DrRocket

Calculations are not based on "real and verified experiments". Calculations are based on a theoretical model. Potentially valid models, like Newtonian mechanics or special relativity are self-consistent and cannot produce self-contradictory calculations. What they can produce are calculations that fail to match experimental results. Such is the case for Newtonian mechanics when speeds approaching that of light are encountered. Thus far no verifiable disagreement between relativity and experiment has been found, but researchers are xconstantly looking. See the controversy surrounding recent announcemenrts from the OPERA group -- but the concensus remains that SR is valid. That link is a confusing mess, and it is rather hard to determine what the author's point is, or if he has one. But SR is just fine and the author apparently fails to understand the theory. It is not possible to discredit special relativity with a calculation alone.

You will not find this topic discussed in most calculus courses. You will more usually encounter it in a linear algebra class. Any good text with " Linear Algebra" in the title should have what you need. Linear algebra is commonly encountered after introductory calculus, but calculus is not really a prerequisite. If your text is an applied mathematics text, then the topic may be presented under "matrix theory" and in that context the rule for matrix multiplication is simply a definition.

The "rule" is the result of the correspondence between matrix multiplication and linear transformations expressed in terms of a selected basis. This then carries over naturally to the composition of two linear transformations. Your book should show you this in a section on linear transformations. It is a fairly gory exercise in the bookkeeping of indices.

Uncured silicone rubbers might work. Some of them are fairly transparent. They exhibit marked flow thinning at high shear rates.

There is also a notion of Planck length, and just as much rank speculation in the popular literature. All currently accepted theories, including quantum field theories, model spacetime as a manifold which implies no minimal unit for either space or time. Any statements to the contrary are pure speculation. Attempts to formulate discrete models have not, thus far, been successful -- but people are still trying. That said, no one really has a clue what happens at the Planck scale. If you like this post, punch timo's "+" button.

This makes no sense. Swansont's point is that, given that special relativity is mathematically consistent, it cannot be shown to be wrong by a thought experiment. Any demonstration that special relativity is wrong must come from a disagreement with enpirical data. exactly the same statement applies to Newtonian mechanics -- it is shown to be wrong because it disagrees with effects seen at high speed and with the observed constancy of the speed of light. So, the question that ought to be asked is whether special relativity is really mathematically consistent. Now special relativity can be mathematically formulated in terms of Minkowski space and transformations that preserve the Minkowski metric (ortochronic inhomogeneous Lorentz transformations). This is straightforward, and will be consistent so long as the structure of mathematics -- the Zermelo-Fraenkel axioms plus choice -- are consistent. Cohen showed that the Axiom of Choice is independent of the ZF axioms, so the question boils down to he consistency of those axioms. Godel's theorems show that one cannot prove the consistency of ZF by means of formal first order logic within the structure of ZF itself. But those theorems do not rule out metamathematical proofs, and such proofs of consistency have been devised. So, if you are willing to either believe that mathematics itself is consistent, perhaps by accepting the meta proofs, then special relativity is also consistent. The short answer is that Swansont is correct and special relativity cannot be disproved by a thought experiment.

no

Folklore has it that it is true, by act of the legislature, in Mississippi and/or Iowa. Elsewhere pi is transcendental, hence irrational.

Get a life. You need better fantasies. http://retardmedia.com/uploads/2007/11/Catherine-Zeta-Jones-1(05).jpg

Nobody knows for sure.

Google is your friend. http://en.wikipedia.org/wiki/Muons "The muon was the first elementary particle discovered that does not appear in ordinary atoms. Negative muons can, however, form muonic atoms (also called mu-mesic atoms), by replacing an electron in ordinary atoms. Muonic hydrogen atoms are much smaller than typical hydrogen atoms because the much larger mass of the muon gives it a much smaller ground-state wavefunction than is observed for the electron. In multi-electron atoms, when only one of the electrons is replaced by a muon, the size of the atom continues to be determined by the other electrons, and the atomic size is nearly unchanged. However, in such cases the orbital of the muon continues to be smaller and far closer to the nucleus than the atomic orbitals of the electrons. A positive muon, when stopped in ordinary matter, can also bind an electron and form an exotic atom known as muonium (Mu) atom, in which the muon acts as the nucleus. The positive muon, in this context, can be considered a pseudo-isotope of hydrogen with one ninth of the mass of the proton. Because the reduced mass of muonium, and hence its Bohr radius, is very close to that of hydrogen[clarification needed], this short-lived "atom" behaves chemically — to a first approximation — like hydrogen, deuterium and tritium." Note that muons decay rather quickly so this atom will not hang around very long.

Google "cardoid"

Sine waves do not have parabolas in them. You can write down a Fourier series for a square wave, but the convergence is rather poor -- Google "Gibbs phenomena". There are more sophisticated summability methods that can provide better convergence (Google Cesaro summability) but they are more complicated. In any synthesizer you will be limited to a finite number of sinusoidal components, so the wave generated will always be continuous, and in fact infinitely differentiable. You can't get a true square wave in that manner. If your synthesizer uses square waves as the basic building blocks then you may be concerned with ways to generate arbitrary functions with trains of square waves -- Walsh functions. This involves the general theory of orthogonal functions and thus Hilbert spaces. There is no simple "formula", but the general theory does not, at the most rudimentary level, require much more than calculus. You might take a look at Fourier Series and Orthogonal Functions by Davis or Mathematical Methods in Engineering and Physics, Special Functions and Boundary Value Problems by Johnson and Johnson.

The issue is not whether we exist eternally in the present. That is a triviality. The issue lies with the assertion that there is a global notion of "present", hence time, valid throughout the universe. There is, in fact, no such notion. Your emphasis was on the word real, which merely comes for free with a Lorentzian manifold. I had already, several times, pointed out that a time coordinate is purely local, an approximation taken from the tangent space to the spacetime manifold at av given point. Global time does not result in singularities. Wherever did you get such a notion ? You can get a global notion of time from a homogeneous and isotropic spacetime. It is not really time, but it suffices for many purposes in cosmology. But the spacetime in which we live is neither homogeneous nor isotropic. The Wheeler Dewitt equation has absolutely nothing to do with the issue at hand. Nor does quantum theory, as we lack a unification of quantum theory with general relativity. Relativistic quantum theories compatible only with special relativity are not relevant. Nope. I am not confused at all by your misleading use of buzz words, and misuse of the associated concepts.. But, unlike you, I understand those concepts. You did not say the universe is almost flat. You said Which is a reflection of local isotropy, not homogeneity, and which has nothing whatever to do with curvature. Almost flat is not flat. And the difference between "almost flat" and flat makes all the difference in global spacetime topology. Moreover "almost flat in all directions" from a single point does not imply flatness either. Curvature does not work that way. Better read up on differential geometry. While you are at it you might try to learn the differences among homogeneity, isotropy and flatness.

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So, you are not really interested in what someone else thinks of your paper. What you want is to be told that that piece of uncritical junk is just wonderful. Well, it isn't.

Which is obvious from any physical theory in which light propagates at a finite speed. So what ?

Pretty much sucks. You have cherry picked references from the lunatic fringe (e.g. Randin Dean) and ignored objective mainstream debunking of parapsychology. If I were grading the paper yopu would get a 0 for critical thinking. But who knows ? If your professor is a nut job you might get an A.

What you are describing is not a change in he speed of photons, or of an electromagnetic wasve in a vacuum. You are describing a change in speed of propagation of an electromagnetic signal in a medium. Even a piece of glass does that -- Google "index of refraction". This has nothing to do with any violation of special relativity or of the definition of a light cone. SR works quite well without your proposed modification. If you want to "see into the past" just look up at the stars at night. Or watch re-runs of Leno.

No it is comparable to the fact that up and down are measured relative to your present location, which is neither up nor down -- a trivial fact.

nor mathematics

That is quite saimple, as the context is general relativity, and your claim is for a universal present. Spacetime is a curved Lorentzian manifold. There is no global chart for that manifold. There is therefore no universal notion of either "time" or "space". Both are local, not universal, concepts. The fact that time is a local real variable is completely irrelevant. We are talking about a Lorentzian manifold and all coordinates are therefore described as real quantities -- that comes with the definition of a smooth manifold. That is what general relativity is all about. owl and Mystery111 As a local concept "the present" is essentially a definition, not really a tautology as there is no logical sentence. But "the time is now" is pretty much a tautology, so you did capture the flavor. The fact that our direct experience is with "now" is a triviality. But the assertion thast was made was for a universal notion of "the present" which requires a universal notion of time. In general relativity there is no such thing as a universal time (or even universal space). Both time and space are local concepts. "Time here" and "time there" have no clear meaning. Now I know that you work daily with synchronozed clocks. You can do that in special relativity. Special relativity deals with flat spacetime. General relativity is locally just special relativity, and that local approximation is very good indeed, so long as you deal with small gravitational effects and modest distances. But it fails on very large scales and in the presence of high spacetime curvature -- hence is not valid for the universe as a hole or in highly curved regions such as associated with black holes. Note that "almost flat" or "flat on the largest scale" which may or may not describe the universe are clearly not true on small scales (else there would be no gravity", and hence are irrelevant to the specific issue that has been raised. .

That is not homogeneity. It is isotropy. "to 100,000th of an error" is meaningless. It is also irrelevant to the issue at hand. More buzz words that you don't understand.

Unfortunately the key piece of information required is contained in this sentence describing the figure in your attachment -- "The line passing through our solar panel appears is oriented at approximately 24:6 degrees west of due north." -- which makes no sense. I have long thought that proficiency in the English language should be a prerequisite for calculus. It appears that it should also be enforced as a requisite for those teaching calculus.

If you understand more advanced mathematics then read the Feynman lectures, but read between the lines as you do so. Those lectures contain a great deal of what is known about physics when read in a sophisticated manner. After that you are ready for advanced books on specialized topics in physics. There are a great many of those. A very good set are the several volumes on theoretical physics by Landau and Lifshitz. Goldstein's Classical Mechanics and Jackson's Classical Electrodynamics are well-known standard texts. Thirring has a nice series on mathematical physics. Gravitation by Misner, Thorne and Wheeler is an excellent treatment of general relativity. Quantum Field Theory in a Nutshell by Zee is well regarded. Those should keep you occupied for at least several weeks.