Markus Hanke

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.

Why do you keep posting this nonsense all over the forum? You have already been told to stop several times. Reported.

So do crumple-horned snorkacks. Is that why Luna never found them? Nope. No words.

Great thanks +1 PS. I accidentally hit the downvote button instead of the upvote one (touchscreen)…I tried to correct it, but it’s now displaying an upvote in red color. Not sure what that’s about, but it’s definitely meant to be an upvote.

The word “apple” is just an arbitrary label in a particular language (English) - just precisely what such labels refer to is usually given by common consensus of the speakers of said language, and that consensus is usually rooted in ordinary everyday experience, and organically emerges from there over time. In English, when people speak of an “apple”, they refer to a piece of fruit that is sharply delineated from its environment - an apple can be on a table, on a plate, hanging from a tree branch, be located in my backpack, can be held in my hand etc etc. It is a label that is to some degree independent from its external context, so all these differing instances of the same fruit can be called “apple”. Of course you can decide that “apple” should refer to the fruit itself plus the “topmost layer” of atoms on a table. The problem here is of course that you then need different labels to refer to this situation in different contexts that don’t involve table tops - for example, if the fruit hangs on a tree, you can no longer call it “apple”, because there’s no “top-most layer of atoms on a table” (what does this even mean?) present there. So it wouldn’t be an “apple”, but must carry a different label instead. Also, the aforementioned layer of atoms wouldn’t be a “layer of atoms” anymore, but “part of an apple”, whereas the layer immediately underneath (let’s assume they can be neatly separated), would still be “atoms”. If you don’t delineate labels carefully, things become messy quite quickly. But to make a long story short - labels carry no physical significance, so their choice is entirely arbitrary, so long as the labelling scheme is internally self-consistent. This is why you can use a completely different language to talk about the same physical situation.

It does, thank you I did notice though that you haven’t mentioned Schaum’s Outline in your literature list - is the omission deliberate, ie is there something about the text I should be weary of? I’m just asking because I happen to have that text in my possession already - I haven’t read it yet, but it looks good at first glance.

Yes, thank you I’m clear about index symmetries…it was rather about that strange notation where indices are written vertically aligned one atop the other. Turns out it’s just sloppy notation.

Ok, that clarifies it! I was simply wondering if that was just sloppiness, or whether there is any significance to the notation. Thank you

Ok, so basically this is just a sloppy way of writing them?

Yes sure - but if both indices are written in one vertical line, as in \(B^{\mu}_{\nu}\), how do you know which is which?

Ok, thank you As it happens I have both Schaum and Synge/Schild in my personal library, but haven’t read either one of them yet. Just leafing through them, they both look pretty good. Still, it will be helpful to see what other recommendations people here might have!

Thank you joigus. Unfortunately though this isn’t quite what I meant, but the fault is entirely mine - I didn’t make my question clear enough, and the additional clarification I just added came too late for your post, and somehow got tacked onto Lorentz Jr’s post instead of the OP. It’s all gone a bit messy here. Apologies for causing you lots of typesetting work. My question was actually about vertical alignment of indices - how does \(B^{\mu}_{\nu}\) formally relate to (eg) \(B{^{\mu}}{_{\nu}}\), with the emphasis on vertical alignment or lack thereof of tensor indices? Or is there perhaps no difference at all?

Yes, I have indeed - but only to the extent of ordinary exterior calculus in the context of GR, as presented in (eg) MTW. I’d definitely like to deepen my understanding and skills in the area of geometric algebra, since it is a very powerful formalism; many aspects of physics can be cast into that language. But that’s a future project - at present I need to recap my tensor calculus. But thank you for the links, I really do appreciate that +1 PS. I think I need to add some clarification to question (1) in my OP (which, for some reason, it won’t let me edit?). I’m good with raising and lowering indices, thus the relationship between the latter two notations \[B{^{\mu}}{_{\nu}}=g^{\mu \alpha} g_{\nu \beta}B{_{\alpha}}{^{\beta}}\] isn’t the problem. What I’m wondering about is specifically the notation where two upper and lower indices are vertically aligned; thus I’m wondering how the above relates to \(B^{\mu}_{\nu}\).