtain the overall S-matrix. It is obvious that each individual Feynman diagram is CPT invariant, but it is not obvious whether or not a large ensemble of resulting S-matrices necessarily has this symmetry, especially not after renormalisation. So I think CPT invariance holds only locally, same as Lorentz invariance can and does only hold locally. I think it is precisely this failure of CPT symmetry to hold over larger ensembles that gives rise to the second law of thermodynamics. The other problem is that from what I remember, renormalisation is a procedure that in itself really only works locally. In any case, it would be practically impossible to actually do the maths for anything more than a few particles at a time, in a small region of space.

No I haven’t, but it sounds very interesting. Have you got a link or a reference to this? Really? But what do you do if the object in question is of higher rank or dimension, such as e.g. the rank-4 Riemann tensor? It’d be a bit awkward to write that out as a matrix

This is a really good question, and, I think, one that we don’t have the final answer to. I have been thinking about this for some time as well (see what I did here...) The situation is complicated, because what seems to be happening is that, while CPT symmetry and T-symmetry are formal properties of the laws themselves, they don’t appear to necessarily apply to physical systems described by these laws. For example, each individual microscopic interaction between elementary particles is CPT invariant, but a large thermodynamic ensemble of the same particles isn’t (due to the second law of thermodynamics). Or take the GR field equations - they are trivially T-symmetric, but some solutions of these equations (spacetimes that contain event horizons) are not. I think the basic problem is that we do not have a unified description of the fundamental interactions with gravity, so it is very difficult to see just what is really going on, on a fundamental level.

Very nicely presented, thank you for the time and effort spent on that post! +1 (I’d give more if I could) The answer is of course no, you don’t need time. Neither do you need space. All you need in order to capture all relevant information/dynamics of the system is a suitable network of relationships, as represented by your S-matrix (which should transform as a tensor). This is intuitively obvious for spacetime and gravity, which is explicitly about relationships between events, entirely independent of how we embed these into a spatiotemporal coordinate system (given some basic conditions). The big question, however, is this - can this be generalised for any physical system or model? In other words, is it conceivably possible to formulate all of physics entirely independently from spatiotemporal embeddings, using a suitable ‘language’ such as (e.g.) tensor networks, algebraic topology, etc? I have a feeling it just might be. If it is, then we must seriously ask ourselves in what sense space and time can possibly be ontologically fundamental properties of the world. It is probably obvious by now that I am personally of the opinion that space and time are not fundamental in the sense we usually take them to be; I think they are artefacts of our perception of reality, rather than fundamental properties of reality itself. I think what is fundamental to the ontology of reality are only networks of relationships, which is why I kept going on about that during the discussion on ‘change’. I think it is actually much like functions in mathematics - the fundamental ontology of a function is a relationship between two or more sets; you can then choose to embed these relationships in a spatiotemporal coordinate system of a suitable kind, and draw them as a plot. But the plot is in no way fundamental to the function - only the domain(s) and codomain(s) and their relationships are. The plot is simply a more or less arbitrary way to visualise that information, to embed it spatially, similar to how a computer monitor spatially embeds the non-spatiotemporal information in a computer’s memory banks. Note that such embeddings are not entirely arbitrary though, since the structure of the underlying network constraints (but perhaps not uniquely determines) how it can be represented; so there will be privileged embeddings for specific underlying structures, in terms of how many axis are needed, how they are arranged etc. I am hoping that this perhaps translates into physics - for example, it might eventually be possible to show that we perceive the world as (3+1)-dimensional simply because that is the privileged (or perhaps even only possible?) scheme to consistently embed the underlying network of relationships for gravity and electromagnetism, being the two interactions that are directly relevant on our length scale. So the big question then is - is there a single underlying network that is fundamental to reality? What are its nodes, and how can it be described and represented? There actually seems to be an issue with this. None of the fundamental laws of our universe determine a unique time orientation, except the second law of thermodynamics (the Standard Model is CPT invariant, and GR is T-symmetric). However, because in our (0,3) toy universe there are no spatial degrees of freedom, so there is no consistent notion of entropy in the thermodynamics sense, and hence the second law simply does not exist in this universe (obviously, since there are no ensembles of particles). So even though there is plenty of time here, there is nothing that could pick out a unique ‘future’ direction on any of the axes. There is nothing to even stop ‘future’ and ‘past’ being oppositely oriented on different time axes. Also, the differential equations describing the physics in this universe would contain multiple derivatives with respect to the three time axis, and no spatial derivatives; that makes them elliptical, so what little dynamics there are in this universe would be deterministic, but nonetheless entirely unpredictable. I don’t even see how there could be a consistent notion of causality. Note also that this universe as a whole, even though it consists of only time, is an entirely static block universe

That’s because there isn’t any such ‘problem’. How could there be? The energy-momentum tensor is a local quantity, whereas the self-interaction of the gravitational field is non-local, so of course it doesn’t form part of aforementioned tensor. It can never be a tensorial quantity, because all tensors are local, and any concept of ‘gravitational energy’ is necessarily non-local and observer-dependent. You can, however, define such a quantity as a pseudo-tensor (such as the Landau-Lifshitz pseudo-tensor, or the Einstein pseudo-tensor); this allows you to write down a combined conservation law. The self-interaction of the field is instead encoded in the structure of the field equations themselves; that is why they are non-linear. This is perfectly well understood, both physically and mathematically, so there is no ‘problem’ here.

I’m not an expert on the engineering applications of thermodynamics, so others here are more qualified to address this. I’m more of a theory guy - so in what way do you think this is related to the second law? I don’t see a direct connection, and most certainly not anything that would put the law into question.

I have to say this is one of the worst papers on GR I have ever seen; it is just full of errors and basic misconceptions from beginning to end. Only goes to show that what is on arXiv is to be taken with a grain of salt - it’s a pre-print server after all.

Nice point I never thought of this. Exactly, you got it What distinguishes time from space in the metric signature is only the fact that they have opposite sign - whether the sign itself is plus or minus is arbitrary. My original (3+0)-dimensional tea cup universe had metric signature {+,+,+} OR {-,-,-}...a purely time-like (0+3)-dimensional universe would have a metric signature of {-,-,-} OR {+,+,+). Hence, their metric would be precisely identical. In theory at least, unless I am overlooking something, these universes should be indistinguishable, at least geometrically. Crucially, in both cases they are static and stationary - so not only is change<>time (IMHO), but also time<>change. Interesting...this isn’t really true in German though, at least not directly. There are, in fact, two nouns for ‘change’ - Änderung, which is the concept of non-homogeneity, of something having different attributes as a comparative relationship; and then there is Veränderung or Abänderung, which is the process of making something different. Only the latter has a connotation of implying time, at least in my opinion.

Oh I see I never bothered to fact check this...just come across it here and there. A somewhat more scientific one - it is possible to construct (mathematically) a geometric body that has finite edge length, and infinite surface area enclosing zero volume.

Here's another one - it is actually illegal to bring a Furbie into the Pentagon. It's in fact a criminal offence, punishable by lengthy prison sentences

Does this have any connection to the fact that a fully stretched out (standard shop-bought) slinky is precisely 82 feet long?

Yes, perhaps. My original impulse was actually to just use a function or a field that does not explicitly depend on time, but I thought it would have been too abstract. Hence the tea cup in a 3D universe. But perhaps my first impulse would have been better, as those things have an explicit set-theoretical definition. I think it must be that our thought processes are fundamentally different. For me, it is difficult to understand how someone can not see a notion of spatial change in that picture, just as it is hard to see why change should be in any special way connected to time. But perhaps that's just me and my autistic mind But then again, the concept of change without reference to space or time is fundamental to some models in physics, notably attempts at quantum gravity that are background-independent, such as LQG and CDT. Even in ordinary quantum physics, time does not play the same central role as it does in the classical world. So I'm certainly not the only one finding value in it. Perhaps it is best to leave it at this, since my intention hasn't been to convince anyone of anything; it was mostly to explore the meaning of 'time' a bit more, and hopefully get the reader to go beyond what might seem obvious at first glance.

In a way yes, in the context of Theravada. Ajahn Chah was a bit of a reformist, in that he attempted to cut through all the elaborate ritual, magic spells etc etc that tends to proliferate once any system of thought becomes a folk religion. He wanted people to go back to doing the actual practice, rather than attach themselves to form and ritual. He was very clear on that anyone can find liberation in this lifetime, given sufficient dedication, effort, and the right way of practice. He was also a simple village boy by background, so the way he taught was very down to earth and no-nonsense - in fact his style was very similar to that of the Zen masters of old. Unfortunately, in recent years, as the Ajahn Chah lineage became more popular, it too began to ossify and ritualise again. But I suppose that's the way it goes, once anything becomes institutionalised. This where one needs to take responsibility for one's own practice, and try to focus on what actually matters. I take from the tradition what I find useful, and come up with ways to peacefully coexist with the rest. I will be relocating to Thailand for that, I have already been formally accepted into a monastery of one very well known and respected (in Thailand) teacher within the tradition. He is one of the original students of Ajahn Chah, and the Thai people consider him fully enlightened. I figured if I do this thing I might as well do it right, and learn the original form. It will be a challenge, as I do not as of yet speak any Thai, but hey...life becomes boring if it is too easy So I'll be back and forth between Thailand and Ireland for a few years, but, once the 'junior monk' period is over, I am hoping to be based somewhere in Europe, as all my family is here of course. Absolutely. For me these are the most prominent forms of suffering, as my body is still largely cooperating; but being on the autism spectrum can be tough, and then of course you have all the other usual vicissitudes of life.

No, the second law follows directly from fundamental considerations of statistical mechanics and some basic maths, and as such is not in contention. It also follows more or less directly from \(Z_2\) symmetry and unitarity, so it is fundamentally motivated by manifest symmetry considerations. Essentially, in a world where unitarity holds for quantum systems (as is evidently the case in our universe) you can't not have the second law of thermodynamics, if that makes sense. I am not sure what the point of all this really is, since the setup you have there is not an isolated system in the thermodynamics sense - so what does it have to do with the second law at all? Note also that the second law is a global statistical statement, and as such it does not contradict temporary (even long-lasting) local decreases in entropy. What exactly are you attempting to show here?

Yes, I agree with this. It's just that the act of observation isn't what the discussion is about - I think it is fair to say that we all agree that in the absence of time, there can be no observation. This is not in contention (but it actually has interesting implications, if you think about it...but that's for another thread). So that leaves us with this: if change in itself (not its observation) does not need to involve a process, and thus does not reference time either implicitly nor explicitly, how can it be defined? Here is where I would argue that the best way to do so is via set-theoretical considerations, as detailed earlier. Essentially, I understand it as a relationship between elements in a set (or even between sets of comparable type). Surely that is not an unreasonable (albeit admittedly counterintuitive) position? To be honest, having slept over this whole thing, I am not sure whether I should. Since the tea cup universe has proven so controversial, the implications of a purely temporal universe might set off a riot Perhaps a shout-out to the other participants here is in order - how do you feel about this discussion? Should we continue on, or agree to disagree (which is fine)? I think even from the little bits we have been talking about, it is quite clear that time does not just equal change.

No, the law of reflection does not hold at relativistic velocities.

Yes, the point is precisely to arrive at a suitable definition for the notion of 'change'. I for one would say that 'change' is implied by the non-identity of entities, and is thus not a spatiotemporal concept. It is the failure of a set of entities to be homogenous. So by this definition, a universe that has no time dimension(s) must be perfectly homogenous in all aspects, purely on account of there being no time? In what way is 'change' ontologically identical to 'observation of change'? That's like saying a tree is the same as the act of observing it - your are postulating the identity between a static entity and a (quite separate) process, which is dubious at best. P.S. Very interesting discussion so far, fair play to everyone But wait...the real fun starts when we raise the stakes a little - what happens when we have a universe that consists of only time dimensions, e.g. a (0+3)-dimensional universe? And would you believe it when I said that such a universe might be entirely indistinguishable from a (3+0)-dimensional one

Ok...and that's all I've been doing, really. But no one has been saying this...? I don't really understand where this statement is coming from - neither the definition of functions nor of derivatives involves any notion of changing variables. Functions are formally defined as relationships between the elements of domains and codomains (which are sets), and derivatives are functions that involve limits evaluated at a single point, so they are also relationships between elements in sets. This is the textbook definition. So we are talking about relationships here, not processes or actions of any kind. This is what I have been trying to point out all along.

On a manifold that has both spatial and temporal dimensions, these will indeed by inseparable, and in the sense that they both make an appearance in the metric with opposite signs. A purely spatial 3D universe wouldn't be locally Lorentz invariant since the metric signature must be either {+,+,+} or {-,-,-}, so GR does not apply here. GR does not in any sense establish concepts of space and time - it is simply a constraint on the form local geometry can take, given local sources of energy-momentum and appropriate boundary conditions. All I can say here is that GR - as being a tensor equation - does not demand any specific number of dimensions nor metric signature (i.e. mix of spatial and temporal parts) to be valid. So there is nothing from stopping you e.g. to write GR with 17 spatial and 6 temporal dimensions, the field equations would look exactly the same. Whether what is described then bears any resemblance to our own universe is a different question. But again, this is actually irrelevant, because I didn't demand GR (or any other specific law) to hold in my toy universe. I find it hard to explain my thought process on this point, as I lack both the formal philosophical background knowledge and the necessary vocabulary to do so. Essentially I am of the opinion that while observation requires existence in the ontological sense (something needs to exist first before anyone can observe it), the reverse is not true in my opinion - I don't see any logical reason why the absence of observers should imply that nothing can exist. Hence, while the tea cup universe is undoubtedly physically unreasonable, it is not philosophically inconsistent. Again, the ontology of existence does not, to me, necessarily imply persistence - but persistence always implies existence. Of course my toy universe is not physical - I never claimed that it is. It's simply a philosophical thought experiment. There can be no observer whatsoever in such a universe, neither internal nor external, and I did not postulate one in my original example. This is simply a universe with a single tea cup in it, and otherwise complete empty (vacuum). I also do not require it to be embedded in anything. My point was this: 1. Consider the hypothetical universe as an ordered, uncountably infinite set which consists of all physical locations/points (just like ordinary spacetime manifolds, only in 3D) 2. Each element in the set be of type boolean, i.e. either of value 0 (meaning it is vacuum) or 1 (meaning not vacuum) 3. Not all elements of the set are of value 0, because of the tea cup 4. Because not all elements within the set have identical value, this implies a concept of 'change' 5. Since the elements of the set are abstract entities (it is irrelevant what they physically correspond to), the notion of 'change' introduced here is purely a relationship between elements of an abstract set, and thus neither spatial nor temporal in nature Whether or not this change is observed by anyone is irrelevant, because it makes no difference to the structure of this mathematical set, or the relationships between its elements. I am simply saying that there is nothing special about 'time', so far as change is concerned - you can have change with respect to time, as well as change with respect to any other quantity. Change simply isn't spatiotemporal, it is an abstract, relational concept - in my opinion. So change does not logically imply the existence of time, and conversely time also does not logically imply any change (you can have a complete empty universe without any processes taking place, that nonetheless has temporal dimensions). Why? When the early universe formed, there was no one to observe that (it was too small and dense to contain any observers, it didn't even contain particles), but the change obviously still happened. Of course that was a change with respect to both time and space, but nonetheless is happened without observers, so the above statement is clearly inconsistent. Again, the example was specifically about a 3D universe that contains only a tea cup, and nothing else. Postulating an observer would have been pointless, since without a temporal dimension, no act of observation is possible. This does not however negate the existence of the tea cup and the vacuum - which aren't identical.

Schrödinger hated cats He was very much a dog person, which is why he came up with such a cruel thought experiment... Besides - my own cat definitely exists on both sides of the door simultaneously, as it is impossible to keep her out of the kitchen. Indeed. And that’s the crucial point - the cat should be in a state of superposition, but when we look at it, it never is. And this is true for any observation we make, be it on a quantum system, or on something macroscopic - we never observe any superpositions, only definite outcomes. So how does the system prior to observation, which demonstrably is in a state of superposition, get to take on precisely one definite state when we look at it? This is essentially what is called the ‘measurement problem’.