If you're reading what I say then why do you keep going off on tirades like the one quoted here that have nothing to do with what anyone is saying in this thread? You're mixing semantics with physics. Space-time curvature certainly has implications for geodesic worldlines. Whether or not you want some aspect of that "motion" doesn't change that. I have no misunderstanding. It's you who needs to listen to your own advice here: coordinate systems have no real existence and the metric components describe how stretched a given coordinate system is. The metric components and their gradients are coordinate-dependent. That's why you can't just make unqualified statements like "When light curves it's because there's a gμν gradient across the space it's moving through." Do you see the problem now? You are talking about a coordinate-dependent quantity as if it had "real existence". We keep warning you about this sort of thing. You don't seem to be able to tell the difference between your ideas and other people's theories. What you've said above is not the view GR adopts. Again, you can say that if you want. But that's not what general relativity says and it's not a basis for dismissing the possibility of curved but homogenous spaces in theories like GR that are different from yours. No, I said that when space is curved you were forced to adopt descriptions of trajectories that were not straight lines in a coordinate system. I didn't say anything stronger than that. In fact I have explicitly stated that whether a trajectory curves is not defined in a coordinate-independent manner and so is not a good measure of the presence of curvature in space-time. Prometheus said the same thing even earlier. You don't seem to have noticed that nobody else is approaching general relativity with the same single-minded focus on light that you are. No, I am citing a well known result of Riemannian geometry. What's Z[sub]0[/sub] and where do Friedmann, Lemaître, Robertson, or Walker apply your definition? No, WMAP, if we accept the validity of GR, is telling us which is being realised. It is not telling us that the other solutions are somehow non-physical. There's nothing physically unreasonable about them. And what would that change? I've already read your quote several times now. Einstein nowhere says anything as specific as "there is gravity only where space is inhomogenous". My reading the quote again is not going to make such a statement magically appear. This is misleading. What we now call the "coordinate speed" of light is coordinate dependent (as its name suggests) and can be numerically different from the constant c. The phrase "speed of light" nowadays specifically refers to the speed of light in an inertial coordinate system, which is always c in GR. And? Huygen's principle basically just says that we can treat an illuminated object as a collection of point-like sources. By the way, notice that when he talks about light curving: We easily recognize that the course of the light-rays must be bent with regard to the system of co-ordinates, if the \(g_{\mu\nu}\) are not constant.This is almost exactly what I said above. Note that I didn't exactly have to go scouring through Einstein's paper to mine out this quote. It just happened to directly precede the reference to Huygen's principle. He dedicates the whole of section 16 to what we now call the Einstein field equation.
Tirades? Nothing to do with what anyone is saying? I'm disappointed that you attempt to retrieve your situation by calling me names. I can only reiterate that curved spacetime is the effect rather than the cause. See my penultimate response below. But if you prefer, change what I said to a gradient in the properties of the space it's moving through. These aren't my ideas, I'm quoting Einstein at you, the view that GR adopts isn't Einstein's view. Re time, see A world without time for Einstein's view. It isn't my theory, and I don't dismiss possibilities, I take note of the evidence like that provided by WMAP. Noted. But please do note that Prometheus exhibited great confusion between curved space and curved spacetime. The whole reason we're having this discussion is because others don't share my approach, which is in line with Einstein's. He said time and time again that the speed of light varies, and yet Prometheus dismissed Einstein as an idiot, saying "we know better". It's still circular reasoning. Light doesn't curve because "spacetime is curved". And you have evaded the scientific evidence I raised, concerning the parallel-mirror light clock readings at different altitudes. We know about vacuum impedance and c, those clock readings are hard scientific evidence, the clock clocks up the motion of light, so when the clock goes slower the light goes slower, just like Einstein said. Vacuum impedance, and they don't. Come off it. This discussion is a spin-off from "Is the universe flat?" The answer is yes, WMAP provides the evidence, so those other solutions are non-physical. Einstein is describing a gravitational field as inhomogeneous space, that's quite enough for anybody. And nowadays you're out of line with Einstein. See the article here on the Baez website, and look at the General relativity section. See this: Since Einstein talks of velocity (a vector quantity: speed with direction) rather than speed alone, it is not clear that he meant the speed will change, but the reference to special relativity suggests that he did mean so. This interpretation is perfectly valid and makes good physical sense, but a more modern interpretation is that the speed of light is constant in general relativity. Then look at the last line: Finally, we come to the conclusion that the speed of light is not only observed to be constant; in the light of well tested theories of physics, it does not even make any sense to say that it varies. This article acknowledges that Einstein was talking about a variable speed of light, says it makes sense, and then says it doesn't make sense. It contradicts itself, the position it takes is untenable. That's the position you're taking. It's to do with refraction, but nevermind. And it's very similar to what I said above when I said "When light curves it's because there's a gμν gradient across the space it's moving through". And you castigated me for it. Yes, what we now call the Einstein field equations. The interpretation has shifted. Read the whole thing and note he's giving the equations of motion, not the equations of curved spacetime. NB: I'm moving house this Friday, so I won't be able to post much for a week or so.
People in glass houses.... As for the rest of your post it boils down to "Only I know what Einstein and the universe really mean!". Shame you can't provide any evidence or justification for any of that which doesn't also boil down to "Because I say so". I asked you to explain why you're the only one who understands Einstein, despite you having a below undergraduate level knowledge of any of his work. Hell, I don't think you could even get on an undergraduate physics degree at a university in the top 20. You want to think you've got some grand insight but your mistakes are immediately pointed out and you avoid challenges to justify your position. You're all talk with nothing to say.
AlphaNumeric, ok you are a mathematical genius, and I agree with the mathematics that you have presented. But, getting bogged down in this type of Matrix mathematics of stacking vectors, scalars, and other matrixs will move lot of people out of the room of conceptual understanding; which helps to develop what is really going on! I was wondering however, why you did not include the Spectral Theorem; In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. By the spectral theorem, since the moment of inertia tensor is real and symmetric, there exists a Cartesian or Spherical coordinate system in which it is diagonal, having the form \(M \sim \left( \begin{array}{cccc} M_{11} & 0 & 0 & 0 \\ 0 & M_{22} & 0 & 0 \\ 0 & 0 & M_{33} & 0 \\ 0 & 0 & 0 & M_{44} \end{array} \right) \) Alphanumeric, let's not discuss this type of mathematics, and continue in discussions about who has more math experience than who; in this posting. You have studied more math than I. In general, I don't like the way this type of mathematics is evolving over time. The math that you are pushing is what is keeping lot of independent researchers from getting a basic understanding of what General Relativity is all about. There is another approach to understanding GR by simplying some of the mathematics and working through the concepts that are presented by the subject, such as: Euclidean and Non-Euclidean Geometry, Curvature of spacetime, Spacetime Expansion, Vacuum Energy, Heat Radiation Gravitation, Isotropy, Anisotropy, Homogeneity, and Inhomogeneity of the gradient of spacetime, as well as the Thermodynamic considerations of the Ideal Gas Vacuum Energy of Spacetime; without the mathematics that will cause you to leap off of a building trying to understand. I was forced to learn math to do physics. You seemed to have gone the other way around! I never posted anywhere that the Schwarzschild Radius, or Event Horizon "Area" was a real singularity or a fake singularity. Schwarzschild Radius Vector \(r_{S} \mathbf{e}_{r} = \2 (\frac{m_{net}G} {c^2}) \mathbf{e}_{r} = (\frac{\frac{1}{2}\ m_{net}c^2} {\frac{1}{4} \frac{c^4}{G}}) \mathbf{e}_{r} = (\frac{T_{ab}} {\frac{1}{4} \frac{c^4}{G}})= \frac{G_{ab}}{2 \pi} \). Spacetime Metric for Non Expanding Spherical Net Inertial Mass Body at the Schwarzschild radius Event Horizon \(s^2 = \({r_{S}^2} + \ {r_{S}^2}\({a}^2 + \ b^2 \sin^2(a_0) \)\). Vector for Non Expanding Spherical Net Inertial Mass Body at the Schwarzschild radius Event Horizon \(s \mathbf{e}_{r} = \({r_{S}\mathbf{e}_{r}} + (\ {r_{S}{a})\mathbf{e}_{a}} + ({r_{S}{b}}) \sin(a_0) \mathbf{e}_{b} \)\). The Statement that the above equation represents maximum curvature and minimum space is a correct statement. Although you can enter the Event Horizon and exist at a radius that is less than the Schwarzschild Radius \( (d < r_{S} = \2 (\frac{m_{net}G} {c^2} \))\). The Schwarzschild Radius \( r_{S}\) describes how a specific quantity of matter and energy are distributed evenly throughout the radius, diameter, area, and volume of the spherical system; where this sphere is filled with mass-energy. The Stress Energy Tensor \({T_{ab}}\) is distance dependent and represents this mass energy and the Ideal Gas Elastic Expansion Energy of Spacetime due to the mass-energy \(m_{net}c^2\) in that local vicinity. And what this means specifically is that the the maximum curvature of spacetime is governed by the Schwarzschild radius which is directly proportional to the net inertial mass of the system. What makes this Stress Energy Tensor \({T_{ab}}\) a strange term is that it is an energy term which is a scalar in three dimensions of space; however this same Stress Energy Tensor \({T_{ab}}\) it is at the same time Schwarzschild Radius distance dependent; existing in three dimensions of space and one dimension of time. So Stress Energy Tensor is an energy term that is distance dependent; and that distance can be described by a vector, scalar or Tensor. The distance vector can be discussed because a Cosmological "Dark Force" vector is present; the two vectors cancel and the Energy term remains a scalar. Once again, this Schwarzschild Radius is not a singularity. This is a fundamental law for the amount of curvature caused by the mass, and energy in a local region of space, and the presence of the mass and energy in that space causes space and time to behave differently that if there were no matter present. In this scenario, one can choose an inertial or non-inertial coordinate system that is at the Mean Center of the system, the Center of Mass of the system, or the Schwarzschild Radius Event Horizon surface area of the system. Although you stated that the Einstein Field Equation is Reference Frame or Coordinate Frame independent and you state that Tensors solve this coordinate dependent problem, I do not completely agree. There is a relativistic mass and energy that must be dealt with, if this system is moving at relativistic speeds, if this system is at rest or moving slowly we have the current Einstein Field Equation; which in my opinion does not take into account this relativistic motion. You may disagree?? Once we start to speak of things happening inside and outside of the Schwarzschild Radius we have to use the following equations. And here is where things get bizarre in terms of singularities. And can be represented by the Schwarzschild Metric. \(s^2 = \(\frac{d^2} {1 - (\frac{r_{S}}{d})}\) + \ d^2\({a}^2 + \ b^2 \sin^2(a_0))\). The above Swarzschild Metric is given without the addition time and the Field Equation component, which I will add below. \({(s_{S})^2} = (1 - (\frac{r_{S}}{d})){\ (ct)^2} - s^2 = (1 - (\frac{r_{S}}{d})){\ (ct)^2} - \(\frac{d^2} {1 - (\frac{r_{S}}{d})}\) - \ d^2\({a}^2 + \ b^2 \sin^2(a_0))\). The above equations predict a singularity at \( (d = r_{S} = \2 (\frac{m_{net}G} {c^2} \))\). The above equations will also allow you to predict that strange things occur when \( (d < r_{S} = \2 (\frac{m_{net}G} {c^2} \)) \). I will agree with you here, there is something strange going on here at (r=0, not \( r_{S} = \2 (\frac{m_{net}G} {c^2} \) \)). This type of rationale leads to a vacuum energy, and a separate heat radiation energy that produces an energy that travels faster than light.
What I've been talking about is not some kind of super advanced stuff only a genius can grasp. The difference between vectors and matrices is something I learnt in high school. Basic indices I learnt in my 1st year of university. Every physics degree in the world covers indices, vectors, matrices, coordinates etc. This is the bread and butter of physics, not the genius level fringe. You demonstrate you don't understand it so saying "I agree with you" is just vacuous and an attempt to get me to back off. I'm not going to agree to that. You have written books on relativity, claiming you have a great understanding of it. Now you're showing you don't and you lack even a basic understanding. I'm not going to let that go and let you go back to just making up arm wavy nonsense. This stuff is the fundamental core of GR and you're horrific at it. Looks like someone has done some Googling! You don't know the difference between a vector and a matrix but you know about the spectral theorem? Oh please! Diagonalisation of matrices is stuff covered in high school too, this isn't ultra advanced. No, its keeping hacks like you from it because you're obviously unwilling (and I suspect unable) to put in any effort to grasp this stuff. You're getting high school algebra wrong! If you're unwilling to put in some effort to understand this stuff you have no business writing books on this stuff! I've spent the better part of a decade studying this stuff, including doing research on it, and this stuff doesn't keep people from GR, it is GR. Without formal structure GR is just words and arm waving, as any area of science would be. If you think I'm being overly mathematical then you demonstrate you haven't looked at anything other than pop science books, which is Farsight's problem too. Go to a library and look at a GR textbook and you'll realise just how naive you are. Without mathematics you have no way of logically and coherently using those concepts to build a formal structure which follows from the postulates and which can predict things. It's clear you just want to spout buzzwords and you're not willing to put in the time and effort to understand them (and I honestly believe you lack the brain power to grasp them anyway). Most of those things were conceived because of mathematics! It took one of the greatest mathematicians in history, Gauss, to conceive of non-Euclidean geometry. There is no sharp dividing line between maths and physics, there is a smooth transition between the two and even in the more abstract and formal corners of mathematics there are things with application to physics. Anyone who has studied theoretical physics knows this. Some of the most fundamentally important things in physics bear the name of people who were mathematicians by education and by job title. Maxwell equations, Navier-Stokes equations, Newtonian (anything), Gaussian (something), Euler (anything), Kelvin, Laplace (soemthing). Do you want me to run off a list of just some of the things which those mathematicians contributed to physics? In more recent times some of the people who have won the 'Maths Nobel Prize', the Fields Medal or the Wolf Prize have done so for contributions to theoretical physics. Witten, Penrose, Hitchin, Atiyah, Donaldson. Yes, if you're doing experiments and building new bits of technology you're not going to be doing as much maths as some other physicists but the blurring between what is maths and what is physics happens a lot more than you realise. You said it has maximum curvature. Still wrong. You're just ignoring corrections. You obviously don't care whether or not you are wrong, you just want to carry on spouting your crap. Besides, I thought you just said you wanted to move away from the mathematics? You can't have it both ways. It seems like you don't want to talk about mathematics to mathematicians but you still want to spout equations because it makes it look like you understand this stuff to those who don't. Also wrong. You've dropped the 'd' parts, which you can't do. I'll get onto that in a moment as you make it even worse further down in your post. As before, equating vectors and matrices. Wrong. \(T_{ab}\) in a SC black hole solution is not SC radius dependent. In fact its almost a vacuum configuration because it has only one contribution, a point mass at a single location. You clearly haven't understood (and likely not even looked at) how the SC metric is derived. Then you're wrong. General covariance is a provable property of the EFEs and anyone competent at the mathematics of GR could do it. You really need to look up the properties of tensors and the covariant derivative. Changing coordinates has nothing to do with changing the system. Relativistic motion is taken into account in the EFEs, they are Lorentz invariant. GR was designed to deal with relativistic energy and mass, something which Newtonian models have no concept of. You're demonstrating you don't understand even the conceptual basics of GR. The 'd' in the usual formulation of the SC metric are not numbers, they are differential operators. \(ds\) is not "s times by d" but an infinitesimal change in the value of s. The whole \(ds^{2} = g_{ab}dx^{a}dx^{b}\) formulation of GR is to say that "If you make small changes in your positions in space-time by \(dx^{a}\) then you'll sweep out a tiny line in space time of length \(ds\)". This is a generalised Pythagorian formula, \(c^{2} = a^{2}+b^{2}\). You can't just cancel the 'd', the only way to get rid of them is to integrate, \(s = \int ds = \int \sqrt{g_{ab}dx^{a}dx^{b}}\). Only once you've done the integration can you then square things back and get \(s^{2} = ....\). Your whole 'divide through by d' shows you don't even know basic calculus. A derivative like \(\frac{dy}{dx}\) refers to the small change in y for a small change in x. It doesn't mean you can cancel the 'd' and get \(\frac{dy}{dx} = \frac{y}{x}\) as an identity. This is, once again, high school stuff. You said you wanted to talk about non-Euclidean geometry, this is non-Euclidean geometry, Riemannian geometry is the generalisation of the concept. And now it turns out you haven't a bloody clue. Every step of the way you fall flat on your face! Given what I just said this is completely wrong. You should be damn ashamed of yourself for conning people into buying your books on a topic you not only don't grasp properly, you obviously don't grasp at all. Your understanding is worse than a high school students! You are a completely dishonest individual.
I don't have a physics degree and I have to agree with AlphaNumeric here. Some physics problems I do in my head, but only because I have mastered both the math and the specific application of it. Entry-level GR textbooks, suitable for 3rd-year physics undergraduates include: R. M. Wald's General Relativity http://www.amazon.com/General-Relativity-Robert-M-Wald/dp/0226870332 Misner, Thorne, and Wheeler's Gravitation http://www.amazon.com/Gravitation-Physics-Charles-W-Misner/dp/0716703440 (the following book seems to cover the fundamentals but I haven't reviewed it myself yet) Hobson, Efstathiou and Lasenby's General Relativity: An Introduction for Physicists http://www.amazon.com/General-Relativity-Introduction-Physicists-Hobson/dp/0521829518 // And same advice, expanded, for actual physics students: http://www2.warwick.ac.uk/fac/sci/physics/teach/module_home/px436/books/
Well, point out where anyone other than you talks about instantaneous action-at-a-distance or gravitons in classical GR. Otherwise yes, you are posting tirades that have nothing to do with what anyone said. You say: No, there's a difference. You left out the bit I bolded: "with regard to the system of coordinates", presumably because you didn't realise it was important. Einstein was well aware that light bending was coordinate-dependent. He didn't give a coordinate-independent (what you'd call "physically real") account of light bending. This is not supported by Einstein's paper. Which property? Be specific, and explain exactly where Einstein gives a quantitative account of it in his paper. Evidence tells you the way the universe happens to be configured. It does not tell you that it would be a contradiction in terms for it to be arranged some other way. You do not seem to appreciate the difference. Has the possibility even occurred to you that maybe we really do know better than Einstein? Or have you just ruled that out as impossible because we're not Einstein and we can't think for ourselves so there's no way we could have our own useful opinions? For heaven's sake, nobody is saying that. Or at least, no-one is saying anything that simplistic. In fact, we have repeatedly told you that light bending is not defined in a coordinate-independent manner in GR so you shouldn't rely on it as an indication of the presence of space-time curvature. You have already admitted that "evidence does not distinguish between interpretations". Are you taking that back now? Then why did you say you were using the word "homogenous" in the same way Friedmann, Lemaître, Robertson, and Walker were using it? No he's not. He said space was inhomogenous. Period. He does not say gravity is inhomogenous space. That he once uttered "space" and "homogenous" in the same sentence might be enough for you, but it obviously isn't enough for "anybody". Nobody else sees any need to interpret that quote by Einstein the same way you do. Did you even read the text in the middle where Baez explains himself? In GR you can get the coordinate speed of light to be any number you want it to be just by messing around with coordinate systems. That's the "speed of light" that Einstein said varied. But the speed of light in a locally inertial reference frame is always c in GR. Baez dedicated most of the text you cut out to explaining this distinction. He was not contradicting himself. If you actually follow your own advice and read Einstein's paper (but in detail this time) and compare it with a more modern textbook on GR you'll find this example is pretty characteristic of the difference between the way Einstein viewed GR and the way we view it now: Einstein put a bit more emphasis on some coordinate-dependent quantities, while nowadays we're a bit more interested in invariant quantities. That's all. The differences between the development in Einstein's paper and eg. the GR course I followed a few years ago are minor compared with the views you are describing. I look at Einstein's paper and overall I see very familiar material throughout. I cannot say the same about the way you have been describing GR. To me it looks like you've taken a few incidental remarks by Einstein and blown them completely out of proportion, while ignoring the meat and potatoes of his work. ... because physicists don't make a habit of naming things after themselves.
It's been a tough week or so playing handyman with the house move, but we're all settled in and I can allow myself the luxury of a little physics. Bliss. Now, where were we? No tirades here pryzk. Yes he did. See section 22 of the 1916 book Relativity: The Special and General Theory: "In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light)". It's clear enough. And if I may reiterate, Einstein is talking about speed because he’s repeatedly referring to the special relativity postulate, and his actual words were "die Ausbreitungs-geschwindigkeit des Lichtes mit dem Orte variiert" which translates into the speed of light varies with the locality. He's talking about a curvature of rays of light. A physically real curvature. Vacuum impedance, but Einstein doesn't give any account of it anywhere to my knowledge. Evidence doesn't. Appreciating what Einstein said does. Of course it's occurred to me. But you don't give a reasoned explanation to support your case. So explain why light curves. No. But you're still evading that scientific evidence. A light clock clocks up the motion of light, so when the clock goes slower the light goes slower. This evidence is staring you in the face. Because homogeneous means what it means. Others do. And it isn't going to go away. Yes. Baez isn't the author by the way. No it isn't. Because we define our second and our metre using the motion of light, then use them to measure the speed of light. The article contradicts itself. It says a variable speed of light makes perfect sense, and doesn't make sense. I beg to differ. If they were minor we wouldn't be having this discussion. Those are not a few incidental remarks to be airily dismissed. I would urge you to stop ignoring what Einstein actually said and think carefully about the differences between the points I've highlighted and what you've been taught. Granted.
I'm pointing out that you have a habit of giving replies that have nothing to do with the point you're supposed to be replying to. In a reply to me you denounced the idea of action-at-a-distance and gravitons, despite the fact I said nothing about either. Check back through the thread if you don't believe me. Then why did you present your own opinion as if it were Einstein's? When you talk about vacuum impedance, you've just admitted you are not talking about Einstein's theory. Science does not advance through blind adherence to what Einstein says. You keep falling back to this argument and we keep giving you the same reply: science is not a religion. Einstein is not God and you are not His prophet. No thinking person will be moved by that kind of appeal to authority. You are failing to have an impact because this is what all your arguments boil down to. We have, many times: the quantities and effects Einstein puts emphasis on in your quotes, such as the coordinate speed of light, light "bending", and "homogeneity" of the \(g_{\mu\nu}\), are coordinate-dependent. By your own standard, you would not consider any of these "physically real". Einstein was apparently confused by this at one point. The really silly thing in this argument is that you actually agree with the modern mainstream attitude, but you've misidentified which quantities are coordinate-dependent in GR and which aren't. It doesn't, at least not in the simplistic sense you are asking. In GR light travels along geodesics, which are the analogue of straight lines in curved spaces (such as space-time in GR). By your own admission, any evidence Einstein's interpretation of GR could account for, the modern interpretation could account for too. So your appeal to evidence in this case is affirming the consequent. Yet everyone, except you, takes one look at the sphere and says "yup, that's homogenous". Including Friedmann, Lemaitre, Robertson, and Walker. So despite what you say, you can't possibly be using the word "homogenous" in the same sense the rest of us are. Even more don't. <classroom>Oh - yes - it - is!</classroom> No, not because of that. The speed of light in an inertial frame is still invariant by any previous SI definition of the metre. It did not suddenly become true in 1983 when the most recent definition of the metre was introduced. The sole purpose of that definition was to give the speed of light in metres per second an exact integer value. It assumes the validity of invariance of c. We wouldn't be using it otherwise. No, that article explains two different points of view and explains why, nowadays, we'd go for the latter. There's no contradiction if you pay attention to the actual explanation. The difference I see between what's in Einstein's paper and what was in my GR course are minimal. I would have little, if anything, to debate with Einstein. The difference between what you are saying and what's in either a modern GR course or Einstein's paper are so large I don't recognise it as GR at all. They are, if you know GR in mathematical precision. You can't do or interpret the math of Einstein's theory, so those quotes are all you have to go on to guess what GR is about. But they're not all prometheus, AlphaNumeric, rpenner, Guest, and I have to go on. We don't need to guess what GR is about, we've all studied the details of the mathematical construction of the theory. When we read those quotes by Einstein, we're reading them in context. It makes a difference, you know. As indicated above, I have things to think about that you've never learned. I'd urge you to learn them. Just look at the evidence: everyone with a mathematical understanding of GR flatly disagrees with you without even blinking. Shouldn't you at least see that as maybe hinting you're missing something important in not learning it?
Farsight is displaying the typical crank tendency to denounce any mainstream physicist, except when it suits his purpose and then suddenly said physicist must be followed unquestionably. Part of Farsight's problem is that he doesn't understand the specifics, the details and the nuances of Einstein's work (or any other physicist's work for that matter) and thus can only cling to or reject concepts wholesale, he can't break them down into specific results and evaluate them one at a time. Farsight doesn't have any understanding of how coordinates are used in GR. This is clear from how he constantly misinterprets what Einstein said and what we say to him. Farsight, you constantly accuse us of thinking of certain things as 'physical' when you claim we shouldn't. What you haven't realised is that you haven't grasped the subtleties of what is and what isn't 'physical' in relativity. You have yet to explain/justify how you've got the true understanding of Einstein's work when you haven't got any working understanding of relativity, you certainly can't do the mathematics, you don't read any actual textbooks (written by Einstein or anyone else) and you couldn't pass a relativity exam if your life depended on it. How it is you have the one true understanding of Einstein's work when you're almost entirely insulated from it due to your own lack of competency at maths and physics? This is a bit like when you claimed to be more competent at electromagnetism than pretty much anyone on the planet. I asked you to demonstrate electromagnetism had Lorentz invariance and you couldn't. You couldn't do a homework problem aimed at undergraduates yet you claimed to be a world beating expert. You've just moved on from claiming to be knowledgeable in electromagnetism to being knowledgeable in relativity. In neither case have you demonstrated even knowledge expected of 1st year students. Feel free to prove me wrong, I'd love to see you try.
I actually find Farsight incredibly interesting. I don't think I've ever seen someone with such delusions of grandeur. My colleagues and I get an awful lot of e-mails from people who know very little but want to talk a lot, and I honestly think Farsight tops them all. I'm not sure if this means congratulations should be in order?
But it was no tirade. It was pointing out that when an object falls down it's because of the environment it's in. People typically call it curved spacetime, but Einstein didn't. He referred to inhomogeneous space instead. I didn't. You asked me, I told you. I made it clear that Einstein didn't talk about vacuum impedance. You're the one appealing to authority here pryzk, not me. You're using the authority of what you've been taught, which is the modern interpretation of GR, to dismiss Einstein along with scientific evidence of light clocks at different altitudes. They clock up the motion of light. When one clock goes slower than the other, it can only be because the light goes slower. That's it. Come off it. Light bends in a gravitational field. That's physically real. And read this Einstein quote: In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light). He's talking about a curvature of rays of light. That's real too. No I don't and no I haven't. And you're digressing. In the modern interpretation of GR, not in Einstein's GR. The distinction is the whole point of what I'm saying. Or should I say what Einstein was saying: "If we call the velocity of light at the origin of co-ordinates co, then the velocity of light c at a place with the gravitation potential Φ will be given by the relation c = c[sub]o[/sub](1 + Φ/c²)". "On the other hand I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of constant gravitational potential" "I arrived at the result that the velocity of light is not to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis". "the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned." And remember that c and geschwindigkeit. It's speed, not a vector-quantity velocity. The speed varies because the space is inhomogeneous and that's why light curves. Geddoutofit. You're accusing me of a logical fallacy to try to squirm out of facing up to what Einstein said and moreover the scientific evidence of those light clocks. They back up what he said. Light doesn't go slower because spacetime is curved. Light curves because space is inhomogeneous. It means what it means. And take a look at The FLRW metric. See where it says The FLRW metric starts with the assumption of homogeneity and isotropy of space. The invariance of c is invalid, and is not in accord with Einstein and the scientific evidence of those light clocks. Have you read what I've said about the definition of the second and the metre? We use the motion of light to define them. Think it through. You're in denial pryzk. The article says this: Einstein went on to discover a more general theory of relativity which explained gravity in terms of curved spacetime, and he talked about the speed of light changing in this new theory. In the 1920 book "Relativity: the special and general theory" he wrote: . . . according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [. . .] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Since Einstein talks of velocity (a vector quantity: speed with direction) rather than speed alone, it is not clear that he meant the speed will change, but the reference to special relativity suggests that he did mean so. This interpretation is perfectly valid and makes good physical sense, but a more modern interpretation is that the speed of light is constant in general relativity. It later says this: Finally, we come to the conclusion that the speed of light is not only observed to be constant; in the light of well tested theories of physics, it does not even make any sense to say that it varies. It contradicts itself all right. You refuse to see Einstein talking about the variable speed of light and inhomogeneous space. You're reading them out of context! And your argument is merey an argument from authority. You have no evidence and no reasoned argument. It's simply "we know best". What you and others have been taught is not evidence. Look at the evidence of those light clocks. You've been very evasive about them.
Why are you assuming I'm just going along with what I've been taught? Just because you can't do the math and actually prove results doesn't mean the rest of us are in the same boat. For example, here's how to prove that inhomogeneity of the \(g_{\mu\nu}\) is coordinate-dependent, just from what Einstein says in his paper: First of all, on page 163 Einstein calls \(g_{\mu\nu}\) the "fundamental tensor" and identifies it as a "covariant tensor of the second rank". On page 159, Einstein asserts that covariant tensors of the second rank transform according to equation 11, which implies that, under a coordinate transformation \(x^{\mu} \rightarrow x'^{\rho}\), the \(g_{\mu\nu}\) transform according to: \(g'_{\rho\sigma} \,=\, \frac{\partial x^{\mu}}{\partial x'^{\rho}} \frac{\partial x^{\nu}}{\partial x'^{\sigma}} g_{\mu\nu} \,.\)You can either take Einstein's word for this or derive this property from the definition \(\mathrm{d}s^{2} = g_{\mu\nu} \mathrm{d}x^{\mu} \mathrm{d}x^{\nu}\). In terms of the gradiants of \(g_{\mu\nu}\), the gradiants \(\partial'_{\alpha} g'_{\rho\sigma}\) are: \( \partial'_{\alpha} g'_{\rho\sigma} \,=\, \partial'_{\alpha} \Bigl( \frac{\partial x^{\mu}}{\partial x'^{\rho}} \frac{\partial x^{\nu}}{\partial x'^{\sigma}} \Bigr) g_{\mu\nu} \,+\, \frac{\partial x^{\mu}}{\partial x'^{\rho}} \frac{\partial x^{\nu}}{\partial x'^{\sigma}} \partial'_{\alpha} g_{\mu\nu} \,. \)Then using that \(\partial'_{\alpha} \equiv \frac{\partial}{\partial x'^{\alpha}} = \frac{\partial x^{\beta}}{\partial x'^{\alpha}} \partial_{\beta}\): \( \partial'_{\alpha} g'_{\rho\sigma} \,=\, g_{\mu\nu} \frac{\partial}{\partial x'^{\alpha}} \Bigl( \frac{\partial x^{\mu}}{\partial x'^{\rho}} \frac{\partial x^{\nu}}{\partial x'^{\sigma}} \Bigr) \,+\, \frac{\partial x^{\beta}}{\partial x'^{\alpha}} \frac{\partial x^{\mu}}{\partial x'^{\rho}} \frac{\partial x^{\nu}}{\partial x'^{\sigma}} \partial_{\beta} g_{\mu\nu} \,. \) If the gradients \(\partial_{\beta} g_{\mu\nu}\) in the unprimed coordinate system are all zero, then the gradients in the primed coordinate system are \( \partial'_{\alpha} g'_{\rho\sigma} \,=\, g_{\mu\nu} \frac{\partial}{\partial x'^{\alpha}} \Bigl( \frac{\partial x^{\mu}}{\partial x'^{\rho}} \frac{\partial x^{\nu}}{\partial x'^{\sigma}} \Bigr) \,, \)which is in general non-zero, unless the coordinate transformation is linear or affine. Conclusion: I can give the metric components a gradient just by performing a non-linear coordinate transformation. Consequently anything derived from the metric gradients (such as the bending of the path of light rays) is also coordinate-dependent. So no, I'm not saying the quantities Einstein is referring to are coordinate-dependent just because that's what I've been taught. I can work out for myself what happens to them if a coordinate transformation is applied, like I just did above. Try again. So what's your reasoning? Anything Einstein talks about must be coordinate-independent? :bugeye: No, in GR. Period. Einstein derives the equations of motion of a point particle in a gravitational field from the action principle in section 9 of the paper linked to above. The result, equation 22, is the geodesic equation: \( \frac{\mathrm{d}^{2}x_{\tau}}{\mathrm{d}s^{2}} \,+\, \{\mu\nu,\,\tau\} \frac{\mathrm{d}x_{\mu}}{\mathrm{d}s} \frac{\mathrm{d}x_{\nu}}{\mathrm{d}s} \,=\, 0 \,. \)If you've ever done even a little Riemannian geometry this equation is immediately familiar. The only difference is the notation used for the Christoffel symbols (\(\{\mu\nu,\,\rho\}\) is the same thing as \(\Gamma^{\rho}_{\mu\nu}\)). Are you still wondering why people read about Einstein's theory and come away with the impression he's doing Riemannian geometry? Quote-mining Einstein does not make the coordinate speed of light coordinate-independent. I've already shown (and not just asserted, actually shown) you above that the metric gradients are not coordinate-independent. Want me to do the same for the speed of light? No, I'm accussing you of committing a logical fallacy because you are committing one. Specifically, having an explanation for an observation does not automatically make it the only possible explanation. I don't see why I should even have to explain this, if you think evidence doesn't distinguish between interpretations of GR. What about it? I'm not denying that they start from the assumption of homogeneity and isotropy of space. I'm telling you that in one family of FLRW solutions, space is a 3-sphere. You can't have it both ways. If you agree with the wiki article you quoted, you are forced to agree that the sphere is a homogenous space. Yes. Did you read my reply? I don't know about you, but the last time I measured the length of something, I probably used either a ruler or a tape measure. I didn't use light sensors and expensive timing equipment. We would not define the SI metre as the distance light travels in a certain time if we didn't believe that definition of the metre would have all the same properties (length contraction, etc.) as a metre long ruler. Denial? Are you even trying to understand that article? You've merely repeated it again leaving the actual explanation cut out. Think about it. The idea of the speed of light varying had to make at least some sense from a certain point of view. Einstein wouldn't have thought twice about it otherwise. The person who wrote that article is, among other things, having the courtesy to sympathize with Einstein's point of view and *not* dismiss him as a complete idiot. Nope. See above. Unlike you I am not reduced to choosing whose word I want to take for anything. I can prove any assertions I make if necessary. I'm pretty sure that you, on the other hand, wouldn't even know where to begin at that. I've got nothing to evade. You've already admitted that evidence doesn't distinguish between interpretations. You are contradicting yourself if you try to claim the evidence of light clocks distinguishes between interpretations.
This is Farsight's problem. He doesn't understand the details, either due to lack of brain power or unwillingness to read said details (both IMO) so he assumes we don't. Thus whenever we point out he's mistaken on the details he assumes we're just parroting things we don't understand. The fact every single person whose ever done a GR course can do this stuff doesn't fit into his worldview. Ironically he then tells us to accept Einstein unquestionably. The fact we demonstrate working understanding of Einstein's work and he doesn't also doesn't fit into his worldview. For instance : Farsight, you cannot be so detached from reality to be unaware of the hypocrisy you're displaying there. You've endlessly said "Listen to Einstein, listen to Einstein!" and now you're whining that przyk is supposedly making an argument from authority? Perhaps this explains your seeming ability to function in the real world, you are practising the kind of cognitive dissonance a creationist would be jealous of. Certainly I don't see how you could hold a job down if you endlessly proclaimed yourself a world beating expert in electromagnetism and to have bested Einstein, you must be able to flick into 'sane mode' when you have to work or interact with people in the real world. You never did answer my question as to whether you proclaim at the breakfast table to your kids that you're smarter than Einstein, but I guess the Facebook page you had them make about your book answers that question for me....
I'm not assuming it, it's the conclusion I draw from your unwillingness to acknowledge what Einstein actually said. Yes he does, but a mathematical proof that inhomogeneity of \(g_{\mu\nu}\) is coordinate-dependent doesn't get to the heart of the matter. It isn't an issue. It isn't an issue because in extremis one light ray closely parallel to another doesn't "see" that the other light ray bends. But you still can't see the underlying reality. No. Electrons are "made of light" as demonstrated by pair production. Park the neutrinos for a moment, and free-neutron beta decay and low-energy proton-antiproton annihilation illustrate the same principle for other particles. So we're like that ray of light. Yes, and that's an equation of motion. Through space. It isn't an equation for curved spacetime. Try showing me where Einstein says it is. When you can't, you might at last appreciate the distinction between Einstein's interpretation of GR and the modern interpretation. Not at all. But I am still wondering why you can't see that it's the motion that's curved. Oh really? Take two atomic clocks, one above the other. You know that atomic clocks are light clocks? They employ the hyperfine transition and microwaves. Now get this: everybody, and I mean everybody, sees the lower clock running slower than the other. So please, don't try to hide behind some mathematical "proof" that airily dismisses what Einstein said as quote mining, and moreover dismisses the patent scientific evidence. Yep. Because I've shown you that we're made of the damn stuff. You're the one doing this, not me. Aaagh! WMAP tells us space is flat. So that family of solutions is wrong. Yes I did, and it's clear that the crucial point didn't sink in. We define the second using the motion of light. And the metre. That's why we always measure the local speed of light to be the same old 299792458 m/s. The ruler is made of matter. That's made of electrons and protons. And we can make electrons out of light in a lab. And we can turn protons and antiprotons into light in a lab. So your ruler is made of light too. I read it. And I understand it perfectly. The "explanation" is garbage. Here it is: If general relativity is correct, then the constancy of the speed of light in inertial frames is a tautology from the geometry of spacetime. The causal structure of the universe is determined by the geometry of "null vectors". Travelling at the speed c means following world-lines tangent to these null vectors. The use of c as a conversion between units of metres and seconds, as in the SI definition of the metre, is fully justified on theoretical grounds as well as practical terms, because c is not merely the speed of light, it is a fundamental feature of the geometry of spacetime. Einstein talked about a variable c and gave the equations of motion through space. He didn't say c was constant, and he didn't give the equations of curved spacetime. Again I refer you to http://arxiv.org/abs/physics/0204044. The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with. And this person, be it Carlip or Gibbs, ends up saying Finally, we come to the conclusion that the speed of light is not only observed to be constant; in the light of well tested theories of physics, it does not even make any sense to say that it varies. It varies, pryzk. Einstein was right, and those light clocks prove it. Clocks clock up motion, not "the passage of time". That's what clocks do. And when one light clock goes slower than the other, it's because the light goes slower. Just replace the clocks with tape reels to get it. Two identical tapes rolling in tamper-proof boxes, sychronised by you and handed to me. I bring them back a while later and one's wound 1000m of tape and the other one 999m. Not because the flow of time was reduced. Because one tape was rolling slower. That's the end of it. See above. No, I'm not. Instead you're still ducking this issue. Now come on, address it. I want your next post to address either those light clocks or those tape reels, and nothing else. Anything else is an evasion. Come back to the other points after you've demonstrated that you are not ducking this issue.
Argument from authority!! I guess the hypocrisy is lost on you. The issue is you have no idea what he wrote and you don't understand it so you must dismiss it. On top of attempting an argument from authority you're now doing an argument from personal incredulity. Says you. Evidence is lacking. Your 'work' on the subject was rejected. It's an equation through space-time, the indices cover all 4 space-time coordinates. It also includes Christoffell symbols and thus applies to both flat and curved space-times, in any valid coordinates. It is the geodesic equation through space-time, its literally homework for an undergrad to derive it. Once again you're arguing from a position of ignorance. You have no idea what is actually in GR, you can only spew out the tiny superficial pop science quotes you've managed to memories or find using Google. And yet every person whose sat a GR course, passed exams on it, has qualifications in it, has done research in it, has published work in it, lectures it, reviews it for journals, every single person where one or more of those apply, has rejected your claims, both about your own work and about GR. You're living in a fantasy land. You're the one who refuses to discuss anything with me, who runs away from a challenge about your work. When push comes to shove and someone steps up to challenge you you run off. You refused to take my monetary bet about your work being published, despite all your talk about it being Nobel Prize worthy. Now you're refusing to discuss anything with me, either in this thread or in a thread where you (or I) get suspended if you're deemed to have been dishonest or avoid direct relevant questions. The person constantly and perpetually ducking relevant issues is you.
The only thing you can conlude from that is that I don't just blindly follow Einstein or what you tell me to believe about Einstein. In any case, since I have just proved to you that I am perfectly capable of investigating for myself whether a quantity in general relativity is coordinate-dependent or not, I've proved your conclusion was wrong. What are you talking about? I've just proved, right in front of you, that the \(g_{\mu\nu}\) can have zero gradient in one coordinate system and non-zero gradient in another. That contradicts the idea that the \(g_{\mu\nu}\) gradients are phyical, coordinate-independent quantities. I prove that and all you can say is "this isn't an issue"? Then you agree with me. Light bending is coordinate-dependent. You've just described a situation in which we can eliminate the bending in the description of light: compare it to another nearby light ray. I wasn't attempting to say anything about the underlying reality. I've just told you that inhomogeneity of the metric components isn't a good candidate for that. No it doesn't. This is another converse error fallacy. Pair production does not require electrons to be "made of light". Proof: the standard model is perfectly able to predict pair production, and it doesn't need to assume electrons are made of light in order to accomplish this. Nope, it's an equation of motion in space-time. This is blindingly obvious if you actually follow the derivation of the equation (which, be honest, you haven't, have you?). Throughout Einstein's paper, in all the quantities which have a Greek letter index on them (such as \(x^{\mu}\)) the index runs over four dimensions, including time. The few exceptions are explicitly stated. In fact, not only is this an equation in space-time, but this is absolutely necessary in order for GR to be able to reproduce Newtonian gravity. The equation of motion of a particle in a Newtonian gravitational field is: \( \ddot{x}^{i} \,-\, g^{i} \,=\, 0\,, \)where each dot above a quantity represents a time derivative (so \(\ddot{x}^{i}\) is an acceleration), and \(g^{i}\) is a 3-vector representing the acceleration due to gravity (for example its norm is about 9.8 m/s[sup]2[/sup] near the Earth's surface). In Newtonian gravity the acceleration of a particle depends only on the gravitational field \(g^{i}\). On the other hand the geodesic equation in space is: \( \ddot{x}^{i} \,+\, \Gamma^{i}_{jk} \dot{x}^{j} \dot{x}^{k} \,=\, 0\,. \)Here the acceleration of a particle depends not only on the values of the \(\Gamma^{i}_{jk}\), but also on its velocity \(\dot{x}^{i}\). If matter obeyed this equation, a particle at rest would not accelerate due to a gravitational field, and it's acceleration would increase quadratically with velocity. This is not what Newtonian gravity predicts, so the geodesic equation just in space is incapable of reproducing Newtonian gravity. On the other hand, the geodesic equation in space-time is \( \ddot{x}^{\rho} \,+\, \Gamma^{\rho}_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} \,=\, 0 \,, \)where, this time, all the indices take on four values including time, and a dot (like in \(\dot{x}^{\rho}\)) indicates a proper time derivative. For non-relativistic velocities, \(\dot{t} \equiv \frac{\mathrm{d}t}{\mathrm{d}\tau} = \gamma \approx 1\), and \(\dot{x}^{0} \equiv c \dot{t} \approx c\). So if you can arrange things in such a way that the Christoffel symbols \(\Gamma^{i}_{00}\) are the only non-zero ones, then with this approximation the geodesic equation for the space-like components reduces to \( \ddot{x}^{i} \,+\, \Gamma^{i}_{00} c^{2} \,=\, 0 \,, \)and you recover the Newtonian equation of motion just with the identification \(\Gamma^{i}_{00} c^{2} = - g^{i}\). This is how you recover Newtonian gravity from the geodesic equation. You specifically need the time-like components in the geodesic equation for this to work. Einstein knew this. He gives his own derivation of this equation in section 21 entitled "Newton's Theory as a First Approximation" of his paper. The equation I posted above appears in a few different forms on pages 194 and 195. Notice how he keeps referring to components like \(\Gamma^{\tau}_{44}\), \([44,\,\tau]\), and \(g_{44}\). He does this because he's working in a four dimensional space. Throughout Einstein's paper, 4 designates the time-like component (it's more conventional to label it 0 nowadays). This is explicitly stated on page 151, and he explicitly states that he's working in space-time at the end of page 153 as well as frequently alluding to it in section 4. This is important since it's in these sections that Einstein establishes the conventions he uses throughout the rest of his paper. Also, in case you're wondering, the reason I have a factor of c[sup]2[/sup] and Einstein doesn't is because Einstein followed the convention of working in units where c = 1, as stated in a footnote on page 154. So you're completely wrong when you say the geodesic equation is just an equation in space. As I've just shown, if it were, GR wouldn't even be able to recover Newtonian gravity. I've already explained that to you: the concept is coordinate-dependent. No you haven't. You've used the same sort of argument you always do: you give the example of pair production and then choose to interpret it a certain way. That doesn't prove anything. And this is pretty irrelevant to the speed of light being coordinate dependent. But to answer the question: you get the formula for the speed of light that you copied earlier from the weak field approximation, where you set \( \mathrm{d}s^{2} \,=\, -\, c^{2}(1 + h) \mathrm{d}t^{2} \,+\, \mathrm{d}x^{2} \,+\, \mathrm{d}y^{2} \,+\, \mathrm{d}z^{2} \,, \)where \(h = 2\Phi/c^{2}\) and \(\Phi\) is the Newtonian potential (this is the metric you use to recover Newtonian gravity, by the way). Set \(\mathrm{d}s = 0\) and you find that the "speed" of light is \( \sqrt{\Bigl(\frac{\mathrm{d}x}{\mathrm{d}t}\Bigr)^{2} \,+\, \Bigl(\frac{\mathrm{d}y}{\mathrm{d}t}\Bigr)^{2} \,+\, \Bigl(\frac{\mathrm{d}z}{\mathrm{d}t}\Bigr)^{2}} \,=\, c \sqrt{1 \,+\, h} \,=\, c \sqrt{1 \,+\, 2\Phi/c^{2}} \,\approx\, c ( 1 \,+\, \Phi/c^{2} ) \,, \)which is the equation you copied in one of your posts above. (See? Not only do I know this result, I also know where it comes from.) Now pick a certain point \(\bar{x}_{0}\) in space, and let \(h_{0} = 2 \Phi_{0} / c\) be the value of \(h\) at that point. Then do the coordinate substitution \(t \mapsto \sqrt{1 + h_{0}} t\), and the speed of light in this new coordinate, at the location \(\bar{x}_{0}\), is c again. Another way of seeing the same result is that the Newtonian potential is only defined up to an overall constant, so you can pick any point in space as your origin and choose to set \(\Phi = 0\) there by convention, which according to the above formula makes the speed of light c there. This is equivalent to what I did above since, as a first approximation, the rescaling \(t \rightarrow \sqrt{1 + h_{0}} t\) is mathematically equivalent to resetting \(\Phi \rightarrow \Phi - \Phi_{0}\). The fact that the speed of light is always c in a locally inertial coordinate system, by the way, is trivially true by the fact that a locally inertial coordinate system is defined as one in which \( \mathrm{d}s^{2} \,=\, -\, c^{2} \mathrm{d}t^{2} \,+\, \mathrm{d}x^{2} \,+\, \mathrm{d}y^{2} \,+\, \mathrm{d}z^{2} \;, \)locally, and in which all metric derivatives are zero at a point. What are you talking about? You are presenting an explanation for something and then committing the fallacy of acting like that rules out any other possible explanation. This is a textbook example of completely broken logic. \(P \rightarrow Q\) does not imply \(Q \rightarrow P\). It's you, not me, who is resorting to such arguments. Practically every time you claim some evidence "proves" some explanation you give, you are committing this fallacy. No, they're not being realized. I can't believe you still don't understand the difference. And this doesn't in any way address the point I raised: Friedmann, Lemaitre, Robertson, and Walker started with the assumption of homogeneity of space and ended up with the sphere, without flinching. How do you explain that if Friedmann, Lemaitre, Robertson, and Walker didn't consider the sphere homogenous? I just want to understand how you can believe you agree with these guys when they obviously considered the sphere homogenous and you don't. You don't seem able to give a straight answer to this. No, it doesn't explain anything. Michelson and Morley did not use those definitions of the metre and the second. At the time the standard metre was defined as the length of a bar of platinum kept in France. They still got an invariant c. So the invariance of c doesn't depend on us defining our units in any particular way. So point to a mathematical model that 1) reproduces all the predictions of the standard model and 2) models fermions as made of light. Without such a model, don't make such grand claims. By the way you're making a similar fallacy to the one Robert Close makes, which I explained [POST=2698800]here[/POST]. Measuring invariance of c does not require everything to be made of waves obeying the massless wave equation. It is sufficient if matter is governed by physics whose mathematical expression is Lorentz invariant. The standard model is a case in point of this. The whole thing, and not just the electrodynamic sector, is Lorentz invariant. Since we're made of stuff governed by the standard model, this alone predicts that we will measure an invariant c just using rods and clocks, no matter the details of their internal composition. Well it looks fine to me. So I can only conclude you didn't understand it. Do you know what a null vector is? I've just shown above that I can locally make the speed of light anything I want it to be just by rescaling the time coordinate. In fact I don't know why you pointed out that c is invariant by definition according to the modern definiton, because that's also detrimental to your position: if we can make the speed of light invariant by definition, we're obviously not forced to accept that it varies. How many times do I need to explain that, because you yourself said "evidence doesn't distinguish between interpretations", I have nothing to duck. I'm not going to address anything until you present a coherent case. I originally confronted you with the fact that all the recent evidence for GR tested the modern interpretation. You wormed your way out of that with "evidence doesn't distinguish between interpretations". So either explicitly take that back and properly address the point I made, or quit complaining when I throw your own excuse back at you. All I'm asking is for you to make your mind up what your position is and stick with it. I'm not going to subscribe to a double standard. Can evidence distinguish between interpretations of GR or not? Pick one and then try to act like you actually believe what you said. Anything else is evasion. Come back when you show you're not ducking the fact you're trying to set a double standard.
Apologies for being away for so long. Oh boy. I said I want your next post to address either those light clocks or those tape reels, and nothing else. Anything else is an evasion. Come back to the other points after you've demonstrated that you are not ducking this issue.. And you're still ducking the issue. You think this massive mathematical smokescreen gets you off the hook? Geddoutofit. You've proved nothing. Absolutely nothing at all. Look: You haven't elminated it at all. You're measuring that curvature with something curved, so you don't see it any more. Come on pryzk! Forget what the standard model proves or predicts. Pair production is a real experiment, where we really do make an electron and a positron from light. Annihilation is another real experiment, where the electron and positron disappear, and the result is light. That's it. I will reiterate: there is no motion through space-time. You cannot move through the block universe. And again, the motion is through space, not through space-time. I said address either those light clocks or those tape reels, and nothing else. Anything else is an evasion. Now address the issue. Noted.
And so on... Farsight: simply saying it doesn't make it true. When I first read your posts on this forum I wanted to give you the benefit of the doubt because you seemed keen. However, it seems you're after nothing more than to perpetuate your own delusions of grandeur. Your personality has a dangerous mix of arrogance and naivety. This coupled with the fact you seem oblivious to so much of the subject at hand (general relativity), it's nigh-on impossible for people to converse with you in a meaningful way. If you think you know better than everyone, why on Earth do you keep coming here? What do you gain from this, other than shattering your reputation and helping indicate that the author of the self-published book "RELATIVITY+" is as arrogant as he is unqualified?
I'm not sure if they have forgotten or have never learned the basis of mathematics. They present a law as a theory but they try to demonstrate as if it would be a theorem. For them it is scurry between the notions of law, theory and theorem.
