DrRocket

The only method that I have ever seen be successful is to actually cponstruct proofs for yourself. There is no recipe for proofs. For any given theorem there may be several proofs. The mark of a good proof is clarity and valid logic. Elegance is nice but that comes with practice and increasing insight. Some people catch on quickly. Some never do. Elementary set theory is a good place to start. Good luck to you.

There are a large number of particles in this system that we call the universe.

Proof [math] 10 \equiv 1 [/math] (mod 3) QED

To learn calculus properly -- not to simply become proficient at symbol manipulation but rather to understand what limits, integrals and derivatives really are -- it helps to learn from an instructor who has a deep understanding of mathematics. Discussions with fellow students who are learning the same material can also be very helpful. That usually means a university. That same comment applies to linear algebra, though less so. It is possible to lose sight of the forest of linear algebra in the clutter imposed by the trees (matrices). It is quite easy once you understand mathematical abstraction, but a good instructor can help you do that. However, recognize that at some level, and at some stage of advancement, all learning depends on one's ability to teach oneself. So, a great deal depends on you, and the relevant qualities are impossible to judge at distance in a forum such as this.

Better try harder.

You have missed the point, and the effect of bending in bchanging the drag coefficient is both secondary and unimportant. The bending of a sheet of paper will reduce the drag coefficient, but the steel plate will still fall faster in air. The net downward force on the falling square will be, to a standard ballistic approximation [math] f= mg - C \rho v^2A[/math] where [math]A[/math] is the surface area presented [math]\rho [/math] is the air density and [math] C[/math] is a shape dependent drdag coefficient which can be taken as constant over the small range of velocities that will be encountered in this situation (all speeds are well below mach 1 for instance). Neglecting any bending the constant [math]C[/math] will be the same for paper, cardboard or steel. So the acceleration will then be [math] a = \frac {f}{m} = g - \frac {\rho v^2 A}{m}[/math] which increases with increasing [math]m[/math] showing clearly why steel falls faster than paper or cardboard in air -- assuming that you don't make the steel so thin that its mass is not appreciably greater than that of the cardboard or paper sheets.

If you drop the object it falls down. If you put it on a frictionless ramp, it follows the ramp. Acceleration is a vector -- there is a direction associated with it. Note also that the object on the ramp travels a greater distance than the one that is simply dropped, before reaching the ground. Now ask yourself what conservation of energy tells you about the final speed. Would you expect the magnitude of the acceleration to be the same in both cases ?

Real scientists. Full data set methodology and analytical codes available. http://www.berkeleyearth.org/

Sounds like great cover for a surreptitious kick. They'll never convict you if I am on the jury.

Amplify the photosensor -- turn up the gain. Your problem is more easily solved with electronics than with optics.

What do you mean IF ?

Start by writing out the definition of a convergent sequence. Ask yourself if it is possible for a convergent sequencve to have two distinct cluster points. Prove your answer.

Unfortunately I can't buy those explanations -- that virtual particles are "immune" to the event horizon. The question is one that requires simultaneous consideration of strong gravitational fields -- highly curved spacetime -- and the behavior of elementary particles, in particular virtual particles -- quantum field theory. We do not have a fully unified theory of quantum gravity that would, if it were to exist, cover questions of both gravitation and the quantm behavior of particles. The closest thing that exists is quantum field theory in curved spacetime (in which I am most definitely not expert), and that theory is rather problematic itself. But to address questions such as the one posed, it is all that is available. Unfortunately in this theory one no longer has the notion of "particles", and hence I find the answers given in the links to be pure handwaving, and invalid handwaving at that. There are times when honest science requires an answer of "I don't know". Ref: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics by Robert M. Wald

First you need to review the definition of convergence. Divergence is a failure to converge. The condition that you state would imply divergence but is somewhat stronger (assuming that you could prove this for any [math] \ell[/math], which you will not be able to do). This sequence has two cluster points, but no limit. You might try writing out the first few terms of your sequence. The pattern that emerges ought to suggest to you a reason why the sequence fails to converge.

Yep. It has been shown directly experimentally, besides the every day application in particle accelerators.. http://en.wikipedia.org/wiki/Hafele-Keating_experiment

A mathematicians revenge (on the chemist who once presented me a chart containing nothing but chicken wire structures -- but you will do as a proxy): http://en.wikipedia.org/wiki/Ultrafilter

You have misconstrued the big bang hypothesis. Based on what has been observed and general relativity Hawking and Penrose showed that the universe in the distant past was extremely dense and highly curved. The theory actually predicts that spacetime is singular, which in this case means that timelike geodesics cannot be extended indefinitely into the past -- i.e. there is, within the context of general relativity, no "before" the big bang. This is generally taken to mean that general relativity is not a fully accurate model all the way back to t=0. It is thought that the model is pretty good from about t= +10^-33 sec onward. No one has the slightest what happened at t=0. But the big bang WAS NOT an explosion of matter within space over some brief period of time. The big bang represents an initial point or surface (a Cauchy surface) in the spacetime manifold of general relativity from which everything subsequent to it has evolved (according to the laws of mechanics). It was essentially the beginning of both time and space. There may eventually be other explanations, probably arising from a future theory that can combine both gravitational and quantum phenomena. No such theory currently exists, but there is a lot of ongoing research -- and a lot of speculation in the popular literature. General relativity is sufficiently well supported that the basic idea of the big bang is on solid footing, so long as you don't try to go all the way to t=0. There are a lot of finer points yet to be resolved, and lots of room for further research. But there is little point in arguing against the basic idea. You would do well to read the "Cosmo Basics" thread. http://www.scienceforums.net/topic/33180-cosmo-basics/ You would also do well to ignore the bull that some folks will try to feed you.

You have got to be kidding.

This ought to amuse the Somali pirates.

EC theory does not "replace curvature with torsion". GR makes the assumption that spacetime is torsion free. That implies that there is a Levi-Civita connection for which the metric is preserved by parallel translation. So you get a nice metric theory, but one that cannot handle intrinsic spin. EC theory makes no such assumption. The result is a more mathematically complex theory. But it is a theory that is indistinguishable from GR with current measurement technology -- the differences in most circumstances are too small to measure. There is still curvature in EC theory, but you don't have it in terms of a metric. The geometry is not Riemannian or pseudo-Riemannian. The singularity theorems of GR do not, in general, hold. I'm not sure what counts as "concrete". GR made cosmology into a science, and provided a framework for study of "the universe" -- the spacetime manifold. Without that it is difficult to be precise about what one means by "the universe". With GR that can be made precise, models constructed, and (as shown by Hawking and Penrose) deep insight gained into the origins, evolution and large scale structure of that universe. GR is really a radical new perspective on what is meant by "space" and "time". One gets a hint in special relativity, but also a distortion because of the use of a single global coordinate system in SR. In GR you are forced to recognize that there is no such thing as global space or global time. What clocks measure is "proper time", which (using units where c=1) is just the arc length of the world line of the clock between events, using the spacetime metric. What is perceived as space is dependent on the local observer. Similarly with time -- "time here' vs "time there" can be meaningless. That is a huge philosophical change. Along with GR you get new phenomena -- an effect of gravity on light, black holes. These have no counterpart in Newtonian physics. You also get some puzzles. Conservation of energy becomes problematic. GR does not handle gravitational energy. Most importantly, GR is purely deterministic and incompatible with quantum theories. Theories uniting gravitation with quantum phenomena have stubbornly resisted consistent formulation. The fact remains that Newtonian gravity makes very accurate predictions under most circumstances, even in astrophysics, and the effects of GR are subtle. You don't need GR to handle typical engineering problems. You can launch a satellite, fix a toaster or pour concrete without any knowledge of GR at all.

Would this not imply that there is rather a lot of this stuff accumulating at the center of the Earth ?

While there appears to be quite a bit of evidence supporting the idea that there exists in the universe a rather large quantity of matter that does not participate in the electromagnetic force, it remains only a working hypothesis. So there is very little to actually refute. There is a pretty clear anomaly involving galactic rotation rates that can be explained by "dark matter". But it can also be explained by modifications to gravitational theory -- MOND cum TeVeS for instance. Dark matter remains basically a name for the unknown solution to a problem. What, if anything, it is remains a mystery, though things like massive black holes have been pretty much eliminated. "Dark energy" is similar. There is good evidence for the accelerating expansion of space. The obvious culprit, vacuum energy, is over-predicted by a factor of about 10^120 by QED calculations. So, while no one has a clue what is going on, it has, in the long tradition of physics, been given a name -- dark energy. You cannot refute it, because there is nothing to refute. This truth unfortunately won't sell books, so you see a lot of hype and distortion in popularizations. "Beats the hell out of me" doesn't make a best seller.

The foundational axioms are: 1) The laws of physics are the same in all inertial reference frames, and 2) the speed of light is the same in all inertial reference frames. It can also be shown, and we discussed this in another thread, that if information can be sent at superluminal speed in one frame, then one can find other frames in which causality is violated. A superluminal neutrino, which is detectable, would provide a violation. Now, relativity might well survive failure of causality. But it can't survive falsification of the basic axioms. That tosses out the whole thing on a logical basis. If [math]m^2<0[/math] then somebody will have to figure out what in the hell that means, without hand waving, and I think that will be quite a challenge. They will also have to explain how neutrinos are produced in the experiment, since with imaginary mass the minimum speed should be c, and energy decreases with increasing speed. Note that appearance of tachyons in a theory is generally viewed as evidence that the theorist has screwed up. But that is secondary. The fact that special relativity is supported by a mountain of evidence actually makes the problem more difficult. When Newtonian physics gave way to relativity, it was because it failed to agree with observation. But in its place we had a theory that both matched experiment and reduced to the Newtonian theory for velocities <<c. One simply has the Lorentz group in place of the Galilean group. Here we have a case where a purported massive particle (with positive real mass) is claimed to have exceeded c. A logical consequence of the 2 axioms of relativity is that this is impossible. What is under attack is the very foundation of the theory. There is no competing theory that is "almost relativity except for neutrinos". Note also that the various quantum field theories are relativistic. They are formulated to be consistent with special relativity, so you have to be very careful in using them to either explain or resolve this issue. Reasoning can become circular in a hurry. The nature of the challenge makes it a really big deal. But it also makes it very very unlikely that the experimental results are valid, precisely because of the empirical evidence that has been amassed in support of relativity. Note that the inferences of the experiment are purely statistical. They did not measure the transit time of any individual neutrino. Basically they have a time history for a bunch of initiation events and a time history for a smaller bunch of reception events. The difference between the means suggests superluminal transit, but there are a lot of issues to be considered. BTW I had a conversation with a high energy physicist last night, He told me that a paper was prepared for formal publication but that a third or so of the "authors" refused to sign. So there is a lot more to be done before confidence should be placed in the experiment -- no surprise there. This deserves serious review. Because of the potential implications. But even more importantly because the likely invalidation would involve subtle considerations that will be important in future experiments.

Seems like rather a fine point. It is certainly a small interaction and not covered by any element of the Standard Model. I would have expected to see it noted that neutrinos interact weakly, which is covered by quantum theory.

In the unlikely event that the result of the experiment is validated and a neutrino has actually exceeded c, then relativity does get tossed out. This would not entail just a minor tweak in response to an isolated troubling bit of data, but rather result in the direct refutation of one of the central axioms of special relativity which carries over to general relarivity, invalidating the logical structure on which the theory is built -- either the speed of light is not constant (and maybe more profoundly no speed is constant), or the laws of physics vary with reference frames. When you further consider that the point of quantum field theories is consistency with special relativity, you see the deep implications. On the other hand, the implications of the usual set of axioms for relativity have otherwise been spectacularly successful, so I think the likelihood of the experiment being upheld is very small. But the implications cannot be minimized in the event that it is upheld.