joigus

I agree with main arguments developed by @Genady, @studiot, and @Lorentz Jr. I particularly liked Studiot's summary. I would call his argument about closed and open sets --as well as those that are neither open nor closed-- a "topological approach." A crash course in topology would include concepts such as, Topology: Existence of an inclusion relation in a set, \( \subseteq \) --contains--, \( \subsetneq \) --does not contain. => neighbourhoods of a point. Limit point --o accumulation point--: A point in a set that has neighbouring points also in the set that are arbitrarily close to it. Interior of a set: All its point are limits points of the set --if I remember correctly--. Boundary of a set: The set of all the limit points of its exterior Closure of a set: The union of the set and ist boundary ... etc. With these rigorous topological definitions, when applied to the real numbers, we can prove they constitute a topological space, and, eg, the set \( \left[0,2\right]=\left\{ x\,\textrm{in}\,\mathbb{R}\,\textrm{such that}\,0\leq x\leq2\right\} \) contains its boundary --and it is, therefore, closed; while the set, eg, \( \left(0,2\right)=\left\{ x\,\textrm{in}\,\mathbb{R}\,\textrm{such that}\,0<x<2\right\} \) does not contain its boundary --and it is therefore, open. https://en.wikipedia.org/wiki/Topological_space

c can have any value you want just by choosing the length and time units accordingly, as @Eise told you. Stop blaming your misunderstanding on others. I also told you.

-1. You would be well advised to stop insulting people and clean up your own house. Neither one of us is "dominant in the world of science." As to you, you are nowhere near the world of science. Reported.

It's a good start for General Philosophy though.

Owls are amazing in this particular department. http://www.instantshift.com/2014/12/12/hidden-camouflage-owls/

My most heartfelt respect. On my part: FSM is no vice-anything. It reigns supreme over all things, real and unreal. It swallows nonsense and spits out nonsense too, only funnier. What do you wish to discuss?

I don't know what you mean by "it's a physical parameter, and not a mathematical one." A physical parameter, in the usual sense of the term, most definitely it is not. A physical parameter is any quantity that we can vary either freely, or subject to some specified conditions. Eg, the magnetisation of a medium of given magnetic susceptibility, etc. In the context of relativistic physics, c is a universal constant, not a parameter. Theoretically, it is derived from principles of electromagnetismf. Experimentally, it is measured. If you mean otherwise, you should say so. Because I've been studying these things in excruciating detail for many years, I can tell you you're using the poor-man's version of boosts. The grown-up version of it is, \[ \boldsymbol{x}_{\Vert}'=\frac{\boldsymbol{x}_{\Vert}-\boldsymbol{v}t}{\sqrt{1-v^{2}/c^{2}}} \] \[ ct'=\frac{ct-\boldsymbol{v}\cdot\boldsymbol{x}/c}{\sqrt{1-\left\Vert \boldsymbol{v}\right\Vert ^{2}/c^{2}}} \] \[ \boldsymbol{x}_{\bot}'=\boldsymbol{x}_{\bot} \] Where you have to decompose position 3-vector \( \boldsymbol{x} \) as, \[ \boldsymbol{x}=\boldsymbol{x}_{\Vert}+\boldsymbol{x}_{\bot} \] \[ \boldsymbol{x}_{\Vert}=\frac{\boldsymbol{x}\cdot\boldsymbol{v}}{\boldsymbol{v}\cdot\boldsymbol{v}}\boldsymbol{v} \] \[ \boldsymbol{x}_{\bot}=\boldsymbol{x}-\boldsymbol{x}_{\Vert} \] So the expression in the numerator is actually not a positive 3-scalar, but a 3-vector projection in some inertial frame. You don't understand anything, and what's worse, you don't ask. So Markus's noble attempt to help you, my attempt to close down possible loopholes, and other members' attempts to walk you through the logic of Lorentz transformations, is --most unfortunately-- to no avail. Pitty. Good day.

β=v in any system of units such as light-years per year, light-seconds per second, etc. That is, any system of units in which c=1 . I thought you understood that, @Abouzar Bahari.

LOL. Yeah, on second thought that's not that clear, what I said. Sometimes I say things just to see how people react. I suppose you need to probe the other one somehow. Especially online, where a lot of usual clues are missing.

Yes, exactly, @Genady and @sethoflagos. Consciousness, we don't know what that is. In particular, we don't know whether it's an emergent phenomenon or there is perhaps basic physics we still don't know about involved in it. So it might be a bit adventurous to try and guess whether it makes a case for reductionism or not. On the other hand, things like Zipf's law: https://en.wikipedia.org/wiki/Zipf's_law IMO make a good showcase --if not a robust case-- for non-reductionist elements having to do with patterns we see across different phenomena. The explanation, it seems, is purely statistical. If that's the case, one could argue that it doesn't matter too much what constituent elements the ensemble is made of, and regularities appear because of the clustering of data for which the underlying law can be very different in nature. Very similar to the linear power law between internal energy and temperature in ideal gases, with specific gases having different specific heats depending on whether the molecule is monoatomic, diatomic, etc. => A statistical reasoning that gives you a pattern irrespective of the reductionistic model, but a concrete microscopic model that completes the parametrics of the problem --typically the constants. The question is not an easy one in general. There are cases --like the power law between metabolic rate and mass of a multicellular organism-- where the power law can be guessed at from some kind of reductionist first principles. See, eg, https://www.science.org/doi/10.1126/science.276.5309.122 Of which there is criticism too.

It's OK. There's a way to hold your own proudly in these anti-reductionist times. Just say you're a reductionist, only not just a naive reductionist.

Reductionism has been very heavily criticised in science in the last decades. One good reason for this is that there are emergent aspects of natural laws that seem impossible to fathom by simply looking at the basic law and its constituent 'parts.' For example, in recent decades there's been a lot of discussion about universality of certain power laws, which would occur no matter what constituent elements make up the 'stuff.' If such were the case, reductionism would take a big blow. I would say the increasing relevance of this concept 'emergence' has a lot to do with why increasingly scientists are crossing out their names from the list of devotee reductionists.

Oh, boy. I'm speechless, speechless I tell you. Nice touch. So do I. I'm going nowhere in no time.

Fingers crossed.

In a nutshell, it is the contention that you can understand a system by analysing its parts and the relationships between them. Assuming that it makes sense to consider it as made up of distinct parts, that is. https://en.wikipedia.org/wiki/Reductionism

I'd say that given that both \( \omega_{\textrm{Earth}}R_{\textrm{Earth}}\ll c \) --slow-rotating Earth-- and \( \frac{GM_{\textrm{Earth}}}{R_{\textrm{Earth}}}\ll c^{2} \) --not very intense gravitational field--, you're quite safe using Lorentz transformations that factor out into a boost --jump to a constant-velocity frame-- and a slow rotation. For finer effects you would want to consider GR --Lens-Thirring effect, and such. Does that answer you question?

Now that I think about it, it's also dangerous when you're in a Euclidean context too.

Yes, that was exactly my point. Swapping indices cavalierly is dangerous when you're in a non-Euclidean context. If you want to correlate observers and you need boost + rotation to do that, transpose Lorentz transformations do not coincide with given transformation. BT not equal to B. @Genady, what's your code for that? I use, \left.B_{a}\right.^{b} which produces, \[ \left.B_{a}\right.^{b} \] Never mind. I've just realised while I was making myself a sandwich! LOL

You bet. You can't imagine how many people I've met who don't know this. As well as the reason why one set of coordinates is co-variant, and the other contra-variant. It's all in the language of differential forms, but that's not gonna be a problem for the likes of you.

Exactly. Wrong notation! Ambiguous, as you rightly said.

As to index gymnastics, I'm very fond of Anomalies in Quantum Field Theory, by Reinhold A. Bertlmann --the famous mathematician of John Bell's article. The first third of the book has a lot of it, because he deals with gravitation a lot. Not in detail, but you can check the calculations page by page as a good gymnastics.

In other words, one should never write the tensors with vertical alignment. That could lead to errors if you have rotations mixing in in you brew. Some people do it, I know.

Vertical alignment is exactly what I'm talking about. If you have a 2-index tensor, the first index is the column index, and the second one is the row index. Unless I'm missing something, that's what the vertical alignment is all about.

This is because the first and the second indices generally act on different indices irrespective of whether they are covariant --they transform with the same matrix as the basis members-- or contravariant --the transform with the inverse matrix. For the Euclidean case, this is of no importance, but as you well know, for Minkowski, it matters. Consider the Lorentz transformations, 1) Boost in the t-x plane: \[ B=\left(\begin{array}{cccc} \gamma & \beta\gamma & 0 & 0\\ \beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right) \] \[ \left.B_{\mu}\right.^{\nu}=\left.B_{\nu}\right.^{\mu} \] 2) Rotation in the x-y plane \[ R=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & \cos\theta & -\sin\theta & 0\\ 0 & \sin\theta & \cos\theta & 0\\ 0 & 0 & 0 & 1 \end{array}\right) \] But, \[ \left.R_{\mu}\right.^{\nu}\neq\left.R_{\nu}\right.^{\mu} \] The way I do it in LaTeX is, \left.B_{\mu}\right.^{\nu} The Schaum series perhaps? PS: Sorry, Markus. I think I swapped co- and contravariant indices in my answer. Let me fix it. I meant, \[ \left.B_{\mu}\right.^{\nu}=\left.B^{\nu}\right._{\mu} \] for the boost, and, \[ \left.R_{\mu}\right.^{\nu}\neq\left.R^{\nu}\right._{\mu} \] for the rotation. I hope that helps.

Both DM and DE are consistent with GR. Sigh.