Acitnoids: I note your post 79. Well said. As ever you're evading the simple point and posting abuse to distract from the discussion. Please address it: a gravitational field is inhomogeneous space. That's what Einstein said. So if space is homogeneous there's no gravitational field at all. And no spacetime curvature, so light travels in straight lines and the universe is flat. End of story.
See his 1920 Leyden Address: "This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)..." The assumption of isotropy and homogeneity means that the Reimann and Ricci curvature is zero. Anything else is a contradiction in terms. No. You counter the simple point. Yes it does. Any non-zero curvature means space is not homogeneous. AdS is a non-real solution I'm afraid. And de-Sitter space is a mathematical space. Whilst we can talk of a de-Sitter universe, the real space within that universe is flat. You haven't corrected me at all. All you've done is demonstrated a penchant for mathematical abstraction which bears no resemblance to reality. Lots of times. Until you understand it. I'm not making any. You are. And Prometheus. LOL! I'm not the one getting defensive here. You are. I am familiar with it. You're just slinging mud because you cannot elucidate an adequate response to the simple point. Homogeneous space means no gravity, no spacetime curvature, and a flat universe.
Er, you've been claiming gravity = inhomogenous space. Alphanumeric just gave you an example of a well known solution to GR which is homogenous yet has non-zero curvature. How is that evading your point? He's confronting you head on with a counterexample!
He merely gave an example of a "possible solution" which is not supported by evidence. The evidence supports a flat universe. Once you understand what Einstein said about a gravitational field being inhomogeneous space, you then understand that there is no spacetime curvature in homogeneous space. Then once you understand electromagnetism, you understand the difference between curved space and curved spacetime, and you learn to distinguish between mathematical abstraction and reality.
And who precisely determines what is this so-called "reality"? And having decided whom, tell us what sort of activity they perform in order to arrive at (what I would prefer to call an approximation of) this "reality". And having figured that out, please give us a precise definition of the above activity performed by those that carry it out, and specifically in what context.
I would disagree with your interpretation of what he said. In special relativity the metric is constant, flat, isotropic and homogeneous. GR says that these are not universelly true, in that its possible to have space-times which are none of these but that doesn't imply specific cases can't have some of those properties. I asked you to demonstrate your claim, not just reassert it. There are well known counter examples to your claim so you need to show why they are not counter examples, ie explain where the flaws in their derivations are. So that's a no then, you can't point out the flaw in the derivation of the FRW metric. And you base this on what, your own interpretation of a wordy quote by Einstein? If you were to bother to look up the formal quantitative definitions of 'homogeneous' and 'isotropic' you'd see that its possible for a metric to satisfy those conditions and still have curvature. Isotropy is roughly rotation invariance and homogeneity is translation invariance. AdS, Minkowski, dS and the FRW metrics are all examples of space-times which have such symmetries but only Minkowski and a particular case of the FRW metric have zero curvature in the Ricci sense and only Minkowski in the Riemannian sense. Here is the explicit proof that AdS is homogeneous and isotropic and yet it doesn't have zero curvature. Its an example of a 'maximally symmetric vaccum solution', the other cases being the afore mentioned dS and Minkowski. Do you think that all the people who've ever studied GR got it wrong, that none of them ever spotted this error you believe exists? Metric invariances are fundamental to GR (as any invariance is fundamental to the relevant model) so its not like this is 'out there' in complexity. Did you miss where I and Kurros mentioned the FRW metric, which is what cosmologists use to model universe expansion? Besides, you didn't say anything about physical space-times (as any physical one will not be perfectly isotropic or homogeneous due to matter distribution), you said that if a space is homogeneous then its got zero curvature 'end of story'. That simply isn't the case, it is possible to have non-zero curvature in a homogenous space-time. I'm absolutely certain it'll be lost on you but I'll give the more quantitative phrasing of this too. Homogeneity means an invariance under \(x^{a} \to x^{a} + y^{a}\) for any x,y choices, ie \(g_{ab}(x) = g_{ab}(x+y)\). It does not then follow that \(R^{a}_{\phantom{a}bcd} = 0\), as can be seen by looking at any of the examples you've been given. So you want to go down the road of "Well such situations never really occur in the real world"? The real universe isn't exactly homogeneous since matter is clumped all over the place so to even talk about actual homogeneity you're having to consider an abstract example, rather than a real one. Then the question is "Does g(x) = g(x+y) imply R(g) = 0? and the answer is no. Hey, its no skin off my noise if you want to delude yourself and waste more time and money on vanity publishing and adverts in magazines because you couldn't pass peer review. Those people who can actually do GR and use the FRW metric to understand the universe will carry on just fine. Yes, just like it was a mistake for us to do PhDs in string theory because you were going to bury it and win a Nobel Prize.... Please Register or Log in to view the hidden image! Tell you what, you write up your argument as to why the use of the FRW metric by cosmologists is completely flawed (since it has non-zero curvature and is derived on the assumptions of isotropy and homogeneity) and I'll type set it for you so you can submit to a reputable journal and then we can see what kind of views astrophysicists or GR researchers have. What do I have to be defensive about? I'm providing plenty of examples to back up what I'm saying and you have yet to present any argument other than "Because I say so". I asked you to show the derivation of the FRW metric is wrong but you ignored me. I know you want to think you're viewed as a threat to mainstream physics or string theory or to my ego but that really isn't the case. If you can't answer my simple questions then you'll never manage to convince the mainstream research community to put large chunks of GR and cosmology in the bin and pick up your work. Several explicit counter examples isn't 'elucidating an adequate response'? Do you use a difference definition of 'adequate' to most people? Besides, despite you asking others to provide detailed explainations when they disagree with you you're unwilling to do the same when you disagree with someone. Can you provide something more than just assertions? Perhaps if you could do tensor calculus (ie could understand GR on more than a pop science level) you'd be able to convince yourself of how there are cases where \(g_{ab}(x^{c}) = g_{ab}(x^{c}+y^{c}) \Rightarrow R^{a}_{\phantom{a}bcd} = 0\) is false. If you want to convince me your claim is true I'd like to see you prove it using tensor calculus, as that's what I'd ask anyone else. I want you to prove \(R^{a}_{\phantom{a}bcd} = 0\) if \(g_{ab}(x^{c}) = g_{ab}(x^{c}+y^{c}) \). Can you do this?
So? The simple existence of solutions in general relativity describing homogenous but curved spaces is enough to contradict your point. They show that if we happen to be living in a homogenous but flat universe, that's just a matter of circumstance. It's not a general feature of GR. The citation you provided doesn't support that. I read this: This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)...as Einstein saying we'd have to learn to view space-time as being endowed with physical properties, and accept that it could be inhomogenous and anisotropic in those properties. That in itself was a new idea at the time. It isn't quite the same as saying gravity *is* inhomogenous space. In any case, the definitive statement of what GR is and isn't are the principle of general covariance (the mathematical formulation of the equivalence principle) and the Einstein field equation. There's nothing in the Einstein field equation that equates curvature to inhomogeneity. I don't think anyone is having trouble with that distinction. Personally I'd be careful about lecturing others on the role of mathematics in physics if I were you. Not too long ago you endorsed some work by Joy Christian that purportedly contradicted Bell's theorem. That paper really was "just math" in the worst possible sense (in one way of formulating Bell's theorem, it would require Grassmann variable-valued probabilities, which is meaningless). Just make sure you yourself can tell when math is and isn't meaningful before you take it upon yourself to lecture others on the subject.
No it isn't. Here's why: you have a square carpet of 16 square metres. How big is this carpet? One solution of √16 is 4. Another solution is -4. The latter however is a non-real solution. The solution might exist, but the underlying reality does not. You've misread it. He said 'empty space' rather than spacetime, and he was describing a gravitational field as inhomogeneous space. Ergo no gravitational field means space is homogeneous. The citation does support the point. See the paper I posted earlier which discusses the equivalence principle and curvature. In any case what really matters is what a gravitational field is: inhomogeneous space. Gravity isn't some magical mysterious action-at-a-distance "force". Even Newton knew that. And gravitons remain hypothetical. So we go back to Einstein and understand that an object falls down because of the space it's in. And when it doesn't, that space is homogeneous. Hence in a homogeneous universe, there is no gravity, and no spacetime curvature. It's that simple. You had trouble with it. See above, where you said: we'd have to learn to view space-time as being endowed with physical properties, and accept that it could be inhomogenous and anisotropic in those properties. It's space that's inhomogeneous, not space-time. See this paper for more. I'm making a simple point on Einstein and the gravitational field, and referring to scientific evidence and reality. If you want to dismiss this and refer to mathematics instead that's up to you.
Scientists. They propose theories and/or conduct experiments to test theories, and they demonstrate that the universe is flat. Either contribute usefully to this discussion, or go away.
He said what he said. He was talking about space, saying when it isn't homogeneous we call it a gravitational field. Ergo no inhomogeneity means no gravity means no space-time curvature. It really is that simple. And WMAP says the universe is flat. You're just flailing about to avoid the obvious. I know what homogeneous and isotropic mean, and I also understand the distinction between a mathematical space and real space. It isn't explicit proof. It's a mathematical proof which departs from GR and misunderstands spatial curvature. This is the key sentence: Fundamentally, the key concept behind the idea of de Sitter space is that it involves a variation on the spacetime of general relativity in which spacetime is itself slightly curved even in the absence of matter or energy. Not all of them. Just those who are taught the "modern interpretation" from MTW instead of studying the original and thinking for themselves. No. I've already said I'm familiar with it. You are still confusing space with space-time. There are no physical space-times. Space is physical, not space-time. Motion through space is physical too. We plot this in Minkowski spacetime, but it's a mathematical space, not a physical space. In similar vein worldlines and light cones are abstractions rather than physical. Then when space is inhomogeneous we observe curvilinear motion and talk of curved space-time. But when space is homogeneous, that curvature has simply gone. Again you're evading the simple point by throwing up mathematics. We all know what homogeneity means. Clutching at straws. No inhomogeneity means no gravity means no Reimann curvature. No, you're throwing everything and the kitchen sink at this to avoid conceding the simple point I'm making. The one originally voiced by Einstein and backed up by WMAP. Here it is again: homogeneous space means no gravity hence no spacetime curvature hence a flat universe. I can't make it any simpler.
Yes, he said what he said. Any other tautologies you want to throw out there? What he meant is something which obviously isn't entirely clear cut since you read it differently to others and given you have zero experience with GR and differential geometry I don't really think you're in the best position to interpret Einstein since you don't know any specific results which might counter your interpretation. So giving explicit examples with explainations is 'flailing' and your avoidance of any justification isn't? I don't think you do. I've asked you to go into details and you've failed to (just like when I ask for details of your own work). Do you even read and think about people's posts or do you just reply to certain words you can Google? You are in no position to tell people about spatial curvature since you don't know any differential geometry and have no hands on experience with GR. Why are you in a position to interpret Einstein's work correctly compared to anyone else? Besides, this whole "Well that doesn't occur exactly in the real world" is pointless, as any mathematical model is not going to be exactly right, there'll always be minor variations or deviations. As I said (and which you failed to respond to properly) you can't go down the road of "AdS is not exactly physical" because neither are homogeneity and isotropy. You made a claim about them, which means you're referring to things which are close to but not exactly like things seen in the real world. AdS space is just another example of such a concept, so you really have no grounds to ignore it given you made a claim about precisely the same sorts of things. You made a false claim and now you're making excuses, at least be man enough to admit it. That isn't a departure from GR at all, its an example of a solution to the governing equations of GR. Perhaps if you got your knowledge from things other than pop science books you'd see that its not too hard to have a working grasp of these concepts. If all you read are pop science books then you rely on someone else giving you their interpretation of the details, you have no access to the details themselves and thus you can't examine them for yourself. Some of us actually looked at the details and understand them to some degree. Not everyone has the same shortcomings in their knowledge you do, some of us give a damn about intellectual honesty. And yet you didn't think of it when you made your claim, illustrating how a pop science understanding leaves you wanting in terms of understanding. I managed just fine when it came to getting work on space-time published, which cannot be said for you Please Register or Log in to view the hidden image! Now we're wandering into territory Wikipedia would class as 'original research' (not that I'd call your work 'research'). I'd ask you to provide justification but given your 'research' has been turned down by numerous journals its far to say you don't have any viable justification to provide even if I asked. We? You mean people who do the details in relativity, people who can do tensor calculus and differential geometry, people who understand curvature. 'We' implies you're one of us, you aren't. And by the way, in case you missed it in a previous thread, I do have a job doing physics and the fact my PhD was in string theory was a bonus in regards to my application. Every single model ever is to some degree an abstraction from the real world, you're making excuses. Where did I say they were physical? Put down the straw. You have no experimental evidence of this and, as explained, since 'homogeneity' is itself only an abstract concept not perfectly existing in the real world if we consider abstractions like the differential geometry of valid solutions to the Einstein field equations then your claim is false. A metric can have curvature and translation invariance, they are not mutually exclusive. You talked about me flailing but why is it you can't provide anything other than assertions? I've told you what you need to provide, a demonstration a shift in position implies vanishing Riemann tensor but you avoided it. To demonstrate I'm able to back up my position (so I'm not asking you to do something I haven't done myself) how about a simpler example of a translation invariant metric which has non-zero curvature? AdS space is hyperbolic, which is a geometry less commonly examined than its opposite, dS space which is akin to spherical geometry. Spherical geometry is something you can draw lots of nice intuitive pictures for. A sphere is a space which has constant positive curvature, equal to the inverse of the radius of the sphere. Consider a 2d sphere in 3d space (like the surface of the Earth, idealised to being a perfect sphere) and draw a bunch of curves all over the surface, which you then measure the lengths of somehow (ie you compute \(s = \int_{C} \sqrt{g_{ab}\dot{x}^{a}\dot{x}^{b}} dl\) where \(\dot{x} = \frac{\partial x}{\partial l}\)). Pick a point on the sphere and notice that to move from that point to another (ie a translation) you can view as a rotation about the centre of the sphere. Since the sphere is rotationally symmetric obviously you don't change the lengths of the curves, no matter what curves you originally drew. Thus the space is homogeneous due to metric invariance. But it has non-zero curvature! A circle, a plain simple circle, is a counter example to your claim homogeneity implies vanishing curvature. 'Throwing up mathematics' isn't evading the point. I know you don't understand it and your 'work' lacks any mathematics and you know none of the details relevant to your claims but that doesn't negate my argument. Compare that to what you are 'throwing up', blind assertions on real phenomena you have no experience with and abstract concepts you don't understand. If you consider my argument weak then yours is dead. I'm clutching at straws by doing something other than blind assertion? Pray tell, how would you describe your 'argument'? You've provided neither theory or experiment. I like how you try to frame the fact I can provide multiple arguments and examples as somehow a bad thing. I've yet to see you throw anything to back up your assertion. You're again confusing the different types of curvature. Having the Ricci scalar R=0 doesn't imply the Riemann tensor is zero. The Schwarzchild black hole space-time has R=0 but not \(R^{a}_{bcd} = 0\). This example, well known to Einstein, demonstrates your interpreting his words incorrectly. The fact you don't understand what people say in reply to you doesn't mean they didn't understand what you said. If you could do the quantitative stuff we'd be able to do everything clearly and precisely, no 'flailing'. But because you haven't got any understanding of the quantitative stuff we're stuck having to explain things to you in terms you can understand and it results in a Catch 22. Anything precise but complex you refuse to accept on the grounds you don't understand it (though you'll claim its irrelevant to avoid admitting to that) and anything devoid of precision but down on your level you'll refuse to accept on the grounds of not being rigorous and justified properly. But then I suppose that's part of the reason cranks about the quantitative stuff, its easier than putting in effort to learn and understand and it allows you a way of ignoring anything which doesn't match your preconceptions/pet theory.
That's a false analogy. Solutions to the Einstein field equation that are not realised in nature are not analogous to carpets with negative length. They're analogous to hypothetical carpets with real, positive lengths that are different from the length of the carpet that happens to be in front of you. The fact you brought up the issue of experimental evidence completely undermines your analogy. Cosmologists couldn't just ignore homogenous but curved solutions by ruling them as non-physical, the same way you could ignore the possibility of a negative length carpet. They had to make measurements to determine which particular solution was being realised. Conversely, when was the last time someone measured the length of a carpet to see if it was positive or negative? So? You're splitting a hair that's completely tangential to the point I'm making: the definitive statement of what GR is and isn't is its mathematical formulation, and there is nothing in that mathematical formulation that relates curvature or gravitation to inhomogeneity of anything. This is an issue because when we talk about general relativity being an experimentally supported theory, we mean that the mathematical formulation you'll find in GR textbooks is experimentally supported. It's the equations, after all, that are used to do the calculations whose results are compared with observations. So anything anyone - even Einstein - says about gravity is only experimentally supported to the extent it has something to do with those equations. Do you realise your entire argument basically amounts to "gravity is inhomogenous space because Einstein said so"? Even if Einstein meant what you think he did, physics isn't a religion and even Einstein isn't Gospel. So even if Einstein ever believed "gravity is inhomogenous space" - which is doubtful because it contradicts the mathematical formulation of his theory - we're not required to go along with his original interpretation of anything, just because he's Einstein, anyway. Science doesn't work that way. No it doesn't. I've given you an alternative interpretation of that citation. The difference is that my interpretation is also consistent with the mathematical formulation of general relativity. Yours isn't. This isn't something you can argue based on Einstein quotes. You'd need to know the mathematical formulation of general relativity to make a judgement here. Why are you bringing this up? This is well known and taken for granted now. Why are you bringing up gravitons at all? I never said anything about gravitons. We're just discussing classical - not quantum - gravity here. No gravitons. What does that have to do with whether I understand the distinction between mathematical abstraction and reality? Would you also use this quote as evidence I can't cook my own dinner? Where did I say I was willing to consider "maths" instead of evidence? Math is great for abstractions and it's quantitative, which is great for experimental verifiability. It's not one vs. the other. Yes, math isn't reality. Yes, the map is not the territory. No, we don't need you to explain that to us. No, that doesn't give you an excuse for treating the math as an afterthought or outright ignoring it.
Yes they are. Closed timelike curves are a case in point. They "offer the possibility of time travel", which is even more nonsensical than a negative carpet. And now they've made the measurement, accept it. The universe is homogeneous on a large scale, and it's flat. So you can now kick all those notions of the universe having the topology of a chocolate teapot into the long grass. I'm not splitting hairs, this is absolutely crucial to the point I'm making. If a light beam is moving through space, then if that space is homogeneous, there isn't anything to make that light beam deviate from a straight line. If however the light beam traces a curvilinear path, we then talk of curved space-time and we say there's a gravitational field there. But what's really there? What caused that curvilinear motion? Ask Einstein: inhomogeneous space. Inhomogeneous space is what a gravitational field is. You're retreating behind mathematics to dismiss a simple point and to dismiss Einstein. No it doesn't. Because as you agree, we can forget about action-at-a-distance, and about gravitons, so what's left? Why does a light beam curve downwards? Answer: because the properties of the space it's moving through are not uniform in the vertical direction. This space isn't curved. If it was, thrown balls would follow the same arc regardless of speed. And you can't say "because the light is moving through curved space-time" because nothing moves through space-time. That curvature is curvilinear motion through space. Einstein gave the equations of motion. Through space. And that space has a gμv gradient in its properties from top to bottom. He said what he said, it's patently obvious that he said what he said, and that he meant what he said. Look in the mirror, pryzk. You're dimissing Einstein here, and my simple point, and glossing over the distinction between curved space and curved space-time. For what? Because you are utterly convinced that what I'm telling you contradicts the mathematical formulation? You're exhibiting religious conviction here. Really. Stop it, sit yourself down, and think it through for yourself. True. Science works through the scientific method. And we're not required to go along with the current interpretation either. Here's your interpretation: I read this as Einstein saying we'd have to learn to view space-time as being endowed with physical properties, and accept that it could be inhomogenous and anisotropic in those properties. That in itself was a new idea at the time. It isn't quite the same as saying gravity *is* inhomogenous space. In any case, the definitive statement of what GR is and isn't are the principle of general covariance (the mathematical formulation of the equivalence principle) and the Einstein field equation. There's nothing in the Einstein field equation that equates curvature to inhomogeneity. It's wrong. Because you don't understand the limitations of the equivalence principle as per the paper I linked to, and you don't know the difference between space and space-time. You thus do not appreciate that curved spacetime is synonymous with inhomogeneous space, as per this paper. There you go again. It's what they call the crystal spheres defense. Don't be facile. You don't know the difference between space and space-time, so you don't understand why inhomogeneous space results in the curvilinear motion that we label as curved spacetime. Glad to hear it. I'm not. I've said repeatedly that mathematics is a vital tool for physics. My point is a very simple one. It started with this post. Read back through the thread and you'll find that Alphanumeric threw up a smokescreen of math to try to deny the bleedin' obvious.
Then accept it instead of resorting to outraged abuse. There you go again resorting to your elitist defence that is no defence at all. And it's patently obvious I read peoples' posts. I've made no false claim, and no excuses either. It's a departure. Here's another quote from the wikipedia page: "A constant scalar curvature means a general relativity gravity-like bending of spacetime that has a curvature described by a single number that is the same everywhere in spacetime in the absence of matter or energy". In GR a concentration of energy results in what we call a gravitational field and curved spacetime. In AdS we have a gravity-like curvature in the absence of energy. Crystal spheres. You get your interpretation from a text book. And if you gave a damn about intellectual honesty, you would have confirmed that prometheus was wrong. Your text book mathematics leaves you wanting in terms of understanding the physics. LOL, you mean NS-NS superpotentials of Type IIA and Type IIB string theories and four dimensional gauged supergravities descending from non-geometric string compactifications? Try to stay on topic. You don't understand space-time curvature. You don't understand gravity either. Remember the last time you tried to explain it? It was a shambles. You didn't, but in post #86 you said "physical space-times". WMAP does. You have no experimental evidence of curvature in homogeneous space. We all know what homogeneity is. Stop banging on about it. So can a sphere. But that doesn't mean the universe has curvature. I don't need to provide that, I just need to provide references and simple logic. You need to counter it without resorting to spurious demands. You're lost in maths. A sphere is a sphere. A mathematical space isn't space. But OK, scale this up a dimension to a hypersphere and apply it to the universe, and you're inventing curvature in the large-scale homogeneous space of the universe where none exists. The universe isn't curved, it's flat. So your sphere is mathematical hyperbole without foundation. More's the pity that Einstein didn't have the courage of his convictions with respect to cosmology and entertained this idea. You have no argument. Just a pile of mathematical abstraction that bears no semblence to reality and no relevance to this discussion. A gravitational field is inhomogeneous space. Homoegeneous space means no gravity, and no spacetime curvature. Hence the universe is flat. Going all round the house talking spheres and fiction just doesn't cut it. I've proved references and facts and simple logic. You haven't countered them, all you've provided is abstraction. Not black holes again. Stop digressing, and stop pretending I've interpreted Einstein incorrectly. I haven't. I'm no crank. That's why I've got you nailed down, which is why you're resorting to abuse instead of addressing the argument. But of course, string-theory quacks do attempt to trash intelligent civil discussions and turn science forums into a physics-free zone.
So what is it exactly that makes homogenous but curved solutions nonsensical? Why did cosmologists bother with measurements at all if those solutions were a priori impossible? Yes, fine, no problem. The universe is homogenous and it's flat. That's what the evidence shows. But you're not just claiming that. You're going further and claiming that they're synonymous. The evidence doesn't show that. Correlation doesn't imply causation, especially when you only have one data point. Good thing I never said the universe had the topology of a chocolate teapot then. In future, could you restrict your attention to stuff I did say? Tautologically true. But he did not say "gravity is inhomogenous space" or "gravity/curvature of space(-time) are synonymous/equivalent" or any variant. He did not literally pronounce those sentences. That's exclusively your interpretation of what Einstein said. I don't think there's anything wrong with it. Yes I do. The fact I use one term instead of the other doesn't mean I don't know the difference. Substitute "space-time" with "space" in my quote above if you like, and then explain what's wrong with it. Space is physical, and it can even be inhomogenous. What, specifically, is wrong with that? I think I'll pass on those papers. I explained my own position, which to summarise here is: Your idea that gravity = inhomogenous space contradicts the mathematical formulation of GR. Generally speaking, the math is what's tested when we compare GR's predictions with observations. Consequently, the math takes precedence over anything anyone - even Einstein - says about gravity where there is a disagreement. I don't agree with your interpretation of what Einstein said anyway. We can construct homogenous but curved solutions to GR, demonstrating that, in GR, curvature and inhomogeneity are not synonymous. You have greatly exaggerated the significance of the one piece of evidence you have presented, as explained above and in earlier posts. These are the sort of response I'm seeing: These replies are inadequate and dismissive. To answer the last one: I'm not exhibiting any religious attitude (if you've forgotten, that would look more like [POST=1357580]this[/POST]). The first two points are based on the mathematical formulation of general relativity, which you have obviously never bothered to learn. You are therefore not able to fully appreciate my perspective. Of course you don't stand a chance of changing my opinion if you can't address the points it's based on! Making up excuses isn't helping your case either. Hence why I don't see any point in reading the papers you linked to: your attitude doesn't warrant it. If you're not willing to make an effort to understand my position and to address it properly, why should I waste my time on yours? If you want to keep telling yourself that I'm just getting caught up in mathematical abstractions or that I'm deliberately using math as an obfuscation technique, then be my guest. But in that case, you make rational debate impossible with you. It looks like I'm not the only person who has reached that assessment.
Come on Farsight, even you aren't so thick as to only read half a sentence before replying to it. Or are you that desperate to come up with a retort? The first reply in your post would seem otherwise, since I went on to explain that while he said those words the point he was making was not what you think it was. Instead you quote half the sentence and do the "So you agree with me then!" routine. No reasoned argument, logic or evidence either. Says who? AdS and similar negative Ricci scalar spaces are considered all the time in cosmology and by people doing relativity. Clearly people doing GR don't think AdS is a departure from it. This comes back to my point of what makes you someone whose in a good position to know what is or isn't part of GR? You assert things which you have no experience of, in contradiction of people who do have experience. Is AdS space a solution to the Einstein field equations? Yes. Then its relevant to general relativity. Is AdS space phenomenologically relevant? Yes. Then its of interest to cosmologists. To claim its a departure is just a denial of the facts. And we're back to another of my points, that you don't understand the difference between different kind of curvatures. The Riemann tensor is \(R^{a}_{bcd}\). It defines the Ricci tensor by \(R_{bd} = R^{a}_{bad}\). This defines the Ricci scalar by \(R = R^{a}_{a}\). Constant scalar curvature means R is constant. This doesn't mean \(R^{a}_{bcd}\) is constant. Examples of R being constant but \(R^{a}_{bcd}\) not being include the Schwarzchild metric, dS space-time, AdS space-time and most Einstein manifold, whose definition is \(R_{ab} = \lambda g_{ab}\) and thus \(R = 4\lambda\), of which specific examples include Calabi-Yaus. You're quote mining things you don't understand. No doubt you'll say something like "You're throwing the kitchen sink again" or "Stick with real things, not your string theory stuff" but that doesn't negate the fact you have made claims about GR which simply are not true. The bit I've just quoted of your post is a statement about GR in general, not specific physical systems. There exist in GR solutions to the field equations which have constant scalar curvature but non-constant Riemannian curvature. If you could do basic tensor calculus you'd be able to work it out for yourself, rather than mindlessly parroting other people's wordy explainations you don't understand. No, I get it from a working understanding of differential geometry and years of hands on experience doing precisely this kind of mathematical physics. At best you're simply wrong about my understanding and at best you're projecting your own short comings when it comes to understanding physics. Given your mindless parroting of even wordy explainations I'm inclined to lean towards the latter case. Though I'm not specifically pointing the finger at you I'm sure plenty of cranks do project in that regard. They can't understand the textbooks, all they can do is parrot it so they think everyone else must be too and then they convince themselves its okay to not understand it, no one else does. I hate to break it to you but some of us don't parrot it mindlessly, we understand it. No, that isn't a title of any paper I've had published but its quite close in topic. Come on Farsight, can't you even manage to Google my name properly? And that's a pretty flimsy attempt at an insult, pointing out I have an understanding of complicated stuff and it passed peer review. I could reply in kind and quote a title of a work you've had pass peer review but well.... there isn't one. How about you demonstrate the explicit step by step derivation of how no metric with Poincare invariance can have non-zero Riemannian curvature, which is what you're claiming (in fact you're claiming something stronger but I'll be kind). That's what the point of contention is. I've provided explicit examples and explanations and I've requested you provide something other than just assertions to back up your claim. Nothing yet. Its funny, you ask people to explain themselves when they disagree with you but you refuse to do likewise when you disagree with someone. A case of 'Do as I say, not as I do' hypocrisy. Again, other people in the research community who do seem to think I do well enough to get work passed peer review and various qualifications. Remember, you're not arguing with some personal view of mine, you're arguing against the entire GR and differential geometry community here because you're claiming things in direct contradiction to major areas of research in cosmology and GR. I know you think you're an expert in various areas of physics (you've explicitly stated electromagnetism in the past, do you think the same in regards to relativity/space-time?) but you're in a group of one in that regard. Until you can demonstrate you understand anything about relativity which can't just be got from 30 minutes on Wikipedia there's no reason to think your ability to evaluate your own or anyone else's physics capabilities is anything other than abysmal. But that was in response to the fact you suddenly tried to back peddle and restrict the issue to phenomenological space-times, not just valid solutions to the EFEs. Until you'd been given an explicit example of a valid GR space-time which contradicted your claims you were making entirely general claims. No, WMAP shows that the deviations from homogeneity occur around the 1 part per 100,000 level. Clearly the universe isn't perfectly homogeneous or it'd be filled with diffuse gas. Galaxies, stars, planets and people are deviations from exact homogeneity so to even discuss it you have to allow for a small amount of abstraction since it doesn't exist perfectly in the real world. No doubt you'll try to twist this so you can continue to try to avoid admitting a complete failure of understanding of GR on your part. So you know its an abstract concept but you're willing to talk about it, but the similarly abstract concept of AdS space you aren't? Picking and choosing to avoid admitting a mistake there FS. So you admit a sphere has curvature and rotational invariance? That's, as I explained, another counter example to your claim since embedded rotational invariance is the sphere version of homogeneity. Do you want me to draw you a picture? Seriously, I'll do it if needs be. Your only reference has been to a quote of Einstein which you interpret differently to everyone else and the interpretation you pick means Einstein is contradicting his own work. And its not like all I've resorted to is 'spurious demands', I've given references, examples and equations, typically of things covered in basic geometry courses and certainly in introductory GR courses. And yet you're unaware of them and even when shown them you don't understand. No, I just have a firm grasp of it allowing me to be precise with my reasoning and logic. You really need to come up with a new response, any time anyone throws in a single iota of algebra you whine about how they are just 'lost in maths' or detached from reality. Unlike you physicists realise they have to make precise statements in order to end up with clear cut predictions and models. Without the quantitative side there's no way to use the precession of Mercury's orbit to distinguish between Einstein and Newton. Crunch the algebra in what is little more than a homework question and you'll find Newton predicts the wrong amount, Einstein the right. The details are essential and all you do with your "Oh you're using maths, that's not physics!" excuses is show how little you understand how science works. It's close to flat and even if somehow you could determine that the mean scalar curvature is zero that doesn't mean the curvature is locally zero (since matter clumps together, breaking the uniformity) or that the mean Riemann tensor is zero. In fact you can measure the amount of mass in a system by considering totals of the Riemann curvature tensor, in the same way you can compute the charge of something using Gauss's law and the Maxwell tensor. I suggest you don't spend so much money on your next round of advertisements for your book and you put £100 or so towards a few good textbooks on introductory geometry and relativity. I'll be happy to give some suggestions. Other than examples, including many used by cosmologists and the relativity community. Compared to your...... your...... repeated assertion :shrug: I really think you need to precisely define what you regard as abstract and what is valid discussion. You reject spheres and AdS space yet you accept homogeneous space when the very fact you and I exist is evidence perfect homogeneous space doesn't. And none of that has any bearing on the fact you're making repeated statements about geometry. Can, in principle (ie for a particular distribution of matter and energy in the universe), homogeneous space have non-zero Riemannian curvature? Yes. Can in principle, homogeneous space has non-zero scalar Ricci curvature? Yes. Do any of these, including homogeneous space, exist in reality? No. Thus the entire argument is a hypothetical one and your claim is false. Which way do you want it, false on the grounds of homogeneity not existing in the real world or false on the grounds of hypothetical counter examples which satisfy all GR requirements exist? I'll let cosmologists know they haven't actually been doing anything for the last 50~100 years, Farsight has asserted the results and models they thought worked and tallied with observation actually didn't. Please Register or Log in to view the hidden image! Tell you what, why don't you write up your 'argument' and submit it to a cosmology or GR journal. I'll even help you make sure your typesetting is right so you've got no excuses, its judged on its 'merits'. Sound good? After all, if I know so little about relativity as you claim I do then it's only right I default to those whose business it is to do cosmology. Sorry, I know how giving yet another counter example to your claim just gets in the way of your self delusions. Let's see, you have done the following things : * Written a pet 'theory' claiming to explain large sections of physics which you've never studied. * Claimed to be more of an expert in electromagnetism than pretty much anyone in the world * Claims your work is worth a Nobel Prize * Spammed a great many forums with your pet theory * Made claims about how your work would be taught in schools in the not too distant future, as well as killing string theory * Failed to provide a single working model of any phenomenon in the real world, despite being asked many times over the space of years. * Don't know the different between a mathematical axiom and a physical postulate * Submitted to many journals and been rejected for publication by all reputable ones * Spent your own money on vanity publishing your work * Spent your own money taking adverts out in physics magazines for said vanity published book. * Made a television appearance on ... whatever the hell that show was, which covers other topics like alien abductions, 9/11 being an inside job and other bat shit crazy stuff. I would say that makes you a crank. I've made the same offer as I made to you before you submitted your 'Relativity+' to journals, I'll help you present your work to journals in such a way as to ensure it is evaluated only on its scientific merit, not irrelevant things like presentation and typesetting. I am happy to defer the discussion to professors in GR who review for reputable relativity journals. Would you care to take me up on that offer? I have absolutely zero worry that they'll agree with you. I know you want to convince yourself that people view you as a threat to their physics world view or are 'nailed down' by you but that isn't the case. You've not an average crank, they tend to just dissolve away after their initial "OMG my work is amazing. Where do I collect my Nobel Prize?" is slapped down while you've started pouring your own money into keeping your ego going. Please, if you have children don't waste money which you could use taking them on holiday or put towards university or help with a mortgage or any other good fatherly use of the money. Yes yes, your work will bury string theory, I'm wasting years of my life doing it and it'll not be of any use getting a job. Funny how none of that came to pass. It would seem I've spent my time doing more physically useful stuff than you Please Register or Log in to view the hidden image!
I only saw this after I've posted but its worth noting. If you scroll up to Ben's list of 'Farsight's greatest hits' it quickly becomes apparent just how long Farsight has been peddling the same stuff. That part of the thread is more than 3.5 years old and Farsight's talking about "You'll cringe when you find out I'm right!", the very humble "Because this is real physics, the best physics you'll ever see!" (which saves me having to go find links to back up my list of cranky things Farsight has done) and last but by no means least "nobody beats me at arm wrestling, and I’ve never lost a fight". Now if Farsight could only get onto the well known 'physicist arm wrestling tour' he'd have a Nobel Prize in no time..... And it also highlights another of Farsight's trends, which he's shown here too. Whenever someone points out an error of his he'll ask them to explain themselves. Nothing unreasonable about that but when they do and they can provide detailed examples/explanations and even, shock horror, some algebra he comes out with "Stay on topic". He's done that with my examples of AdS, dS, spheres and Calabi Yaus all being counter examples to his claims. And yet when someone turns around and says "Okay, let's see you provide evidence and details, not just arm waving" suddenly its okay to steer a little bit away from the topic. If the point of contention is "homogeneous is synonymous with zero curvature" surely nailing down the definitions and implications with unarguable logic, aka mathematics, is precisely on topic? Well not according to Far-lets move the goal posts when backed into a corner-sight.
Farsight, Thanks to your link I just read your first post in this thread. These are my thoughts on what you said. I'm sorry but this is pure speculation. Are you trying to say that I can travel to the center of Jupiter without any gravitational consequences? I think you are missing a few important points here. The universe is only inhomogeneous on localized scales. This is why we have atoms, dust, moons, planets, stars and galaxies. These warps in space-time tend to average out over the grand scale of the universe and this is the shape they're currently trying to measure. So far as they can tell, the shape of the universe is as close to being flat as can be measured (ex. WMAP). I can't see why anyone would ever recognize that a void exists at the center of all massive objects, that is until you can prove I won't be crushed in the pursuit of occupying that space. Contrary to your belief, space-time is more inhomogeneous at the center of a massive object than it is on its surface. Ultimately this means that a "flat universe" can have multiple warps and bends in it (on localized scales) and still be considered flat. This is old news and I think your quote from Einstein sums it up nicely. . P.S. Don't forget, Einstein once thought that the universe was static. This was, according to him, "his biggest blunder".
Acitnoids, this thread is nothing if not interesting, and I'm definitely out of my element in commenting, but when Farsight wrote I think what he meant (or maybe rather a more appropriate thought experiment) was to consider all of the mass of the planet to reside on the shell (i.e. Newton's sphere) -- that way all of space on the interior of the planet possesses zero gravitational gradient. This is where I get confused, though, because I would not call this area "flat", but rather uniformly curved. The lack of gradient implies "homogeneity", so this example actually exposes the difference between the two concepts I believe. Maybe this isn't right... Can someone clarify this? For example, if a space is globally subjected to a uniform gravitational field (setting aside the plausibility of this), is it considered to be curved?
Quite. I've read the rest of your post, and it's an outpouring of hubris and abuse that essentially says "I can't be wrong because that's what's in the textbooks". You have still utterly failed to address the issue. And no thanks re the offer.
