It's a point which has been made before but I'll reiterate it since it relates to something else which has been said. A geodesic path can be found by solving the geodesic equation \(\ddot{x}^{a} + \Gamma^{a}_{bc}\dot{x}^{b}\dot{x}^{c} = 0\) where \(\Gamma^{a}_{bc}\) you construct from the metric. Thus if you know the metric you can work out its geodesics. Your claim is that if the energy in GR is smoothed out to be uniform then the only geodesics for light (ie null geodesics) are straight lines. Let's consider this.... The Einstein Field Equations quoted in this thread are \(R_{ab} - \frac{1}{2}R g_{ab} = 8 \pi T_{ab}\) but this isn't the most general, which is actually \(R_{ab} - \frac{1}{2}R g_{ab} + \Lambda g_{ab} = 8 \pi T_{ab}\). \(\Lambda\) is the cosmological constant and is, by definition (due to being a constant of integration in the derivation of the equation), a constant in space and time, ie uniform. The simplest uniform matter distribution is nothing, ie \(T_{ab} = 0\). Thus you have \(R_{ab} - \frac{1}{2}R g_{ab} + \Lambda g_{ab} = 0\). Now let's derive some stuff about such solutions... Taking the trace in 4D we get \(R - 2R + \Lambda = 0\), so \(R = \Lambda\). Thus if you put a homogeneous constant energy into space and nothing else you end up with non-zero Ricci curvature! Since \(R = g^{ab}R_{ab} = g^{ab}R^{c}_{acb}\) the Riemann curvature must be non-zero too, if \(\Lambda\) is non-zero. Depending on the sign of \(\Lambda\) you get either negative curvature (AdS space) or positive curvature (dS space). In each case null geodesics will be curved. In the case of dS space the positive curvature means two geodesics can intersect twice, just like great circles on a sphere will intersect twice. In the case of AdS space they might never intersect, even when they are not parallel (which is a non-trivial thing to define in curved space). Properties of geodesics, at least locally, with regards to intersecting one another, are covered in introductory textbooks on differential geometry and GR. I happen to be reading the mechanics textbook by Arnold at present and he covers it too. Farsight, I can do into quantitative detail if you wish. The mathematics I've given in this post are the tiniest tiny tip of a huge iceberg, all of which disagrees with your take on what Einstein said. Mathematics, general relativity and Einstein's own work disagrees with you and just because you're profoundly ignorant of all that work doesn't mean it doesn't exist. przyk, Prom, myself and others have shown you the start, if you had any intellectual honesty you'd at least wind your neck in a bit until you got around to reading up on this stuff. I'm sure we'd all like it if you could engage in a 'discussion' where you did more than just assert things, instead showing some understanding. There's no conspiracy of silence against your work other than your own avoidance of criticism. I really should start counting the number of relevant, direct posts of mine you've ignored, that'd be a quantification of how much you avoid honest discussion.
They talk about a profile. There's no one value of refractive index, it's graded, something like this analogy. Light bends more closer to the planet. No. Don't be specious. Of course I have. Come off it. You're attempting to put up complexity as a reason to avoid considering the simple case. I beg to differ. Black holes are static. From the POV of observers in the universe at large time dilation at the event horizon goes infinite, so nothing moves. That's where Einstein's variable speed of light takes you. Black holes don't spin, and a quantum fluctuation takes all of eternity so Hawking radiation never happens. You're in for a shock when you read about mass ratios. But meanwhile have a read of http://www.scienceagogo.com/news/renormalization.shtml If you removed the word "over" I'd be happy with that. He's talking about a gravitational field here, saying that where a gravitational field is, space is inhomogeneous. As ever, he doesn't mention curved spacetime. Yes, here's what I actually said in entirety. Pay attention to the last line. You're in italics: I read this as Einstein saying we'd have to learn to view space as being endowed with physical properties, and accept that it could be inhomogenous and anisotropic in those properties. No problem. That in itself was a new idea at the time. It wasn't totally new. Einstein is talking about an "aether" here. It isn't quite the same as saying gravity *is* inhomogenous space. But it is the same as saying a that a region of space where we find a gravitational field is a region of inhomogeneous 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. This isn't your interpretation of what Einstein was saying, this is your given reason for disregarding it. I don't. I use scientific evidence as evidence, like the Shapiro delay and the GPS clock adjustment, and what parallel-mirror light clocks tell you. Light goes slower where gravitational potential is low. The evidence for that is right in front of your face. The Einstein quotes support my position and show I'm not just some "my theory" guy. No. Einstein didn't get everything right. But you dismiss the things he did get right in favour of "the modern interpretation", and you dismiss patent scientific evidence. And the only justification you have is the argument from the authority of your textbook teaching and consensus. I don't dismiss anything that's backed by scientific evidence. But the WMAP evidence indicates that the universe is flat, and it sounds to me as if you dismiss this. You didn't, prometheus did, it's his thread. Go with the flow. When space is flat light travels in straight lines. Thus the universe is not a hypersphere like Einstein once suggested. Volume change profile and local volume change. If the energy is homogeneous, light travels in straight lines. So where's the curvature? Is it upwards, downward, leftwards or rightwards in this homogeneous isotropic space? Or none of the above? This is the crux of our discussion, pryzk. I think we should focus on it. This is getting ridiculous pryzk. Go ask what Einstein meant by the curvilinear motion of light. I've given my views on KS coordinates previously. You don't get it to bend. You moved. Your motion does not alter other things in the universe, merely your observations. Drop it. I've already answered this. The quantitative evidence is for Einstein's theory of general relativity, and Einstein said a gravitational field is inhomogeneous space. See http://arxiv.org/abs/gr-qc/0510072 and note that evidence does not distinguish between interpretations. Pot, kettle. You're turning a blind eye to Einstien and scientific evidence, and to The FLRW metric starts with the assumption of homogeneity and isotropy of space. And might I add that your posts are now so huge, and so laden with distraction and trivia that it's crystal clear you're attempting to bury the discussion because you know you've lost the argument. My advice is this: stop digging.
You display no sincerity, and no honesty, you will not correct prometheus on his blatant errors, and all your ever attempt to do is stifle discussion. I've long since given up on you. But on this instance I will respond to your post. The metric is not space, or motion through it. It's an artefact of measurement. Do not treat it as the fabric of reality. Let's not rely on Lambda or "by definition", because this was the area where Einstein had some difficulty. He added it to maintain a static universe, and we now know that the universe is expanding. You might claim that this expansion is the source of curvature, but the moot point remains that at any time, the space within that universe is flat. The flaw to this is that if there is no energy, there is no space. This is the wrong concept. The Ricci tensor describes volume change. Space and energy are the same thing, when you put energy in you put space in. Until you do this you have no volume. Start with no space, inject space, and you have a ball of space. If this is internally homogeneous on a large scale there is no overall gravity within this space. It's Euclidean. It's flat, as borne out by WMAP. And it didn't suffer a gravitational collapse when it was very dense, and isn't suffering one now. So if it isn't infinite, (and as far as we know there are no infinities in nature) then it has some form of boundary. Beyond which there is no space. As to how one tackles curvature near this boundary is debateable. You're conflating a mathematical space with actual space. You know, the dark bits between the stars that sustains waves and fields that Einstein said wasn't empty. So let's take a look at de Sitter space and anti-de Sitter space. Wiki will do: In the language of general relativity, de Sitter space is the maximally symmetric, vacuum solution of Einstein's field equations with a positive (repulsive) cosmological constant Λ (corresponding to a positive vacuum energy density and negative pressure). When n = 4 (3 space dimensions plus time), it is a cosmological model for the physical universe; see de Sitter universe. In the language of general relativity, anti de Sitter space is a maximally symmetric, vacuum solution of Einstein's field equation with a negative (attractive) cosmological constant Λ (corresponding to a negative vacuum energy density and positive pressure). There is no such thing as negative energy, just as there is no such thing as negative mass or negative length, or negative volume, so anti de Sitter "space" is off the table. De Sitter "space" attempts to describe the evolution of the universe, fair enough. But there is a problem. Energy has the dimensionality of pressure x volume. It isn't called stress-energy for nothing. Not only is the energy positive, but the volume is positive, and the pressure is positive too. That's why space expands. Geodesics don't have properties. Things that move have properties, as does the space they move through. And as ever, yours is an erudite argument from authority with so much intellectual arrogance that you feel you can totally disregard what Einstein said, the scientific evidence, and the salient points of this discussion. I give the scientific evidence, the Einstein quotes, the robust logic, and the honest discussion. As ever you give abuse and dismissal. Thanks but no thanks.
I have previously stated, and will do so again, that if I do not follow through with my offer then you can request, with my consent, that I be suspended for a month. Actually, make it 3 months. If I don't follow through with my offer this post hereby gives you my permission to ask for a suspension of my account, with my consent, for said period of time. How much more do you want? Like where? Repeated offers to enter into a discussion on topics you bring up are attempts to stiffle discussion? All I ask is a few reasonable ground rules, is that too much to ask? Is requesting people I interact with have the honesty to answer direct relevant questions being unreasonable? I didn't. You seem to have trouble understanding how mathematics is used by physicists. No one thinks that a metric is a physical thing but it describes physical things. 'Hot' is not a physical thing but it describes physical things. Concepts are, by their very nature, not physical things but they can be associated to physical things. 'Happy' is not a physical thing but we can attribute the property of happiness to someone. You haven't ever grasped the interaction between maths and physics and hence you misunderstand what those of us who have say to you. Unfortunately you blame others for your short coming. So suddenly Einstein isn't the gospel truth and you're not going to follow him? Trying to have your cake and eat it. Multiple times now you've had it explained to you that the dividing line between 'space' and 'time' in space-time is not unambigious, it depends on your description, ie your choice of coordinates. Even in special relativity this occurs, where a Lorentz boost alters the dividing line. An observer moving inertially will say "I am not moving in space" but someone moving relative to him will say "Yes, you are!". Since both of them see light move in the same way (ie always at light speed) their views are equally valid, yet they disagree on how each of them move in space and time. Now we've moved beyond relativity into your own pet theory. While its true that space-time can contain energy (such as in the form of gravitational waves) by virtue of its curvature that doesn't mean they are synonymous. Your statement means that \(T_{ab} = 0\) is impossible in relativity. All vacuum solutions, including the special relativity Minkowski case, are counter examples to your claim. Now it is you who is disagreeing with Einstein. Hopefully the fact you can disagree with him while not throwing out all his work will make you realise the mistaken statements you've been saying to Prom. Unless you're blind to your own hypocrisy and self contradiction. Either prove that this is the case within general relativity or admit this is just your own pet take on things. You won't be able to do the former, since you don't know anything formal in GR and thus lack the necessary skills, tools and understanding to formalise such an argument mathematically, as well as the fact its not true. Thus we're left with you needing to admit its just your uninformed take on things. Can you admit to that much? And how does one 'inject space'? With a syringe? No, you just fail to grasp how terminology in relativity is used, the formalism underneath and the vast gulf between your understanding and what relativity actually involves. I know you struggle (if not completely fail) to understand the abstract concept of 'a space' and thus dismiss it and assume others struggle too but that isn't the case for some of us. The fact what I and others say about space, space-time and 'a space' doesn't gel with your understanding is not because we don't understand them but because you don't. Now if all Prom, myself, przyk etc had ever done on the matter is the same as you, just read pop science books and have a below A level understanding of mathematics, then this would come down to a "I say... you say...." stalemate but that isn't how it is. Unlike you our understanding has been put to the test many times over the years, in exams, in coursework, in projects, in seminars, in conferences, in journals and (at least in my case) in vivas. And we, unlike you, have met those challenges and passed. Hence if we don't understand these things it means the people who evaluated our understandings don't. And likewise the people who evaluated them. The end limit of this line of logic is that pretty much the core of the GR community 'don't understand this' while you, somehow, have managed to understand it. Is this a fair evaluation of your views or not? If so, how is it you've got the right understanding and no one since Einstein has? If I'm mistaken then explain who does have the understanding and explain how it hasn't transferred to people like Prom and myself, despite us having (or had) personal interactions on a daily basis with professors in this stuff. Can you provide something more than just assertions? You canonise Einstein then you disagree with him. You complain mainstream people don't open their minds then you complain when they consider something you disagree with. You sure do want to have your cake and eat it! If you could provide a bit more of a rigorous argument it'd be worth publishing. But I guess you already tried that.... Energy is called stress energy? So does that mean energy is call stress-(stress-energy)? I suspect you're thinking of the stress-energy tensor. It's called that because its a tensor defined in regards to several physical things, which in their nice simplistic cases relate to energy and stress. It's sometimes called the energy-momentum tensor, since matter can have energy and momentum. Or the stress-energy-momentum tensor. 'Stress' would be included if you were looking at the tensor in regards to a fluid or deformable material, as in fluid mechanics. I would also point out that \(\Lambda\) is not the only way to introduce negative energy into a system. There are various conditions on \(T_{ab}\xi^{a}\xi^{b}\) used in general relativity, known as the energy conditions (like weak and strong), as certain results depend upon the assumption of how 'nice' the matter is. A much simpler way of considering negative energy is the Kerr-Newman black hole and the Penrose process. The GR extension of \(-m^{2} = -E^{2} + \mathbf{p}^{2} = \eta_{ab}v^{a}v^{b}\) is \(-m^{2} = g_{ab}v^{a}v^{b}\). The \(g_{00}v^{0}v^{0}\) then relates, on some level, to the energy terms. For a Schwarzchild black hole \(g_{00}\) is negative above the event horizon and positive below it. The sign flip isn't an issue because nothing can get back up out of the black hole. However, for the K-N case the sign flip happens on the surface of the ergosphere, not the event horizon, changing the metric signature from (-+++) to (+-++). This sign flip means its possible to steal energy from the black hole's rotation by firing 'negative energy' bullets. The Penrose process is the method of using this to extract as much energy from a spinning black hole as possible, all thanks to the sign flip on the energy related term of the metric. You're attempting to use everyday thermodynamical processes to describe something which isn't like a gas. Can you formalise any of this within general relativity to show your claims necessarily follow from general relativity? If not then you have no grounds for said claims other than you wanting it to be true. Its human nature (thanks to evolution) to hope new things behave like old things, its how we deal with new situations but that sense evolved while dealing with a very very small corner of the universe. Its naive to think everything in the universe can be framed in terms of everyday experiences and concepts. It's surprising how often I have to say that to cranks, cranks who keep telling me not to close my mind.... Geodesics do have properties, as they are well defined things, even if they aren't physical. We can most certainly talk about the trajectories of objects, either under a force or moving without propulsion. That's how we design rockets to put satellites into orbit, we know the paths they'll move along if we accelerate them in particular ways. That's part of the power of physics, it doesn't matter if the satellite in question has a mass of 10kg or 10,000kg, it'll move along the same trajectory if given the same initial velocity from the same initial position. Thus we can talk about orbits, defining different orbits by initial position and velocity. A geodesic is the generalisation of that, not just restricted to orbital mechanics but anything which moves without propulsion. You're showing, again, you lack of grasp of how physicists go about doing physics and the general ethos they employ. Sorry, we employ. I've given book references, lengthy explanations, equations and offers to discuss things further. That isn't an argument from authority, that's an argument from understanding. An argument from authority is "This person knows a lot therefore he is right", which is precisely what you're using with Prom in regards to Einstein. Citing works by others isn't an argument from authority if I'm citing their work not just their name. Given you throw out so many of them you should really learn about fallacious arguments. You're the guy who claimed to be a world leading expert in electromagnetism, a killer of string theory and whose work is/was worth more than one Nobel Prize, all based on nothing more than your view of yourself. As such I hardly think you're in a position to be calling anyone else intellectually arrogant. Besides, compared to you I think its pretty valid to consider myself a more competent, informed, well read and accomplished scientist. As I mentioned before, this isn't as case of "I say.... you say....", other people have evaluated my capabilities and work and I've met plenty of standards in science. You've been evaluated by fewer scientists but each and every one of them has found you lacking. By any measure of scientific achievement, knowledge or ability I am better than you and while it might be seen as arrogant to say it in such blunt terms it is, unlike your views about yourself, supported by evidence. How many times are you going to say this to people in this thread? You've been corrected on it by at least 3 (inc myself) people now. No one is 'totally disregarding' what Einstein said, they are disagreeing with your interpretation it because your interpretation doesn't square with what others have said, what Einstein said elsewhere, what general relativity says and what our own understanding leads us to conclude. Unlike you we don't rely entirely on layperson translations and quote mining, we actually take the time to develop working understandings of the details and thus can see single quotes within a larger context (even when the quotes themselves might be taken out of context), something you have a lot of difficulty with. Continuing with this "You're totally disregarding Einstein!" line does you no favours other than to appear like you're all out of viable arguments (as if you had one in the first place). Such as? Such as? You give your interpretation of said quotes. You dismiss the mathematics, you can't get much more robust logic than that. I'm pretty sure if we put it to a vote on this forum the outcome would be you're considered dishonest. I'd imagine similar outcomes occurring on other forums you've plastered your crap. I give lengthy explanations of your mistakes too but you use the fact I don't sugar coat my responses to you as an excuse to ignore them. For instance, you ignored my earlier post about how the 'standards' you mention cannot be scientific standards as you failed to meet any of them. You ignored my list of reasons you're a crank, after you claimed you aren't, because you couldn't retort it. The fact you'll then repeat mistakes/lies I've corrected you on shows you either have no wish to learn from your mistakes or that you do learn but you keep telling lies if you think it'll further your goals. For instance, your attempts to insult Ben and myself for doing 'worthless' string theory PhDs, despite the fact Ben is now in a very well paid job and I'm still doing physics. I guess being honest and accepting that doesn't fit into your rose tinted view of the world.
I hate to say it, Farsight, but each of your posts not only fail to persuade me that your position is credible, they make me go so far as to doubt your own ability to conduct rational thought. Your behaviour in this thread is not only unscientific, but arrogant and naive. If I want to be totally honest, many of your posts are downright embarrassing.
Good. Neither do I. So why did you say it was enough to consider only static gravitational fields, given we live in a dynamic universe? No, I've already conceded that the analogy works in the simple case. I'm telling you that a special case does not prove a generality. Implying it does is a logical fallacy. I'm not denying that Schwarzschild black holes are static. I'm telling you that not everything is a Schwarzschild black hole. No, he isn't saying that. He's just saying space is inhomogenous. He doesn't even use the word "gravity" in that quote, let alone say anything as strong as "where there's gravity, space is necessarily inhomogenous". I honestly have no idea how you managed to torture that interpretation out of Einstein. Yup, to break it down, That was my interpretation and your response to it, exactly as I just said. As you say yourself, this isn't part of my interpretation of Einstein. I said as much [POST=2670673]here[/POST]: So my point still stands. Your reply to my interpretation of Einstein was "No problem". That you took issue with some other stuff I said doesn't change that. No, I've reminded you that the space being flat does not imply that space-time is flat in general relativity. I also explained that "correlation does not imply causation" - i.e. showing me one example of a universe that happens to be flat and homogenous does not necessarily imply that this is mandated by GR. If it were, you wouldn't even need evidence. Homogenous but curved solutions to GR shouldn't even exist, and you wouldn't need WMAP to rule them out. Citation? There's a relation between the Ricci tensor and scalar and volume deformations compared with flat spaces. But both tensors are curvature tensors. They're both defined as contractions of the Riemann tensor. Incidentally, since you brought up The Foundation of the General Theory of Relativity (here), I'd be interested in seeing you explain what you think Einstein is talking about in section 12 of his paper. The answer is "none of the above". First of all, the point of curvature in GR isn't that it makes light bend, because as I've already explained that's tricky if not impossible to define in a coordinate-independent manner in GR. The point of curvature in GR is that it implies the impossibility of mapping space-time with globally inertial coordinate systems. But if you want to talk about light: light always travels along (light-like) geodesics in GR. Light rays follow what appear to be straight lines locally, at least as described in locally inertial coordinate systems, but globally you'll get behaviour not associated with flat space, eg. you could emit light in two different directions that cross paths later. For a specific example, consider the case where space has spherical topology. The sphere is homogenous (there are no priviledged points), and locally on scales smaller than the radius of curvature, light just follows straight trajectories. But if you emit two light flashes in any directions from some point on the sphere, they'll separate but meet again at the opposite side. I can work out the details if you really want them, but unless you want to argue that the sphere isn't homogenous, you've got an example of a homogenous but curved space in which the curvature has a (non-locally) measurable effect on light. Your intuition that light appearing to follow a curved path automatically breaks homogeneity is wrong. No you haven't. You've told me that an unrelated quantity is described by ten functions and you've entered your own little digression about what you think the metric really is and isn't, which is not what I asked you for. You haven't explained why Einstein uses a ten component metric given that only metrics associated with four dimensional spaces have that number of components. Yes. And derives solutions to GR from those assumptions, most of which are curved in space-time. That contradicts your position. How many times do we need to tell you this? If you think my posts are longer than what you'd like to reply to, then here are what I consider the most important questions I'd like to see decent answers to from you: If homogeneity of space is equivalent to curvature in space-time in GR, why does GR allow solutions such as FRW that are both curved and homogenous? On what basis are you dismissing them? Why does Einstein refer to "space" being described by ten \(g_{\mu\nu}\)s, and not some other number? How do you reconcile this with the fact that only metrics associated with four dimensional spaces have ten components? What do you think section 12 of The Foundation of the General Theory of Relativity is about? Why do you keep presenting the same quote by Einstein as proving your position when even you have admitted there is "no problem" with the way I interpret it? Why do you present this paper as evidence for your position when the authors only considered a special case? EDIT: Since AlphaNumeric brought up the cosmological constant, here's another one: if Einstein believed gravity only existed in the presence of inhomogenous space, why did he need to introduce the cosmological constant in order to predict a static universe? What, you think you can read my mind now? Well since you aren't correct, maybe you should consider the possibility you don't understand my psychology or perspectives as well as you think you do. By the way, these statements contradict one another: How can I be attempting to both bury and perpetuate the discussion at the same time?
As przyk points out, you can't use that argument because you have repeatedly refused to take me on. You can't use the "You're being disingenuous" argument because if I don't stick to my word I've stated I'm to be suspended. I believe the reason you refuse to take me on is that one of the ground rules I've requested is that direct relevant questions are to be answered honestly and if the person is deemed to have dodged the question or being dishonest then said suspension is handed out. I suspect you know you can't meet such a requirement, you are unable to demonstrate any real working knowledge, your work can't stand up to scrutiny and your 'arguments' are heavily dishonest. You know this and thus you avoid 'discussions' you can't shirk away from when you realise you've put your foot in it. Come on Farsight, how about we have a thread just you and me* comparing the relative 'merits' of our own areas of work, you with 'Relativity Plus' and me with string theory. We can compare formulations, derivations, applications, predictions, models and future potential. Or we could be a bit more specific and discuss specific things to do with your work, like the difference between a mathematical axiom and a physical postulate. Or curvature in geometry and relativity. Or perhaps you'd like to talk about my thesis area, seeing as you've previously made comments about it (as attempts to insult me) . The 'is the universe flat' thread showed you consider things like deformations of tori, curvature of tori, the FRW metric, Ricci flat spaces. All of those things arose in my PhD so I'd be happy to discuss them, with you or anyone else, its really quite interesting. How many times are you going to ignore a genuine offer to discuss your work in an insult free environment before you realise the hypocrisy of your "You're all too afraid to take me on!" comments? Put up or shut up. * In the formal debate forum the set up is the main debate thread is only for the debaters and there's a comments thread for everyone else. I'd be happy to have such a set up here in the physics forum, assuming its okay with the mods.
Your comments noted pryzk. I'm going to prune our discussion down a little. Then read a few words more: 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) has, I think, finally disposed of the view that space is physically empty. Note gravitation and the gμν. I'm not torturing this at all, not a bit. There's an important issue here. Solutions are said to "exist" even when they are non-real solutions. The simplest example of this is the negative carpet. We have a square room with an area of sixteen square metres, and there are two solutions to √16, four and minus four. We rule out the latter because we know that a carpet cannot measure -4m by -4m. No such carpet exists. When it comes to GR, Einstein described a gravitational field as inhomogeneous space, such that when light passes through a region with a left-to-right gμν gradient it veers left, the light path indicating curved spacetime. You dispute this, and contradict Einstein in order to assert the reality of homogeneous curved solutions which I say are a contradiction in terms. That's why I rule them out, even before WMAP vindicated my position. We can determine our speed through the universe via the CMBR dipole anisotropy, we can map space via SDSS, we can gauge the expansion of the universe over time via standard candles, and WMAP is telling us that on a large scale space is flat. So we actually have those things that you say are impossible. How can I get this across? It's like you're saying "forget Einstein, it's impossible to say what size carpet we need because there's more than one solution". My reasoning isn't wrong, yours is. The sphere might be homogeneous, but the space within that sphere is not. Light curves, just as it does in the inhomogeneous space that is a gravitational field. And as I've said previously, it's important to distinguish between curved space and curved spacetime. I rule out this solution because I'm confident that the electromagnetic field is curved space, and we do not observe electron drift. How many times do I have to tell you that the scientific evidence provided by WMAP contradicts most of those solutions? Who would you rather put your trust in, what you've been taught, or me, Einstein, Newton, and the scientific evidence. Re Newton, see Opticks queries 20 and 21: "Doth not this aethereal medium in passing out of water, glass, crystal, and other compact and dense bodies in empty spaces, grow denser and denser by degrees, and by that means refract the rays of light not in a point, but by bending them gradually in curve lines? ...Is not this medium much rarer within the dense bodies of the Sun, stars, planets and comets, than in the empty celestial space between them? And in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great bodies towards one another, and of their parts towards the bodies; every body endeavouring to go from the denser parts of the medium towards the rarer?" He got the density back to front, but like I said, nobody's perfect. Understanding of both curved spacetime and curved space vindicated by WMAP and the fact that the universe expanded rather than collapsed. In short: gravity is a pressure gradient within space, and the universe is a ball with an innate pressure, largely uniform, that causes it to expand because there is no space around it. As I said previously, he's giving a fluid solution. Space isn't empty, it isn't nothing. It sustains waves and fields, and matter is made of such. If there's no gradient in gμν space-time curvature vanishes. I also think it's about the equivalence of electromagnetic four-potential and gμν but that Einstein missed a trick there. One for another day perhaps. That's misrepresentation and you know it. I only said "no problem" to your initial sentence as per the bolded section of this post. You don't give an interpretation of what Einstein said, but instead give reasons for disregarding it. Because this case is quite sufficient to demonstrate Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime and is in line with Einstein, Newton, and the hard scientific evidence. Because he modelled the universe like a dust cloud that collapses to form a star, and needed something to prevent the collapse.
Now that's a thought. But I think one thread will be confusing, it has to cover so many different topics. Perhaps we should discuss say mass in the context of relativity+ v string theory.
So where does he say a gravitational field only exists where space is inhomogenous? That's not in your quote. No, we define length as a positive quantity. You don't have this issue if you state the problem correctly: the length of a square carpet is the positive square root of its area. I already gave you an example of a curved homogenous space. Do you think the sphere is a "contradiction in terms"? On that note: What, exactly, is the distinction? Note that if you do manage to make a distinction, that just reduces your entire argument to semantics. You wouldn't be arguing against the mainstream, you'd just be defining homogeneity differently than everyone else does. For example, when we say the FRW solutions are homogenous in space, we mean that they're homogenous in the same sense that the sphere is homogenous. In one family of FRW solutions, the spatial section is a 3-sphere. So if you've got some non-standard idea of homogeneity, then the resolution to your problem with the FRW solutions is simply that they're not homogenous by your definition, and not that they're somehow self-contradictory or unphysical. So? Evidence doesn't tell you what general relativity is or isn't. General relativity had already been formulated nearly a century before WMAP was put into orbit. Evidence just tells you whether you can keep general relativity or you have to throw it away. GR specifically predicts that space would have curvature if any of the matter density, radiation density, or cosmological constant were different than they happen to be. Evidence doesn't change what predictions GR makes. I don't agree with your interpretation of Einstein (or Newton, for that matter), I have no reason to trust you, and you said this: about the evidence. So what's left? No he's not. Here's what your own source said: Einstein was not giving an exact solution of the Einstein field equation. He was clearly talking about the metric. So, again: why did Einstein speak of ten \(g_{\mu\nu}\)s when the metric only has that number of components in a four dimensional space? You're close. He's referring to a well known result in Riemannian geometry, which states that when the Riemann curvature tensor is zero, there exists a coordinate system in which \(g_{\mu\nu}\) has no gradient (ie. is constant). If you can do that, you can also pick coordinates in which the metric takes the form of the Minkowski metric (\(g_{\mu\nu} = \eta_{\mu\nu}\)). In other words, when the Riemann curvature tensor is zero, you can map the whole of space-time with an inertial coordinate system. What? That initial sentence was the whole of my interpretation of Einstein. You replied "No problem" to it. The next sentence, "That in itself was a new idea at the time", is not something I was attributing to Einstein. Sorry, but over-generalising is a logical fallacy, as I've previously explained. Why didn't Einstein just model the universe as a homogenous fluid? Wouldn't that have solved his problem without him having to introduce an extra parameter into his theory?
That seems unfair. For one, we would want to hear you trace your claims to the axioms of the respective theories which is liable to traipse through other topics that you here want to place off-limits by stipulation. That negates the purpose of deriving the claims from the axioms and at the end, we are left with a bunch of sterile assertions, unconnected to hypothesis or observation. That does not make for a rational discussion worthy of a thread. For another, narrowing the discussion is unscientific, as a purported general theory which explains one tiny phenomenon and ignores other phenomenon which are better modeled with precision by a different general theory is white elephant, not a better general theory. A concrete example is given by MOND enthusiasts who claim their model is superior to dark matter in explaining galactic rotation curves, but don't begin to address the superiority of dark matter in explaining cosmology and gravitational lensing and the Bullet Cluster. At a minimum, the wider phenomenon renders the question of "MOND or dark matter" as a false dilemma. Thirdly, for making the discussion specific, there are various topics buried in your proposed stipulation of limiting the topic to mass in the various theories. Here are some of the more obvious: Is the inertial mass of a fundamental particle fundamental ? If not, why do all electrons (experimentally unresolved as composite) and all protons (experimentally resolved as composite) appear to have the same masses? Why are there three observed generations of leptons and quarks, which mainly differ from each other in inertial mass? How does your theory predict the spectrum of massless particles currently observed? What is the mechanism or mechanisms for giving observed particles inertial mass? How do those calculations work in detail? What are the precise inertial masses of the observed particles (and any unobserved particles your theory requires) ? How many handles are there on your theory to tweak these results? Why are the unobserved particles required by your theory with inertial masses below or close to accelerator limits currently unevidenced? What is the relationship between inertial mass and gravitational mass in the Newtonian limit? How does that relationship fare with experiment? Are the fundamental particles point-like and massive? If so, why are they not black holes as described in General Relativity? None of these look off-limits to me. So curiously, by attempting to limit the topic, you just open a web of long-standing questions. Naturally, this list looks formidable without even straying into other topics like checks on exact and approximate calculation techniques and how cosmology and LHC results compare with the theories or specialized topics like how theories overlap in predictions or calculation techniques. Or how well the theories recapture Newtonian mechanics, quantum theory, General Relativity, QED or QCD in the appropriate limits and boundary conditions.
Perhaps we could narrow it down to just the parts of your work and string theory for which working models of physical phenomena have been derived, as you often complain that string theory makes no testable predictions and/or is unfalsifiable. Can you elaborate a little more on what you mean by 'discussing mass'? Remember, whatever we do agree to discuss one of the first relevant questions I'll ask you for anything you make a claim about is to show you can reach such a conclusion from a set of initial postulates through clear unambiguous logic/reasoning. If your planned 'discussion' about mass is just going to be throwing out wordy arm waving then we should talk about something else. After all, wordy arm waving doesn't provide justification for anything so you'd be backing yourself into a corner immediately if that's all you have. After all, I want to discuss the specifics of your claims/work, not give you a platform to monologue stuff we're already heard from you. Personally I think it would be more appropriate for the issue of what can be addressed by your work to be considered first. After all, if you can't demonstrate you can accurately describe a system there's absolutely no reason to think you can give more philosophical view points on it with any justification. First you would need to show your work indeed leads to a description of a system before you could claim it then explains it, as well as describes it. Are you willing to do such a demonstration? Pick a single real world phenomenon which you believe your work is able to describe and explain. Clearly state your starting premises, then give the step by step argument which leads from those postulates to description of your chosen system and finally show your description is an accurate reflection of reality. Once that's done then you'd have justification for then taking your analysis further and saying your work explains the phenomenon in question on a level deeper than just raw dynamical description. You've had plenty of opportunities to stand on a soap box and give us your take on superficial aspects of physics, I am not (and I imagine neither are others) terribly interested in you doing more of the same. I want to discuss the specifics of how you get from your assumptions to your conclusions. Its those I've repeatedly tried to engage you in discussion on. Much of the criticism you level at string theory I level at your work so I'm happy to provide detailed explanations of how string theory retorts said criticisms. I've repeatedly said string theory is superior to your work in every single way and I'd like to discuss that with you. The devil is in the details and, despite my repeated requests, you've thus far not provided ones relevant to your claims. The point of my offer/challenge is to get you to discuss your work. If you're unwilling to answer direct questions on it and only want to restrict it to superficial arm waving on 'mass' then you're unwilling to meet my terms (which I've previously clearly stated). The point of scientific discussion and peer (not that you're my peer in science) review is that you're asked questions you might not want to be asked. If you're unwilling to answer relevant direct questions then you're not willing to do science. I'll answer, to the best of my ability, any relevant question relating to string theory, my own academic work or my views on science in general. I don't plan to ask you 50 different things at once, I plan to ask you about specific things until you've satisfied my interest on the matter. Hence saying "Let's limit it to mass, else there's too much to talk about" is unnecessary. You can ask a spread of questions if you want, I have some specific ones in mind. Besides, if you're as on the ball as you want people to think (world expert in electromagnetism, remember?) and you're so sure my time doing a PhD resulted in nothing more than a worthless black mark on my CV then surely answering my questions will be like shooting fish in a barrel?
Einstein Field Equation Explained The Most General version of the Einstein Field Equations is given by \(R_{ab} - \frac{1}{2}R g_{ab} + \Lambda g_{ab} = 2 \pi (\frac{T_{ab}} {\frac{1}{4} \frac{c^4}{G}}) = G_{ab}\). In the Super Principia Mathematica, Robert Louis Kemp describes the Einstein Field Equation as a metric that describes how localized matter (mass) and energy affects space and time in any general localized vicinity. The Equation on the right describes the Source of Curvature \(G_{ab} = 2 \pi (\frac{T_{ab}} {\frac{1}{4} \frac{c^4}{G}})\) for any Isolated Net Inertial Mass body system and is the measure of the most maximum curvature and minimum distance Geodesic in spacetime for that Net Inertial mass system. This minimum distance equation on the left is directly propotional to the Net Mass, Rest Energy, and Schwarzschild radius \(r_{S} = \2 (\frac{m_{net}G} {c^2})\) of the isolated system. This Source of Curvature can also be called the matter Energy term. The Equation on the left describes how space and time are moved or affected do to the pressence of the matter and energy in that Location. The Riemann/Ricci Curvature Tensor \(R_{ab}\) is the measure of the most minimum curvature and describes the Maximum Distance Geodesic of the system. Any distance beyond this Riemann/Ricci Curvature Tensor \(R_{ab}\) is considered flat space. That Flat space can still expand but matter (mass) does not curve space beyond this Maximum Distance Geodesic. The isolated system that the Einsten Field Equation describes is observer dependent, Spherical, the Net Inertial Mass \(m_{net} \) is conserved, the Energy Content \(E_{Rest} = m_{net}c^2 \) is conserved, and the system is Adiabatic; meaning no additional heat energy other what is in the system due to the mass interaction is added or lost by the system. The Ricci Flow Curvature Tensor [\( - \frac{1}{2}R g_{ab} + \Lambda g_{ab}]\) is the measure of the elastic expansion of curvature of spacetime and is a measure of the difference between the Mininum Geodesic and the Maximum Geodesic of the System. This term is what you would get if you imagined placing a bowling ball on a rubber sheet. This term is the curving of the sheet. And it must be kept in mind that this curving is curving in three dimensions, so from every vantage point on the surface of the sphere any observer will experience this three dimensions of spatial curvature and one dimension of time during the expansion. Now we have our Spacetime Metric Term \(g_{ab}\) in General Relativity although this term refers to a volume, most use this term to mean any spatial term such as (distance, area, or volume). The mathematics seem to fall out correctly however if we assume that the The Spacetime Term is a Volume on the surface of a three dimensional sphere where \((_{ab}) \) describe that at any location at any radius of a sphere there are tangetial and orthogonal compontents that can be represented by the latitude and longitude on the surface of a sphere. Also what makes the Spacetime Volume Metric [/B] \(g_{ab}\) term unique is that it can be used to describe a hypersphere, and the advanced mathematics of multi-dimensional space. \(g_{ab} = \frac{V_{ol}}{4 \pi} \({4 \pi}sin^2({\frac{\phi}{2}})\) \). The Scalar Curvature \( - \frac{1}{2}R g_{ab} \) is an inverse square distance term that is Heat Radiation Graviation Dependent. This term is Heat and Temperature dependent and is independent of any Matter and Energy in that location. The more mass the system has the hotter the system gets the larger this term gets; however as the distance relative to the Schwarschild Radius get bigger this equation falls off with distance squared; and once again for clarity this term is temperature and inverse square distance dependent, and independent of the Net Inertial Mass of the system. Thus when there is no matter and energy present The Net Inertial Mass \(m_{net} = 0 \) , and, the Energy Content \(E_{Rest} = m_{net}c^2 = 0 \), then \(R_{ab} - \frac{1}{2}R g_{ab} + 0 = 0 \). The above equation describes a system that is filled with a heat and radiation that is homogeneous and Isotropic, and no inertial matter "Rest Mass" is present; however Curvature is present. This means that even heat radiation also curves spacetime just like matter does. The Vacuum Energy Inverse Square Distance term \( \Lambda g_{ab}\) is the measure of a homogenous, isotropic, Vacuum Energy that is separate from the energy of Newton's Inertial Mass Graviation and varies inversely with distance from the center of the system, and inversely with the Net Inertial Mass of the System. This term is also refered to as the Cosmological Constant Term, however there is nothing constant about this term at all. What is cosmological is that this term means is that there is an Aether Energy out there; and lots of it!! If we say that there is no Vacuum or Aether Energy in the universe then this term becomes zero \( \Lambda g_{ab} = 0 \). And the Einstein Field Equation becomes \(R_{ab} - \frac{1}{2}R g_{ab} = 2 \pi (\frac{T_{ab}} {\frac{1}{4} \frac{c^4}{G}}) = G_{ab}\). The above equation describes a system that is filled with Matter and Heat; but there is no Vacuum Aether Energy Present. Based on this thread, Alphanumeric has a really good grasp of the mathematics of General Relativity, Farsight seems to be throwing darts to see which ones stick. Keep working hard Farsight don't stop! The ones that keep hitting the rock will eventually cause it to fall!
Okay, I'll bite because you're either a crank or a troll or both. Is that you? The fact you've written a pop science book attempting to explain things doesn't mean your work is valid. Is it published in any journals anywhere? Or did you circumvent journals? Calling your book after Newton's great work only serves to make you look more like a crank. Those people competent at physics don't need to call books by such names, they let the merit of the work inside speak for itself. Anyway, lets see how much you understand. describes the Einstein Field Equation as a metric that describes how localized matter (mass) and energy affects space and time in any general localized vicinity. Please demonstrate that \(G_{ab}\) is "is the measure of the most maximum curvature and minimum distance Geodesic in spacetime for that Net Inertial mass system". The EFE are not about minimal distances only. They are obtained from the Einstein-Hilbert action by using the principle of extremising. Typically the notion of least action involves 'shortest paths' but that isn't always the case, particularly in general relativity where a number of systems take the longest path. How is it proportional to those? It might depend on things likey mass and energy but that isn't the same as being proportional to them because they all combine together. In addition they contribute to a tensor expression, which includes notions of directionality. No, its the Ricci curvature tensor. It's constructed from the Riemann curvature tensor via \(R_{ab} = R^{c}_{acb}\). If you're confusing the two then you clearly have less experience with GR than would be gained from an introductory course in it. This makes your "I've written a book and named it after Newton's greatest work" all the more laughable. Nonsense. \(R_{ab}\) is not a distance, distances are scalar quantities, not rank 2 tensors. While \(R_{ab}\) can be used to construct things like tidal forces and accelerations someone might experience this is done by contracting with vectors. Flat space implies \(R_{ab} = 0\) but \(R_{ab} = 0\) (ie being Ricci flat) does not imply flat space. There's infinitely many examples of spaces which are Ricci flat but not Riemann flat (\(R^{a}_{bcd} = 0\)), such as Calabi Yau manifolds. A tensor is not a geodesic. A geodesic is a path which satisfies the relevant equations of motion from the EH action. A path is not a rank 2 tensor. Ricci flow is something entirely different. Now you're just putting your own labels on things in an attempt to add buzzwords to your nonsense. Prove it. No, they don't. The metric is not a volume or area of length. You can use it to determine such things for objects within the space-time but that isn't the same. The volume of some region \(\Omega\) is \(V = \int_{\Omega} \sqrt{|g|} \) where g is the determinant of the metric. That's quite different from the metric being volume. The metric is originally constructed in regards to lengths via \(s = \int_{C} \sqrt{g_{ab}dx^{a}\dot{x}^{a}\dot{x}^{b}}d\lambda\). If you don't understand what the metric is and how its used you shouldn't be pretending to others you understand. Either you're ignorant and don't realise it or you're profoundly dishonest trying to scam money out of people by just making stuff up about general relativity and charging people to buy it. I don't for a nanosecond think you know how to do any of the mathematics. That's just pulled from nowhere. Or more likely your backside. Anyway, I have to get ready for work so I'll skip to the end... And you're just another faker like Farsight. Both of you are quite ignorant and both are dishonest enough to try to charge people money for the crap you've made up which couldn't pass scientific review.
AlphaNumeric, I can see that you have more knowledge and are a better student of General Relativity than myself. So I will not be able to answer all of your questions, since I am a Troll and a Crank!! But I will attempt to answer this basic question. Source of Curvature \(G_{ab} = 2 \pi (\frac{T_{ab}} {\frac{1}{4} \frac{c^4}{G}})\) Once again for any Isolated Net Inertial Mass body system the Source of Curvature is the measure of the most maximum curvature and minimum distance Geodesic in spacetime. This Source of Curvature is a closed Geodesic. A closed geodesic for a sphere is the circumference of the sphere Source of Curvature as a closed geodesic \(G_{ab} = 2 \pi ({r_{s})\). This minimum distance and maximum curvature equation is directly propotional to the Net Mass, Rest Energy, and Schwarzschild radius. \(r_{S} = \2 (\frac{m_{net}G} {c^2}) = (\frac{\frac{1}{2}\ m_{net}c^2} {\frac{1}{4} \frac{c^4}{G}}) = (\frac{T_{ab}} {\frac{1}{4} \frac{c^4}{G}}) = \frac{G_{ab}}{2 \pi} \). The above equation describes how space and time are warped or curved do to the pressence of the matter and the Energy Content in that Location. Energy Content \(E_{Rest} = m_{net}c^2 \). As stated earlier the "Tensor" indices (\(G_{ab}\)) gives the directionality and describe that at any location at any radius of a sphere there are tangetial and orthogonal compontents that can be represented by the latitude (\({a}\)) and longitude (\({b}\)) on the surface of a sphere. 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\({a}^2 + \ b^2 \sin^2(a_0))\). I won't entertain the comment on Ricci "Geodesic" Flow Tensor. The Ricci Flow Curvature Tensor \( [R_{ab} - G_{ab}] = [- \frac{1}{2}R g_{ab} + \Lambda g_{ab}]\) is the measure of the elastic expansion of curvature of spacetime and is a measure of the difference between the Mininum Geodesic and the Maximum Geodesic of the System. You should look this term up on Wiki. Best
Not all systems have closed geodesics. A sphere is a simple case which does, due to its uniform positive curvature. A space with negative curvature will not have any closed geodesics. That equation is flat out wrong. \(G_{ab}\) is a rank 2 tensor, \(2\pi r_{S}\) is a scalar. You've made the same mistake I've previously pointed out. If you didn't understand what I said and you don't understand why its wrong then you have no business claiming to grasp any general relativity because you're making a mistake in basic linear algebra, never mind GR. It does not. You have no understanding of how to do tensor calculus at all. You're doing a more abstract version of saying "4 seconds = 10 kilograms". Units don't match therefore its wrong. If tensor rank doesn't match then its wrong, no ifs, no buts. You don't appear the grasp the differences between coordinates, metric tensor components and lengths. You should look up 'tensor' 'coordinates' 'metric' 'volume form' 'rank' and get yourself a basic introductory book on linear algebra. Besides, its a bit daft you admit I'm well versed in this stuff and then tell me to look it up on Wikipedia. You clearly know I'm familiar with this stuff so at the very least you should be a little more effort into your replies rather than fall back on Farsight's method of baselessly saying "I know this, you don't".
