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Everything posted by DrRocket

  1. The error was not minor and it was repeated and reinforced. I can recognize incompetence even when you cannot or chosoe not to. Try working on your so-called "expert". You migh consider practicing what you preach.
  2. Prove it. Shouting is no substitute for sound theory and logic.
  3. That is a nice simple concise explanation. If only it were so clear as you imply or not dependent on a number of choices. It has NOTHING to do with any compact sets, contrary to your "explanation". There is no "of course" about it. DH strikes again. Let us first consider the hyperbolic functions in the context of the question posed by the OP. That context is in terms of real-valued functions of a real variable. [math] cosh \ x = \frac {e^x+e^{-x}}{2}[/math] [math] sinh \ x = \frac {e^x-e^{-x}}{2}[/math] [math] tanh \ x = \frac {sinh \ x}{cosh \ x} = \frac {e^x-e^{-x}}{e^x+e^{-x}} = \frac {e^{2x}-1}{e^{2x}+1}[/math] [math]coth \ x = \frac {cosh \ x}{sinh \ x} = \frac {e^x+e^{-x}}{e^x-e^{-x}} = \frac {e^{2x}+1}{e^{2x} -1}[/math] Some observations: 1. All of the above functions are injective (one-to-one) 2. cosh is an even function which maps the real line to real numbers [math] \ge[/math] 1. 3. sinh is an odd function which maps the real line to the entire real line 4. tanh is an odd function which maps the real line to the real numbers of absolute value less than 1 5. coth is an odd function with a pole at 0 that maps the real line to the real numbers of absolute value greater than 1 Note that tanh and coth have disjoint images. Hence artanh and arcoth disjoint domains. It thus makes no sense to talk of them as differing by a constant, even though as we shall see, their derivatives have a common algebraic expression. Now consider the inverse functions arcosh, arsinh, artanh and arcoth. artanh is defined on [math]x \in \mathbb R : |x| < 1[/math] So suppose that [math] y=arcosh \ x [/math] then [math]x = cosh \ y[/math] and we can solve for [math] y [/math] as follows: [math] x = \frac {e^y+e^{-y}}{2}[/math] [math] x + \sqrt {x^2-1} = \frac {e^y+e^{-y}}{2} + \sqrt {\frac {e^{2x} +2 +e^{-2x}}{4} -1}[/math] [math] = \frac {e^y+e^{-y}}{2} + \sqrt {\frac {e^{2x} +- +e^{-2x}}{4} }[/math] [math] = \frac {e^y+e^{-y}}{2} + \frac {e^y-e^{-y}}{2}[/math] [math] = e^y[/math] Hence [math] y = arcosh \ x = ln(y+ \sqrt{x^2-1}[/math] for [math] x \ge 1[/math] Similarly [math]arsinh \ x = ln(y+ \sqrt{x^2+1}[/math] for [math] x \in \mathbb R[/math] With a bit more algebra you find that [math] artanh \ x = \frac {1}{2} \ ln(\frac {1+x}{1-x}) \ \ |x| <1 [/math] And [math] arcoth \ x = \frac {1}{2} \ ln(\frac {x+1}{x-1}) \ \ |x| > 1 [/math] Differentiating these expressions you find that [math] \frac {d}{dx} artanh \ x = \frac {1}{1-x^2} \\ |x| <1[/math] [math] \frac {d}{dx} arcoth \ x = \frac {1}{1-x^2} \\ |x| > 1[/math] So, while the expression may appear to be the same they have no common domain and it makes no sense to say that they "differ by a constant." This is the answer to the question as posted by the OP. For some reason the theory of functions of a complex variable has been injected into the discussion, so it seems necessary to explain the situation with respect to the extension of the real-valued functions of a real variable to the case of mermorphic funcctions of a complex variable. Now let us consider the extension of the hyperbolic functions to functions of a complex variable. It is obvious that there are meromorphic extensions simply because the exponential function, by virtue of the usual power series definition, extends from the real line to a holomorphic function on the entire complex plane. Since the exponential function is periodic with period [math] 2 \pi i[/math] we see immediately that sinh, cosh, tanh, coth are not injective when viewed as complex-valued functions of a complex variable. In particular they do not have an inverse unless one restricts the domain so that the restrictions are injective. As a practical matter this means restricting the domain to strips of width [math] 2 \pi [/math] parallel to the real axis in the complex plane (you could conceivably take smaller strips from several of these strips, but no one in their right mind would want to do that). One can also view the functions as being defined on a tube. A similar problem arises with the ordinary trigonometric functions of a real variable, so this is not an overwhelming problem, but it does call for care. Let us make the usual choice and take as our domain [math] D = \{ z \in \mathbb C : - \pi < Im(z) \le \pi \}[/math] With this restriction of the domain we apply the same definitions for the various hyperbolic functions, simply replacing "x" with the complex variable "z". The symmetry observed continues to hold but the notions of "even" and "odd" lose their utility. What we do see is that 1. All of the above functions are injective (one-to-one) 2. cosh maps D to the complex plain and has a single zero at [math] z= \frac {i \pi}{2}[/math] 3. sinh maps D to the complex plain and has a single zero at [math] z =0[/math] 4. tanh maps D to the complex plain and has a pole at [math] z =\frac { \pm i \pi}{2 }[/math] and a zero at [math] z =0[/math] 5. coth maps D to the complex plain and has a zero at [math] z =\frac {\pm i \pi}{2} [/math] and a pole at [math] z =0[/math] Now we work in analogy with the case of the real-valued functions of a real variable to solve for expressions giving us the inverse functions. We will restrict our domain to D less the set of points at which the specific functions that we consider have no poles. We look at the artanh and arcoth functions that precipitated the question and proceed formally (meaning that some steps in the calculation will be justified only later): Let [math]y = tanh \ z = \frac {e^z - e^{-z}}{e^z+e^{-z}} [/math] [math]\left( \frac {1+y}{1-y}\right )^{\frac{1}{2}}[/math][math] =\left( \dfrac {1 + \frac {e^z - e^{-z}}{e^z+e^{-z}}}{1-\frac {e^z - e^{-z}}{e^x+e^{-z}}} \right )^{\frac{1}{2}}[/math][math] = (e^{2z} )^{\frac {1}{2}} = e^z[/math] From which we will eventually conclude that [math]y = artanh \ z = \frac {1}{2} log\left( \frac {1+y}{1-y}\right)[/math] Similarly [math]y = arcoth \ z = \frac {1}{2} log\left( \frac {y+1}{y-1} \right)[/math] [math]= \frac {1}{2} log\left( \frac {-(1+y)}{1-y} \right)[/math] Now the claim is that artanh and arcoth differ by a constant additive factor, so one might want to say that [math] = log\left( \frac {-(1+y)}{1-y}\right) = log \left( \frac {(1+y)}{1-y}\right) \ + \ log(-1)[/math] Which, of course would require putting some meaning on the formal manipulations and on [math] \ log(-1)[/math] Remember that we have restricted the domain of our hyperbolic functions to strips in which the exponential function is bijective onto the entire complex plane. That is equivalent to what in the theory of functions of a complex variable is called "choosing a branch of the logarithm". The logarithm can be defined as an analytic function, but only by deleting from the complex plane a half-infinite ray extending from the origin. In essence this is because the "argument" of a complex number is only defined modulo [math] 2 \pi [/math] Our choice corresponds to deleting the negative real axis. So we cannot extend the logarithm analytically to make sense of [math] log (-1) [/math] Nevertheless with our choice for the domain there is a unique value of [math]z[/math] for which [math] e^z=-1[/math] and that value is [math] z= i \pi[/math]. With the (arbitary) choices made one can now show that with the (decidedly non-compact) domain that we have chosen for which tanh and coth are bijective that on the complex plane. artanh and arcoth then are defined on the complex plane, and with our choice of the domain of tanh and coth with demands that the argument lie between [math]-\pi[/math] not inclusive and [math]\pi[/math] inclusive, differ by only the additive constant [math] i\pi [/math] . This is the result of these various choices, and is likely not particularly obvious to the casual observer.
  4. Maybe you ought to read what you wrote. You very clearly addrressed coordinates "in general relativity". Your statement was just plain wrong. No, you didn't. I guess we are back to that reading thing. Not really. I do have limited patience with incompetent "experts", requiring time and effort to clean up the resulting mess that would confuse neophytes.
  5. Wrong. Stress waves propagate at approximately the speed of sound in the material -- which is a LOT less than the speed of light. Take that.
  6. timo is on target in recommending a standard paper textbook over online resources. There is a reason that classic textbooks are classics. Assuming that you are correct in assessing yourself as "gifted", and I have no reason to believe otherwise, then learning basic algebra and trigonometry is the right thing to do after you have learned geometry. There are many suitable textbooks, and none at that level that I would call "classic". Any book with a title like "Algebra and Trigonometry" or "Pre-Calculus Mathematics" should be suitable. The old SMSG (School Mathematics Study Group as I recall) paperback books seem to be available at a very reasonable price and ought to do the job -- see Alibris.com or Amazon.com A good introductory calculus text is almost a non sequitar, but Mike Spivak's Calculus is pretty good. You might try that one when you are ready for calculus. Non-calculus based physics texts and classes do exist, but they are pretty much a waste of time. Learn calculus either before studying physics or concurrently with studying it for the first time. Calculus was invented for the purpose of making sense of mechanics. The Feynman Lectures on Physics remains the classic introductory physics text, but Halliday and Resnick's Fundamentals of Phsics (latest edition by Walker, Halliday and Resnick) is also good. It will help you a lot if you can find someone with a deep knowledge of physics and mathematics to serve as a mentor and someone with whom you can discuss your ideas. Even an intelligent study partner at your own level will help.
  7. So, what have you done to try to solve these problems ?
  8. This statement is pretty much self-contradictory. Maybe you ought to read up on what "local" means in mathematics and, by proxy in the case of general relativity, physics. There is no reason to believe that a local coordinate system is valid as anything beyond an approximation at any location other than the single pointat which it is based. There will be some neighborhood in which a coordinate patch can be used to approximate the spacetime manifold, but there is absolutely no a priori reason to thing that it applied at some remote point on the manifold. In regions in which the curvature is small the extent of the region in which the coordinates are a good approximation may be quite large. One can also create curvilinear coordinate systems that apply to somewhat large areas even in the presence of significant curvature. But you can also be badly misled -- as with the initial belief from Schwarzschild coordinates that the event horizon of a black hole is singular (it is not). This works as a local approximation. which is the best that you can do. But any IMU in free fall will produced the measurements that you describe, whether or not it is near a "gravitational body". So your test is no different from the test used in general relativity to identify a local Lorentzian reference frame. The real problem with the inertial frame of Newtonian mechanics and of special relativity, is that it does not exist. It is an excellent local approximation and extremely useful for most engineering applications. But there is really no such thing as a true global inertial reference frame. First you need to have a global coordinate system, but suppose that you have found one of those (good luck). A reference frame is just a global coordinate system attached to some physical feature (the observer). It is pretty easy to see that any two inertial reference frames are uniform motion with respect to one another. So, once you have one inertial reference frame you have a test to determine if any other frame is inertial. The wrinkle is that you can't find the first one, which must of course be inertial to an arbitrarily strict level of precision. How about one attached to the Earth ? -- nope the Earth revolves around the sun. How about one attached to the sun ? -- nope the sund revolves about the center of the galaxy. How about one attached to the center of the galaxy ? --- nope the galaxy is a complicated dance within the local group. Etc, etc, etc. So the idea of a global inertial reference frame is just a convenient fiction. It is only because they are local that inertial frames work in general relativity. And in general relativity a local reference frame is one attached to a body in free fall -- it is locally Lorentzian. Note also that gravity does not make an appearance in the local Lorentzian frame of general relativity -- a rather huge difference with the case of Newtonian mechanics in which the gravitational law is expressed in such a frame. The big difference between the concepts of a local inertial frame in general relativity and the global frames of special relativity and Newtonian mechanics is that the former can be found to exist. I agree. There are several people who could answer the question. It is a good idea to see responses from more than one such person, as the variation in perspective among those who can give adequate answers to such questions can lead to a deeper understanding than one can get by relying on just one good source. Unfortunately, DH is not one of them.
  9. Detonation is distinguished from deflagration in that combustion is initiated by adiabatic compression. A full Chapman-Jouget wave detonation proceeds at the local speed of sound in the advancing wave front. In typical solid explosives that speed is on the order of kiometers/second with pressures (rough order of magnitude) of a million psi. That is very much faster than the speed of sound in the parent material, due to the extreme pressure and temperature in the wave front. While deflagration-to-detonation transition does occur, detonations in bulk materials most commonly require a detonating input. That is the function of a blasting cap. Primary explosives that are used to create the required input are very specialized and very sensitive. Liquid explosives, nitroglycerine for instance, operate under the same basic principles. They tend to be somewhat shock sensitive, though not normally so sensitive as depicted in motion pictures. I would not advise creating violent bubbles in nitroglycerine, though that would be a poor way to intentionally initiate detonation. You might get a reaction and you might not. If you do, you will not like it. Nitroglycerine is often transported by eduction in water emulsion. A water hammer in the emulsion line is a bad thing. The general rule with explosives is to do only those operations that are safe under conditions that are known to be safe. That means not introducing energy into liquid explosives without knowing beforehand the likely result. It also means only handling sensitive explosives for some pre-established purpose, a purpose with a potentially useful outcome. In particular, people who would blow bubbles into nitroglycerine to see what happens aren't allowed to handle NG in the first place.
  10. The only hard and fast rule in mathematics is that one must be clear. The question is predicated on a potentially ambiguous mathematical sentence. It is not a matter of what is "right". It is a matter of writing and expression that is easily understood. It does not matter what convention you were taught in kindergarten. When in doubt go the extra inch to make yourself clear.
  11. Define "representation". Define "inflation". While you are at if you need to also clearly define what you mean by "information" and "amount of information".
  12. At your age Don Zagier had a double major in math and physics at MIT. Who is telling you that you can't learn because you are too young ?
  13. I know people who were fired because they wrote a book when they should have been doing research. A book proves very little. I have seen several pertinent arguments, addressing various different reasons why Cox's statements were off base. There is more than one reason why Cox is wrong.
  14. Angles? Angles ? All I see are curves.
  15. Givien all this discussion, you need to define what you mean by "0". The symbol "0" is used in mathematics for many analagous concepts. The most common is the integrer 0. It sometimes used as the symbol for the additive identity in an abstract abelian group. It is sometime used as the symbol for the additive identity of the abelian group that is part of a ring structure. Similarly "-0" might mean the additive inverse of the abstract abelian group element "0". It might also mean "-1 x 0" in the context of a ring. The proofs are very similar in all cases, but it would help understanding for the context of your question to be made clear.
  16. That depends on where you think the fence is. I'm not sure that there is one. On the other hand, there might be a wall, and that wall might divide two limited or equally erroneous views. It is a bit difficult to judge from excerpts alone. I have no idea what that quote from Mould is supposed to mean. I am a bit puzzle by French as well. the sentence "We can meaningfully discuss a displacement and all its time derivatives within the context of the Lorentz transformations" is mysterious -- basically I don''t know what he means by "within the context of" . If he means that one cannot handle accelerations by reference back to inertial frames and Lorentz transformations in analogy with the handling of accelerating frames in Newtonian mechanics, then his statement is patently wrong. If he means that when you do that what pops out does not look like the expressions one sees in elementary treatments of relativity, then his statement is obvious. But here is how I look at it: Special relativity works only in the context of an inertial reference frame. Exactly the same statement applies to Newtonian mechanics. But in Newtonian mechanics one an handle acceleration and accelerating reference frames, simply by referring back to some inertial frame. This results in pseudo-forces, such as the Coriolis force. One can handle acceleration relative to an inertial frame within the context of special relativity. That is done quite regularly -- see for instance Introduction to Special Relativity by Wolfgang Rindler, or perhaps the sources that you reference with which I am not familiar. Just as with Newtonian mechanics and pseudo-forces, the form of transformations between non-inertial frames will be different in appearance from that anticipated by those familiar with Lorentz transforms. No surprise. In either case what you see is the natural distortion that comes from trying to express well-understood principles in coordinate systems that are not naturally suited for that expression, though they may be convenient for some particular special purpose. The big difference between the relativistic case and the Newtonian analogy is that Newtonian mechanics, in all forms, applies to a single global reference frame. The underlying assumption is that spacetime is flat, spatially Euclidean and time is universal. Relativity is fundamentally different. General relativity makes no such assumption. Spacetime in general relativity is a 4-dimensional Lorentzian manifold of topology and geometry that are determined by the distribution of mass/energy. There need be no single chart that covers the manifold, space and time are local notions only, and special relativity is only a local approximation -- i.e. special relativity applies on the tangent space, not on the spacetime manifold. One does not pass from special relativity to general relativity by some bizarre coordinate choice (only locally could this be done) and therein lies the fundamental distinction between the special and general theories. One can say that Newtonian mechanics approximates special relativity in the sense that the equations give similar predictions at low velocities. One can say that special relativity approximates general relativity for small gravitational fields. But the two notion of approximation are fundamentally different. The relationship between Newtonian mechanics and special relativity is in some sense just happenstance. The equations differ by virtue of terms that are small under certain assumptions. The relationship between the special and general theories on the other hand is of a fundamentally geometric nature -- the difference between a manifold itself and the local approximation near a point by its tangent space ("all manifolds are locally flat'). The special and general theories are not two different theories but rather are part of a single geometric picture. So, yes one can approximate the physics as closely as one one would like using the special theory and the piece those local approximations together to produce a more global picture -- but that is nothing but a description of manifold theory. When you do that you are really just using general relativity. I think that this may illustrate why excess reliance on the "equivalence principle" can lead to difficulties in seeing the larger picture. As I said elsewhere, that principle was of great utility in helping Einstein to discover general relativity, but it is not particularly essential to a more modern and cleaner development of the theory. Einstein's genius comes through in the actual discovery of general relativity. But his path was, understandably, a bit tortured and characterized by fits and starts. Once discovered, there is much to be said for more modern, abstract, and elegant approaches to its development. Einstein was not very well schooled in geometry and relied on Marcel Gross to help him -- not a criticism. But is does suggest that there may be more elegant paths by which one could build upon the work of Riemann (on whose work that of Einstein in certainly based) to arrive at the same end point, but perhaps with an alternate perspective that brings greater clarity to some aspects of the theory.
  17. Whoa !! Yes controversy sells. But the value of your blog also lies with your integrity, not with sensationalism. If you were to airm for sensationalism you would be joining the camp of the likes of Cox and Kaku and that would be a shame. The number of visits is less impressive to me than quality of opinons and scientific integrity. "Few but ripe" -- C.F. Gauss Where can I get a ticket ? Harry Hill ? Is that British humor ? What is British humor ? Is there such a thing ?
  18. Frankly in my opinion the content is in the books. Audios are just an interesting and entertaining sidelight.. But different people learn differently and whatever works for you is the way for you to go.
  19. Of coourse. But it illustrates my point that what constitutes a practical Faraday cage is a function of the specific application and one's immediate concerns. A Faraday cage is not so specific as some might think, but in practical applications is a rather qualitative description. That is part of the point. In fact, since a true Faraday cage has no penetrations, I can think of no examples of such in "real life". It is not black and white, just lots of shades of gray. So when you are told that something is a "Faradat cage" it is a good idea to take a look and see what the term means in the particular case at hand.
  20. Those lectures are online as are the Robb Lectures that formed the basis for the book QED. Both are excellent. But they have nothing to do with The Feynman Lectures on Physics. There are some audios available of the lectures themselves, but without visuals I would think that the book is much better source for those wishing to study physics seriously. This is not to say that there is no value in hearing the lectures being given by Feynman himself complete with the New York accent. Compared to many other science books the lectures are quite reasonably priced, and are availabe in soft cover versions that are even more reasonably priced. At any price they are welll worth the money.
  21. There are various types of popularizers. There are a few great scientists -- Steven Weinberg, Richard Feynman -- who have given lectures and written books for popular consumption that do a great service and accurately reflect the beauty and wonder of science. There are great scientists, Stephen Hawking, who have written excellent books (A Brief History of Time) but later succumbed to commercialism (The Grand Design) and pursued popularizations apparently for money. There are hacks, Michio Kaku leaps to mind, who peddle all sorts of tripe for personal gain and ego satisfaction. I laud the former and decry the latter. I pick and choose among the works of those in the middle. Brian Cox seems to be in the process of characterizing himself, and based on his response to valid criticism I am leaning toward putting him in the bottom third. Good popularization (category 1) serves the very valuable purpose of educating intelligent laymen as to some of the beauty of science, though a full appreciation requires depth that is impossible in a popularization. It also serves to help the public appreciate the importance of science to the progress of civilization, progress which is impossible without the financial support of that same public. Bad popularization sells books and feeds egos but ultimately damages the reputation for intellectual integrity that characterizes a true scientist. It titillates the public but ultimately damages everyone by giving false impressions of what science has accomplished, what it may accomplish and what lies outside the realm of the possible. Hurray for Feynman and Weinberg. A pox on Kaku and (now) Cox.
  22. In which case he ought to acknowledge the mistake and move on. By defending nonsense he does nothing but deepen the confusion he engendered in the minds of the lay public and degrade any understanding that he may have imparted. Not to mention making himself look like an egotistical fool. Don't believe ANYTHING because of who said it. Base your belief on what was said.
  23. elmotat, unsurprisingly, hit the nail on the head. One more thing, since you asked this in terms of general relativity. The notion of reference frames applies to special relativity, not so much to general relativity. Special relativity is the local version of general relativity, and is a useful approximation so long as gravitation is not very important. It is in local coordinates that one can speak of comparison of time registered by clocks that are spatially removed from one another, hence of "time dilation". In general relativity time and space are co-mingled. What clocks measure is called proper time, and that is connected to the world line (space and time history) of the clock. So different clocks that see a different position-time history will register different time intervals. This is where acceleration enters the picture. A clock that is in free fall in a gravitational field has a world line that is what called a "geodesic". In general relativity geodesics are lines of maximal proper time. So, given two clocks that meet at two points in space and time, one moving in free fall and one subjected to forces other than gravity (and that is what is meant by accelerating in general relativity) the clock in free fall will always register the longest time interval between the points at which the clocks meet. Bottom line is that anything subject to any force other than gravity, i.e. not in free fall, is "accelerating" in general relativity. In special relativity, gravity is neglected, and "accelerating" is sensed in the usual manner, but then you have to be very careful about what is and is not an inertial reference frame -- special relativity applies only in inertial reference frames. This is a bit stilted, as I am trying to explain ideas that require the mathematics of differential geometry without actually using differential geometry, but hopefully the main idea has survived.
  24. The ideal version of a Faraday cage comes from the simple fact that one sees in an introductory electomagnetism class that there is a zero internal electric field for a closed comductive surface on which is placed a static charge. That is then extended from the DC case to dynamic fields under the assumption that the surface remains closed and there is zero relaxation time for any disturbance to the charge distribution. Note that thus far nothing has been said about frequency (in the vernacular we are dealing with "DC to daylight".) Once you dispense with these unrealized idealizations, enter the real world, and introduce real materials, with real relaxation times, real resistivity, real seals and real penetrations, etc., frequency of the fields of interest and details of the construction of the "cage" become very important, as do the tolerances imposed on shielding effectiveness as a function of frequency. You have now entered the world of electromagnetic compatibility engineering, and it gets complicated fast. A cardboard box is a fairly good shield against visible llight. It is pretty useless against radio signals or x-rays.
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