Why two mass attracts each other?

Discussion in 'Physics & Math' started by hansda, Mar 19, 2013.

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  1. Farsight

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    You started the ad-hominems. Not me.

    I've been exposing misconceptions here Markus, and you've been fighting shy of that exposure. You won't even respond to the fabulous post #158. Oh and by the way, talking of misconceptions, you said locally you will always measure exactly c. That's wrong I'm afraid. Once you take on board Einstein's variable speed of light and understand general relativity properly, you will understand this: if you're at a place where the coordinate speed of light is zero, you don't measure anything. Ever. You can't make a stopped clock tick by sticking a stopped observer in front of it.
     
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  3. Farsight

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    Doubtless you wish that were true, but it isn't.

    Gotta go.
     
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  5. OnlyMe Valued Senior Member

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    Farsight.., Einstein was presenting an analogy in that instance. He did not presenting a definitive definition. He was speaking in analogy when he said, spacetime is the ether of GR.., as within the context of GR light propagates through spacetime. Once again and as przyk has pointed out at least indirectly, the Leyden address and his initial presentation in 1916.., to his peers, were directed toward different audiences.

    Generally speaking his audience at Leyden were for all intents and purposes, rooted in having been thought physics from the perspective of the existence of the aether and Newton's mechanics. These were students and some of the staff (not all of which had strong backgrounds in cutting edge physics). His references of analogy were intended to bridge the gap between their earlier education and understanding of the world and a model relatively new to the achedemic setting they were in.

    You have to at least try to understand the context within which anyone makes any statement. That becomes even a greater issue when dealing with historical accounts were not only the culture of science was different, but the shear weight of experimental evidence has changed. Einstein set the stage for and ushered in a sea change in how the scientific community has come to understand the world, but lacking knowledge only available after the fact, he is not the foremost authority on GR. The child of his imagination and insight has grown far beyond his dreams.
     
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  7. Guest254 Valued Senior Member

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    Apologies for such a late response Farsight, very busy week!

    I think I might have a way to get to the bottom of this. I think your continued run-ins with people on this forum stem from your notion of "understanding" a scientific theory and other people's notion of "understanding" a scientific theory. I think the following might help.

    Let's pretend that (heaven forbid) the entire population of the planet was wiped out tomorrow, apart from you and a collection of 10 intelligent but scientifically uneducated adults (e.g. they've not heard of Einstien). In addition, let's assume that all literature was wiped out, so there's no books, papers, articles to which you can refer. How would you pass on Einstein's legacy?

    You can no longer answer questions with "because Einstein said", or do a quick google and point people to Wikipedia. If you claim that the Riemann curvature tensor has something to do with tidal forces, you'll have to explain to these people what on earth the "Riemann curvature tensor" is and demonstrate its relationship to tidal forces. If you want to talk about about specific solutions to the Einstein equations, e.g. Schwarzschild, you'll have to convince everyone why they're solutions (not to mention the fact you'd have to explain what the field equations are where they come from).

    All the professionals I know of could do these things, and they don't have anywhere near your level of confidence. I guess I could make a similar post with electromagnetism in place of general relativity, since you've told us you understand (your usage of the word) electromagnetism better than Dirac. I'm positive Dirac could have reproduced a good amount of it from the ground up, because he understood (my usage of the word) the theory.

    Genuinely interested in your response - I've said this before, but I think you're a fantastically interesting character!

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  8. OnlyMe Valued Senior Member

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    There is a flaw in your reasoning! The 10 naive "others", in the abscence of the availability of another source of information on the subject, would be Farsight's idea audience.., because as long as his answers to their questions are consitent they would have nothing to compare them to and whether they were good answers or out right fantasy, the 10 would have no way to challenge his rendition of "the way things are". At least for the few hundred to few thousand years it would take to recover from scratch.
     
  9. brucep Valued Senior Member

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    It's obvious, to me, that you understand GR. Probably studied GR in graduate school. To bad it isn't obvious to Shortsight.
     
  10. Guest254 Valued Senior Member

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    Whilst this hypothetical bunch might be uneducated, they are *intelligent*. For example, such people are unlikely to accept that GR predicts that the perihelion advance of mercury is precisley X without some thoroughly quantitative details! I'd also like to think Farsight would be honest!
     
  11. brucep Valued Senior Member

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    All you ever do is make irrelevant unsubstantiated assertions. Scientific irrelevance = unsubstantiated assertions. Pathetic.
     
  12. brucep Valued Senior Member

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    Do you want some of the details on how that result [all Einstein orbits naturally precess] is derived from the metric?
     
  13. Guest254 Valued Senior Member

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    That's ok brucep, but a very kind offer.
     
  14. AlphaNumeric Fully ionized Registered Senior Member

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    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...

    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'?

    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.

    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. :

    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 \(R^{a}_{bcd}=0). For metrics in general relativity they have to solve the field equations and can have elaborate forms but thanks to the definition of smooth manifolds meaning that they look flat up close we can use "normal coordinates" to show that any GR metric is also pseudo-Riemannian, as they locally become the Minkowski metric in normal coordinates. A space-like slice of the kinds of pseudo-Riemannian metrics seen in general relativity will give rise to Riemannian metrics on the resultant sub-manifold, as seen in such things as the first and second fundamental forms, which are ways of quantifying curvature on such slices, as well as metric pull backs. But the link doesn't make this restriction, it doesn't mention pseudo-Riemannian anywhere. Sorry, am I getting a bit too technical for you? None of this is particularly advanced, it's undergrad stuff and since these are just the specifics of things your link brought up there shouldn't be anything wrong with raising the discussion a little, right? After all, you do claim to understand quantum field theory better than Dirac and have done multi-Nobel prize worthy work, a little undergrad stuff which formalised notions of curvature, which you have been 'explaining' to everyone (for years!) shouldn't be an issue, right? Anyway... As a result of all of this there's numerous mistakes and failures to be precise in that link. It says [tex]ds^{2} = g_{11}dx^{2} + 2 g_{11}g_{22}dx dy + g_{22} dy^{2}\) 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?
     
  15. brucep Valued Senior Member

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    That is nice. I'm going to save that for future reference. You're the interesting character.
     
  16. Undefined Banned Banned

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    That's ok, I never confused an infalling object with the photon or light from that object. So that's not a problem in my naive understandings so far.

    You explain that the photon follows a geodesic which is curved more extremely at that "trapped" location I mentioned. But that is a theoretical construction involving some "extra long" spaceTIME path, isn't it? If the space is whatever it is, and the photon is moving as it does, then that geodesic is being created by the photon's motion, not by the space itself. BUT if the photon is trapped and can't in fact actually move either way, then there is no null "spaceTIME" geodesic to trace out in spaceTIME, is there, because it's not moving through space and it is creating no time because it is trapped there and and effectively stationary in both space and time? Or are you suggesting that space is moving through the photon so that photon acquires some sort of "induced relative to space" imaginary motion which creates some sort of imaginary time for an imaginary coordinate theory interpretation of the physics that is happening to the trapped photon there?
     
  17. Markus Hanke Registered Senior Member

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    No, I am not suggesting anything except what I already said to you - light traces out null geodesics in space-time, which are not determined by any mechanics of motion, but by the causal structure of space-time itself. Light never moves at anything other than c, which automatically means that it always stays on the hypersurface of the light cone relative to each and every point along its trajectory. And geodesics constraint in that way are precisely null geodesics.
    I have not said a word about "space moving through the photon" and "imaginary time" - where are you getting this stuff from ? Certainly not from my post, because I never stated any such thing.

    So once again - light traces out curved null geodesics in a curved space-time. It can get "trapped" only in the sense that these geodesics become closed curves, i.e. in the sense that the photon moves in elliptical orbits around the black hole without ever reaching the far away observer. Still, at any point on such trajectories it nonetheless propagates at exactly c. It is very simple and straightforward.

    Just out of interest - did you actually look at the interactive visualisation I referenced ?
     
  18. Undefined Banned Banned

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    Yes I looked. I am still trying to work out how that "causes" anything? What precisely is "the causal structure of spacetime" you refer to? Without light tracing a path in space there is no null geodesic to follow because the null geodesic does not "exist" in space or time but only in our theoretical construction of "a spacetime" in our mathematical models of motion in space, yes?

    Your elliptical orbit explanation assumes your theoretical use of a "spacetime model" mathematical interpretation is actually what happens physically there? I naively understand no more than that from your explanation so far about it.

    By the way, I am still not clear what null geodesic a rock sitting on Earth's surface is "following" if it is not "freefalling" anywhere? Maybe if you could clear that up for me I might understand better your other explanations regarding "causal structure of spacetime" interpretation of the physical behavior of motion in space?
     
  19. Markus Hanke Registered Senior Member

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    Thanks brucep. I should, however, clarify that I have no formal education in any field of science. I never even went to graduate school. I'm a bush pilot; I fly supplies and people into remote communities and airstrips that no other local pilot wants to go, because those airfields are little more than mudbaths with a few rocks strewn in. And even that I never formally learned; where I come from there is little to no official regulation by the "authorities" ( ha ! ), so I taught myself how to "bush pilot" in a 22-year old single engine Cessna without a working AH when I was a teenager.

    Anyway, my understanding ( or, according to Farsight, lack thereof

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    ) of GR and differential geometry is the result of a life-long interest, and years of self-study from printed textbooks, and online material such as MIT OpenCourseware. When I study I do it right - I don't go for quotes, but sit down with pen and paper and actually do all the calculations, so that I truly understand it. And if I don't get it right, I'll do it again the next day, and again and again, until such time when it clicks and I get the meaning and mechanics behind it. This of course requires the ability to recognise when you're wrong on something, and then go back and fix it.
     
  20. Markus Hanke Registered Senior Member

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    This refers to light cones. At any point of space-time you can draw a light cone; whatever is either within or on the cone is causally connected ( time-like ), wheras anyhing outside the cone is not ( space-like ). The cone itself is the boundary between what can be causally connected and what not; this is where light is. Null geodesics are geodesics which lie on precisely this boundary, the light cone.
    This is why it is referred to as "causal structure of space-time".

    Yes. You see, I give all my answers in the context of classical General Relativity; all I am trying to achieve is to explain how classical GR models these things. For yourself you can of course decide that GR is not what physically happens, but then I can't really help you. Just think of me as a "talking GR textbook".

    A rock sitting on the surface of the earth follows a time-like world line, not a null geodesic. Only massless particles such as the photon trace out null geodesics.
    Regardless, I know what you mean. The world line of such a rock, just like any other body, is a world line through space-time, and it would actually be a pretty complicated one since the rock is not at rest. It rotates with the earth, which moves around the sun, which moves....you get the idea. The rock's world line reflects all that motion, and as a world line in space-time it also has a component in the (ct) coordinate "direction", i.e. its curvature has a component along the time axis. Physically this would manifest as gravitational time dilation, though the effect is very small in the case of a rock on the earth.

    The point is simply that all gravitational effects have a common origin if considered in the context of space-time. I understand that trying to picture the geometry of and on a 4-dimensional manifold is very difficult, and can lead to many apparent problems. Gosh, I have been there myself ! I only truly understood it once I gave up trying to "visualise" things, and started to do the maths. Then it all became clear.
     
  21. Markus Hanke Registered Senior Member

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    Farsight, you have not answered any of the points in post #353. Specifically you need to answer this : in the GR field equations

    \(\displaystyle{G_{\mu \nu }=\kappa T_{\mu \nu }}\)

    what range of values do the greek indices \(\displaystyle{\mu, \nu}\) cover ? What does that range of values represent ? Do you know the difference between the use of greek and latin letters in a tensor equation ? Do you agree that these are Einstein' equations ?

    We'll take things slowly from there, then.
     
  22. Undefined Banned Banned

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    Thankyou for your replies, Markus Hanke, they are very helpful to improve my naive understandings through discussing.

    Light cones are just representations and recognition that light and its information or exchange has a limited speed, right? So that is just a reflection of what happens in space across distances which may or may not be within reach of a particular light signal. It doesn't actually "cause" anything in "time". It just makes motion of light across space the "determinant" of any "time" interval between any two spatially separated objects or locations, yes? These "cones" don't exist except as models. The final determinant is the light travel in space which "results" in such limitations across space which our "spacetime cones" only model but not actually create? That is my naive understanding of all that "theoretical construction" of the "light-motion-across-space" effects on physical objects and event timings and such?

    I have not "decided" one way or the other yet. I just want to go naively beneath the jargon and models to try and understand what physics actually happens which we are "modeling" with GR etc theories and "explanations" based on theories which assume things that may not "exist" as such in physical space and motion interactions over distances which may have lightspeed limitations for their "causality". Such limitations to "casuality" does not in itself make GR or any other theory so far an actual representation of what exists and happens. I understand naively that one can become technically fluent in GR modeling. But what I am trying to find is naive understandings which are independent of the theories and assumptions which depend on "abstract coordinate constructs, as you called them. My naive understanding so far is that "light cones" and "causal structure" and such "terms" are just modeling what space and motion interaction limitations are already in nature due to lightspeed limit on information exchange and object motion/momentum exchange across space and such before any "theory and model" interpretations are made about them?

    I already recognized in my earlier posts the other Earth motions through solar system and galaxy and the rest of the universe. My question allowed for what other Earth motions there are, I just wanted to cover the case where a rock sits on a non-rotating planet's surface and the geodesic or whatever it "traces" only with respect to its component due to the gravitational field of the planet, nothing more than that component. That is the only way I will ever understand naively what the rock on the Earth (or a photon trapped above the bh horizon I described before) is actually doing with respect to the gravitational body involved and nothing else. Once I can naively understand that "component" of geodesic then I can progress to more complex cases.

    By "4-dimensional manifold" you mean the "spaceTIME" manifold, yes? But that is exactly what is still not clear to me because I can understand that space and objects/light moving across space creates all the information we "model" as "spacetime" 4-dimensional manifold. But that "4-D manifold is only "mathematical construction" of the theoretical model, it is not existing in physical action unless there is objects and light motion across space distances which we then represent and analyze through mathematics from the information we get from measuring space distances and motions of objects and light. It is from the information about "space" and "motion" that we derive our 4-D manifold mathematical modeling and interpretations as space and "time", isn't it?
     
  23. Markus Hanke Registered Senior Member

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    No problem. I will always try to do my best to answer genuine questions with regards to this subject.

    I am not certain if I understand correctly what you are trying to say here, but the first sentence is a good and correct assessment - light cones are a direct representation of the fact that light propagates at a constant, limited speed.

    That is perfectly fine and acceptable. The question of the precise relationship between a physical model such as GR and "reality" is actually a rather interesting one, which has wide-ranging philosophical implications; but that's probably for another thread. My beef with Farsight is not about the validity of GR as a model of reality, it is about his failure to acknowledge GR for what it is mathematically, which is a model formulated on (3+1) dimensional space-time. Whether or not this represents reality is a separate issue altogether.

    All we can really do in the context of physics is extract numerical predictions from its models ( e.g. light deflection, orbital precession, Shapiro delay etc etc ) and compare those to what we actually observe. So far as the predictions of GR, based on space-time as it is, stack up very well against "actual" figures from the "real world", if you know what I mean. At the end of the day that is really all we can expect from our models.
    Like I said before - I am not a GR fanatic. I would be the very first person to congratulate anyone who comes up with a more powerful, more wide-ranging model, even if it does away with geometry completely. Unfortunately we don't really have any such candidate models yet ( I am not particularly fond of M-Theory, but do try to keep an open mind ).

    Ok, sure. Even if we only consider relative motion with regards to the earth's surface, you can still think of it as a free fall scenario. Basically, the rock tends to describe a curved world line in a curved space-time, as before; however, the interaction with the surface of the earth counteracts the free fall, accelerating the body upwards. That is why an observer on the rock measures a gravitational field ( which is equivalent to local acceleration, as you probably remember ), and the body does not move. Because it doesn't move, the world line ( not space-time ) of the rock has no components in the spatial directions ( in other words - it remains stationary in one spot, which we can choose to consider the origin of our coordinates ), but it still has components and curvature in along the time axis ( in other words - it "ages" ). Physically this is what gives you gravitational time dilation.

    To be honest this is a scenario which, though very basic, is actually really hard to visualise, so I appreciate the difficulties one may have with this. However, it is of course possible to do the maths here, and you will find that it yields just what one observes - a stationary rock on the earth's surface, which, as compared to some far away observer, experiences a well defined amount of gravitational time dilation.

    Yes, indeed.

    Sorry, I am not entirely certain what you mean to ask here. You are correct in that our Riemann manifold combines space and time into just one construct; therefore you don't get "just motion" as we experience it in our daily lives, but rather static 4-dimensional world-lines, which represents objects at all times of their existence. A 4-dimensional space-time encompasses not just all points in space at a given time, but all points in space at all times, in just one manifold.
    Sometimes, just for myself, I picture it as a sheet of clear ice, and the world line of a particle or object is like a crack running through the sheet of ice. There actually are attempts to model elementary particles as topological defects in space-time, just like in the ice analogy.
    Remember in this context that GR is a classical model, so it does not account for any uncertainties introduced by quantum physics. Everything in GR is well determined, there are no quantum effects.

    One question I can not answer for you is why we as human beings perceive only "slices" of that space-time, i.e. why we only perceive three dimensions out of the four. For us it appears as if we see a rapid succession of "slices" out of that space-time, i.e. we can only perceive one moment in time, the "now", like in a film projector. This still gives us a total of four dimensions ( since it is just a so-called "foliation" of hypersurfaces parametrized by the time coordinate, which is mathematically equivalent to 4-manifold ), but we perceive only three of them at a time. The easy answer is probably that our sensory apparatus is simply incapable of perceiving anything more than three dimensions, so our mind cannot visualise a 4-dimensional manifold for lack of a suitable model.

    In either case, I am probably confusing you now, so I better stop

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    Last edited: May 9, 2013
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