Markus Hanke

Where and how does the kinetic energy in the system’s COM appear in the field equations, exactly? The dynamics of a system of moving masses, as in this example, most definitely has a gravitational influence - evidently a stationary system and a non-stationary one have very different spacetime geometries. I’ve tried to explain this in some detail in my previous post. My point though was specifically that you won’t find any explicit “mass” or “kinetic energy” terms in the vacuum field equations that somehow increase when there’s more motion, and that act as gravitational sources. That’s how it works in Newtonian gravity, but in GR it’s much more subtle. In fact, if you want to work out the geometry of the vacuum region outside the orbiting bodies, you’ll find that there are in fact no source terms at all. All you have is a rather complicated system of coupled differential equations for the relevant components of the metric, but nowhere does this system contain any terms for either masses or momenta (or energies of any kind). This being a set of differential equations, you then have to supply boundary and initial conditions to obtain a specific solution - and it’s only here that a “description” of the 2-body system enters the picture. The form these boundary conditions will take will be the position of the bodies in question at a specific time (in your chosen coordinate system), two free parameters, and the velocity 4-vectors of each body at that time, plus perhaps some global constraint or another. Starting with this initial “freeze frame”, the solution to the system of DGLs then describes the future evolution of this system - this will be a 2-parameter family of metrics, and these parameters are just the rest masses of the two bodies. So yes, motion (in vacuum) most certainly has a gravitational influence - but it enters as a boundary/initial condition, and not as a source term containing energies of any kind. The difference is subtle but very important. Also, because the system of equations is coupled, you cannot mathematically isolate the effects of motion from the effects of the other boundary conditions; so you can’t write the overall metric as the sum of a purely static metric, plus a metric purely for motion (which is, I believe, what geordief’s question was originally about). The effects of mass and motion are both intertwined in the overall metric, you can’t neatly separate them.

So if I understand you right, you are looking for a mechanism in spacetime that somehow acts on a clock to make it go slower? If so, then we are not talking about the same thing, because I’m looking for a mechanism for spacetime, in the sense of some system of dynamics the classical limit of which can be written as a semi-Riemannian manifold endowed as with a connection and a metric (ie GR). SR’s Minkowski spacetime is then just the local limit of that. Euclidean geometry is also a mathematical abstraction - one that ignores time, and doesn’t gel very well at all with the Standard Model. Why is Euclidean geometry scientific and physical, but Minkowski geometry is not? So far as I am concerned, the idea of the world being Euclidean is pretty much explicitly ruled out by existing observational data. Sure, but that’s just as true for my IPU (Invisible Pink Unicorn). The point is that we have no reason to think that either of these things exist, or even need to exist. My understanding of the ether in LET is that is has no physically detectable consequences, ie there is no experiment - local or global - that you can perform to detect the presence of this ether. This seems to imply to me that it cannot provide the mechanism you are looking for. No, I don’t think there needs to be any “something” at all, other than the choice of how those twin clocks move. To give a Euclidean analogy - it’s like connecting the same two cities by a group of airplanes. Let everything be the exact same - departure time, type of plane, ground speed, weather conditions, cruising altitude, laws of aerodynamics etc etc - except the route they take. One of the planes flies along the shortest possible route (great circle segment), whereas the others choose different routes, no two of which shall coincide. What is the mechanism that “makes” the total flight time different for each of these planes? Is there an ether that “drags” on those planes (without being detectable!), thus making them arrive later than the shortest-route flight? Why would you expect planes that fly along different routes to always take the same amount of time, in the first place? The only physically relevant mechanism here is that we need to apply measurable acceleration at some point in order to not make all the flight path coincide - and once you do, the flight times must necessarily diverge. This is entirely irrespective of the kind of mathematical description you give this situation. Actually, the point I was trying to make was simply that this idea postulates an unnecessary mechanism that then requires an entire set of new mechanism to make it work.

Yes, the motion does of course “contribute” to the geometry of this spacetime, as I said in my post - but not really in an intuitive Newtonian way of making it “stronger”, which is an ill-defined concept in GR. This is the classic “does a body become a black hole if it moves fast enough” question, which I don’t think we need to go over again. The rest (!) mass of the orbiting bodies doesn’t change, irrespective of how they move, and in any case we are looking for a vacuum solution here, so \(T_{\mu \nu}=0\) everywhere outside these bodies. Thus the equation you are solving is simply \(R_{\mu \nu}=0\), without any source terms at all. The angular motion makes an appearance only as part of the initial/boundary conditions once you solve that system of DGLs. So in practical terms, what would be the difference between the spacetime surrounding two bodies momentarily at relative rest at some distance wrt to each other, and the same system with the two bodies orbiting around a common center of gravity? In the former case you have a stationary spacetime that looks more and more like Schwarzschild the further away from the two bodies you get, and eventually becomes Newtonian; in the latter case you get a gravitational radiation field, where frequency and amplitude depend in some way on the rest masses of the binary system, the distance between the orbiting bodies, and the angular frequency. In the full (non-linearised) description these dependencies are pretty non-trivial, especially in the region very close to and in between the orbiting bodies. When you are located further away from the binary system, the amplitude will fall off with distance, again in a specific way. So the crucial difference is that the former case is stationary (i.e. the components of the metric do not depend on time, thus there is a time-like Killing field on this spacetime), whereas the latter is not. It’s a difference in symmetries, more so than degrees. What is true though is that the amplitude of the wave field at a given distance from the binary system will depend on the angular frequency of the bodies - the faster the bodies orbit each other, the higher the amplitude will be at any given distance. So if you wanted to, I guess you could - in this very particular sense - say that the gravitation of the binary system gets “stronger” if they orbit faster, since you’ll have more powerful tidal forces at a given distance. Personally though I would say that such a concept isn’t helpful, because it very easily lends itself to misinterpretation and various misconceptions. In general there just isn’t any way to meaningfully say that one metric is somehow “stronger” than another metric, because you are comparing tensors, not numbers. They are just different. But I guess you can always pick out specific local measurements and compare those. What you can absolutely not do though is take a known metric in a specific coordinate system - such as Schwarzschild -, and simply replace rest mass by relativistic mass to reflect relative motion, while leaving everything else the same, and expect the result to be a valid metric again. That doesn’t work, because such an operation is not in general a valid diffeomorphism.

Ok, but in order to discuss this we need to clarify just what “mechanism” actually means here, or else we won’t know whether or not anyone is denying its existence. So how do you define this term, exactly? It isn’t a straw man, it was merely my interpretation of what it is you are trying to say, because that is not at all clear to me. Since I evidently missed the actual point you were trying to make, it would be helpful if you could summarise it clearly and concisely. They are a description of your physical scenario, and thus both mathematical and physical. The physical difference is the path these clocks are taking. I disagree. The ether you are referring to here is undetectable and has no physical consequences whatsoever. I might as well postulate an Invisible Pink Unicorn in my back garden, for all the difference it makes. It’s philosophically permissible precisely because it has no detectable consequences - but it’s not science, because it isn’t amenable to the scientific method. Besides, if you do this, you are going to have to explain a “mechanism” for why that ether has the properties it has, why it behaves the way it does, and where it came from - so in the end you haven’t actually explained anything, you have just kicked the can down the road. Again, I think you need to define for us exactly what you mean by “mechanism”.

Indeed!

Ok, let me explain. My understanding of what you are trying to say is that we are missing a deeper mechanism of how gravity works, and why it is the way it is. We have an equation that allows us to describe any given scenario of classical gravity, but we don’t know why the equation works, or why it looks the specific way it does. We can’t even be 100% sure yet that the equation we are using is indeed the best possible description of classical gravity - I think it is at least conceivable that one of the very many possible modifications of GR might eventually turn out to be better. I’m not prepared to categorically rule this out, though for various reasons I think it is unlikely. But regardless, I agree with you on the basic premise - we do not know yet what the underlying mechanism might be, so our understanding of gravity is still lacking in that regard. It’s an epistemic description, but not an ontological one. As you put it, there’s time varying metrics etc, but no underlying mechanism to “make that happen”. What I question though is why you seem to hold GR accountable for not providing such a mechanism. It was never designed to do this, since it is simply a generalisation of SR, which in itself proposes no mechanisms either - it simply arose from the empirical observation that Minkowski spacetime provides a model that fits very well to available observational data. Newton also never provided any “implementation” of how his forces work - he simply posited them as a convenient computational tool, not as an ontological description of reality. So I feel that pointing out that what you are looking for is outside the remit of GR is a valid criticism. I also don’t agree that current practices make no attempt at find such underlying mechanisms, because that is what the whole quest for quantum gravity is ultimately all about - and it is an area of research that is ongoing and very active. Naturally in such a quest there’s going to be very many dead ends, especially since the domain in question is beyond our technological capabilities to probe it directly, essentially making is fumble in the dark here. Within current constraints (technology, funding, politics etc etc) I actually think we are doing pretty much the best we can in that regard. Our efforts aren’t perfect, but they are all we can muster right now. I think you are demanding way too much from contemporary physics - you seem to basically say that “if it isn’t a fully fledged ontological explanation, then it’s not science”. I cannot agree with this. I think any epistemic description of an aspect of the world that allows us to make predictions by way of computation is valuable, at least as an intermediary step, even if it is not explicitly ontological in nature. Of course we want such models to approach the status of an ontological explanation over time, but that is not going to happen all at once. You start with something purely descriptive, and then keep refining it; occasionally you might need to change your paradigms; and in the end we might get to something that approaches ontology. I understand physics as being a process that will take time. My guess is that the day will come when GR will be understood as merely the semi-classical description of something much more fundamental, but we are not at that stage yet. Mind you, there is also no guarantee that there actually is a fundamental ontological explanation - perhaps spacetime just is what it is and can’t be further reduced, in the same way that the specific and irrational numerical value of pi can’t be reduced to any more fundamental “mechanism”. I sincerely hope for this not to be the case, but I think we are also not in a position to categorically rule that out yet. Obviously, because they are using the same laws, but different boundary conditions. Sharing the same laws does not mean that everyone measures the same thing irrespective of the situation they find themselves in - it means only that they agree on what each of them measures. To be specific, the law in question is the total amount of proper time physically accumulated on a clock that propagates from one event to another. This is not just some nebulous theoretical concept, but it is what you physically see accumulated on a clock that you are holding in your hand while travelling. That total time is calculated as \[\tau =\int _{C} ds=\int _{C}\sqrt{g_{\mu \nu } dx^{\mu } dx^{\nu }}\] as you probably know already. This law is the same for everyone, since it is written in covariant form. What is not the same for everyone is the path C that you need to put in as a boundary/initial condition in order to evaluate that integral - it naturally depends on the pair of events that are being connected, and it also depends on the spatial trajectory that is being travelled. So once you evaluate that line integral, you are (in general) going to obtain different numerical results depending on your choice of C, even though the same law was used by everyone involved. That is just how line integrals work. Once again you can ask why this is so - why does this law look the way it is, ie why is it a line integral and thus dependant on a choice of path? But SR/GR cannot answer this question, because it is outside their remit. Hopefully we can eventually come up with a more fundamental model that can explain why this law looks the way it does, but at present we don’t have that yet. Of course not. These are gauge symmetries we are talking about. If they were always global, then the entire rich phenomenology of the universe around us would disappear - there wouldn’t be the kind of particle zoo we find, and there wouldn’t be any gravity. PS. When I speak about a “deeper mechanism” in the context of context of gravity, what I mean specifically is why, in the classical limit, curvature and gravitational sources are locally related via an equation of the general form G=T, as opposed to some other relationship. Part of the reason is clearly mathematical consistency, since the properties of G and T themselves already rule out most alternative forms of the equation. Another already known constraint comes from topological considerations, to do with the conservation of certain topological quantities. So we are not completely clueless as to the “why”, it’s just that the bigger picture is incomplete. Note also that once the local relationship between sources and curvature is established, and boundary conditions are set, then what happens in all the rest of spacetime is a foregone conclusion on account of the basic requirements of continuity and differentiability. So once you know what the deeper link between sources and curvature is, you don’t actually need any other “mechanisms” - it’s then simply a matter of logical consistency, because you can’t randomly glue any old geometry onto any old source distribution. The only true mechanism here is that initial (purely local!) link from sources to geometry.

The problem I see with this statement is that relativity itself wasn’t ever meant to be a “metaphysical principle” - right from the outset it served a very practical purpose, namely being a descriptive model that doesn’t suffer from the internal contradictions and conflicts with observational data that Newtonian physics did. In that it has been pretty successful, and crucially it allowed us to reduce our reliance on some rather dubious concepts - such as for example Newtonian forces. I mean, think about it - a Newtonian “force” is a supposed thing that cannot be directly observed or detected (we only ever see its effects), that is entirely non-local, somehow acts instantaneously across arbitrary distances, and there is no underlying mechanism that might explain how it could possibly do all these things. Metaphysically speaking this is entirely ridiculous by anyone’s standards. Yet it works to some degree, and thus to this day we teach it in our schools. If you are looking for an underlying metaphysical principle in relativity, then it would be that of symmetries - turns out that the fundamental objects which SR deals with (Lorentz transformations, tensors, spinors etc) are representations of the Lorentz group, whereas GR is a gauge field theory with GL(4) as its fundamental symmetry group. Of course Einstein himself didn’t know that at the time. Symmetries are also the metaphysical principle underlying many other areas of physics, most notably HEPP. Seems to me these things are all pretty useful!

I don’t think there’s an easy answer to this. However, at least part of any possible approach should be to educate our youngsters about how to skilfully relate to digital media - I mean specifically to teach them skills that help to recognise and properly relate to misinformation. I’m talking about general media skills here, which is quite a separate thing from having expertise in any particular area. “Media Skills” should be part of any school curriculum, IMHO. Ultimately we’ve got to understand that misinformation and crackpottery has always been present, and will continue to always be present. So the question isn’t how to eradicate this, but how to help people relate to it properly.

Absolutely not my area of expertise, but my amateur-ish opinion on this is that you will never be able to come up with any kind of objective measure of a civilisation’s development, simply because it relies on values that are not universal, but contextual. In my opinion, the best measure of a civilisation’s development is in fact one that is explicitly subjective - people’s self-reported general sense of well-being. Note that this is not the same as happiness, wealth, or even “feeling good” - someone might be living in a democratic state of affluence and plenty, and yet not be well in themselves (this is in fact depressingly common). Conversely, someone may live in simple and basic conditions, yet still have a strong sense of general well-being in their circumstance. Being well is the culmination of all the many factors that contribute to your basic needs being met, and you still having time to pursue other things in life as well - it is the coming-together of material, intellectual, environmental, and spiritual balance. All the traditional concerns such as economy, politics, healthcare, education etc etc contribute to this, but not in ways that are easily measured and broken down. So if you want to know how a civilisation is doing, ask its citizens if they are well - not happy, rich, healthy etc. If you make their governing body explicitly responsible for the overall sense of well-being experienced by the people, then I think this would be much more conducive to a more balanced world overall.

Not reliably of course - I would hazard a guess and say that evolutionary pressures on other inhabitable worlds will be broadly similar to our own, so any sentient race that evolves there will likely evolve a reality-model that is also broadly similar to ours. Based on what we see here on Earth, nature tends to come up with similar solutions for similar problems. Nevertheless, even small differences might help us get a better understanding of our own concept of reality, and how it might relate to a possible ding-an-sich external reality. As an aside, I would also conjecture that the more different a species’ reality model is from ours, the harder it would be to establish mutual communication. Arguably, if the models are sufficiently different, there might come a point at which no meaningful communication is possible at all, because we’d share too few fundamental categories.

I can’t answer this, as evolutionary biology isn’t my area of expertise. It’s a difficult subject also because the autism spectrum is so broad - there are some like myself, with very few to no situational support needs, and then there’s a sliding scale of increasing severity right up to forms of autism that make independent living (never even mind independent survival) practically impossible. So it’s hard to generalise. I’m speaking only for myself now, and perhaps those with similar profiles and predispositions as me. I think there might be an evolutionary advantage precisely because people like me don’t fit into the mainstream. For example, my sense of purpose, meaning, and well-being is not contingent upon social acceptance and belonging - things like how many friends I have, social gatherings and occasions, belonging to a certain group (or not), being around other people etc etc are simply of very little importance to me. This might at first glance sound like an evolutionary disadvantage, but think about it - it frees up enormous amounts of time and energy that can then be re-invested into other pursuits. I don’t know if there are statistics about this, but I bet that, among people who have made important contributions in their fields - the arts, sciences, literature etc etc - a disproportionally large amount might be found to be on the spectrum, or at least have autism-like traits of some sort or another. This is because such people are more likely to engage deeply in pursuits not directly concerned with survival and procreation (which is what social preoccupations are ultimately geared towards). I think society benefits from this kind of archetype - the ones who can stand on the sidelines, look back onto the mainstream from a more neutral and wider external perspective, and pursue “higher” things and unusual ways of thinking. I think there’s an evolutionary advantage for the group as a whole in having such individuals, because they function like a mirror that reflects back the forest when all you yourself are usually able to see is the trees, due to your own day-to-day involvement. Such individuals are often simultaneously despised (because they don’t fit in), and valued (for their contributions, often only posthumously), and sometimes burned at the stake; but whatever the case may be, their perspective is an important one. These are just some of my own thoughts, I’m making no claim to any academic truths here. Yes, but it’s not just that - it’s a theory of the world, including the physics side of things. When we are building models in physics, then these are necessarily models of aspects of how the world appears to us. They are thus models of aspects of another model, namely the reality-model that our minds create for us. We all tend to agree on certain aspects of that generated reality simply because we all share a similar sensory apparatus (plus its extensions), and a roughly similar neurophysiological brain structure - thus the boundary conditions are similar, meaning the resulting reality-model is also roughly similar. The reality-model of an organism that evolved under sufficiently different boundary conditions may potentially be quite different from ours - an example from sci-fi literature that comes to mind are the heptapods in Ted Chiang’s “Story of Your Life”, whose minds do not employ the principle of temporal sequencing in constructing their reality-model. I know it’s just a story, but it’s an interesting example. So what happens if the boundary conditions vary? I wrote about autism and social “mind-blindness” above - so what is actual reality here? Are social relations and intuitions real, irreducible aspects of the world - or are these contingent add-ons that your neurotypical brain artificially generates, and it is actually my own mind-blind autistic self who sees things as they really are? Or how about this - in addition to being autistic I am also a synesthete. Words to me have colour, texture, size, spatial orientation, and sometimes temporal extension. These, to me, are not associations (e.g. sky=blue), but intrinsic properties of the words themselves (so for me sky=off-white, smooth and cold like marble, angled backwards and to the right), like spin and charge for an elementary particle. For me this is so intrinsically normal that I am pretty much unable to imagine what experience would be like without these attributes - I only know intellectually that most people can’t experience this the way I do. So who perceives “actual” reality here - is the concept “sky” really smooth and cold, and you are all just blind to that? Or does my brain adds this on randomly? Who’s right and who’s wrong? Or is the entire concept of “reality” just a constructed idea, the meaning of which is strictly contextual? Now think about the wider implications for physics - it makes models of a model. But how do we know, within how the world appears to us, what is an actual part of exterior reality, and what is an add-on by our brain? How can we distinguish, in the absence of having an external reference in the form of other reality-modellers against whom we can compare our reality? Do (e.g.) time and space really exist in the way we experience them, or are they just convenient representations to impose order onto a set of data, like the windows on the GUI of your computer? Are there other ways to structure that same information? Or are there aspects of exterior reality that are not being represented in our model at all, not even by deduction or induction, perhaps because they are irrelevant to our continued evolution? Does the way we do science thus say more about ourselves and how or brains make reality appear to us, than exterior reality? I think these are important questions to consider not just in philosophy, but also in the foundations of science - just focussing on the model, while ignoring what the model is actually about, and who constructs it, might be misguided and eventually come back to haunt us. I don’t feel this is spoken about enough in the physics community, or even taken seriously.

I don’t know about “fundamental”, but ultimately this reality-modelling machine is a result of the process of evolution. What this means is that its function is not at all to neutrally reflect “external reality as it is”, but rather to present us with a model of external reality that is specifically geared towards survival and procreation, and as such will be filtered, distorted, and pre-digested accordingly, with this goal in mind (pun fully intended). For example - out in the jungles and savannahs where we originally came from, if you encounter other members of your species with whom you compete for limited resources, it is advantageous for you to have available a model that allows you to (at least to some degree) predict their intentions, mind-states, and possible future actions. Likewise with the flight path of an arrow, the weather, the behaviour of water in a river etc etc. If you have good models available that take sensory inputs, processes them, and generates something that allows for predictions of how your environment will evolve into the immediate future, you’ll simply have much better odds to do well and thrive, evolutionary speaking. So it’s actually not a surprise at all that things are as they are. As a little aside: I, as being on the autism spectrum, am missing a part of this reality-modelling machine - when I encounter another human being, I am socially blind; I generally have no intuitive concept whatever about what kind of mind-state that individual might currently have, I might as well be looking at a stone statue. I don’t immediately know their intentions, nor can I easily tell how they will behave in the next few seconds. All I can do is make educated guesses based on experiences gathered during previous interactions I have had with people; but this takes an active and conscious effort, and sometimes I get it quite wrong. It’s called “mind-blindness”. This is part of the reason why autistic people often struggle with social interactions.

All you can really meaningfully say here is that the geometry of these two spacetimes (two stationary masses vs two masses in relative motion) will be different - in particular, in the latter case of relative motion, some or all of the components of the metric will be explicitly time-dependent; it’s essentially a GR 2-body problem (which, btw, can only be solved numerically unless one of the masses is very much smaller than the other). But I think what you are getting at is ultimately whether relative motion in vacuum is in itself a source of gravity, and the answer to that is no, it isn’t. Its presence does, however, have an impact on spacetime geometry, in the sense that it will make the situation less symmetric and thus more complicated. But since you can’t in general meaningfully compare tensors (“tensor X is greater than tensor Y…”), all you can really say is that the spacetimes are different. This “difference” is in itself a non-trivial concept, because you can’t as a rule of thumb tell if spacetimes are different just by looking at the metric - for example, the Schwarzschild metric and the Aichlburg-Sexl metric look very different, but they do in fact describe the same physical spacetime. So there are a lot of subtle issues here. When you are not in a vacuum, ie in the interior of some mass-energy distribution, the situation becomes more complicated, because now the energy-momentum tensor explicitly contains terms that can be interpreted as momentum density. So for example, the motion of plasma currents in the interior of a star will have a gravitational effect within that interior region, as compared to some otherwise identical star without moving currents. But since the field equations are non-linear, it is not possible to neatly separate out these effects and isolate them from the other source terms, such as pressure, strain etc. They all interact and interplay. It should be noted though that in the exterior of the bodies the energy-momentum tensor vanishes, so the masses of the bodies and their relative motion only enters the field equations in the form of boundary conditions.

I would say that the defining characteristic of quantum systems isn’t so much the discrete spectra of some observables, but rather the fact that there are pairs of observables that do not commute. That’s something we don’t find in the classical realm.

Lol I can’t speak for others here, but I found that, whilst the vast majority of crankhoods and crackpotteries out there leave me large cold, there are some things that just tend to grate my gears. Relativity denials are an example of something I find hard to ignore, for some obscure reason which I don’t understand myself; so yeah, I tend to get sucked into those threads in particular, for better or worse. This is in some sense a vulnerability, because the topic pushes my buttons, so sometimes it’s hard to walk away from threads where there’s no further benefit to be gained from continued participation for neither myself nor the OP. It’s a strange thing - you know it’s time to walk away, but there’s that childish ego-based impuls towards having to get the last word in. I very rarely regret things I post here, but when I do then it’s usually connected to not having walked away when I should have.

I would look at LTs as a self-consistent way to choose new labels for events in the same spacetime - it’s much like looking at the same physical situation from a different perspective.

I think one should also mention that none of the other fundamental interactions (strong, weak, EM) are invariant under rescaling, so a “shrinking matter” type of model is not compatible with known physics.

I’m not so sure about this, because it doesn’t seem clear to me at all that/why there should be ‘something’ that is ontologically distinct from an interaction. If there is, then we have never observed it directly - any perception, any measurement, any experiment we can perform always boils down to interactions, at the most fundamental level. Even if there is ‘something’ there, then all we can ever see is the interface it exposes to its environment - and this tends to be highly contextual, especially in the quantum realm. Based on human intuition we tacitly and naturally assume that if there’s an interaction, there needs to be ‘something’ there that interacts, but I’m not so sure. But of course, these are just philosophical musings of mine (even if they do, as you correctly observe, gel well with Rovelli et al), so I might well be entirely wrong

Nice way to look at it. Though I would perhaps even go a step further and say that physics models describe how things relate to other things, wherein the term ‘things’ is to be understood in its most general and abstract meaning. So perhaps it would be far better to look at reality as a network of interactions and relationships, rather than a collection of ‘stuff’ that’s doing things. It’s a bit like the concept of motion - it’s a very useful concept in order to describe certain aspects of the world, but it has no fundamental, ontological reality in and of itself, unless viewed as a relationship between things. I’d like to suggest that perhaps other aspects of reality are similar, though in less obvious ways.

I don’t think this is even possible at all - CPT symmetry is fundamentally implied by local Lorentz invariance, and vice versa as well. You couldn’t have either symmetry without the other, and I’d think this is quite irrespective of the details of the model involved. I must admit this made me smile I’m just an ordinary guy who does all this as a hobby, purely as a matter of personal interest…not sure how it came to happen that the real experts are asking me technical questions now

It means that Λ is invertible, which implies that the aforementioned frames A and B are symmetric, see above and below. “Symmetry” means that you apply a transformation to an object in order to obtain a new object; and then apply the inverse transformation to the new object; you end up again with the original object. That’s exactly what you have demonstrated here - ⅛ x 8 = identity. Thank you for confirming this for us (once again). Likewise in physics - you Lorentz-transform a frame A into a frame B; and then you reverse-transform B back into A using the inverse of the original transformation matrix. That’s how symmetry is defined. Physically, it means that all inertial frames experience the same laws of physics, irrespective of their state of relative motion. No, your just repeating this nonsense does not make it any less wrong. Two frames A and B are symmetric iff B=Λ(v)A A=Λ(−v)B=Λ(−v)Λ(v)A=IA Physically speaking, this means simply that all inertial frames experience the same laws of physics, irrespective of relative motion. Once again, here is the experimental evidence for that, which is clear and unambiguous. You are perfectly entitled to your own misconceptions, but not your own physical facts. What I have shown you is elementary linear algebra, and it is not in contention by anyone except people who have agendas that are incompatible with actual science. Then you have wasted your time, because evidently you don’t even understand simple linear algebra - or, more likely, you don’t want to understand it. That being the case, you are really not in any position to argue about SR. The proof has already been provided. Several times, in fact. So far as anti-relativity sentiments are concerned, this was a very underwhelming and childish attempt, I have to say. With this kind of approach you will never be taken seriously by anyone who has even just cursory knowledge of the subject matter. Needless to say you have utterly failed to convince anyone here on this forum. And honestly, given the overwhelming amount of experimental and observational evidence for SR (a small selection of which I have linked above), I will never understand why people like yourself are even wasting your time with this. You might as well argue that a round shape isn’t in fact the best shape for the wheels on a car - this debate has been settled long ago. You aren’t making any kind of valuable contribution to science, you know. Had you used those 10 years you mentioned to actually learn real physics and maths, you might have been able to contribute something of value. It’s a missed opportunity. We really are done here now. Good luck to you.

The parameter v is not a scalar, nor is it a vector, since it does not transform like either of those kinds of objects. It’s simply a real-valued parameter of the transformation matrix, which can take either positive or negative values. To see why, you need only consider the geometrical meaning of the general Lorentz transformation - it’s simply a combination of a boost and a hyperbolic rotation. As such, the transformation parameter can also be expressed as a hyperbolic angle (called rapidity) - and since a rotation about a point of origin can always be either clockwise or counterclockwise, the rotation angle can and does carry a sign. So it’s really simple - you start at a point A, and hyperbolically rotate your coordinate system by some angle ϕ to arrive at a new point B; you then perform the same rotation in the opposite direction, ie by the angle −ϕ , and arrive back at A. That’s just what it means for a linear transformation to be invertible (=symmetric), and that’s exactly what the Lorentz transformation does in spacetime. This is all just elementary linear algebra. Several people here have already shown you that they are symmetrical - including a formal mathematical proof. If you choose not to believe us here, you can find different proofs of their invertibility in pretty much any decent textbook on Special Relativity; here is another online one. And here you will find a long list of experimental results that show that Lorentz invariance does indeed hold in the real world. So where do we stand with this thread? We have explained to you why the transformations are symmetrical; we have shown you formal proofs that they are symmetrical; and we have provided experimental evidence that the whole theory matches up with real-world experimental data. I think we’re done here.

“Mechanics” probably isn’t a good word here, but there are at least three levels - there’s the classical domain of the familiar Newtonian and Einsteinian physics; there’s quantum mechanics that concerns itself with the evolution of quantum system where the number of particles involved does not change; and then there’s quantum field theory, which provides the best currently known description of elementary particles, their properties and interactions.

It’s neither a scalar not is it a vector - it’s a parameter of the transformation matrix, and as such it can be positive or negative. However, this is totally irrelevant, since you need only show that the matrix itself is invertible, which is what I have done already.

Lorentz transformation matrices are always invertible: \[\Lambda(v) \Lambda(-v)=\Lambda \Lambda^{-1}=I\] What I have shown you in my post is one of the standard methods to formally proof this; there are many other ways to provide the same proof. Therefore, all Lorentz transformations are necessarily symmetrical. No you have not.