joigus

The \( f^{0} \) comes from the 0 component of 4-momentum by differentiating wrt coordinate time. It's not to do with the usual gamma factor in Lorentz transformations. It's to do with a time-dependent gamma factor. The energy of a particle is \( mc^{2}\gamma \), but this \( \gamma \) is a function of time. The power gain/loss for the particle is the time derivative of its energy, so that, \[ \frac{d}{dt}\left(mc^{2}\gamma\right)=qu^{\nu}\left.F_{\nu}\right.^{0}=q\left(c\gamma F_{00}+\gamma v_{k}F_{k0}\right)=q\left(c\gamma F_{00}+\gamma\boldsymbol{v}\cdot\boldsymbol{E}\right) \] \( F_{00} \) is zero, as F is a 2-rank antisymmetric tensor. Its non-diagonal elements are the electric and magnetic field. The zero component of the time derivative of 4-momentum is thereby the power. It's all beautifully wrapped up in space-time formalism. I've re-instated the gamma factors, but I might be missing some c factor and perhaps my initial definition of F (the electromagnetic tensor) had the wrong sign. I'm sorry that I'm missing the main point in relation to physics and reality at the moment. Please, let me come back tomorrow and try to catch up with the finer points.

The Lorentz force does have a time component, though. When you write down the complete --covariant-- form of the equation it gives, \[ f^{\mu}=qu^{\nu}\left.F_{\nu}\right.^{\mu} \] as a 4-vector equation. Where \( \left.F_{\nu}\right.^{\mu} \) produces all the components of \( \boldsymbol{E} \) and \( \boldsymbol{B} \). These equations decouple into, \[ f^{0}=q\boldsymbol{v}\cdot\boldsymbol{E} \] \[ f^{k}=q\left(\boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B}\right) \] and where I think I may be missing a gamma factor. The 0-th Lorentz equation gives you the power gain or loss, and the spatial equation is the conventional Lorentz equation.

Very small curvature in both cases, never mind Riemann tensor components or curvature scalar (some kind of average of all Riemann components). It can be estimated by means of Newtonian gravity.

It certainly doesn't cramp my style.

Equivalence relations are at the basis of categorical thinking, or Aristotelian categories. We're hardwired to think in terms of categorical thinking. When we can't place the category, when that category does not close in mathematical terms, it's loosely defined, we feel confused. I think it was Wittgenstein that worried a lot about that problem.

Correct. https://en.wikipedia.org/wiki/Self-energy Is it just a manner of speaking due to electrons not being isolated 'things' in any meaningful sense?

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel One has to be careful even with the natural numbers. The ocean of natural numbers admits arbitrarily many more drops! The real numbers are even more counterintuitive.

Just stretching/shrinking space without time being involved... I see this difficult to reconcile with known physics. Physics with matter --massive-- is not invariant under scale transformations. Time-dependent scale transformations would make this even worse. I don't see how you could save conservation of charge, for example, if space is actually shrinking at small scales... Exactly!

You mean shrinking space? Or also time? What about mass, electric --and other charges-- etc? Would they be shrinking in your picture? x-posted with @studiot

This is very deep. If I understood it correctly, it's like what Swansont said about phonons. There are other examples: Defects in a crystal, different 2-dimensional modes that live on the surface between insulators (topological insulators.) These things live in a context: Surface phenomena, modes in a lattice, etc. They are not 'real' in the sense that you can take, say, a phonon and separate it from its context, and study it in isolation from everything else. You can't take any of these instances, isolate them, and study them independently of the embedding context. There are contingencies --I think that's the right word-- that define their 'being there.' If you dissolve the contingency, you dissolve the 'thing.'

I would call it a lemma or proposition --small theorem-- that's easy to prove.

To me, the funny thing about it is that every so often the mathematical model does suggest to us that the previous model in terms of inalienable properties A, B, C, etc. does suggest that we'd better drop say, C, no matter how cherished a property of this 'reality' it is. In QM, this 'C' could be 'position' or 'momentum.' In SR, it could be position and/or velocity, and in GR it could be the concept itself of an inertial system, or coordinates by themselves. For lack of a better term, I would define this as a very ordered process of 'letting go' of anchors to what we think to be real. x-posted with @exchemist Agreed. But I think that's more an over-simplification that some people do when they don't understand time-tested principles like operationalism --the theory should have a counterpart in laboratory operations--, Ockam's razor --the theory should never create arbitrarily complicated constructs, and should be as logically simple as possible; it should produce falsifiable propositions --Popper--, etc. I recognise that as a risk, but I think an acquaintance with some principles of the philosophy of science generally operate as a good antidote. If one is well-versed in the history of science, I think one can minimise the risk.

This is probably the best starting point for this question --no pun intended. If x, y are in R, and x \( \neq\) y. Then, Either x>y or x<y. Assume x<y. A property of the real numbers is there always exists a z in R such that x<z<y if x and y are different. So there isn't such a thing as 'next number' in R. x-posted with @Genady

In the short term, yes. I've got friends who trusted me with their dog every now and then when they were away. I'm told I'm pretty good with pets when it's just for, say, a month, or some weeks. In the long term, no. I can't afford taking care of a pet on a daily basis. Having a pet is an enormous responsibility. It seems like it was yesterday when you mentioned it. CPT symmetry suggests they would. When you include magnetic monopoles in Maxwell's equations, you have both electric charges q, and magnetic charges g. For every particle with charge q there must be the corresponding anti-particle of charge -q. So for every particle with magnetic charge g there should be an anti-particle of magnetic charge -g, or else CPT symmetry would not hold. We think CPT symmetry holds. It would be a total re-think of quantum field theory if CPT symmetry failed. For more details, ask @Markus Hanke.

You're confusing a group with a representation of a group. Any Abelian group can be represented by addition of parameters. Example: \[ e^{x+y}=e^{x}e^{y} \] The Lorentz group is represented multiplicatively on wave functions, but additively on other objects. Again, take a look at, https://www.amazon.com/Theory-Application-Physical-Problems-Physics/dp/0486661814 and it will dawn on you. There's nothing else I can do for you here. I'm sorry my help wasn't enough.

Mr. @Abouzar Bahari, I'm reinstating all the negative points you're giving everybody, just out of spite on your part, apparently. You've already gotten your answers, non-scientifically. Thank you. That's all I wanted to read from you.

I understand you got a new cat. Is that right? I love pets, but I can't be trusted with them. Neither do I. I'm just wondering where we can get from just nothing. Maybe something's in store for us all.

Thank you, @Genady. I must confess I didn't see your argument back then. If anything, the fact that we both concur on the same argument makes our case only more poweful.

A little bit more for you to ponder about, Mr. @Abouzar Bahari. You might want to take a look at, https://www.amazon.com/Theory-Application-Physical-Problems-Physics/dp/0486661814 The fact that the transformations are symmetric --in the sense you mean them not to be-- is not an exclusive property of Lorentz transformations. AAMOF, Galilean transformations must comply with the same property you are in denial of. So, for slow velocities v, your argument is in trouble too: \[ x'=x-vt \] \[ ct'=ct \] \[ y'=y \] \[ z'=z \] So, even Galilean transformations are inconsistent with what you say. There is a powerful theorem that guarantees that the only relativity principles that can be consistent with a F=ma (second-order evolution equations) formulation of dynamics are the Galilean principle of relativity or the Einstein principle of relativity. I rest my case. Or do I?

Agreed. Interesting, at the very least. P(N)=И P2(N)=N => P2=I Is there an anti-nothing? I mean, an anti-nothing worth distinguishing from the usual nothing? Always do. Irrespective of how much respect they deserve. They're there for a reason. Plus... people can be touchy. Arguments are not.

You do not understand what parity is. Parity is an involution, by definition. That is P2 = I. That's because, if you change the sign of the coordinates, and then you change it again, you must get back to where you started. Lorentz transformations, OTOH, are not. There is no reason why Λ2 should be the identity. Every symmetry transformation should be a representation of a group, for consistency. So Λ(v)Λ-1(v)=I. As it happens, Λ(v)Λ(-v)=I, so Λ-1(v)=Λ(-v). End of story as to the mathematics. If you are willing to present an experiment that contradicts this mathematics, that would be great. https://en.wikipedia.org/wiki/Involution_(mathematics) Are we done here? Another thing, Mr. Bahari. You can keep giving neg-reps every single time I take pains to explain to you why your idea cannot be right, if you are so inclined. I would like to see a reason why you're doing so, other than you thinking I'm 'illiterate.'

It's the standard dummy number, maybe it stands for 'Nature,' maybe it stands for 'nothing'... I stand for nothing. I like it.

What makes you think that's the relevant question, and not: "The table consists of the lower atoms of the apple? Distinctions are in our mind. Our most sophisticated distinctions are in our theories. Nature doesn't know about them. A hydrogen atom in the apple's molecular structure certainly doesn't "know" whether it's apple or table. Ultimately, there is no meaningful way to say "this hydrogen atom is an apple atom." Distinctions are in our mind, Nature doesn't know about them. Isn't this philosophy?

reported as well. Are you reporting yourself? No wonder you can't handle a sign inversion.

LOL. Cats are known to collapse wave functions. That's their job in this world.