DrRocket

I think that Discovery Magazine and Julian Barbour are equally ignorable.

The large scale structure of space time is a book co-authored by Hawking and C.F.R. Ellis. It is a study of the large scale structure of spacetime as described by the general theory of relativity using sophisticated methods of differential geometry. It includes many proofs and a thorough discussion of the singularity theorem that applies to the big bang. It is definitely not a popularization. As this thread was started with Hawking radiation as a topic, it is worth mentioning that the book does not discuss Hawking radiation. Hawking radiation is predicted using quantum field theory on curved spacetime and that is beyond the scope of the Hawking and Ellis book. That is the explanation that one finds in popularizations. It is nice, neat, understandable. It would be even better if it were correct. Unfortunately the prediction of Hawkiing radiation is based on quantum theory on curved spacetime. That is a theory that is somewhat shaky but one clear feature is that one looses the very notion of "particle" , real or virtual. Another unfortunate fact is that virtual particles come with quantum field theories (on flat spacetime) while the event horizon comes from general relativity. General relativityand quantum field theories are not compatible. So it is rather dicey to predict the existence of virtual particles using quantum field theory and then predict their behavior near an even horizon using a theory that is incompatible with the theory that predicts their existence. Final note: While Hawkiing's aargument for Hawking radiation is quite deep and ingenious, and while it is generally thought to be probably correct in its basic prediction, that is by no means to say that the existence of Hawking radiation is a certainty. It is a very small effect and there is at present no experimental confirmation of it.

This represents a rather superficial understanding of the derivative, the integral and calculus in general. The gist of differential calculus is that one can learn a lot about a fairly general class of functions by studying their local behavior in terms of linear approximations. For instance when the best linear approximation at a point is a flat line neither increasing nor decreasing in either direction, the function itself has a local maxima or minima. The gist of elementary integral calculus is the development of a concept called the Riemann integral which is obtained as a limit of a sum each term of which is the product of the length of an interval and a value of a function at a point on that interval. This is generalized and made much more clear in more advanced classes on measure and integration and the theory of the Lebesgue integral. The net result is concept of "area under the curve" and a means whereby that area can be computed exactly for functions that have "nice" expressions that allow exact values of the integral to be computed. (While calculus books are full of examples chosen to be "nice" there are in fact relatively few functions for which the anti-derivative is easily found in closed form.) Integral and differential calculus are united by the Fundamental Theorem of Calculus which, for functions which are derivatives (not all functions are the derivative of anything) relates the integral over an interval to the values of the anti-derivative at the end points of the interval. While (poor) elementary calculus classes emphasize symbol-pushing and "evaluation" of integrals and derivatives, that approach quite commonly results in little or no understanding of the subject of calculus itself, or the conceptual framework involved in its development and use. To understand the basis for calculus one needs read a good book on real analysis -- any of the previously recommended books are adequate and there are certainly others.

That would be a reasonable assumption, since if a group is generated by any single element it is, by definition cyclic and there is, up to isomorphism, only one such group of a given order. The alternative interpretation of the question would be to determine the elements in each minimal generating set for each group of order 60. This is a much more involved project as one would have to classify all finite groups of order 60 -- I suppose that Sylow theorems would be a good place to start. The project does not appeal to me.

Measure theory is not the right theory for this question. I deals with a notion of "size" of sets, something akin to length, area, volume, etc. Measure theory is used in the general theory of integration and when specialized to positive measures of total mass 1, yields the modern theory of probability. The notion of a point or number being close to another point or number is usually embodied in the topological notion of a "neighborhood". In the case of the real line or a the typical Euclildean space one has a notion of distance, a metric, that serves to define the topology and to measure "closeness" of distance of one point relative to another. So, you might look to an introductory topology book for more ideas related to your question. Maxwell Rosenlicht's Introduction to Analysis has a nice elementary treatment of metric spaces, and Lecture Notes on Elementary Topology and Geometry by Singer and Thorpe goes further and presents a very nice introduction to more general concepts of topology and geometry. 0 and 1 are not "right next to one another". In fact there are infinitely many numbers between then -- uncountably many in fact. One might thing of "relativity of size" of numbers via the notion of ratios, but any sensible comparison would still view 100000 as being larger than 0.00001 if the two are compared using the same criteria.

If you ever do attain a perfect understanding of the neutron, please let others know immediately. It will be the first example of a perfect understanding of ANYTHING in the history of science. Until then we will just have to muddle along with models of increasing accuracy, but never perfect.

The series [math]\displaystyle \sum_{n=0}^\infty \frac {z^n}{n!}[/math] where [math]z \in \mathbb C[/math] converges abslolutely and uniformly on compacta. It can be shown to define a periodic function, which we call the exponential function. [math] \pi[/math] is defined to be the period of this function divided by 2i. It can be proved that [math]\pi[/math] is not only irrational but in fact transcendental. Therefore its decimal representation does IN FACT neither terminate nor repeat. There is no snide "supposedly" about it. One can also show that in Euclidean geometry the ratio of the diameter of a circle to its circumference is precisely [math]\pi[/math]. This is a matter of logic and rigorous proof. There is no "supposedly" involved here either. You would do better to learn your mathematics from Walter Rudin in, for instance, Real and Complex Analysis than from Dan Brown and the Da Vinci Code.

In which case the notion of a "reference frame" is purely local. Not really. Special relativity is general relativity in a flat spacetime in which the spatial part is Euclidean. In that unique case there is a single Lorentzian chart that covers the entire spacetime manifold. However, in any spacetime that has mass and therefore gravity (our universe for instance) this condition is not met. It is quite possible to handle acceleration in special relativity. But to do it you ultimately refer everything to some inertial reference frame. It does address your argument if you think about it. You are attempting to shift among reference frames that are not in uniform motion with respect to one another. It is impossible for both of them to be inertial and universal.

All true. What is also true is that you cannot precisely draw a line of rational length either which is a corollary of your last sentence.

The equations of special relativity, particularly the "time dilation effect" of the Lorentz transformations apply only when used in an inertial reference frame. You can analyze acceleration in special relativity, but first you need find an inertial frame in which that acceleration is determined. In the twin paradox there is only one inertial frame in evidence -- that of Steve. It is pretty clear from simple considerations of special relativity that Steve sees a greater time elapse than does Ariana, at least in his reference frame. Since the same physics takes place in all reference frames Steve's perspective is valid, and relatively easily understood. You can take Arianna's perspective, but to do that you need to reference things back to some inertial reference frame and modify the transformation laws -- a real mess. If you do that you will get the same answer, plus a headache. One can also use general relativity. First, put Steve in orbit around the Sun, neglecting the Earth. That is what is really intended by the stay-at-home epithet. Then Steve is in free fall and therefore his world line is a spacetime geodesic. Arianna on the other hand has a world line that is not a geodesic -- she is undergoing forces other than gravity. Now their world lines intersect at two points -- when Arianna leaves and when she returns. The length of their two world lines is (in units in which c=1) the proper time measured by their respective clocks. Because Steve's world line is a geodesic it is the longest spacetime path between those two points, hence Steve measure a greater elapsed time than does Arianna.

Mathematics is not a science. It differs from science in that validity of a theory or assertion is determined by logic starting from a very few axioms that are assumed as "true" without proof. Experiment has no part in determining mathematical "truth". Science, on the other hand, determines "truth" by means of observation and measurement of physical phenomena. While science, and especially physics, make use of mathematics and mathematical models, the validity of those models is determined empirically, by experiment and observation and not by mathematical consistency or pure logic.

That is very deep insight for a fourteen-year-old. You will do well.

Wrong again. The Newtonian theory of mechanics and special relativity are ultimately incorrect. The former was shown invalid by relativity and special relativity applies only in the complete absence of gravity. But the subject was the concept of an inertial reference frame. A concept cannot be incorrect. It can be inapplicable or inappropriately applied, but a conceptual construct is not incorrect unless is inconsistent. The concept of an inertial reference frame is perfectly valid. The Newtonian theory of mechanics and the special theory of relativity are consistent -- they are just not correct in the final analysis. BTW, virtually ALL existing physical theories are probably incorrect. General relativity and quantum field theories -- the current pillars of physical theory -- are incompatible with one another. They cannot all be correct, and it is quite likely that all of them will be revised in some future theory that will be more nearly correct. It is also quite possible that in the usual progress of physical theories as a series of successively better approximations that we may never have a theory that is "correct" in the purist sense that you seem to use the word. You are missing the essence of the issue and the essence of the basic physical theories in your effort to prove yourself "correct". These distinctions are important and crucial to ever reaching a profound understanding of the subject. You are touching on the essence of general relativity. But you cannot apply these ideas in the context of special relativity or of Newtonian mechanics. It is important to recognize the fundamental differences among these theories and to use them only in situations in which they are applicable. Newtonian mechanics and special relativity are both extremely useful theories. But they do not apply to every situation. You cannot force fit them where they do not fit. In Newtonian mechanics and special relativity acceleration is absolute. There is a very simple reason for this. Suppose you have found an inertial reference frame (forget about the problem that they don't really exist in nature and accept the idealization). Then every other inertial reference frame in the universe is in uniform motion with respect to the one that you have in hand. Inertial reference frames are very special. They are so special that acceleration with respect one is acceleration with respect to all of the others -- velocity differences differ by a constant vector, acceleration is the same in all of them. So, when you work in the context of Newtonian mechanics or special relativity, acceleration is indeed absolute. As a practical matter there are frames that are very good, not perfect but very good, approximations to true inertial reference frames. They are good when used with appropriate understanding of their limitations. Don't push them too far. Too far depends on the context. If you want to deviate from that you are left with an alternate theory. The most common alternate theory is general relativity. It has the advantage that it is supported by a lot of experimental evidence. One of the things that you give up in general relativity are global reference frames. You then have to contend with mathematics that is quite a bit more abstract than what you see in Newtonian mechanics or special relativity. Reference frames become local charts. Ordinary derivatives become covariant derivatives. Geometry is no longer Euclidean. The metric is not positive-definite, but rather Lorentzian.

Once again you demonstrate a total lack of understanding of the basic concepts of physics. A global inertial reference is indeed a convenient fiction. It is convenient in the sense that it allows one to use the convenient fiction of Newtonian mechanics for many applications and the convenient fiction of special relativity in other applications. A convenient fiction indeed. But nevertheless a fiction. In general relativity a local inertial reference frame, is nothing more and nothing less than a local reference frame attached to a body in free fall. Yep, that does exist. Where do you think that the "equivalence principle" came from ? It is basically just this simple idea. I suspect that you do not understand manifold theory and hence do not understand what "local" means. But local reference frames do indeed exist and without them one does not have the mathematical theory of manifolds and differential geometry. General relativity is based on the theory of Lorentzian manifolds and gravitation is the result of curvature in that setting. Local coordinate systems are rather important, and in general relativity local Lorentzian frames are particularly important. So, yes the post that I labeled as "nonsense" is indeed "nonsense". So is your reply. Nonsense on top of nonsense.

Nonsense. The Newtonian concept of an inertial frame is the same concept as an inertial frame in special relativity. The kinematical equations describing transformations among inertial frames are different, but the concept of the frames themselves is identical. The concept is valid and very useful. It can hardly be called "incorrect". Witness the success of special relativity and of Newtonian mechanics as an approximation that is quite adequate in a great many situations. What must be recognized is that an inertial reference frame is an idealization, whether one is working with special relativity or with Newtonian mechanics. And to effectively use that idealization one must check the specific circumstances of the situation under consideration.

The hypothesis of the twin paradox is symmetrical from a purely kinematical point of view. But it is distinctly not symmetrical from an overall physical point of view. Acceleration relative to an inertial frame is readily sensed and that distinguishes uniform motion relative to an inertial frame from uniform motion relative to any other frame -- when a car accelerates you can feel yourself pushed back into the seat even though you are at rest relative to the car. Special relativity and the Lorentz transformations apply only between inertial reference frames. It is rather like applying a mathematical theorem -- before you can correctly use the conclusion you must first verify that the conditions of the hypothesis are satisfied. In special relativity that means starting with an inertial reference frame.

In that case my best suggestion is to take a look information theory and the work of Claude Shannon. http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html

Actually there are various uses of those terms, mostly rather loose uses. I though that perhaps you had some specific definitions in mind. But at this stage I am lost.

Kaku is not quite so adept with physics of the possible. You are in over your head. Pick something simpler.

No, completeness of the real numbers is not an axiom. It is a result of the construction of the real numbers themselves -- using Dedekind cuts or equivalence classes of Cauchy sequences. You can show that any two complete Archimedian fields are isomorphic, so the precise method of construction is not important. The least upper bound property is equivalent to completeness of the real numbers with the usual metric that is induced by the order of the reals.

That is one way to do it. Another is to use Dedekind cuts. The method you describe, using equivalence classes of Cauchy sequences, generalizes to a method that can be used to complete any metric space. Connecstor could see all of these things done elegantly were he not so averse to reading the books that have been suggested to him.

I can CONSTRUCT the real numbers, including sqrt of 2 and pi using nothing except the natural numbers and simple set theory. Yep, I figured it out for myself to be true, along with rather a lot more mathematics. You can see this done in print in Landau's Foundations of Analysis or Rudin's Principles of Mathematical Analysis. There is no brainwashing involved. All that is required is to have a brain in the first place and to use it to understand real mathematics.

The number closest to the sqrt of 2 IS the sqrt of 2. There is no rational number closest to the sqrt of 2. Between any two real numbers there is a rational number (in fact there are infinitely many of them). The bottom line is that this is nonsense.

We have a theory of quantum electrodynamics that has given some exquisitely accurate explanations for things like the anonalous magnetic moment of the electron. We also have a very good theory of gravitation that has given accurate predictions of the precession of the perihelion of Mercury, and predicted previously unanticipated phenomena such as gravitational lensing and black holes. I would hesitate to call either theory "deeper" or more profound than the other. What is abundantly clear is that the two theories are fundamentally incompatible -- one is stochastic and the other is deterministic. General relativity is apparently mathematically consistent, or at least is well-defined and no inconsistencies are known. But it does admit singular spacetimes that may well not be representative of reality. Quantum electrodynamics is not so blessed, and I think it would help a bit if it could be put on a more satisfying mathematical basis. Nevertheless it seems to contain in its essence of much of the physics of everyday life -- the physics governed by the electromagnetic force -- which in principle includes chemistry and biology. On the other hand the quantum vacuum is not well understood. Those predictions that are finite, made in an attempt to explain the cosmological constant that is consistent with astronomical observations, are off by a factor of something like 10^120. Moreover, the equations ar so difficult that as a practical matter it is not useful in explaining anything nearly so complex as chemistry. So I think it is more fair to say that, while a great deal is known of modern physics, even more is not known and an ultimate answer to many questions will have to wait for better theories -- better than either general relativity or the various quantum field theories.

And the reason for your unhelpful answer is that "Wolfram Alpha made me do it." ? My "attitude" is that people deserve to have their question answered accurately and in the most straightforward manner available, even when that requires one to read a bit more into the question and the background of the poster than is openly presented. That is fairly common with mathematics questions -- the first step is to determine what the question really is. It also means that inaccurate or misleading responses by so-called "experts" (even if they come via Wolfram Alpha) are not on my list of the particularly useful. I am not likely to lose that attitude. In this case by first lookiing at the case of functions of a real variable, the essence of the issue comes out cleanly. The case of functions of a complex variable can be, and was, addressed, but requires much greater care and is more clearly presented once the theory of the real-valued functions is understood.