Alphanumeric said something moves through space-time along some worldline. You know that's wrong. And Alphanumeric knows that It is very important to be precise about these things.
I find it funny that suddenly you care about details when you repeatedly make the same laughable qualitative mistakes and the only reason you don't make quantitative ones is that you cannot do anything quantitative. It is even more funny you continue to pretend to understand Riemannian curvature, after you made the mistaken conflation of the curvature tensor and the tidal effect, partly due to the fact you don't understand what a tensor is or how the construct the tidal tensor. Or how about that thread a while back with Prometheus where you didn't understand the curvature tensor and how the conceptual splitting of space-time into variation directions impacts the tensor's contributions in various directions. Or how you think the fact the universe isn't perfectly homogeneous somehow undermines the application of the FRW metric to cosmology. Or how you think setting ones velocity using the CMB somehow amounts to fixing ones velocity relative to the universe. Or how you don't understand the role postulates play in physics compared to axioms or theorems or models.
Seeing as you have now acknowledged that you have read my repeated explanations of why details, quantitative details, are important can we all assume you're now going to discuss Riemannian curvature etc using quantitative details? And that you're going to start giving quantitative structure to all your '[Something] explained' work? And that you are aware how a model can sound qualitatively valid but be quantitatively invalid? If the understand this last point then you understand why your work is pointless until you can give it a quantitative foundation.
As for world lines and space-time... Particles which below the speed of light move through space, being at one and only one point in space at any given time (where 'space' and 'time' are defined in terms of slices of space-time by whatever coordinates are being used). This sweeps out a world line in space-time. The world line is not time dependent, it is a fixed curve through a 4 dimensional space (where 'space' is now being used in the general manifold sense). However, the line is parametrisable (as all continuous paths in any hypothetical suitable construct are) by a single variable. For arbitrary geodesics the choice of this parametrising variable can be a little sticky, null and space-like paths have particular details that need to be considered. For particles moving along a time-like curve (ie a curve whose tangent vector is always time-like) a natural parametrisation is available, namely the corresponding time coordinate of the space-time manifold. Doesn't have to be time but due to the fact the curve is time-like we can always find a monotonic increasing function which relates our choice of parametrisation with the time coordinate. So in this sense as we move along the curve via our parametrisation we move through the set of $$(t,\mathbf{x})$$ coordinate locations associated to the particle, sweeping out the world line. It is in this sense that the particle moves through space-time along a world line. The world line, as a collection of points in space-time, is a time-less concept but when associating the line parametrisation with the splitting $$x^{\mu} \to (t,\mathbf{x})$$ then saying a particle moves through space-time along a world line is an entirely sound thing to say.
Anyway, to go back to your reply to me from many pages ago...
I didn't say directly and I didn't say is.
You said equates to it. Equating two things is to say they are equal. What you should have said is that the Riemann curvature tensor allows you to
construct the tidal tensor. The tidal tensor is not curvature, it is determined by curvature. At best you failed to explain yourself and at worst you just don't understand. As usual you ignored my point that you don't understand the relevant mathematics and clearly you have serious failings in your qualitative understanding so telling others how it is is just laughable.
You concede that the details are important and yet this thread has basically been you claiming to understand something which you have zero mathematical grasp of and zero physical experience with, all you have is a highly flimsy qualitative understanding obtained from reading layperson analogies and simplifications someone else wrote.
Please explain why anyone should think you have any insight into this? On what are you basing your (in my opinion and many others opinion) completely unjustified self confident that you understand this stuff? You cannot have distilled it from the quantitative models, you cannot do any of the mathematics. A lacking in mathematical understanding of models others have developed could be excusable if you were doing experimental/observational work on this, thus giving you personal experience and helping develop intuition for such things, but you aren't. Why should anyone think you have all of this 'explained'?
Would you care to address my post
#158 instead of boring us all with your customary carping whining ad-hominems?
Your post 158 is just rehashing things I've already discussed with you. Clearly you either don't remember or don't wish to remember, neither of which reflect well on you.
instead of boring us all with your customary carping whining ad-hominems?
And there's your usual excuse. Am I cutting a little too close to the bone for your liking? Couldn't you think of a retort to my querying why anyone should take you seriously when you don't know the quantitative stuff at all and you have dubious grasp of even qualitative stuff? It's the excuse you always use; I give
valid reasons to question your self assuredness and you fail to retort them, complaining I'm just ad hom'ing. It isn't an ad hom to point out you have, for all intents and purposes in regards to any of this material, zero mathematical knowledge, it is a statement of fact. It isn't an ad hom to point out you have zero experience with actual data in regards to this material. It isn't an ad hom to point out your flawed understanding of such things as the applicability of the FRW metric. It isn't an ad hom to point out the Riemann curvature tensor cannot
equate to the tidal tensor, rather the former is used to construct the latter. It isn't an ad hom to point out your work has zero quantitative structure. It isn't an ad hom to point out that it is hypocritical of you to criticise string theory for, in your view, having no real world applicability/validation while your work has less than string theory. It isn't an ad hom to point out the one and only time, over more than 5 years, you've ever given a direct response to my request you provide a quantitative model derived from your work it was
someone else's result and it was pure numerology of a kind secondary school children can spot the flaw in but which you called (something like) 'astounding'. It isn't an ad hom to point out your reliance on metaphor and analogy is never going to amount to anything even remotely resembling viable physics. It isn't an ad hom to point out that your work was rejected from every single journal you submitted it to.
edit
Actually I just noticed something in post 158. :
The Riemannian metric describes the state of space. And it features pressure and shear stress. These and other terms result in curved motion through that space, to which we apply the label curved spacetime. It's like a car encountering mud at the side of a road. The road isn't curved, nor is the mud. But the car veers left. It's path is curved.
All sorts of problems there due to poor explanation. Having looked at the
link you used I can see why you're getting confused, since the paper was written a long time ago and thus fails to use definitions and terminology in the same way as present. If someone were to read that and assume that what it says in its
older terminology were valid using
present terminology they would reach seriously flawed conclusions. Furthermore there's a number of serious errors or omissions in it. This could be due to the transscriber or it could be because when Einstein wrote that there were a number of mistakes in how people understood differential geometry. One of the errors there is so huge it is either a transcription error by the person who made the website or that article was not written by Einstein at all. I cannot find any other source for the article.
The issue is the difference between a Riemannian space and a
pseudo-Riemannian space. A Riemannian space is one where the metric is positive-definite, ie for all possible vectors X, g(X,X)>0. A pseudo-Riemannian space need not satisfy this, there can be X such that g(X,X)<0 and now the condition is that g is non-singular. The Euclidean metric is Riemannian, it is a diagonal matrix with all +1's down the diagonal, g = diag(+1,+1,+1,...) so g(X,X)>0 for all non-zero X. The Minkowski metric of special relativity is pseudo-Riemannian, since it has (up to arbitrary signature notational choices) the form g = diag(-1,+1,+1,....). Both of these kinds of metric are flat in the Riemann curvature sense, $$R^{a}_{bcd} = 0$$, (the other notion of 'flat' is defined as metrics like the FRW have R = 0 but $$R^{a}_{bcd}\neq 0$$ or the Schwarzchild metric being Ricci flat, $$R_{ab} = R^{c}_{acb} = 0$$, which are weaker conditions than $$R^{a}_{bcd}=0). For metrics in general relativity they have to solve the field equations and can have elaborate forms but thanks to the definition of smooth manifolds meaning that they look flat up close we can use "normal coordinates" to show that any GR metric is also pseudo-Riemannian, as they locally become the Minkowski metric in normal coordinates. A space-like slice of the kinds of pseudo-Riemannian metrics seen in general relativity will give rise to Riemannian metrics on the resultant sub-manifold, as seen in such things as the first and second fundamental forms, which are ways of quantifying curvature on such slices, as well as metric pull backs. But the link doesn't make this restriction, it doesn't mention pseudo-Riemannian anywhere.
Sorry, am I getting a bit too technical for you? None of this is particularly advanced, it's undergrad stuff and since these are just the specifics of things your link brought up there shouldn't be anything wrong with raising the discussion a little, right? After all, you do claim to understand quantum field theory better than Dirac and have done multi-Nobel prize worthy work, a little undergrad stuff which formalised notions of curvature, which you have been 'explaining' to everyone (for years!) shouldn't be an issue, right? Anyway...
As a result of all of this there's numerous mistakes and failures to be precise in that link. It says $$ds^{2} = g_{11}dx^{2} + 2 g_{11}g_{22}dx dy + g_{22} dy^{2}$$ is a Riemannian metric. No, it isn't and for a number of reasons. I'd ask you to give them but I know you'll ignore such a challenge. Firstly we note that the right hand side can be written as $$(g_{11}dx + g_{22} dy)^{2}$$, implying $$ds = g_{11}dx + g_{22}dy$$. Utterly wrong.
Utterly. Clearly the person doesn't even know how to use a metric. A metric defines a line element ds by $$ds^{2} = \sum_{a,b}g_{ab}dx^{a}dx^{b}$$. In this case a,b take values 1,2 and we get $$ds^{2} = g_{11}dx^{2} + g_{12}dxdy + g_{21}dydx + g_{22}dy^{2}$$. Since dxdy = dydx and metrics are symmetric so $$g_{12} = g_{21}$$ we get $$ds^{2} = g_{11}dx^{2} + 2g_{12}dxdy + g_{22}dy^{2}$$. Notice the difference? This isn't a perfect square, unlike the link. It involves the off diagonal terms $$g_{12} = g_{21}$$, unlike the link. The link then asserts this is a Riemannian metric, just because he's slapped some arbitrary coefficients in the expression. As I've already explained a Riemannian metric is one which is positive definite, g(X,X)>0 for all X, which amounts to $$g(X,X) = \sum_{a,b}g_{ab}X^{a}X^{b} = g_{11}(X^{1})^{2} + 2g_{12}X^{1}X^{2} + g_{22}(X^{2})^{2} > 0$$. Yes, the expression given in the link is manifestly non-negative since it is, as I just explained, mistakenly a perfect square but that isn't the reason the link asserts the expression is a Riemannian metric.
The link explicitly states
" it is possible to show that the space - time continuum has a Riemannian metric", which is false, it is pseudo-Riemannian. Putting this together with "
the Riemannian continuum is a metric continuum which is Euclidean in infinitely small regions" gives the link saying the space-time continuum is Euclidean close up. It is flat close up but the structure of Minkowskian (or Lorentzian, depending on personal preference), not Euclidean in the inner product sense. It also says that "
the Euclidean continuum is richer in relationships than the Riemannian.". How can a particular case be richer than the more general one? The Euclidean space has particularly simple parallel line and parallel transport rules but all Riemannian and pseudo-Riemannian spaces can have such things defined on them, its one of the core things in differential geometry. The notion of parallel transport is intimately linked to geodesics and covariant derivatives, since the tangent vector to a geodesic transported along said geodesic obeys the parallel transport condition $$\nabla_{X}X = 0$$. In the case of Euclidean spaces this is trivial but it is much much richer in more general (pseudo)Riemannian spaces.
Now some of these errors can be put down to terminology changing and the increase in the use of pseudo to distinguish from positive definite metrics. The rest might just be errors by the transcriber. Either way if a lay person, such as yourself, were to read that page and think "Well it says Einstein wrote it so it must be sound" and then, as you love to do, kept referencing Einstein (by using that link) to make assertions about metrics and relativity then regardless of where the mistakes came from
there are mistakes. Once again you link to a document someone else wrote which uses a lot of terminology you don't understand and because you don't understand the terminology or know the quantitative details yourself you are unable to identify them. You love to play the "Well what did Einstein have to say about it?" card but this illustrates how you don't really understand what Einstein or anyone else is really talking about, you can only parrot bits and pieces, mindlessly accepting their qualitative simplifications, without the capacity to confirm or correct them.
This is exactly what happened with your "This result is astounding! Oh, it's just numerology?!" case. You didn't learn from that mistake and you haven't learnt from all the previous times you've spouted assertions about domains of mathematical physics completely beyond your comprehension due to you not knowing any of the required mathematics. Remember, this is a statement of fact, just because you don't like being reminded of this doesn't make it an ad hom which you can just ignore. Sure, you could ignore all of these examples of how your inability to understand any details and unjustified self belief lead to you making clanging errors but at the end of the day you're only hurting yourself. If you'd started learning A level maths back in 2008 you could be midway through a degree by now, having a wealth of mathematical methods and new points of view at your disposal. Instead you're still here, posting links you don't understand on topics you don't understand while telling others they're the ones who don't' understand. It's ironic you immediately went on to telling Marcus how you thought he got the space vs space-time thing wrong, that's essentially the same kind of mistake your link made.
You're always saying how you want to 'talk about the physics' whenever someone backs you into a corner and you're needing an out well there's plenty of quantitative detail in this post for you to discuss. You regularly pull Einstein into a discussion and use him to try to justify your 'explanations' and now that has exposed your inability to identify even extremely basic and glaring problems in the material you cite. This further undermines your attempt to present yourself as understanding this domain of physics, you fundamentally lack the ability to assess any source for scientific content, you can only trust it to have no mistakes and that by citing a source you'll be able to avoid having to do any details yourself. This proves otherwise. Now you can either attempt to defend your referencing of an article with the mistakes and problems described above or you can do the more difficult and time consuming, but ultimately infinitely more useful and rewarding, task of using this as motivation to sit down and actually
learn some physics. You'll have to learn some maths too and you'll have to start on pretty basic stuff like GCSE or AS level but it'll be a start. You clearly have the drive and resources to be able to write a book, self publish and self promote it for a lengthy period of time, put that time and money to better use and enrich your mind.
Or failing that you can ignore or dismiss this post as 'too long' or 'ad hom!' or 'let's talk about the physics', unwilling or unable to face up to your mistakes and we can repeat all of this back and fore in a few months time, like always happens. So which is it, defence, education or denial?$$
