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Title: Stephen Crothers: Why Black Holes Don't Exist Source: Stephen Crothers URL Source: http://www.youtube.com/watch?v=A4GFAjX62Yg&feature=player_embedded Published: Apr 14, 2009 Author: Stephen Crothers Post Date: 2009-04-14 12:57:37 by Horse Keywords: NoneViews: 378 Comments: 23
http://www.youstupidrelativist.com/03Intro.html A new language The casual browser and superficial reader typically get the wrong idea that this site is business as usual. So in order to minimize blind-siding the average visitor, I would like to say up front that this site relies on and uses a different language to communicate ideas. For instance, when I say that in Physics or in Science the word space means 'no shape,' I am not saying that this is what the idiots of the establishment mean by space. I am saying the this is what the word space means in genuine as opposed to contemporary 'science.' Whenever I want to represent or depict the official version, I will refer to them as 'mathematicians' or 'mathematical physicists' or idiots or something like that. There are no physicists in mainstream 'Physics.' From Newton to Einstein to Hawking, they are all just mathematicians as far Science and Physics are concerned. This site defines Physics as the science of existence. Physics is first and foremost the science that studies objects, specifically those objects that exist. In contrast, Philosophy is a discipline that studies concepts. The establishment has turned this state of affairs around over the centuries and today it is incongruously in reverse. The contemporary 'physicists' play around with abstract concepts such as energy, mass, space- time, virtual particles, and black holes while the philosophers are trying to answer fundamental questions of genuine Physics, such as 'What is an object' or 'What does the word exist mean?' Physics is first and foremost the science that studies objects. Without objects we have no physical 'anything' to study or to carry out an experiment with. And even if the moron of Mathematics believes that Physics also studies concepts, the idiot has yet to define what a concept is. Therefore, whether objects or concepts, we must get to the bottom of these strategic words because Physics certainly studies one of them. So I caution you that communication will be a major problem. The mathematician comes to this site with the idea that Physics is about time travel and wormholes and virtual particles. He discovers, instead, what he missed in college, what he failed to question his teachers about. Physics is about objects and location and motion, and these words don't mean what they mean in ordinary speech. The mathematicians (known today as 'physicists') have never had to confront these words in high school or college, except perhaps from a casual 'philosophical' perspective in a course they were forced to take at gunpoint. Therefore, they are ill- equipped to deal with the subject matter at hand and are likely to feel frustrated. Here, these words become a major component of every argument. As soon as a mathematician so mentions a buzz word such as object, exist, dimension, position, straight, or continuous, I will ask him to define it. The reason for this is not to cause unnecessary trouble. I'm not here to play devil's advocate. The reason for this is that the mathematician erroneously believes that he has communicated his idea effectively when in fact nothing of the sort has occurred. The mathematician cannot have any idea what he is talking about because he takes these words for granted. He extrapolates the definitions of ordinary speech into a scientific context without realizing that a rigorous definition of the same words destroys his entire argument. I promise to harp on such vital words as object, exist, and concept until it hurts. I will insist until you hate me. The contemporary mathematicians have no idea what these words mean or how important they are to scientific communication. This is where the battle will begin. This is where the battle will end. Language. You seem to have trouble grasping simple, abstract, mathematical theories, Bill. Therefore, we are through with words. Perhaps it's time for something a little more pragmatic. We decided that the time has come to demonstrate our theories to you empirically. 3.0 Mathematical Physics offers irrational physical interpretations for common phenomena Contemporary Mathematical Physics is comprised of three pillars: General Relativity (GR), Quantum Mechanics (QM), and String Theory (ST). The establishment regards these disciplines with absolute reverence, but don't be misled. GR, QM, and ST are actually more like Andersen's proverbial Emperor's Clothes tale. The mathematician points to an invisible entity and interprets for you what you are observing. Suspiciously, the idiot of Mathematics always ends up with his beloved point. An atom is a particle. An electron is a particle. A photon is a particle. An event is a particle and space-time is comprised of events (meaning particles). And a string is made of particles. When you so much as insinuate that his explanation borders on the irrational, the moron urges you to set aside your common sense and intuition and trust his equations and calculations. He throws a list of who's who at you to reinforce his claim with authority. All the celebrities of Nobel and Templeton fame believe in the particle. So how come you don't believe in particles? Who are you to question so much authority? But I have just shown that you can provide different physical interpretations to an equation. So what compels you to switch your intellectual allegiance? Why should you trust his interpretation of a physical phenomenon when it is also based on intuition? The results are on the table. After centuries of raving lunacy, the mathematicians of this world have absolutely nothing to show for their abstract theories. We have yet to see a mockup of space-time or a picture of a single particle of the Standard Model or a sculpture of the 1-D Planck Length theorists claim the Universe is made of. Not a single mathematician on Earth can tell you WHAT light IS. Not a single mathematician working at NASA or CERN can tell you how a simple magnet works or WHAT the invisible lines of force that surround it ARE. Not a single mathematician at Cambridge or at Harvard can illustrate mass or energy for you. And certainly numbers and equations will not help them in any of these quests. Mainstream scientists don't understand the first thing about the scientific method The underlying problem is that the mathematicians have never defined the word science unambiguously and have yet to grasp the nature of the scientific method. The establishment has erroneously concluded that experimentation and Math are necessary components of the scientific method. A mathematician models the path of an invisible particle with an equation, runs a test to prove his hunch, and then boasts that the experiment has proven his theory. [1] Yet, when you go back to check, the infamous particle was merely an assumption. If the invisible thing at the center of his experiment is not a particle, his alleged 'proof' disintegrates before your very eyes. The mathematicians mistake assumptions for proofs and confuse hypotheses with theories. After 2500 years of research, the disciples of Pythagoras and Euclid have nothing to exhibit at their trade shows, nothing to show during their show-and-tells. The mathematicians are still in the Dark Ages as far as their understanding of nature is concerned. They have merely modified the language of Plato and Augustine. The members of the establishment continue to talk about spirits and ghosts and miracles, but today allude to such phenomena using scientific-sounding names like 'singularity' and 'carriers of force' and 'uncertainty.' The irony is that the mathematicians believe that the development of technology proves that the inventions we enjoy today confirm their theories. The language of Physics and of Science is called visualization. In order for the prosecutor and the juror to be on the same wavelength, they must both watch the same movie. If the prosecutor is talking about rocks and the juror imagines trees they cannot possibly be communicating. We don't understand rocks and trees. We see them with our eyes! There is only one way to guarantee that everyone visualizes the same thing: the presenter should be able to make a movie of his theory. A theory is an explanation of how or why something occurred. If the presenter cannot put his ideas on the Big Screen for everyone to watch (the same thing), he is not doing Science. And in order for the theory to be converted into a movie, an even more fundamental requirement is form. Without shapes, the prosecutor has nothing to film and the juror has nothing to watch. The first requirement of science and of the scientific method is to produce the physical objects. We cannot make a film with abstract concepts! We cannot do science with the idiotic words of Mathematics: energy, mass, time, force, or field. These words do not represent physical objects. Science is not about running experiments or proving theories or about Math. Science is about communicating ideas. Afterwards you can infer whatever you like, run experiments in the lab, and reach your own conclusions. In order to communicate ideas precisely, the presenter absolutely needs to define his words rigorously. A precise definition of the words that make or break a theory is the second requirement of the scientific method. Mathematical Physics fails both of these requirements. Not a single mathematician in the world can make a movie of his presentation. What is he going to put on the screen? A picture of energy? A scene where mass jumps up and down? And not a single mathematician on Earth defines the words that make or break his theory rigorously. The contemporary mathematicians: • tell you that their crucial words are primitives (i.e., undefinable) (e.g., point, line, mass, energy, time) • use these terms inconsistently during their talks anyways • replace concepts with objects (e.g., the center of mass with a dot, space-time with a tesseract) • move abstract concepts (e.g., transfer energy, accelerate point particles, bend time, blend orbitals) • describe interactions between abstractions (e.g., virtual particles, field and charge, annihilation of two 0-D particles) ...and then wish you to believe that they are doing Science. The mathematicians of Mathematical Physics are the first to attempt to give a physical interpretation to their equations. When you call their bluff and show that the explanation is irrational, relativists defend themselves by saying that you are raising a philosophical issue and not one that concerns Science (by which they mean Physics). But it was they who introduced 'philosophy' (i.e., give a physical interpretation to the equations) in the first place. In other words, the mathematician wants you to believe that he is authorized to provide a physical interpretation to a series of variables or to a function because he is backed by Math and authority (celebrities). When you question his logic, he accuses you of relying on subjective intuition and common sense at the expense of objective Mathematics. He dismisses your attack as petty philosophy, meaning that it's just your opinion. If this argument fails to persuade you, he produces next a list of who's who from the 'scientific' community who believe in the same idiocy he does (i.e., the argument from authority). These people are backed by Nobel Prizes (which their peers gave to them). Turn around and look behind you. Who supports your version? The mathematicians are convinced that whether a theory is scientific depends on a show of hands! 5.0 A mathematician has no explanation The mathematicians have become the priests of the modern world. We have delegated the task of uncovering the nature of our Universe to idiots that laymen mistake for geniuses. I can find no better word than idiot to qualify an individual who has a Masters Degree or a PhD and offers irrational and fantastic physical interpretations for natural phenomena. Those who invoke such ridiculous concepts as space-time, Big Bang, black hole, parallel universe, field, energy, mass, wave-packet, point particle, time travel, tunneling, warped space, or annihilation to explain a phenomenon of nature should be locked up in a mental institution. The official scientific world is a farce. The establishment has gone completely mad. Actually the official interpretations of the real world have as much to do with the equations the mathematicians back them with as the Bible with the existence of God, but naive people have gradually surrendered themselves to celebrities. Most people stand in awe of the nonsense coming out of the universities and think tanks simply because they don't understand any of it. The result is that we have made a 180º turn. We are back to the Middle Ages as far as understanding of nature is concerned. If, like me, you suspect the physical interpretations of the mathematical sages of our world, the following pages contain material that will provide you with solid arguments in your heads-to-heads with relativists, mechanics, and string theorists. I have simplified concepts and theories of Mathematical Physics and illustrated wherever possible to make the site accessible to the average visitor. You don't need to know Math to realize that relativity is 100% poppycock. A good dose of common sense will do! Poster Comment:I will explore this further.
http://www.youstupidrelativist.com/03Intro.html
A new language
The casual browser and superficial reader typically get the wrong idea that this site is business as usual. So in order to minimize blind-siding the average visitor, I would like to say up front that this site relies on and uses a different language to communicate ideas. For instance, when I say that in Physics or in Science the word space means 'no shape,' I am not saying that this is what the idiots of the establishment mean by space. I am saying the this is what the word space means in genuine as opposed to contemporary 'science.' Whenever I want to represent or depict the official version, I will refer to them as 'mathematicians' or 'mathematical physicists' or idiots or something like that. There are no physicists in mainstream 'Physics.' From Newton to Einstein to Hawking, they are all just mathematicians as far Science and Physics are concerned.
This site defines Physics as the science of existence. Physics is first and foremost the science that studies objects, specifically those objects that exist. In contrast, Philosophy is a discipline that studies concepts. The establishment has turned this state of affairs around over the centuries and today it is incongruously in reverse. The contemporary 'physicists' play around with abstract concepts such as energy, mass, space- time, virtual particles, and black holes while the philosophers are trying to answer fundamental questions of genuine Physics, such as 'What is an object' or 'What does the word exist mean?' Physics is first and foremost the science that studies objects. Without objects we have no physical 'anything' to study or to carry out an experiment with. And even if the moron of Mathematics believes that Physics also studies concepts, the idiot has yet to define what a concept is. Therefore, whether objects or concepts, we must get to the bottom of these strategic words because Physics certainly studies one of them.
So I caution you that communication will be a major problem. The mathematician comes to this site with the idea that Physics is about time travel and wormholes and virtual particles. He discovers, instead, what he missed in college, what he failed to question his teachers about. Physics is about objects and location and motion, and these words don't mean what they mean in ordinary speech. The mathematicians (known today as 'physicists') have never had to confront these words in high school or college, except perhaps from a casual 'philosophical' perspective in a course they were forced to take at gunpoint. Therefore, they are ill- equipped to deal with the subject matter at hand and are likely to feel frustrated. Here, these words become a major component of every argument. As soon as a mathematician so mentions a buzz word such as object, exist, dimension, position, straight, or continuous, I will ask him to define it. The reason for this is not to cause unnecessary trouble. I'm not here to play devil's advocate. The reason for this is that the mathematician erroneously believes that he has communicated his idea effectively when in fact nothing of the sort has occurred. The mathematician cannot have any idea what he is talking about because he takes these words for granted. He extrapolates the definitions of ordinary speech into a scientific context without realizing that a rigorous definition of the same words destroys his entire argument. I promise to harp on such vital words as object, exist, and concept until it hurts. I will insist until you hate me. The contemporary mathematicians have no idea what these words mean or how important they are to scientific communication. This is where the battle will begin. This is where the battle will end. Language.
You seem to have trouble grasping simple, abstract, mathematical theories, Bill. Therefore, we are through with words. Perhaps it's time for something a little more pragmatic. We decided that the time has come to demonstrate our theories to you empirically.
3.0 Mathematical Physics offers irrational physical interpretations for common phenomena
Contemporary Mathematical Physics is comprised of three pillars: General Relativity (GR), Quantum Mechanics (QM), and String Theory (ST). The establishment regards these disciplines with absolute reverence, but don't be misled. GR, QM, and ST are actually more like Andersen's proverbial Emperor's Clothes tale. The mathematician points to an invisible entity and interprets for you what you are observing. Suspiciously, the idiot of Mathematics always ends up with his beloved point. An atom is a particle. An electron is a particle. A photon is a particle. An event is a particle and space-time is comprised of events (meaning particles). And a string is made of particles. When you so much as insinuate that his explanation borders on the irrational, the moron urges you to set aside your common sense and intuition and trust his equations and calculations. He throws a list of who's who at you to reinforce his claim with authority. All the celebrities of Nobel and Templeton fame believe in the particle. So how come you don't believe in particles? Who are you to question so much authority?
But I have just shown that you can provide different physical interpretations to an equation. So what compels you to switch your intellectual allegiance? Why should you trust his interpretation of a physical phenomenon when it is also based on intuition?
The results are on the table. After centuries of raving lunacy, the mathematicians of this world have absolutely nothing to show for their abstract theories. We have yet to see a mockup of space-time or a picture of a single particle of the Standard Model or a sculpture of the 1-D Planck Length theorists claim the Universe is made of. Not a single mathematician on Earth can tell you WHAT light IS. Not a single mathematician working at NASA or CERN can tell you how a simple magnet works or WHAT the invisible lines of force that surround it ARE. Not a single mathematician at Cambridge or at Harvard can illustrate mass or energy for you. And certainly numbers and equations will not help them in any of these quests.
Mainstream scientists don't understand the first thing about the scientific method
The underlying problem is that the mathematicians have never defined the word science unambiguously and have yet to grasp the nature of the scientific method. The establishment has erroneously concluded that experimentation and Math are necessary components of the scientific method. A mathematician models the path of an invisible particle with an equation, runs a test to prove his hunch, and then boasts that the experiment has proven his theory. [1] Yet, when you go back to check, the infamous particle was merely an assumption. If the invisible thing at the center of his experiment is not a particle, his alleged 'proof' disintegrates before your very eyes. The mathematicians mistake assumptions for proofs and confuse hypotheses with theories.
After 2500 years of research, the disciples of Pythagoras and Euclid have nothing to exhibit at their trade shows, nothing to show during their show-and-tells. The mathematicians are still in the Dark Ages as far as their understanding of nature is concerned. They have merely modified the language of Plato and Augustine. The members of the establishment continue to talk about spirits and ghosts and miracles, but today allude to such phenomena using scientific-sounding names like 'singularity' and 'carriers of force' and 'uncertainty.' The irony is that the mathematicians believe that the development of technology proves that the inventions we enjoy today confirm their theories.
The language of Physics and of Science is called visualization. In order for the prosecutor and the juror to be on the same wavelength, they must both watch the same movie. If the prosecutor is talking about rocks and the juror imagines trees they cannot possibly be communicating. We don't understand rocks and trees. We see them with our eyes! There is only one way to guarantee that everyone visualizes the same thing: the presenter should be able to make a movie of his theory. A theory is an explanation of how or why something occurred. If the presenter cannot put his ideas on the Big Screen for everyone to watch (the same thing), he is not doing Science. And in order for the theory to be converted into a movie, an even more fundamental requirement is form. Without shapes, the prosecutor has nothing to film and the juror has nothing to watch. The first requirement of science and of the scientific method is to produce the physical objects. We cannot make a film with abstract concepts! We cannot do science with the idiotic words of Mathematics: energy, mass, time, force, or field. These words do not represent physical objects.
Science is not about running experiments or proving theories or about Math. Science is about communicating ideas. Afterwards you can infer whatever you like, run experiments in the lab, and reach your own conclusions. In order to communicate ideas precisely, the presenter absolutely needs to define his words rigorously. A precise definition of the words that make or break a theory is the second requirement of the scientific method.
Mathematical Physics fails both of these requirements. Not a single mathematician in the world can make a movie of his presentation. What is he going to put on the screen? A picture of energy? A scene where mass jumps up and down? And not a single mathematician on Earth defines the words that make or break his theory rigorously. The contemporary mathematicians:
• tell you that their crucial words are primitives (i.e., undefinable) (e.g., point, line,
mass, energy, time)
• use these terms inconsistently during their talks anyways • replace concepts with objects (e.g., the center of mass with a dot, space-time
with a tesseract)
• move abstract concepts (e.g., transfer energy, accelerate point particles, bend
time, blend orbitals)
• describe interactions between abstractions (e.g., virtual particles, field and
charge, annihilation of two 0-D particles)
...and then wish you to believe that they are doing Science.
The mathematicians of Mathematical Physics are the first to attempt to give a physical interpretation to their equations. When you call their bluff and show that the explanation is irrational, relativists defend themselves by saying that you are raising a philosophical issue and not one that concerns Science (by which they mean Physics). But it was they who introduced 'philosophy' (i.e., give a physical interpretation to the equations) in the first place. In other words, the mathematician wants you to believe that he is authorized to provide a physical interpretation to a series of variables or to a function because he is backed by Math and authority (celebrities). When you question his logic, he accuses you of relying on subjective intuition and common sense at the expense of objective Mathematics. He dismisses your attack as petty philosophy, meaning that it's just your opinion. If this argument fails to persuade you, he produces next a list of who's who from the 'scientific' community who believe in the same idiocy he does (i.e., the argument from authority). These people are backed by Nobel Prizes (which their peers gave to them). Turn around and look behind you. Who supports your version? The mathematicians are convinced that whether a theory is scientific depends on a show of hands!
5.0 A mathematician has no explanation
The mathematicians have become the priests of the modern world. We have delegated the task of uncovering the nature of our Universe to idiots that laymen mistake for geniuses. I can find no better word than idiot to qualify an individual who has a Masters Degree or a PhD and offers irrational and fantastic physical interpretations for natural phenomena. Those who invoke such ridiculous concepts as space-time, Big Bang, black hole, parallel universe, field, energy, mass, wave-packet, point particle, time travel, tunneling, warped space, or annihilation to explain a phenomenon of nature should be locked up in a mental institution. The official scientific world is a farce. The establishment has gone completely mad.
Actually the official interpretations of the real world have as much to do with the equations the mathematicians back them with as the Bible with the existence of God, but naive people have gradually surrendered themselves to celebrities. Most people stand in awe of the nonsense coming out of the universities and think tanks simply because they don't understand any of it. The result is that we have made a 180º turn. We are back to the Middle Ages as far as understanding of nature is concerned.
If, like me, you suspect the physical interpretations of the mathematical sages of our world, the following pages contain material that will provide you with solid arguments in your heads-to-heads with relativists, mechanics, and string theorists. I have simplified concepts and theories of Mathematical Physics and illustrated wherever possible to make the site accessible to the average visitor. You don't need to know Math to realize that relativity is 100% poppycock. A good dose of common sense will do!
Poster Comment:
I will explore this further.
Post Comment Private Reply Ignore Thread
#1. To: Horse (#0) Some truth there, but he asks of science what it cannot give; no dictionary is self-contained, and must leave some words formally undefinable by other words, understandable only by direct, shared experience. What if there is no direct, shared experience to which we can refer to describe a variable in a model? Then yes, using a term like 'wave-particle' seems like nonsense and is misleading. And since all such attempts at macro analogies are misleading, we should just use a completely made up word; it's just a placeholder anyway, like a variable name in a program, and we can give it any name we want. The important thing, to the scientific mind, is whether it is useful in prediction, whether the outputs of the program are useful. There's more than one way to skin a cat. A particular(!) program might be inelegant, like celestial spheres, but it's not wrong, exactly, unless it prevents prediction to within desired tolerances or conflicts directly with some other observations. Science is about the outputs, not the program. But sure, programmers can get absolutely religiously attached to one algorithm or even coding style. Scientists are no better about that than anybody else, and so science advances one funeral at a time. We tend to trust scientists because of many useful things produced, like computers, and these things are worthy of respect, but the ontological scaffolding is hardly sacred. It is disposable. So whenever a scientist uses authority or celebrity to claim crap like "there's no such thing as race" you know he's not talking as a scientist but as a priest. A scientific objection would sound something like "Race is not a useful predictor of X in situation Y to tolerance Z." A priest won't say that, because he wants to assign you both your situations and your tolerances. Expose the Paranoids!!Did you know that there are Paranoids all around us?Even your next door neighbor might be one! Prefrontal Vortex posted on 2009-04-14 13:54:15 ET Reply Trace Private Reply #2. To: Horse (#0) Little more than semantics and existentialism here. Crackpots often insist on inventing their own nomenclatures. TooConservative posted on 2009-04-14 14:19:30 ET Reply Trace Private Reply #3. To: Horse, Prefrontal Vortex, TooConservative (#0) People need to go to Stephen J. Crothers website to understand the reference of the post and the video. In short: the math "doesn't add up." Crothers does a decent job of explaining it; just bypass the formulas and keep in mind that "Ric=0" proves his point. The whole aim of practical politics is to keep the populace alarmed (and hence clamorous to be led to safety) by menacing it with an endless series of hobgoblins, all of them imaginary. - Henry Louis Mencken rack42 posted on 2009-04-14 21:35:18 ET Reply Trace Private Reply #4. To: Horse (#0) You don't need to know Math to realize that relativity is 100% poppycock. All you need is the nonlocality of quantum entanglement. I've already said too much. MUDDOG posted on 2009-04-14 21:42:16 ET Reply Trace Private Reply #5. To: Horse (#0) Anyone that thinks time travel is possible by traveling at the speed of light is not too bright no matter how many degrees they have. People that do investigate things on their own and understand things on their own step by step will accept all kinds of crazy nonsense as truth. These people then point to the establishment as accepting it as their "proof" they are correct. Funny. Reality does not care what the establishment thinks, it never has. God is always good! RickyJ posted on 2009-04-14 21:57:11 ET Reply Trace Private Reply #6. To: rack42 (#3) Crothers does a decent job of explaining it; just bypass the formulas and keep in mind that "Ric=0" proves his point. I wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Just wondering. Dr_Tron posted on 2009-05-10 19:53:08 ET Reply Trace Private Reply #7. To: Dr_Tron (#6) I wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Pulling rank, eh? I've already said too much. MUDDOG posted on 2009-05-10 21:41:08 ET Reply Trace Private Reply #8. To: MUDDOG (#7) Pulling rank, eh? Not at all. If I came across that way - my apologies. Just that you cannot debunk relativity if you don't understand it in the first place. That is what I was trying to convey. Doc Dr_Tron posted on 2009-05-11 0:06:51 ET Reply Trace Private Reply #9. To: Dr_Tron (#8) If I came across that way - my apologies. No problem. I couldn't resist the play on "rank." I've already said too much. MUDDOG posted on 2009-05-11 0:27:20 ET Reply Trace Private Reply #10. To: MUDDOG (#9) LOL! OK. :-) Dr_Tron posted on 2009-05-11 23:07:26 ET Reply Trace Private Reply #11. To: Dr_Tron (#10) Anyone who likes the Christoffel symbols is OK by me. I've already said too much. MUDDOG posted on 2009-05-11 23:22:03 ET Reply Trace Private Reply #12. To: MUDDOG, Dr_Tron (#9) In mathematics and physics, the Christoffel symbol describes curvature in a non-euclidean space, such as geometry on the surface of the globe. Whereas the metric tensor describes a sort of first derivative of the warp, the Christoffel symbol describes a sort of second derivative. At each point of the underlying n-dimensional manifold, it is an array with three dimensions: n × n × n. Each of the n3 components is a real number. Under linear coordinate transformations on the manifold, it behaves like a tensor. The Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are often called Christoffel symbols.[1] The Christoffel symbols may be used for performing practical calculations in differential geometry. Unfortunately, the calculations are usually quite lengthy and complex, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is terse, and allows theorems to be stated in an elegant way, but requires more advanced techniques for practical calculations. In many practical problems, most components of the Christoffel symbols are equal to zero. oh my, and the paki cadillac went over my head. ;P The smooth criminal transition from Bush/Cheney to Obama christine posted on 2009-05-12 1:02:57 ET Reply Trace Private Reply #13. To: christine (#12) (Edited) over my head. ;P That's because the definition you posted is the result of over a hundred years of studying and refining the concept. It's like when Sherlock Holmes presents his conclusions without the intermediate steps. It seems amazing, but when he explains the intermediate steps, it's elementary, my dear Watson. The Christoffel symbols are like that. Their roots are in the study of 2- dimensional surfaces in ordinary three-dimensional space, from the question of how to define the curvature of a surface in space. It's pretty straightforward to define the curvature of a (not necessarily straight) line. A straight line should have zero curvature. The direction of a straight line doesn't change. For a curvy line, the curvature should be the amount it changes direction. In calculus, the direction of a (possibly curvy) line is measured by its first derivative. The curvature should be measured by the change in direction, that is, the change in the first derivative, which is the second derivative. And that's the key to defining the curvature of a surface. It should be something measured by second derivatives in the equations (coordinates) of the surface. Think of it this way. A flat surface in 3-dimensional space is like a tabletop. The direction "outward" (or up) from the tabletop stays the same all across the tabletop; it points toward the ceiling. No change in the outward direction. But now think of the world globe in your study. The "outward" direction at the north pole of the globe indeed points toward the ceiling; but the "outward" direction at the equator of the globe (for a tiny man there!) points toward the wall of your study. The globe is curved -- the "outward" directions vary across the surface of the globe. Mathematically, the "outward" direction at a point of a surface in 3- dimensional space is known as the "normal vector" at that point. And the variation of this normal vector measures the curvature of the surface. If there's no variation of the normal vector, like in the tabletop, there should be no curvature; the surface is flat. If the normal vector varies, like in your world globe, there's curvature of the surface. You have to give a mathematical measure of all this, but that's just how Gauss proceeded 200 years ago in his pioneering studies of curvature. Now let's get to a general surface in 3-dimensional space. The first derivative of the coordinates of the surface gives you a measure of how the surface sits in space at a point of the surface; but the second derivative of the coordinates gives you a measure of how the surface is curving in space. (Technically, the normal ("outward") vector of the surface is given by the "cross product" of the first derivatives of the coordinates, so that the change (or first derivative) of the normal vector involves the second derivatives of the coordinates.) To calculate this changing in the normal vector, which Gauss used as the measure of surface curvature, you can use the Christoffel symbols. They’re defined as follows: When you express the second derivatives of the coordinates of the surface in terms of the first derivatives and the normal vector to the surface, the Christoffel symbols are the coefficients of the first derivatives. The mathematical expression of curvature is then calculated from the Christoffel symbols and is called the Riemann-Christoffel curvature tensor. After this it gets pretty technical (if it hasn't already!). What happens is, first you can generalize to n-dimensional (Euclidean) space, and consider an (n- 1) dimensional hyperspace sitting in there, so you have a normal vector as in our 3-dimensional case, and you take second derivatives of the hyperspace coordinates, define the Christoffel symbols as the coefficients of the first derivatives in the expression for second derivatives, and solve for the Christoffel symbols by using the notion of length in the Euclidean space. Then you calculate the Riemann-Christoffel curvature tensor analogously to the surface case. An interesting special case is a “flat“ hyperspace. The hyperspace is flat if you can find coordinates for the hyperspace for which the "metric" on the hyperspace is particularly simple (Euclidean), which is equivalent to the Christoffel symbols being equal to zero at all points of the coordinates. But there are so many possible coordinate systems you can use, how do you know if there even are any coordinates for which the Christoffel symbols are zero? The coordinates you start with may have nonzero Christoffel symbols. It turns out that no matter what coordinates you start with, if you calculate the Christoffel symbols for those coordinates, then calculate the Riemann- Christoffel curvature tensor from those Christoffel symbols, the tensor will equal zero at all the points if and only if there exists a set of coordinates (not necessarily the ones you started with) for which their Christoffel symbols are zero. So "flatness" is the same as "zero curvature." You can generalize further from hyperspaces in Euclidean space, to "Riemannian spaces," and then generalize to non-Riemannian spaces which have an "affine connection." The tie-in to Einstein's general relativity is he used curvature of space to explain gravity. I've already said too much. MUDDOG posted on 2009-05-12 15:46:02 ET Reply Trace Private Reply #14. To: Dr_Tron (#10) See post #13 above. I've already said too much. MUDDOG posted on 2009-05-12 15:47:54 ET Reply Trace Private Reply #15. To: Dr_Tron (#6) wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Just wondering. What is it about "Ric=0" don't you understand? Please, dispute it because it seems that Physics has become Mathematics. The whole aim of practical politics is to keep the populace alarmed (and hence clamorous to be led to safety) by menacing it with an endless series of hobgoblins, all of them imaginary. - Henry Louis Mencken rack42 posted on 2009-05-14 23:19:15 ET Reply Trace Private Reply #16. To: Dr_Tron (#6) Crothers does a decent job of explaining it; just bypass the formulas and keep in mind that "Ric=0" proves his point. No, what you said makes no sense. I doubt that you have any understanding of what he said. The whole premis of "black holes" is that "Ric=0." You've provided nothing to disprove that. Do you even understand what "Ric=0" means? No, you don't. Because it is impossible. Unless you want to propose a new Physics. I'd be very interested in that. The whole aim of practical politics is to keep the populace alarmed (and hence clamorous to be led to safety) by menacing it with an endless series of hobgoblins, all of them imaginary. - Henry Louis Mencken rack42 posted on 2009-05-14 23:31:57 ET Reply Trace Private Reply #17. To: rack42 (#16) The whole premis of "black holes" is that "Ric=0." Do you mean Rij=0 ? That is the field equation for the vacuum space-time. Or in other words a pseudo-Riemannian manifold called Ricci-flat. And this can only be proved if the local manifold is equivalent to flat space. And this disproves black holes how? (Actually it was interesting he spent so much time on how he was kicked out of grad school and how every theoretical physicist for the past 100 years have been wrong) Dr_Tron posted on 2009-05-17 22:40:39 ET Reply Trace Private Reply #18. To: MUDDOG (#14) Nice writup. You kinda went to warp factor 10 for the average guy about 2/3 in. LOL! It's been a while since I studied Differential Geometry. Brings back memories. :-) How about tangent bundles. :-) (tangent space over all points of the manifold) And we could really get crazy - Cotangent bundles acting as operators or in other words mapping to the tangent space. Note: you started to get into the realm of coordinate free geometry (when you stated "The coordinates you start with may have nonzero Christoffel symbols". And a second note for the lay readers: When you said; "Mathematically, the "outward" direction at a point of a surface in 3- dimensional space is known as the "normal vector" at that point. And the variation of this normal vector measures the curvature of the surface. If there's no variation of the normal vector, like in the tabletop, there should be no curvature; the surface is flat. If the normal vector varies, like in your world globe, there's curvature of the surface." The question is how do you get the normal vector? It is the cross product of pairs of tangent vectors. Dr_Tron posted on 2009-05-17 22:53:16 ET Reply Trace Private Reply #19. To: Dr_Tron (#18) (Edited) Good to see you! The big thing I left out of the previous post was, the Gaussian curvature is only a partial measure of the curvature of surfaces in 3-space. For example, the surface of a cylinder in 3-space has Gaussian curvature zero, even though it's "curved" in the ordinary sense of the word. What makes the Gaussian curvature so special, though, is that it is "intrinsic" to the surface -- that is, it's completely calculable from the metric tensor. That fact impressed Gauss so much that he called it the "Theorema egregium" -- "a most excellent theorem" -- because he had started out defining the Gaussian curvature in the usual manner of looking at the variation of the normal vector, and discovered that the resulting Gaussian curvature actually could be calculated without the normal vector, i.e., without a bigger 3-space in which the surface was "curving." That is indeed an amazing result. And this is why it's the Gaussian curvature, rather than some other measure of curvature, that generalizes so well -- it's not dependent on a larger space containing the surface, in which a normal vector, pointing into some bigger space, determines curvature. (Although the normal vector is a very helpful visual aid!) So, since the Gaussian curvature is intrinsic, it generalizes well to abstract spaces ("manifolds") which need not be defined as imbedded in some other space. So this is part of the reason why, when laymen hear about the curvature of space, it doesn't make much sense because "physical" space doesn't seem to be sitting in some bigger space in which it is curving. But you don't need a bigger space, because the curvature is intrinsic to "physical" space when you use the generalized Gaussian curvature (the Riemann-Christoffel tensor). And the Riemann-Christoffel tensor is indeed a generalization of the Gaussian curvature, not just in the formulas from the Christoffel symbols, but also in that the "surface" Gaussian curvature of the 2-dimensional submanifolds generated by appropriate geodesics is precisely the value of the Riemann- Christoffel tensor evaluated at the two tangent vectors generating the 2- dimensional submanifold. I've already said too much. MUDDOG posted on 2009-05-17 23:25:32 ET Reply Trace Private Reply #20. To: Dr_Tron (#17) Do you mean Rij=0 ? No. You're trying to divert. Bad try. ...with the power of conviction, there is no sacrifice. rack42 posted on 2009-05-18 22:34:03 ET Reply Trace Private Reply #21. To: rack42 (#20) I did some looking. Instead of slogging thru the BS myself I found others had already done so: http://www.physicsforums.com/showthread.php?t=123802 http://www.physforum.com/index.php?showtopic=12491&st=15 Saved me a lot of time. Dr_Tron posted on 2009-05-19 8:06:27 ET Reply Trace Private Reply #22. To: Dr_Tron (#21) (Edited) RE: truth2k that person is, um, I don't know what. Let's say confused. Link to truth2k in any capacity is suspect in the extreme. RE: Other Link I don't see that anything is resolved. Crothers defends his hypothosis: There is, I submit, a fundamental conceptual error with the now commonplace claims for black hole binaries, black hole collisions, and black hole mergers. By what rigorous means do the scientists maintain that the supposed problems are well posed? In other words, by what rigorous arguments do they hold that black hole binaries or black hole collisions are well-defined scenarios? I remark that the black hole is alleged to be a consequence of Einstein’s General Relativity. Assuming for the sake of argument that this is correct, it is evident that the black hole is the result of a solution to Einstein’s field equations for the configuration of a single spherically symmetric gravitating body interacting with a test particle in vacuum. It is not a solution for the interaction of two comparable masses, such as two black holes. Before one can talk of black hole binaries or black hole collisions it must first be demonstrated that Einstein’s field equations admit of solutions for multi-body configurations of gravitationally coupled spherically symmetric comparable masses. This can be done in two possible ways, in principle: 1) by deducing a particular solution, or 2) proving an existence theorem. There are however, no known solutions to the field equations for the interaction of two or more spherically symmetric comparable masses, so option 1) has never been met. In fact, it is not even known if Einstein’s field equations admit of such solutions as no existence theorem has ever been adduced, so option 2) has never been met either. Furthermore, one cannot simply assert by analogy with Newton’s gravitation that a black hole can be a component of a binary system or that black holes can collide or merge. Consequently, all talk of black hole binaries, black hole mergers, black hole collisions, etc. does not deal with well-defined problems at all. The now commonplace claims and arguments by a great many investigators are meaningless. They are without any scientific justification. One cannot test General Relativity by means of a chimera. The responses are in the same vein that "Einstien predicted blackholes and the Schwarschild solution prove it." It does not. It is evident that not of the posters read Crothers hypothesis and don't address the issues. Just keep harping on "Schwarschild radius" and the false conclusions from that because it is obvious that they, and you, haven't read Crothers hypothesis. To me, Crothers hypothesis trumps the "black hole" and "Big Bangers." Look at the MATH. GEEZ. ...with the power of conviction, there is no sacrifice. rack42 posted on 2009-05-23 21:53:37 ET Reply Trace Private Reply #23. To: rack42 (#22) Ok. I will look at the math. And with an open mind. Give me a week. Dr_Tron posted on 2009-05-25 2:28:58 ET Reply Trace Private Reply Top • Page Up • Full Thread • Page Down • Bottom/Latest
Some truth there, but he asks of science what it cannot give; no dictionary is self-contained, and must leave some words formally undefinable by other words, understandable only by direct, shared experience. What if there is no direct, shared experience to which we can refer to describe a variable in a model? Then yes, using a term like 'wave-particle' seems like nonsense and is misleading. And since all such attempts at macro analogies are misleading, we should just use a completely made up word; it's just a placeholder anyway, like a variable name in a program, and we can give it any name we want. The important thing, to the scientific mind, is whether it is useful in prediction, whether the outputs of the program are useful. There's more than one way to skin a cat. A particular(!) program might be inelegant, like celestial spheres, but it's not wrong, exactly, unless it prevents prediction to within desired tolerances or conflicts directly with some other observations. Science is about the outputs, not the program. But sure, programmers can get absolutely religiously attached to one algorithm or even coding style. Scientists are no better about that than anybody else, and so science advances one funeral at a time. We tend to trust scientists because of many useful things produced, like computers, and these things are worthy of respect, but the ontological scaffolding is hardly sacred. It is disposable. So whenever a scientist uses authority or celebrity to claim crap like "there's no such thing as race" you know he's not talking as a scientist but as a priest. A scientific objection would sound something like "Race is not a useful predictor of X in situation Y to tolerance Z." A priest won't say that, because he wants to assign you both your situations and your tolerances.
What if there is no direct, shared experience to which we can refer to describe a variable in a model? Then yes, using a term like 'wave-particle' seems like nonsense and is misleading. And since all such attempts at macro analogies are misleading, we should just use a completely made up word; it's just a placeholder anyway, like a variable name in a program, and we can give it any name we want. The important thing, to the scientific mind, is whether it is useful in prediction, whether the outputs of the program are useful.
There's more than one way to skin a cat. A particular(!) program might be inelegant, like celestial spheres, but it's not wrong, exactly, unless it prevents prediction to within desired tolerances or conflicts directly with some other observations. Science is about the outputs, not the program.
But sure, programmers can get absolutely religiously attached to one algorithm or even coding style. Scientists are no better about that than anybody else, and so science advances one funeral at a time. We tend to trust scientists because of many useful things produced, like computers, and these things are worthy of respect, but the ontological scaffolding is hardly sacred. It is disposable.
So whenever a scientist uses authority or celebrity to claim crap like "there's no such thing as race" you know he's not talking as a scientist but as a priest. A scientific objection would sound something like "Race is not a useful predictor of X in situation Y to tolerance Z."
A priest won't say that, because he wants to assign you both your situations and your tolerances.
Expose the Paranoids!!Did you know that there are Paranoids all around us?Even your next door neighbor might be one!
Prefrontal Vortex posted on 2009-04-14 13:54:15 ET Reply Trace Private Reply
Little more than semantics and existentialism here. Crackpots often insist on inventing their own nomenclatures.
Crackpots often insist on inventing their own nomenclatures.
TooConservative posted on 2009-04-14 14:19:30 ET Reply Trace Private Reply
People need to go to Stephen J. Crothers website to understand the reference of the post and the video. In short: the math "doesn't add up." Crothers does a decent job of explaining it; just bypass the formulas and keep in mind that "Ric=0" proves his point.
In short: the math "doesn't add up."
Crothers does a decent job of explaining it; just bypass the formulas and keep in mind that "Ric=0" proves his point.
The whole aim of practical politics is to keep the populace alarmed (and hence clamorous to be led to safety) by menacing it with an endless series of hobgoblins, all of them imaginary. - Henry Louis Mencken
rack42 posted on 2009-04-14 21:35:18 ET Reply Trace Private Reply
You don't need to know Math to realize that relativity is 100% poppycock. All you need is the nonlocality of quantum entanglement.
You don't need to know Math to realize that relativity is 100% poppycock.
All you need is the nonlocality of quantum entanglement.
MUDDOG posted on 2009-04-14 21:42:16 ET Reply Trace Private Reply
Anyone that thinks time travel is possible by traveling at the speed of light is not too bright no matter how many degrees they have. People that do investigate things on their own and understand things on their own step by step will accept all kinds of crazy nonsense as truth. These people then point to the establishment as accepting it as their "proof" they are correct. Funny. Reality does not care what the establishment thinks, it never has.
God is always good!
RickyJ posted on 2009-04-14 21:57:11 ET Reply Trace Private Reply
Crothers does a decent job of explaining it; just bypass the formulas and keep in mind that "Ric=0" proves his point. I wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Just wondering.
I wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Just wondering.
Dr_Tron posted on 2009-05-10 19:53:08 ET Reply Trace Private Reply
I wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Pulling rank, eh?
I wonder if he understands the rank four Riemann Cristoffel Curvature Tensor.
Pulling rank, eh?
MUDDOG posted on 2009-05-10 21:41:08 ET Reply Trace Private Reply
Pulling rank, eh? Not at all. If I came across that way - my apologies. Just that you cannot debunk relativity if you don't understand it in the first place. That is what I was trying to convey. Doc
Not at all. If I came across that way - my apologies.
Just that you cannot debunk relativity if you don't understand it in the first place. That is what I was trying to convey.
Doc
Dr_Tron posted on 2009-05-11 0:06:51 ET Reply Trace Private Reply
If I came across that way - my apologies. No problem. I couldn't resist the play on "rank."
If I came across that way - my apologies.
No problem. I couldn't resist the play on "rank."
MUDDOG posted on 2009-05-11 0:27:20 ET Reply Trace Private Reply
LOL! OK. :-)
Dr_Tron posted on 2009-05-11 23:07:26 ET Reply Trace Private Reply
Anyone who likes the Christoffel symbols is OK by me.
MUDDOG posted on 2009-05-11 23:22:03 ET Reply Trace Private Reply
In mathematics and physics, the Christoffel symbol describes curvature in a non-euclidean space, such as geometry on the surface of the globe. Whereas the metric tensor describes a sort of first derivative of the warp, the Christoffel symbol describes a sort of second derivative. At each point of the underlying n-dimensional manifold, it is an array with three dimensions: n × n × n. Each of the n3 components is a real number. Under linear coordinate transformations on the manifold, it behaves like a tensor. The Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are often called Christoffel symbols.[1] The Christoffel symbols may be used for performing practical calculations in differential geometry. Unfortunately, the calculations are usually quite lengthy and complex, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is terse, and allows theorems to be stated in an elegant way, but requires more advanced techniques for practical calculations. In many practical problems, most components of the Christoffel symbols are equal to zero. oh my, and the paki cadillac went over my head. ;P
In mathematics and physics, the Christoffel symbol describes curvature in a non-euclidean space, such as geometry on the surface of the globe. Whereas the metric tensor describes a sort of first derivative of the warp, the Christoffel symbol describes a sort of second derivative. At each point of the underlying n-dimensional manifold, it is an array with three dimensions: n × n × n. Each of the n3 components is a real number. Under linear coordinate transformations on the manifold, it behaves like a tensor. The Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are often called Christoffel symbols.[1] The Christoffel symbols may be used for performing practical calculations in differential geometry. Unfortunately, the calculations are usually quite lengthy and complex, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is terse, and allows theorems to be stated in an elegant way, but requires more advanced techniques for practical calculations. In many practical problems, most components of the Christoffel symbols are equal to zero.
The Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are often called Christoffel symbols.[1] The Christoffel symbols may be used for performing practical calculations in differential geometry. Unfortunately, the calculations are usually quite lengthy and complex, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is terse, and allows theorems to be stated in an elegant way, but requires more advanced techniques for practical calculations. In many practical problems, most components of the Christoffel symbols are equal to zero.
oh my, and the paki cadillac went over my head. ;P
The smooth criminal transition from Bush/Cheney to Obama
christine posted on 2009-05-12 1:02:57 ET Reply Trace Private Reply
over my head. ;P That's because the definition you posted is the result of over a hundred years of studying and refining the concept. It's like when Sherlock Holmes presents his conclusions without the intermediate steps. It seems amazing, but when he explains the intermediate steps, it's elementary, my dear Watson. The Christoffel symbols are like that. Their roots are in the study of 2- dimensional surfaces in ordinary three-dimensional space, from the question of how to define the curvature of a surface in space. It's pretty straightforward to define the curvature of a (not necessarily straight) line. A straight line should have zero curvature. The direction of a straight line doesn't change. For a curvy line, the curvature should be the amount it changes direction. In calculus, the direction of a (possibly curvy) line is measured by its first derivative. The curvature should be measured by the change in direction, that is, the change in the first derivative, which is the second derivative. And that's the key to defining the curvature of a surface. It should be something measured by second derivatives in the equations (coordinates) of the surface. Think of it this way. A flat surface in 3-dimensional space is like a tabletop. The direction "outward" (or up) from the tabletop stays the same all across the tabletop; it points toward the ceiling. No change in the outward direction. But now think of the world globe in your study. The "outward" direction at the north pole of the globe indeed points toward the ceiling; but the "outward" direction at the equator of the globe (for a tiny man there!) points toward the wall of your study. The globe is curved -- the "outward" directions vary across the surface of the globe. Mathematically, the "outward" direction at a point of a surface in 3- dimensional space is known as the "normal vector" at that point. And the variation of this normal vector measures the curvature of the surface. If there's no variation of the normal vector, like in the tabletop, there should be no curvature; the surface is flat. If the normal vector varies, like in your world globe, there's curvature of the surface. You have to give a mathematical measure of all this, but that's just how Gauss proceeded 200 years ago in his pioneering studies of curvature. Now let's get to a general surface in 3-dimensional space. The first derivative of the coordinates of the surface gives you a measure of how the surface sits in space at a point of the surface; but the second derivative of the coordinates gives you a measure of how the surface is curving in space. (Technically, the normal ("outward") vector of the surface is given by the "cross product" of the first derivatives of the coordinates, so that the change (or first derivative) of the normal vector involves the second derivatives of the coordinates.) To calculate this changing in the normal vector, which Gauss used as the measure of surface curvature, you can use the Christoffel symbols. They’re defined as follows: When you express the second derivatives of the coordinates of the surface in terms of the first derivatives and the normal vector to the surface, the Christoffel symbols are the coefficients of the first derivatives. The mathematical expression of curvature is then calculated from the Christoffel symbols and is called the Riemann-Christoffel curvature tensor. After this it gets pretty technical (if it hasn't already!). What happens is, first you can generalize to n-dimensional (Euclidean) space, and consider an (n- 1) dimensional hyperspace sitting in there, so you have a normal vector as in our 3-dimensional case, and you take second derivatives of the hyperspace coordinates, define the Christoffel symbols as the coefficients of the first derivatives in the expression for second derivatives, and solve for the Christoffel symbols by using the notion of length in the Euclidean space. Then you calculate the Riemann-Christoffel curvature tensor analogously to the surface case. An interesting special case is a “flat“ hyperspace. The hyperspace is flat if you can find coordinates for the hyperspace for which the "metric" on the hyperspace is particularly simple (Euclidean), which is equivalent to the Christoffel symbols being equal to zero at all points of the coordinates. But there are so many possible coordinate systems you can use, how do you know if there even are any coordinates for which the Christoffel symbols are zero? The coordinates you start with may have nonzero Christoffel symbols. It turns out that no matter what coordinates you start with, if you calculate the Christoffel symbols for those coordinates, then calculate the Riemann- Christoffel curvature tensor from those Christoffel symbols, the tensor will equal zero at all the points if and only if there exists a set of coordinates (not necessarily the ones you started with) for which their Christoffel symbols are zero. So "flatness" is the same as "zero curvature." You can generalize further from hyperspaces in Euclidean space, to "Riemannian spaces," and then generalize to non-Riemannian spaces which have an "affine connection." The tie-in to Einstein's general relativity is he used curvature of space to explain gravity.
over my head. ;P
That's because the definition you posted is the result of over a hundred years of studying and refining the concept.
It's like when Sherlock Holmes presents his conclusions without the intermediate steps. It seems amazing, but when he explains the intermediate steps, it's elementary, my dear Watson.
The Christoffel symbols are like that. Their roots are in the study of 2- dimensional surfaces in ordinary three-dimensional space, from the question of how to define the curvature of a surface in space.
It's pretty straightforward to define the curvature of a (not necessarily straight) line. A straight line should have zero curvature. The direction of a straight line doesn't change. For a curvy line, the curvature should be the amount it changes direction. In calculus, the direction of a (possibly curvy) line is measured by its first derivative. The curvature should be measured by the change in direction, that is, the change in the first derivative, which is the second derivative.
And that's the key to defining the curvature of a surface. It should be something measured by second derivatives in the equations (coordinates) of the surface.
Think of it this way. A flat surface in 3-dimensional space is like a tabletop. The direction "outward" (or up) from the tabletop stays the same all across the tabletop; it points toward the ceiling. No change in the outward direction.
But now think of the world globe in your study. The "outward" direction at the north pole of the globe indeed points toward the ceiling; but the "outward" direction at the equator of the globe (for a tiny man there!) points toward the wall of your study. The globe is curved -- the "outward" directions vary across the surface of the globe.
Mathematically, the "outward" direction at a point of a surface in 3- dimensional space is known as the "normal vector" at that point. And the variation of this normal vector measures the curvature of the surface. If there's no variation of the normal vector, like in the tabletop, there should be no curvature; the surface is flat. If the normal vector varies, like in your world globe, there's curvature of the surface.
You have to give a mathematical measure of all this, but that's just how Gauss proceeded 200 years ago in his pioneering studies of curvature.
Now let's get to a general surface in 3-dimensional space. The first derivative of the coordinates of the surface gives you a measure of how the surface sits in space at a point of the surface; but the second derivative of the coordinates gives you a measure of how the surface is curving in space.
(Technically, the normal ("outward") vector of the surface is given by the "cross product" of the first derivatives of the coordinates, so that the change (or first derivative) of the normal vector involves the second derivatives of the coordinates.)
To calculate this changing in the normal vector, which Gauss used as the measure of surface curvature, you can use the Christoffel symbols.
They’re defined as follows: When you express the second derivatives of the coordinates of the surface in terms of the first derivatives and the normal vector to the surface, the Christoffel symbols are the coefficients of the first derivatives.
The mathematical expression of curvature is then calculated from the Christoffel symbols and is called the Riemann-Christoffel curvature tensor.
After this it gets pretty technical (if it hasn't already!). What happens is, first you can generalize to n-dimensional (Euclidean) space, and consider an (n- 1) dimensional hyperspace sitting in there, so you have a normal vector as in our 3-dimensional case, and you take second derivatives of the hyperspace coordinates, define the Christoffel symbols as the coefficients of the first derivatives in the expression for second derivatives, and solve for the Christoffel symbols by using the notion of length in the Euclidean space.
Then you calculate the Riemann-Christoffel curvature tensor analogously to the surface case.
An interesting special case is a “flat“ hyperspace. The hyperspace is flat if you can find coordinates for the hyperspace for which the "metric" on the hyperspace is particularly simple (Euclidean), which is equivalent to the Christoffel symbols being equal to zero at all points of the coordinates. But there are so many possible coordinate systems you can use, how do you know if there even are any coordinates for which the Christoffel symbols are zero? The coordinates you start with may have nonzero Christoffel symbols.
It turns out that no matter what coordinates you start with, if you calculate the Christoffel symbols for those coordinates, then calculate the Riemann- Christoffel curvature tensor from those Christoffel symbols, the tensor will equal zero at all the points if and only if there exists a set of coordinates (not necessarily the ones you started with) for which their Christoffel symbols are zero.
So "flatness" is the same as "zero curvature."
You can generalize further from hyperspaces in Euclidean space, to "Riemannian spaces," and then generalize to non-Riemannian spaces which have an "affine connection."
The tie-in to Einstein's general relativity is he used curvature of space to explain gravity.
MUDDOG posted on 2009-05-12 15:46:02 ET Reply Trace Private Reply
See post #13 above.
MUDDOG posted on 2009-05-12 15:47:54 ET Reply Trace Private Reply
wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Just wondering. What is it about "Ric=0" don't you understand? Please, dispute it because it seems that Physics has become Mathematics.
wonder if he understands the rank four Riemann Cristoffel Curvature Tensor. Just wondering.
What is it about "Ric=0" don't you understand?
Please, dispute it because it seems that Physics has become Mathematics.
rack42 posted on 2009-05-14 23:19:15 ET Reply Trace Private Reply
Crothers does a decent job of explaining it; just bypass the formulas and keep in mind that "Ric=0" proves his point. No, what you said makes no sense. I doubt that you have any understanding of what he said. The whole premis of "black holes" is that "Ric=0." You've provided nothing to disprove that. Do you even understand what "Ric=0" means? No, you don't. Because it is impossible. Unless you want to propose a new Physics. I'd be very interested in that.
No, what you said makes no sense.
I doubt that you have any understanding of what he said.
The whole premis of "black holes" is that "Ric=0."
You've provided nothing to disprove that.
Do you even understand what "Ric=0" means?
No, you don't. Because it is impossible. Unless you want to propose a new Physics. I'd be very interested in that.
rack42 posted on 2009-05-14 23:31:57 ET Reply Trace Private Reply
The whole premis of "black holes" is that "Ric=0." Do you mean Rij=0 ? That is the field equation for the vacuum space-time. Or in other words a pseudo-Riemannian manifold called Ricci-flat. And this can only be proved if the local manifold is equivalent to flat space. And this disproves black holes how? (Actually it was interesting he spent so much time on how he was kicked out of grad school and how every theoretical physicist for the past 100 years have been wrong)
Do you mean Rij=0 ?
That is the field equation for the vacuum space-time. Or in other words a pseudo-Riemannian manifold called Ricci-flat. And this can only be proved if the local manifold is equivalent to flat space.
And this disproves black holes how?
(Actually it was interesting he spent so much time on how he was kicked out of grad school and how every theoretical physicist for the past 100 years have been wrong)
Dr_Tron posted on 2009-05-17 22:40:39 ET Reply Trace Private Reply
Nice writup. You kinda went to warp factor 10 for the average guy about 2/3 in. LOL! It's been a while since I studied Differential Geometry. Brings back memories. :-) How about tangent bundles. :-) (tangent space over all points of the manifold) And we could really get crazy - Cotangent bundles acting as operators or in other words mapping to the tangent space. Note: you started to get into the realm of coordinate free geometry (when you stated "The coordinates you start with may have nonzero Christoffel symbols". And a second note for the lay readers: When you said; "Mathematically, the "outward" direction at a point of a surface in 3- dimensional space is known as the "normal vector" at that point. And the variation of this normal vector measures the curvature of the surface. If there's no variation of the normal vector, like in the tabletop, there should be no curvature; the surface is flat. If the normal vector varies, like in your world globe, there's curvature of the surface." The question is how do you get the normal vector? It is the cross product of pairs of tangent vectors.
It's been a while since I studied Differential Geometry. Brings back memories. :-)
How about tangent bundles. :-) (tangent space over all points of the manifold)
And we could really get crazy - Cotangent bundles acting as operators or in other words mapping to the tangent space.
Note: you started to get into the realm of coordinate free geometry (when you stated "The coordinates you start with may have nonzero Christoffel symbols".
And a second note for the lay readers:
When you said; "Mathematically, the "outward" direction at a point of a surface in 3- dimensional space is known as the "normal vector" at that point. And the variation of this normal vector measures the curvature of the surface. If there's no variation of the normal vector, like in the tabletop, there should be no curvature; the surface is flat. If the normal vector varies, like in your world globe, there's curvature of the surface."
The question is how do you get the normal vector? It is the cross product of pairs of tangent vectors.
Dr_Tron posted on 2009-05-17 22:53:16 ET Reply Trace Private Reply
Good to see you! The big thing I left out of the previous post was, the Gaussian curvature is only a partial measure of the curvature of surfaces in 3-space. For example, the surface of a cylinder in 3-space has Gaussian curvature zero, even though it's "curved" in the ordinary sense of the word. What makes the Gaussian curvature so special, though, is that it is "intrinsic" to the surface -- that is, it's completely calculable from the metric tensor. That fact impressed Gauss so much that he called it the "Theorema egregium" -- "a most excellent theorem" -- because he had started out defining the Gaussian curvature in the usual manner of looking at the variation of the normal vector, and discovered that the resulting Gaussian curvature actually could be calculated without the normal vector, i.e., without a bigger 3-space in which the surface was "curving." That is indeed an amazing result. And this is why it's the Gaussian curvature, rather than some other measure of curvature, that generalizes so well -- it's not dependent on a larger space containing the surface, in which a normal vector, pointing into some bigger space, determines curvature. (Although the normal vector is a very helpful visual aid!) So, since the Gaussian curvature is intrinsic, it generalizes well to abstract spaces ("manifolds") which need not be defined as imbedded in some other space. So this is part of the reason why, when laymen hear about the curvature of space, it doesn't make much sense because "physical" space doesn't seem to be sitting in some bigger space in which it is curving. But you don't need a bigger space, because the curvature is intrinsic to "physical" space when you use the generalized Gaussian curvature (the Riemann-Christoffel tensor). And the Riemann-Christoffel tensor is indeed a generalization of the Gaussian curvature, not just in the formulas from the Christoffel symbols, but also in that the "surface" Gaussian curvature of the 2-dimensional submanifolds generated by appropriate geodesics is precisely the value of the Riemann- Christoffel tensor evaluated at the two tangent vectors generating the 2- dimensional submanifold.
The big thing I left out of the previous post was, the Gaussian curvature is only a partial measure of the curvature of surfaces in 3-space.
For example, the surface of a cylinder in 3-space has Gaussian curvature zero, even though it's "curved" in the ordinary sense of the word.
What makes the Gaussian curvature so special, though, is that it is "intrinsic" to the surface -- that is, it's completely calculable from the metric tensor. That fact impressed Gauss so much that he called it the "Theorema egregium" -- "a most excellent theorem" -- because he had started out defining the Gaussian curvature in the usual manner of looking at the variation of the normal vector, and discovered that the resulting Gaussian curvature actually could be calculated without the normal vector, i.e., without a bigger 3-space in which the surface was "curving."
That is indeed an amazing result.
And this is why it's the Gaussian curvature, rather than some other measure of curvature, that generalizes so well -- it's not dependent on a larger space containing the surface, in which a normal vector, pointing into some bigger space, determines curvature. (Although the normal vector is a very helpful visual aid!)
So, since the Gaussian curvature is intrinsic, it generalizes well to abstract spaces ("manifolds") which need not be defined as imbedded in some other space.
So this is part of the reason why, when laymen hear about the curvature of space, it doesn't make much sense because "physical" space doesn't seem to be sitting in some bigger space in which it is curving. But you don't need a bigger space, because the curvature is intrinsic to "physical" space when you use the generalized Gaussian curvature (the Riemann-Christoffel tensor).
And the Riemann-Christoffel tensor is indeed a generalization of the Gaussian curvature, not just in the formulas from the Christoffel symbols, but also in that the "surface" Gaussian curvature of the 2-dimensional submanifolds generated by appropriate geodesics is precisely the value of the Riemann- Christoffel tensor evaluated at the two tangent vectors generating the 2- dimensional submanifold.
MUDDOG posted on 2009-05-17 23:25:32 ET Reply Trace Private Reply
Do you mean Rij=0 ? No. You're trying to divert. Bad try.
No.
You're trying to divert. Bad try.
...with the power of conviction, there is no sacrifice.
rack42 posted on 2009-05-18 22:34:03 ET Reply Trace Private Reply
I did some looking. Instead of slogging thru the BS myself I found others had already done so: http://www.physicsforums.com/showthread.php?t=123802 http://www.physforum.com/index.php?showtopic=12491&st=15 Saved me a lot of time.
http://www.physicsforums.com/showthread.php?t=123802
http://www.physforum.com/index.php?showtopic=12491&st=15
Saved me a lot of time.
Dr_Tron posted on 2009-05-19 8:06:27 ET Reply Trace Private Reply
RE: truth2k that person is, um, I don't know what. Let's say confused. Link to truth2k in any capacity is suspect in the extreme. RE: Other Link I don't see that anything is resolved. Crothers defends his hypothosis: There is, I submit, a fundamental conceptual error with the now commonplace claims for black hole binaries, black hole collisions, and black hole mergers. By what rigorous means do the scientists maintain that the supposed problems are well posed? In other words, by what rigorous arguments do they hold that black hole binaries or black hole collisions are well-defined scenarios? I remark that the black hole is alleged to be a consequence of Einstein’s General Relativity. Assuming for the sake of argument that this is correct, it is evident that the black hole is the result of a solution to Einstein’s field equations for the configuration of a single spherically symmetric gravitating body interacting with a test particle in vacuum. It is not a solution for the interaction of two comparable masses, such as two black holes. Before one can talk of black hole binaries or black hole collisions it must first be demonstrated that Einstein’s field equations admit of solutions for multi-body configurations of gravitationally coupled spherically symmetric comparable masses. This can be done in two possible ways, in principle: 1) by deducing a particular solution, or 2) proving an existence theorem. There are however, no known solutions to the field equations for the interaction of two or more spherically symmetric comparable masses, so option 1) has never been met. In fact, it is not even known if Einstein’s field equations admit of such solutions as no existence theorem has ever been adduced, so option 2) has never been met either. Furthermore, one cannot simply assert by analogy with Newton’s gravitation that a black hole can be a component of a binary system or that black holes can collide or merge. Consequently, all talk of black hole binaries, black hole mergers, black hole collisions, etc. does not deal with well-defined problems at all. The now commonplace claims and arguments by a great many investigators are meaningless. They are without any scientific justification. One cannot test General Relativity by means of a chimera. The responses are in the same vein that "Einstien predicted blackholes and the Schwarschild solution prove it." It does not. It is evident that not of the posters read Crothers hypothesis and don't address the issues. Just keep harping on "Schwarschild radius" and the false conclusions from that because it is obvious that they, and you, haven't read Crothers hypothesis. To me, Crothers hypothesis trumps the "black hole" and "Big Bangers." Look at the MATH. GEEZ.
RE: Other Link
I don't see that anything is resolved.
Crothers defends his hypothosis:
There is, I submit, a fundamental conceptual error with the now commonplace claims for black hole binaries, black hole collisions, and black hole mergers.
By what rigorous means do the scientists maintain that the supposed problems are well posed? In other words, by what rigorous arguments do they hold that black hole binaries or black hole collisions are well-defined scenarios?
I remark that the black hole is alleged to be a consequence of Einstein’s General Relativity. Assuming for the sake of argument that this is correct, it is evident that the black hole is the result of a solution to Einstein’s field equations for the configuration of a single spherically symmetric gravitating body interacting with a test particle in vacuum. It is not a solution for the interaction of two comparable masses, such as two black holes.
Before one can talk of black hole binaries or black hole collisions it must first be demonstrated that Einstein’s field equations admit of solutions for multi-body configurations of gravitationally coupled spherically symmetric comparable masses. This can be done in two possible ways, in principle:
1) by deducing a particular solution,
or
2) proving an existence theorem.
There are however, no known solutions to the field equations for the interaction of two or more spherically symmetric comparable masses, so option 1) has never been met. In fact, it is not even known if Einstein’s field equations admit of such solutions as no existence theorem has ever been adduced, so option 2) has never been met either.
Furthermore, one cannot simply assert by analogy with Newton’s gravitation that a black hole can be a component of a binary system or that black holes can collide or merge.
Consequently, all talk of black hole binaries, black hole mergers, black hole collisions, etc. does not deal with well-defined problems at all. The now commonplace claims and arguments by a great many investigators are meaningless. They are without any scientific justification. One cannot test General Relativity by means of a chimera.
The responses are in the same vein that "Einstien predicted blackholes and the Schwarschild solution prove it."
It does not.
It is evident that not of the posters read Crothers hypothesis and don't address the issues. Just keep harping on "Schwarschild radius" and the false conclusions from that because it is obvious that they, and you, haven't read Crothers hypothesis. To me, Crothers hypothesis trumps the "black hole" and "Big Bangers." Look at the MATH. GEEZ.
rack42 posted on 2009-05-23 21:53:37 ET Reply Trace Private Reply
Ok. I will look at the math. And with an open mind. Give me a week.
Dr_Tron posted on 2009-05-25 2:28:58 ET Reply Trace Private Reply