MetaFrizzics

Oh, yeah. There was no water - it was space after all. Those were panties? I thought it was a nose-plug.

Wow. I thought your bollocks actually shrank when you did body-building!

That's the whole point. His case was prejudiced by the behaviour of others, and the stereotypical bigotry of those arresting and convicting him. And the law has a lot of leeway: Its all how cops handle things at the scene, how D.A.s feel about the merits of a case, and how judges feel about their favourite points of law. And lets not forget money or politics. Would OJ or MJ have been convicted? If the girl had been black, dressed as a hooker in the red light district and carrying a razor and he was a white-collar IBM employee would he have been convicted? And yet would his story be more plausible or less in that case?

(1) What if he'd said something ruder, like, "Come here, you idiot!" He wouldn't have been convicted if saying "little girl" is what got him the 'sex-offender' label. But why shouldn't he have used 'little girl' prejoratively (as a mild disapproving insult) since she was really 14 and not 'little'. That is, he could have been saying "the way you're behaving regarding road safety is like a little girl (= uneducated or unable to grasp the danger)." So he may have been punished for using a common epithet for a young woman who is acting foolishly. But in these post women's lib times, talking down to a woman can put an extra prejudice into your case! (2) Now "citizen's arrest" will be impossible in the case of juveniles under 18. If you can't restrain someone after being eyewitness to a crime, because they are under 18, it's one more essential right and priviledge eroded away. Already treating murderers as 'kids' when they are under 18 was a disaster. The mafia just hired underage 'gangstas' to do their killing, and get light sentences. Now, if you see a punk stealing, assaulting someone or even killing someone, but it's only your word as a citizen against theirs, you could now be a sex-offender. Scenarios: 14 year old girl in Walmart tries to grab some videos off the shelf. You hold her and call for security. She says you grabbed her ass and called her "little girl". So much for trying to minimize store theft to keep down prices. 13 year old boy steals car, drives all over the road, then parks and flees. You pull over and grab him. He says he knows nothing about a car, and you offered him hockey tickets and a bottle of wine. Good luck. The cops will like it, because more than ever, only 'they' have the right to participate in law enforcement. People will be more disenchanted than ever. Hmmm. I could get a citation or award for being a hero, or I could have to register as a sex-offender. Think the carrot and stick have been confused here, and I'll have to pass on stopping the repeat offender mugging the old lady.

In that case, Div, Grad, Curl, and All That by Schey might be the better book. It is a small and informal but speedy and complete text on vector calculus meant to supply the missing course for those in the sciences. Typically other textbooks or coursebooks only cover a brief set of basics, leaving the student to find out details on his own via secondary texts on mathematical methods for engineers.

Yes. Originally Newton's theory was not conceived or written as a 'field theory'. Quaternions had not been discovered by Hamilton or used by Maxwell, and there was no 'electromagnetism'. Nowadays Newtonian gravitation is normally modelled in the same way as the electrostatic field of classical electromagnetism. In this way of handling things, there are two fields: the force (F) and the potential energy (E). The 'field' (force) is said to be 'zero'. That is, since the forces balance at all times inside the sphere, and the rate of change of the force is zero in every direction until you reach the edge, the field is flat. By convention this is state is chosen for 'zero' on the y-axis (force). Yet since the 'potential' field (E) has no absolute reference, the value at infinity is usually taken to be 'zero', and the plateau on the graph inside the sphere (spherical shell) is actually a 'negative' value. I prefer in these graphs to set the fields at 'zero' inside the sphere at its geometric centre. I do this by placing the sphere at the origin, ( 0,0,0 ), and letting the x-coordinate represent the position of the test-particle. This allows the sign of the x-coordinate to handle the direction of the pull. In gravity there is only one version of the force (attraction), unlike in electrostatics, and this is a handy convention. The equations for the field values are actually 3-dimensional equations, but since in the case of the sphere the fields are radially symmetrical about the centre of the sphere (have the same value at a given distance in all directions) the equation can ignore the coordinate system, and is simply expressed as based upon the absolute distance from the sphere centre. But one should keep in mind that the reason the force is 'zero' is that the forces pulling in every direction are balanced, regardless of the actual pull from various sources in each direction. The subtle difference in concepts between 'zero' and 'balanced' can be imagined by suspending a ball by elastics stretching outward in all directions and fixed to a rigid cage. Although the force is zero for the ball, there is in this case a difference between the ball floating free, and the ball suspended, and its behaviour is quite different. In this case the causes and explanation for the force being 'balanced' and therefore 'zero' must be kept in mind, for if those causes were unreliable or invalid, the field for the force would not be 'zero'.

Newton's Argument In his Principia Newton claimed that a hollow sphere exerts no force on a particle floating inside (Propositions 70 to 74 of Book I). His argument went as follows: (1) Imagine a test-particle at the vertex of a cone of fixed acuteness. You can picture the cone as a flashlight beam coming from the particle. The cone makes a circle on a flat sheet of uniform density and thickness. Moving the sheet further away makes a larger disk on the sheet. The size of the disk increases as the square of the distance (d^2) and so does the mass, while the gravitational force of the disk on the test-particle decreases as the square of the distance (d^2), cancelling out any change. That is, at any given distance, the cone traces the size disk needed to keep the attractive force constant. (2) Now imagine the particle is between two parallel sheets: You can picture two equal opposing cones beaming out from your particle like a lighthouse beam. (Equal cones can be made by rotating one line passing through the particle around another line-axis through it.) Either sheet or the particle can be moved independantly, and the force from the disks will remain equal and balanced, as long as the sheets stay parallel. (3) Similarly, opposing cones will cut out opposing spherical caps on the surface of a hollow sphere if we put our particle and its cones inside. The arbitrary position of the particle and the varying distance of the caps is compensated by their size and mass. Newton proposed constructing more cone pairs, covering the entire surface and showing that the forces inside balance. (4) Someone might object that the spherical caps marked by the cones aren't really flat disks. Newton responds by shrinking the cones to very small (infinitesimals) so that we can make the caps as flat as we wish. (5) Another might ask how you can cover the sphere surface completely with circles. Again, Newton argues that spaces between disks can be filled with smaller circles and so on. We can iterate the procedure as many times as we wish to closely cover the surface area. This concludes Newton's argument and 'proof' for the Sphere Theorem. Fire away with any questions, and I'll try to answer them.

Thanks! I was falling asleep at the end of that. It should be fixed now. I think the best way to handle that problem is tell you when I'm done. In this case I had one more post, but it's not important. It was better to catch any small errors like the one you noticed. Thanks again.

They should look after you. Any effort at security for movement and activities of Armed Forces is totally defeated by having you ask civilians for a lift.

Historical Note Of course atomic theory was hardly developed beyond Ancient Greek notions at the time Newton published his Laws of Motion and Gravity Theory. In the process of developing his theory, Newton practically invented calculus, and his approach was a basic treatment of solid objects as continuous distributions of mass. The incredibly small size of atoms and molecules meant that Newton's methods were more than adequate for objects the size of billiard balls and larger, involving millions of atoms. Theorem? Historically, the Sphere Theorem has been called such in the literature. Here we don't care about the relative value of name-tags, and we don't wish to debate upon the merits of 'theorem' versus 'method' or 'theory'. No such precision is required, and discussion on that just degenerates into semantics. Instead we wish to focus on the content of Newton's idea, and its physical implications and value. To prevent any misunderstanding, this is a physics theorem meant to be applied to physical problems. This requires more than just some simple math. It also requires a physicist's ability to intelligently apply and interpret the physical significance of the theorem. A little of column A, and a little of column B. There is no gain in treating it as a 'mathematical theorem'. Without the reference to gravity and mass, it is contentless. In fact, there would be nothing to 'integrate' without a focus to direct forces. That is, there is no purely 'mathematical' version of the Sphere Theorem. Another side-point is Gauss' Law: Of course Gauss' Law is a mathematical theorem in the sense that it can be used to describe 'field lines' before we assign those lines any physical significance. But again in this case, Gauss' Law is really not a 'proof' of Newton's Sphere Theorem, which is a physical theorem, and does not exist without a physical application. This is a common confusion regarding both mathematical 'proofs' in general, and 'proofs' of Newton's Sphere 'Theorem' (method) in particular.

Newton's Sphere Theorem: The gravitational net force on a test-particle inside a uniform density hollow sphere is zero, and the particle is free to float in any direction in the uniform field inside the sphere. The gravitational field inside a solid sphere is usually constructed by relying upon this theorem to build a solid sphere out of concentric hollow spheres. (The test-particle is a small mass that is effectively a point-particle relative to the size of the sphere.) There is no expectation that the sphere can block external forces outside from reaching the particle, and in the case of gravity, there is no masking or blocking of force possible. The theorem assumes a uniform field, or else just addresses the contribution to the field from the sphere only. (In Electromagnetic Theory, shielding is possible but not relevant here.) As a consequence of the Hollow Sphere Theorem, a uniform density (or radially symmetrical) solid sphere also acts as if all its mass is concentrated at the geometric centre (gc), and has a unique (linear) Gravitational field inside it. A special case of the Centre of Mass method, the Sphere Theorem is presumed to be exact.

This will be the first of a few threads which will lay the groundwork for a discussion of Newtonian gravity theory old and new. I am splitting the concepts into threads so that each thread can stand alone as a good introduction and reference for the mainstream scientific position. Here we will limit discussion to known and accepted scientific knowledge, and I will present everything at a high-school level so that virtually anyone can understand the concepts with a little effort. I will also refrain from straying from the status quo without the support of clear mathematical proofs or simple thought experiments that are uncontroversial. Finally, for this thread I plan the following outline, to save those browsing from going any further if they are not interested: (1) Newton's Theorem and its Interpretation (2) Newton's Proof (3) Disproof of Newton's Proof (4) Modern Proof (5) Failure of Modern Proof and Significance Please refrain from posting until each section is finished, so you get a chance to see the whole picture. Some of your questions will probably be answered as the thread fleshes out. Just post questions at the end of each part and I will adress them before continuing to the next section.

Wait! You need more female characters, and I have my own costume: Please see if you can fit me in!

Checks and Balances: I have stripped the superfluous anecdotes and comments which were objected to previously. The post was obscured by the 'remotely likely' and rather subtle example of people challenging opinions with an alterior motive: (possibly to provoke emotional reaction or bad behaviour from a poster who is disliked, and so get them blackmarked or banned). When the cards are stacked, its unlikely any poster with an opinion could win for long in such a situation. But let us suppose there is no conspiracy, and the moderators and administrators for the most part simply want to run an organized professional looking forum. So let's stick to the best point in Kirby's post: Giving (unsupported) opinions and discussing issues on the fringe of science is an absolutely necessary part of doing science. "...It's not the side you are on. It is the way you present your opinions.". As much as I strive for the exact same ideal, I just can't agree with this on the face of it. If we're just going to 'stick to mainstream science', and disallow opinion and discussion, why have a forum at all? Why not just post a bunch of stickys: The point is, all good questions and opinions will have little or no support from 'mainstream' scientific papers, and can't be 'reviewed by peers' before being offered. Physics forums ought to be the very places where ideas can be exchanged and criticised in a free and open manner. It is not a 'waste' of cyberspace or an 'abuse' of a science forum to discuss the problems of science. And one should not have to be an expert or have prepared 'evidence' for every thought. If anything, saying something might bring forward a quick correction and save years of misunderstanding or wasted efforts. That is what being helpful is all about. The only thing any person at any level needs to bring to a physics forum is open-mindedness and perhaps humility, or at least enough humility to maintain good manners in the face of criticism or opposition.

Two more great little books on Fourier: (1) A Student's Guide to Fourier Transforms J. F. James (2) An Introduction to Lebesgue Integration and Fourier Series Wilcox & Myers These two are more modern, and contain some useful new methods as well.

But this isn't quite true now. When Newtonian gravitation is reformulated as a field theory, it is quite reasonable to have waves after the electromagnetic model. In fact GR is an example of a 'field theory' version of gravity, albiet not Newtonian. But Minkowski Space is a natural consequence of Newtonian Mechanics. Velocity addition Isomorphic in Newtonian and SRT There is nothing wrong with a modern formulation of Newtonian Space-time that allows for gravitational forces (waves) propagating over time. Galilean & Newtonian Space-time In fact there are several models that postulate gravity as an electromagnetic side-effect.

Here is a popular math site: http://www.artofproblemsolving.com

Not a problem. You are the best judge of how to spend your own time. This would be my purpose too.But it doesn't get more mainstream/introductory than Newtonian gravity. Maybe I could start a Q and A thread on that.

Ill take (b) for $200: Graph for a hollow sphere: The Failure of the Sphere Theorem for a discrete number of particles:

I'm not disputing the basic facts of the 2nd law of thermodynamics. However, the theoretical foundation of the 2nd law is a scandal. Of course Boltzmann's 'irreversibility' of the evolution of his closed system is a self-deception, and attempts to remove the reversiblity of mechanistic models are just hand-waving. As you have pointed out, the only hope for an 'arrow of time' lies in Quantum events. But I wouldn't invest all my eggs in the weak interaction, when we don't even have a GUT. Not only is our best theory QED reversible, it seems impossible to formulate it without particles that move backward in time. The best option we have at the moment is the Transactional Interpretation of QM, which allows backward time travel but controls it with an iron fist and preserves both causality and objective reality. If you can formulate a version of QM that is coherent, complete, and fixes the arrow of time without violating SRT I will be quite impressed.

I could not agree more: But I have two things to add: (1) In extending Feynman's program for QED, both his collaborator and student Carver Mead has taken QED to a new level in his small book, Collective Electrodynamics. (2) The Transactional Interpretation of Quantum Mechanics puts previous interpretations to shame by maintaining the objective coherence of Bohm's work extending Einstein's ideas, and fully modernizing and harmonizing it with Feynman's time-symmetrical approach. As a bonus, the Transactional Interpretation has consequences that involve empirical tests that have already been confirmed.

Please, I think we can all stay calm for a few posts. Its just a theoretical chat in the classical mechanics section. Let me take your concerns one at a time: This is quite true and no real secret.While the 'mathematical proof' for the Sphere Theorem is correct as a mathematical theorem, it naturally breaks down due to the discrete spacial quantization of mass. I'll be happy to start another thread on this do demonstrate the point to anyone's satisfaction. But I won't make the effort either if you are just going to say something like, "well, okay, but so what. It's not important." If you think the Sphere Theorem is an exact and accurate physical theory (or law, or method), then say so now, so there is no humming and hawwing later. The Centre of Mass concept will also be inevitably involved, so lets be clear on this too. It is admitted at this time to be an approximation (albeit a good one) even by introductory physics texts. My point will be that the error terms are inadequately quantified in current theory or at least standard presentations. And I will be happy to show just how wildly inaccurate and unpredictable the CM method is for gravitational forces. But if I do, don't just kiss it off as 'we know its an approximation'. Both concepts fail in special cases (cases where QM effects become important), for different reasons entirely. Well, you are certainly right in that this just begs the question. It depends upon one's view of the causal explanation of Spin. By no means is this 'known' according to current QM theory. It is only described. I think this is a bit unfair: if you read my posts so far more carefully, you'll see that I, like you think current formulations of Newton/Galilean Gravity theories are inadequate. I don't dispute Newton's lack of knowledge of 20th century atomic theory. This also is unfair. Gauss' Law is strictly speaking a mathematical theorem. That it seems to work well with Faraday's 'field lines' in Classical Electromagnetic Theory is not disputed either. As a 'proof' of the physical accuracy of Newton's Sphere Theorem, however, it is a dud. I can integrate the Sphere as well as the next man. The issue is not mathematical accuracy or beauty but whether or not it applies in a given physical experiment. So tell me how you want to handle things, and I'll cooperate if I can.

I think I would say it this way: Previous 'field-theory' formulations of Newtonian gravity are premature and fatally flawed. They have been made without looking far enough into the phenomenae actually extant, and without seeing the implications of fundamental axioms of Newtonian gravity itself. If some of the side-effects and ramafications of Newtonian gravitational theory were analyzed more deeply, a different field theory/formulation would result. So to answer your question, yes I think I am implying more than just the surface statements, and suggesting a second look at just what 'Newtonian' axioms require from a field theory.

The measurement of the 'speed of light' is an illusion caused by an incomplete understanding of the phenomenon being 'measured'. The 'speed of light' does not conform to normal definitions and expectations of a 'speed'.

20th century physics can be described by a 'state'. The 'state' it is in is one of incomplete knowledge. One flaw in the picture many people tend to form of space-time and physics is a too-literal interpretation of Minkowski Space. Since we can easily plot one or two dimensions of space along with a dimension of time, and create a 2 or 3-d picture of events and exchanges in space-time, it appears that motion is just an illusion. One can take a deterministic stance and imagine that everything is fixed actually, and causally determined. Given that physical laws appear to be 'reversable' in time, and the arrow of time has not been satisfactorily defined or 'proven', we can postulate that our perception of the present, and our notion of time is just an illusion, or an artifact of our reasoning process. This has been seriously proposed by many physicists. But just because the drawing on our desk has no motion, it is wrong-headed and illusory to think that motion itself and our experience of time is an illusion. Even Einstein (and many others) was mesmerized by his 4-dimensional representation of space-time. It seemed he had proven to his own satisfaction that 'God doesn't play dice.'. But I think this view was just an inevitable but naive first interpretation of a set of theories that were poorly understood for the first 100 years of their existance (and probably still are). Time, and our perception of past, present, and future, is as real as anything else in our world of experience, and one would have to smoke biker weed to stare at a diagram of Minkowski Spacetime and be fooled for very long. The fact that the arrow of time and our understanding of past/present/future is poorly understood is not a sign that these are illusions, but rather a sign of the provisional and incomplete nature of our physical theories.