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Everything posted by DrRocket

  1. Mallet is renowned for being a nut. http://en.wikipedia.org/wiki/Ronald_Mallett You too can join Spike Lee and contribute to this "research". Or, ....
  2. Read the OP. A set [math] A [/math]is finite if it can be put in 1-1 correspondence with the set [math] S_N = \{ x \in \mathbb N: x \le N \}[/math] for some [math] N \in \mathbb N [/math]. [math]A[/math] is infinite if it is not finite. An equivalent characterization of an infinite set is a set that can be put into 1-1 correspondence with a proper subset of itself. Two sets have the same cardinality if they can be put into 1-1 correspondence. A cardinal number is an equivalence class under this relation. A cardinal number is infinite if any representative of its equivalence class is infinite.
  3. The Joke A mathematics professor, early in a class, is interrupted by a student asking how an assertion made in the course of proving a theorem might been seen to be true. The professor stands back, looks at the statement, pauses a moment, and replies "It is obvious". Then he stands back a moment, remaining silent, and continues to stare at the statement. He continues for several, minutes, then turns an walks out of the lecture hall. After an absence of over 30 minutes, one of the students leaves the hall to find the professor. He finds him, in his office, in front of his chalk board, on which is the statement that was questioned and a plethora of mathematical symbols, muttering to himself. The student returns to the lecture hall to report back. Just before the dismissal bell rings the professor returns, and says, "Yes, it is obvious." Background and follow-up--the punch line improves This joke has kicked around the mathematical community for years. I told the joke to a gentleman who happens to be a professor, now emeritus, in an engineering department at MIT and who is quite well known. He was also a student at MIT. To the joke he replied, "That is not a joke. The professor was Norbert Weiner. I was in that class."
  4. Look at Lagrangian and Hamiltonian mechanics in any book with a title like "Classical Mechanics" or Classical Dynamics". The books by Goldstein or Marion would do nicely.
  5. no yes wrong You reject special relativity and all of evidence that supports it ?
  6. There is a very simple way to resolve the twin paradox using general relativity: Let's put the non-traveling twin at the South pole to take the rotation of the Earth out of the picture. Then the non-traveling twin is in free-fall, here orbiting the sun. So the world line of this twin is a spacetime geodesic. The traveling twin must accelerate and decelerate, he is not in freefall, ergo the world line of the traveling twin is not a geodesic. In general relativity, because of the signature of the Minkowski/Lorentz metric, geodesics maximize arc length between their end points. The twins start at a single point and re-unite at a single point in spacetime. The arc length of the world line non-traveling twin, the geodesic, is longer than that of the traveling twin. The arc lengths are the time, proper time, experienced by each of the twins. Ergo the traveling twin is younger. Kinematically this is true. But special relativity requires more. Special relativity very specifically requires an inertial reference frame. The Lorentz transformations relate to measurements in two inertial reference frames in relative motion. But because B can clearly detect his acceleration (a force tries to push him to the back of his rocket ship) his reference frame is not inertial. This breaks the perceived symmetry and if one wants to use special rather than general relativity, one is forced to use the reference frame of A. Inertial reference frames are special. Given any inertial reference frame, any frame in uniform motion with respect to it is inertial. More importantly the converse is true -- given one inertial frame, the set of all inertial frames is just the set of frames in uniform motion with respect to the given inertial frame. This begs the obvious question as to whether any truly inertial frame exists. The answer is "probably not". General relativity avoids the by replacing a global inertial frame with a local frame in free fall. There are, of course, as with Newtonian mechanics, frames that are sufficiently close to inertial to permit accurate modeling. A lot of special relativity confusion can be avoided by recognizing just how special inertial frames really are.
  7. Try reading Cycles of Time: An Extraordinary NewView of the Universe by Roger Penrose or Endless Universe: Beyond the Big Bang by Paul Steinhardt and Neil Turok. Recognize that these books are very speculative. Or, smile and say "We don't know."
  8. Your comfort is not part of the theory. Length contraction is special relativistic effect that depends on speed, not velocity (once things are in standard configuration with axes aligned). There is no "length expansion"; the length of an object is the maximum in its rest frame. My earlier comment is reinforced by these questions. You need to study a good book on relativity. Trying to "get comfortable" using Newtonian intuition. It will not work.
  9. No, negative acceleration is still acceleration. You don't get time dilation for acceleration and time "contraction" for negative acceleration. Both represent deviations from a free fall (geodesic) path in spacetime, and non-geodesics have smaller proper time than geodesics, between the same points in spacetime. You seem to be trying to understand relativity based on some notion of "common sense" and intuition from everyday Newtonian experience. That is not going to work. There is no substitute for studying the theory using a good text. There are several that might be suitable, depending on your background in mathematics. If you can handle Misner, Thorne, and Wheeler at any level, that is one very good choice.
  10. bold added See some details here. http://www.astro.cornell.edu/academics/courses/astro2201/psr1913.htm So far as I know this is the best empirical evidence for gravitational waves. Direct detection would be better, but this is not bad.
  11. Not a good example for the, valid, point that you are trying to make. Pons and Fleischmann were legitimate chemists in a very fine chemistry department. They were operating WAY out of their depth. But while their science was bad their intentions were good. The hasty public announcement of their (erroneous) results was made at the urging of university lawyers who were concerned about priority, patent rights, and future royalties. It was quite obviously a bad idea, but the motivation had nothing to do with any desire on the part of the scientists to avoid peer review. This particular blunder was not the result of the usual wacko taking his "theory" to a public venue because either he has been turned down in scientific circles or because he is ignorant of scientific journals and the peer-review process. Other schools also quickly followed suit and took awing at the "tar baby". The driving factor in all cases seems to be institutional greed, not an attempt by scientists to avoid peer review. Your fundamental point is a good one. The example chosen is not a good illustration. This might serve better:
  12. A few members of the UN agreed on the ISS.
  13. 1. It is always about money. 2. You still have to get the fuel up there in the first place, and that takes more fuel. 3. You need a mission for the station. The mission of the ISS was to pump money into the former Soviet Union aerospace industry so that their scientists and ebgineers could be occupied and not be dealing with third-world nations desiring long-range missiles and nukes. 4. Any large project has to compete with other large projects for funds. 5. The WORLD can agree on very little, maintain agrement for a protracted period on even less, and effectively mannage nothing at all. The ISS is a perfect example of just how difficult project management is in a multi-national setting. Several planned modules have been canceled, and even so, it is still not complete. The cost has been enormous.
  14. I don't think anyone has managed that. It is however an attempt to do quantum field theory rigorously and looks at the subject from the perspective of a mathematician. I am FAR from expert in the subject. I am basically struggling to figure it out and books by physicists confuse me quickly. In Zee's book by page 13 he is explicitly evaluating integrals that I can prove do not exist (although I can certainly get them as a principle value of a singular integral, but not by Zee's hand-waving methods). So I am (very) slowly reading this book. This may be up your alley, and I think you might like the Glimm, Jaffe book. I am not at all surprised that their names are familiar to you. I would be surprised if they were not. If you do look at the book, I would be very interested in your thoughts on it.
  15. We must be reading different articles. In simple terms they took a cylinder of iron (a ferromagnetic material) hit it with a pulse magnetic field to align the magnetic domains (think atoms) in the cylinder and it started rotating. To conserve angular momentum something inside the cylinder must be rotating in the opposite direction. That something is related tp "spin". What is also interesting is that Einstein actually did an experiment.
  16. Right. Two accurate clocks, co-located, will measure the same time intervals, but if they start out with a difference in "displayed" time of one hour they will always be one hour apart. That is why you can find banks of accurate clocks that show " London time", "New York time", "Los Angeles time" and "Hong Kong time". Dilation refers to two reference frames in relative motion to each other. When there is no relative motion, there is no dilation. It is awkward, but possible to handle acceleration in special relativity. You must first identify an inertial reference frame. Then the time effect with respect to an accelerating object is determined by the instantaneous speed relative to the inertial frame. Then you have to integrate to get the net effect. But when that is done, the difference in time intervals between a "stationary" clock and the accelerating clock are persistent, even though when re-united the two clocks continue to measure future time intervals identically -- just like London time vs Hong Kong time.
  17. OK, let's walk through the original logic: "Because we are measuring light, we can pose that [math]D=CT [/math] (1)" True, but simply because distance = rate x time "From the diagram, using Pythagoream theorem, we can pose that [math] X^2 = D^2 + T^2 [/math] (2)" This is really just a definition of [math]X[/math] as a point in an artifical Euclidean space of Time x Space, but spacetime is not just Time X Space. A map of Delaware and a map of Beirut dan be laid out with the same numerical grid system, but Delaware is not Beirut. "Replacing (1) in (2) we get [math] X^2 = C^2 T^2 + T^2 [/math] (3)" True, but artificial as noted above. "If we pose that X (from the diagram) is the same as S (from Theory) we get [math] X^2 = C^2 T^2 + T^2=S=0[/math] (4)" [math]X[/math] has absolutely nothing to do with [math]S[/math] so you cannot "pose" this. This is the source of the mistake. But it is a profound mistake. Everything above might be technically true, but it is also trivial. All of the conclusions that are subsequently drawn are based on this false premise.
  18. You can't. That is pretty much the point of the Minkowski metric. It is also the wrong question. When you are doing abstract mathematics you need to ask the question "Why can I do this ?" before you do it, not "Why can't I do that?" afterward. Your equations 2 and 3 purport to establish a relationship between position/distance and time in what is really a Newtonian perspective. But that relationship is basically speed, and speed is not universal, except for light. So no such relationship should be expected to exist. The difference with the Minkowski metric lies in the relationship between Minkowski distance and time -- Minkowski distance is proper time (see the thread on proper time). The result is that all 4-velocities, not just for light, are c (usually in relativity one chooses units in which c=1). That is why one can relate distance and time via the Minkowski metric, while you cannot do that via the Euclidean metric. You are trying to understand special relativity and the geometry of Minkowski space, fairly abstract concepts, with a Newtonian and Euclidean mindset. Applying "common sense" will not work. That is a path that is fraught with peril. I suggest finding a copy of Gravitation by Misner, Thorne, and Wheeler, or using the link in my earlier post, and reading Chapter 1 very carefully, several times, to get the coordinate-free perspective of spacetime.
  19. Yes, since in special relativity the factors for time dilation and length contraction are the inverse of each other. In fact 4-velocity, in units where c=1, is always 1. The reason is pretty simple. Take a curve in spacetime. Parameterize it by arc length. Then, as one knows from basic calculus, the speed along the curve is 1. But in Minkowski space arc length is just proper time, so the "parameter speed" is also physical speed.
  20. The objection to "mixing units" is a red herring. The constancy of c allows distance and time to be expressed in the same units. The "mixing" of time and space is the whole point of spacetime. Forget specific coordinates for the moment. Any timelike unit vector can be the time axis for some observer. You have to be able to handle time and space on the same footing. The problem here is that you are using the Euclidean metric (equations 2, 3) in a Minkowskian space. Equation 4 which basically asserts the equality of the Euclidean and Minkowski metrics (you left out an exponent but that is not important) is just flat wrong -- this is the critical mistake. You need to be consistent and use the Minkowski metric (what you called the Minkowski formula) everywhere. It is the metric that determines the geometry. You can successfully mix neither metrics nor metaphors. See sections 1.4 and 1.5 in Gravitation by Misner, Thorne, and Wheeler -- here.
  21. Clarity and content are more important than the choice of pronouns.
  22. "What if" or "Suppose" should serve nicely. What is more important is the merit or interest level evinced by the idea that follows. Some "what ifs" are better than others. A friend who has served on high-level commissions once told me that Edward Teller was famous for going off on non-productive tangents, but could be readily be brought back on track. As he put it, Teller would interject into the discussion something like, "You know, if the moon really were made of green cheese then you could .....". But he said, it was easily handled by responding, "Shut up Edward." Teller the immediately re-focused and all was well. So, pick your "what if" with some care. Pose an interesting hypothetical question. The first word is not important. The first paragraph is critical.
  23. Where does general relativity fit in your picture ?
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