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Everything posted by DrRocket

  1. I doubt that you will get any single well-defined set using that method. You could, for instance simply start selecting points from among the extreme points of a given convex set, say a circle. That could continue indefinitely and you could wind up, in the limit, with any contable subset of the circle. You could do exactly the same thing with a parabola. I have not thought about this in depth, but I think the following might also work. Take an equilateral triangle and consider the barycenters. Then take the barycentric subdivision and the consider the vertices. Continue. I think you will wind up with a dense subset of the interior with no 3 points being colinear. You would have to check this one rigorously as it may be wrong. Edited to correct the last (tentative) idea -- replaced vertices with barycenters.
  2. The "relativity of simultaneity" is a statement regarding the surface of constant time in coordinatizations of spacetime corresponding to observers in relative motion. It is usually somewhat loosely stated in textbooks. But it is something that is derived from the Lorentz transformations rather than something that implies those transformations. I know of 3 ways to arrive at special relativity or its equivalent. 1) The traditional way seen in introductory textbooks, which is to postulate that the speed of light is constant in all inertial frames and the equations are the same in such frames. One then goes through some physical reasoning to arrive at the usual Lorentz transformations. This has the advantage of presenting the physical reasoning that was the historic path to the special theory of relativity. 2) Simply choose a reference frame and by fiat demand that all other reference frames are related to it by the usual Lorentz transformations. If you call the chosen frame the "ether frame" this is the Lorentz Ether Theory. It is not usually taught, but it is equivalent to special relativity as it provides EXACTLY the same predictions. The reason that it is not taught is that there is no particular point to teaching it. It does nothing that special relativity in its usual form does not do, and is philosophically less satisfying. It does make clear that special relativity did not really prove the non-existence of an ether, but rather made its existence or non-existence irrelevant. 3) Start with real 4-space and the Minkowski metric. Then determine those transformations that preserve the metric. One notes that these transformations also preserve "light cones" and one can define a positive direction for time (arbitrarily chosen from two possibilities). The metric preserving transformations that also preserve the positive direction of time (orthochronic transformations) are the Lorentz transformations used in physics. This path is rather formal, but has the advantage of revealing deep connections between geometry and the physical theory. It also has the advantage that it is clearly related to the underlying mathematics and makes the mathematical consistency of the special theory of relativity clear (assuming the consistency of ordinary mathematics). There is a large advantage to understanding all three approaches, as it can make clear what is fundamental and what is not, and because for any given issue one perspective may be more enlightening than the others. It is also worthwhile noting that it is sufficient to find any phenomena that propagates at a fixed speed in all inertial frames. That speed, acall it "x" then plays the role of "c" and one gets the usual Lorentz transformations with "x" in place of "c". This shows that there can be only one invariant speed --- a fact that can be derived in reverse from the Lorentz transformations. The fact that the speed of light in a vacuum fulfills that requirement results from experiment or from Maxwell's equations.
  3. Bold added. Herein lies a major problem inhibiting your understanding. The phrase "time as the "underlying dimension" in which all other dimensions exist" demonstrates a fundamental lack of understanding of the term "dimension". Dimensions do not exist within one another, and in fact the idea makes no sense whatever. To begin to understand dimension you should start with a course on linear algebra in which you will find the notion of dimension of a vector space discussed. Ordinary Euclidean space is a vector space (once you choose an origin) with a bit of additional structure. Following that you should study a bit of topology in which you will find that Euclidean spaces of different dimension are topologically different, and that the algebraic notion of dimension carries over to the topological setting. Basically, a dimension is a "degree of freedom" required to describe something. So, if you wish to schedule a meeting you need to specify where the meeting will be held (3 spatial coordinates) and when it will be held (1 time coordinate). Thus a complete description of your meeting is a point in 4-dimensional space. In special relativity this becomes a bit more sophisticated in terms of how the coordinates fit together and the metric imposed on 4-dimensional spacetime. In general relativity one takes another step up in sophistication and deals with a manifold rather than a vector space. The bottom line is the the comic book concept of dimension has nothing to do with the meaning of the term in mathematics and physics.
  4. To be correct, go read my post. You can derive the Lorentz transformation from the constancy of the speed of light. You can derive the constancy of the speed of light from the Lorentz transformations. It is a matter of taste which proposition you take as being fundamental. They are logically equivalent.
  5. Wrong The singularity theorems of Penrose and Hawking show that, given the observed expansion of the universe and a minimal amount of matter (also consistent with observation) that spacetime is singular in the sense that timlike geodesics are not indefinitely extendible into the past. There is NO singularity that is a part of spacetime itself. There is NO statemet that "our universe sprang into existence as "singularity" around 13.7 billion years ago". In fact current theory has absolutely nothing whatever to say about the actual moment of the big bang (i.e. t=0). It is generally believed that the singular nature of spacetime is a reflection of the failure of general relativity to describe the physics of the very early universe, an era in which quantum effects as well as gravitation played a strong role. There is no accepted existant theory capable of handling quantum effects and gravitational effects simultaneously. Before you criticize a theory, the first step is to understand what that theory actually says.
  6. Thank you very much. The spell checker must have missed that. That negative rep from you means as much to me as a positive rep from Terry Tao.
  7. If you start with "c is invariant", plus the laws of physics are the same in all inertial reference frames, then you can logically deduce the Lorentz transformations that describe length contraction and time dilation. This is the historical train of logic followed by Einstein in the original discovery of special relativity. On the other hand if you start with the Lorentz transformations, i.e. with length contraction and time dilation, then you can deduce that c is invariant. So, yes, time dilation and lenght contraction can indeed explain why "the answer is C in both cases." The two assumptions are logically equivalent. As noted earlier the invariance of c also comes from Maxwell's equations of classical electrodynamics, so on e might say that invariance of c has an established foundation in the physics that was known prior to the special theory of relativity. However, historically the interpretation prior to Einstein was that electromagnetic waves propagated through a universal "aether" that provided an absolute reference frame for Newtonian mechanics. Now we know better.
  8. I am not surprised that you don't understand. Repulsive models for gravity have been proposed, evaluated, and debunked ad nauseaum. Yours is neither the first nor, by a long shot, the most novel. Nevertheless, it remains nonsense. I share your regret that you might ever have been considered a rocket scientist.
  9. That is a good place to start. A very good introduction to Hilbert spaces, not particularly oriented towards quantum mechanics, is Introduction to Hilbert Space and the Theory of Spectral Multiplicity by Paul Halmos. It is a very short and readable book. It can be found on the used book market (Alibris.com, Amazon.com. Abebooks.com) for about $10 if you search a bit.
  10. So what ? I made no claim that all sets that have the property that no three points are colinear are the extreme points of a convex set, only that the the extreme points of a convex set have that property. In fact, take a triangle. The vertices are the extreme points and no three are colinear. But you can also add to the set of extreme points the barycenter (or any interior point) and still have a set of points with no three being colinear. And that set is not the set of extreme points of a convex set. You can also take a shape with vertices which have that property, say a five-pointed star, and take the convex hull of those vertices, a regulat pentagon, and you are right back to the extreme points of a convex set. huh ? The question was posed as a question in dimension 2, the plane. The notion of convexity extends to any vector space, even one of infinite dimension. However, higher dimensional examples do not appear to address the original question. I don't understand your statement regarding every set of 4 points. It is quite easy to find 4 colinear points -- start with a line and pick 4 points on it. If you intended to say that given any cardinal number between 3 and the cardinality of the real numbers, there is a subset of the plane having that cardinality having the property that no 3 points are colinear, then my example of a circle provides the necessary example as the circle or any subset of the circle with 3 or more elements has that property. No problem.
  11. The extreme points of a square are the corner points. No three of them are colinear.
  12. Proof was not requested. Evidence was requested. Evidence was provided. Proof applies to mathematics, not science. There is no such thing as proof in science, only evidence.
  13. A tea pot, virtually by definition and certainly because it contains tea, is composed of ordinary matter. Ordinary matter interacts with photons and is not invisible. Therefore there is no invisible teapot orbiting Neptune.
  14. Bob Smith is an expert on this subject. See his web page,and the links contained there. http://www.uusatrg.utah.edu/RBSMITH/public_html/rbs-home.index.html
  15. You can, of course define e in any fashion that results in the number that is universally recognized as e. However the usual definition comes in terms of the exponential function which is defined by the power series [math] exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}[/math] and [math] e = exp(1)[/math]
  16. The extreme points of any closed convex set would satisfy that criteria, so in particular the points on a circle would be one such set.
  17. rubbish New age science is not science. Imagination in science is distinct from hallucination and fantasy.
  18. SR also makes precise the notions of a past light cone and a future light cone. The Lorentz transformations used in physics are those linear transformations that preserve the Minkowski metric (i.e. preserve the spacetime interval) and that also preserve the past and future light cones (so called orthochronic Lorentz transformations). I think it reasonable to say the the past of an observer is the past light cone of that observer. But if the discussion turns toward philosophy this nice tidy mathematical construction does indeed get all mucked up. You are right, I don't much care. I find that rep points are inversely related to the technical content of a post, and this is not really an exception. Yes "Time is what clocks measure". That is the best that we can do, but it is not entirely satisfactory. Maybe someday there will be an answer with deeper insight, but I have no idea what that answer might be.
  19. That depends on what you mean by "possible". If you are using the word with its common everyday meaning, then the answer is no.
  20. 1) Relativistic mass has been taught at all levels for decades. The recent fashion to treat "mass" as rest mass is indeed fairly recent. In fact the formulas using relativistic mass are more easily remembered than those that use a combination of rest mass and momentum. [math]m= \gamma m_0[/math] and [math]E=mc^2[/math] vs [math] m=m_0[/math] and [math] E^2 =( mc^2)^2 + (pc^2)[/math] Which of course are equivalent as one sees by substituting [math] p = \gamma m_0 v [/math] You also get the formula for force that is famililar from Newtonian mechanics [math]F= \frac {d(mv)}{dt} = \frac {dp}{dt}[/math] 2) Invariant mass is not a concept from general relativity. It has nothing to do with general relativity. In fact in general relativity it can be rather difficult to define the term "mass" at all. Invariant mass is a concept from special relativity. It the concept of mass that is used, usually implicitly and without remark, to describe macroscopic objects, since that is what labratory scales actually measure. http://en.wikipedia....eral_relativity http://en.wikipedia....cial_relativity I have found that one of the surest ways to confuse those just learning a subject is to make positive, but wrong, statements that must be corrected later. Better to tell them that there are several useful notioins of "mass", but that for such and such a particular class we will just use this one (insert whatever you intend to use). That alerts them that there are alternatives, but does not require them to grapple with all of them simultaneously. Mass in special relativity is a particular problem because of the fame of the equation [math] E=mc^2[/math] which is stated in terms of relativistic mass.
  21. That depends on the physicist and the problem at hand. Mass is not a cut-and-dried concept and the various different definitions of "mass" each have a useful place. There are three common uses of "mass" in physics. 1) Rest mass. This notion of mass, [math]m_0[/maath] or [math]m[/math] when taken in a fixed context, is what is most often seen in the context of quantum field theory where the rest mass of an elementary particle is particularly important. It is also commonly seen in elementary texts on special relativity. It has the disadvantage that rarely is a frame available in which all particles are at rest, and therefore is problematic when dealing with systems of particles and in particular with macroscopic bodies composed of elementary particles in which case the sum of the (rest) masses of the particles does not equal the measured mass of the body (at the nuclear level the rest masses of the constiuent quarks are much less than the mass of the nucleons which they comprise). When applied to macroscopic bodies the notion of rest mass has the disacvantage that the rest mass of the composite system is no the rest mass of the particles that comprise it. 2) Relativistic mass. This notion of mass [math]\gamma m_0[/math] or [math]m[/math] when taken in an understood context, was advocated by Tolman as the concept most deserving of the term "mass", probably because the equation [math] F=\frac {dmv}{dt}[/math] holds in special relativity when [math]m[/math] is the relativistic mass. It makes the simple and memorable formula [math]F=mc^2[/math] valid. A significant disadvantage is the difficulty in addressing the energy of particles of zero rest mass, such as photons. It has the advantage that, as it is essentially total energy, the relativistic mass of a system of particles is the sum of the masses of it parts. 3) Invariant mass. This notion of mass is used for systems of particles, quite commonly in accelerator experiments, and is the relativisitic mass of the system when measured in coordinates in which the net system momentum is zero. Invariant mass of a macroscopic object is what is measured on a laboratory scale (and yes a object when heated gains mass in this context). As invariant mass corresponds to total system energy the formula [math]E=mc^2[/math] is also valid when mass is interpreted as invariant mass. While that formula is true, it is simply untrue that the mass that a macroscopic object has in a frame at rest with respect to that object is meaningfully related to the the rest mass of its constituents, but rather is the invariant mass of the system of particles that comprise it. The calculation of invariant mass requires knowing the relativistic mass of those particles. This is conveniently ignored in introductory texts on special relativity. The plain fact is that one ought not be dogmatic and demand that one and only one definition of "mass" is correct. All three of the above definitions are used, and used correctly and effectively, in the proper context with the proper understanding. No one concept is superior to the others in all situations. http://en.wikipedia.org/wiki/Mass_in_special_relativity
  22. In two poorly structured sentences you have managed to take a position that is contradicted by not only general relativity but also by the atomic hypothesis and quantum electrodynamics. Nice job. Lunacy[math].^3[/math]
  23. You are really confused. More reading and less writing might be in order.
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