DrRocket

Think of "2a+3" as "x".

nope. That Wiki article applies to indeterminate forms of limits, which are irrelevant to the question. Even in the theory of cardinal numbers 0 times any cardinal is still 0.

This is :1) absolutely true and, 2) unlikely to be fully appreciated in the environment of your average pub, by your average pub denizen..

The nth term doesn't even tend to 0. This is a classic example of why you can't do naive term-by-term operations with infinite series and expect to reach a valid conclusion.

A line (or even a smooth curve) is 1-dimensional. A surface (locally a plane) is 2-dimensional.

About the only thing that you got right is that Hawking conceded his bet with Preskill. But even with respect to the bet things are not settled, for Thorne, who was also partt to that bet, has not accepted Hawking's rationale and has not conceded.

rubbish

Shapiro delay is a coordinate effect, which is rather like assuming that things are flat when they are not. Actual measurements of speed can never be truly local. But the local speed of light is always c. There is lots of confusion on this, even misunderstandings of some of Einstein's early writings. In some choices of coordinate systems you get light speeds that differ from c, but that is just a coordinate effect. Another way to think about this is in terms of special relativity. In SR light always travels at c, as a fundamental axiom. But special relativity is the localization of general relativity. So given that velocity is a local concept, the speed of light in general relativity is again always c. One difficulty, particularly for experimentalists, is that experiments always deal with coordinate values. For instance you assume that it is fair to compare the time on your wristwatch with the clock on the wall across the room. Strictly speaking, GR does not let you do that. But the approximation that spacetime in the laboratory is flat allows you to resort to special relativity as an approximation and you make that comparison without difficulty. Virtually nothing is lost in this approximation -- and thankfully so or you atomic clock guys would have a hell of a problem comparing clocks. But over large distances, or near black holes, the difference between proper time and coordinate time becomes more important. Let's compare the job of the experimentalist vs the theorist in determining speed: 1) The experimentalist first has to determine what is time and what is space -- he must choose a local coordinate system (nature tends to do this for him). Then he needs a ruler and a stopwatch, and some finite time and spatial arrangement to determine the time required by the moving body to cover a known distance. Then he has to get out a pencil and paper and divide the distance covered by the time recorded on the stopwatch. 2) The theorist just says "the 4-velocity is c", and he is correct because the 4-velocity of anything is c. For light it is easy for both because for light the 4-velocity and the coordinate velocity in Lorentzian coordinates coincide. All of the above depends on the theory being correct. The really subtle problem is translating the abstract stuff into coordinate quantities that are measurable with instrumentation and determining if any disagreement with the theory is due to the approximations made in the coordinates or in solving the difficult equations or if the theory itself is wrong. That takes someone with deeper understanding than mine.

The local speed of light is c, whether the manifold is flat or not. One relatively (pun intended) way to see this is to note that light follows a null geodesic so that infitesinally [math]\Delta x = c \Delta t[/math]. What generates confusion is the use of "coordinate time" (and coordinate space) which is an artifice of the local coordinate system. This can result in a "speed" that appears to be different from c. But coordinate time is not really time and that apparent speed is not real either. All sorts of screwy things can appear to happen due to coordinate effects.

You are misinterpreting several things, particularly how time is handled in general relativity and the relationship to special relativity. Maybe this will help. Time, Time Dilation, etc. Throughout this discussion we will make the assumption that time is accurately described by general relativity. This is purely an assumption, but since general relativity is the best currently available thery of space, time, and gravity it is a reasonable assumption. We also choose units in which the speed of light, c, is 1. This is convenient and common in texts on relativity. By “clock” we mean any idealized device that accurately records time. Special Relativity Special relativity is general relativity on flat Minkowski space, i.e. spacetime in the absence of gravity. Alternately it is a local approximation to general relativity; i.e. general relativity on the tangent space to the spacetime manifold. Minkowski space is \mathbb R^4 endowed with a non-degenerate metric of signature (+,-,-,-) [or alternatively (-,+,+,+) but we choose the former.] A vector x in Minkowski space is called timelike if <x,x> >0, spacelike if <x,x> < 0 and null if <x,x>=0. Note that the line joining two spacetime points having identical spatial coordinates (in some fixed but arbitrary coordinate system) is timelike, and the difference in time coordinates is what would be recorded by a stationary clock at that location. A curve in Minkowski space is timelike if the tangent vector to the curve is everywhere timelike. The arc length of a timelike curve in Minkowski space is the proper time associated to that curve. It can be shown that the proper time of a world line (timelike curve) is the time that would be recorded by an accurate clock having that world line. Clocks record the proper time of their world line and that is all that clocks record. Because the spacetime of special relativity is flat Minkowski space there exist global coordinates, “inertial reference frames” that provide global notions of time and of space. What constitutes “time” and what constitutes “space” are dependent on the reference frame (aka observer). Lorentz transformations preserve the Minkowski inner product, which is invariant (indepenfdent of the observer) and provide a correspondence between the “time” and “space” of one observer to the “time “ and “space” of another observer. Thus in special relativity the global time coordinate of one observer is relatable to the global time coordinate of a second observer, the time at one point in a single reference frame is relatable to the time at a second point in the same reference frame (which is equivalent to an ability to synchronize clocks in a given reference frame) and the time of a “stationary” clock in a given reference frame is the same as the proper time associated with the world line of the clock, which is an invariant quantity. All of this is baked in to the Minkowski metric and the Lorentz transformations that preserve it. (For the sophisticated, one can also consider translations, which also preserve the metric and deal with the Poincare group or the inhomogeneous Lorentz group, further specializing to the orthochronous elements). “Time dilation” results directly from an application of the Lorentz transformation group to the time coordinate of any given reference frame. But it is critical to note that one must start with a reference frane. General Relativity General relativity adds gravity and acceleration to the mix in the form of curvature of the spacetime manifold. In doing this the existence of global time and spatial coordinates are given up. There is no such thing as a reference frame in general relativity, only local approximations – this is the critical difference between a manifold and an affine space or vector space. In general relativity “time” and “space” are only local/approximate concepts. But the Minkowski metric generalizes to the Lorentzian metric tensor, and one can still talk sensibly about timelike tangent vectors and arc length of timelike curves. The reasoning from special relativity aqnd Minkowski space applies directly in general relativity to timelike curves (world lines) on the spacetime manifold, and arc length translates directly to proper time. Clocks still measure the proper time of their world lines. Lacking global time or space coordinates there is no obvious way to compare time at one spacetimepoint with time at another point, and no meaning to “time dilation”. Either the familiar time dilation due to relative motion or the “gravitational time dilation” sometimes encountered. So what do people mean when they talk about time dilation in general relativity ? Note that in general relativity we still have the fundamental notion of proper time. Clocks measure the proper time of their world line, and the only viable operational definition of time is “time is what clocks measure”. So to talk about any sort of time dilation we need to relate it to proper time. Special relativity, as the local approximation to general relativity provides the answer. Conceptually a manifold is very different from a Euclidean or Minkowskian space. But locally a manifold is describable as just such a space – just as flat maps describe the surface of the earth over a small area. QA manifold is pieced together with a set of local “charts” that in the case of general relativity are Minkowskian, together with more complex relationships that describe how the charts are “sewn together”. But locally one can approximate the manifold with a single chart and ignore the additional terms, or approximate them simply. In general relativity speed related and gravitational time dilation are simply approximations, in a local coordinate system, of effects that are only 100% describable in more abstract terms. So, in short, in curved spacetime “time dilation” due to either speed or curvature and indeed comparison of “time” at spatially separated points, is dependent on a somewhat arbitrary choice of a local coordinate system, particularly for precise quantification. So why do we standardize time and coordinate standards at different locations on earth ? First, note that the earth’s gravity is fairly weak, say compared to that of a black hole near or inside the event horizon. Spacetime in the vicinity of the earth is nearly flat. A local chart is very accurate over earthly distance scales, as are minor corrections made in terms of the local coordinates for actual curvature (gravity). The local charts determine “coordinate time” and it is coordinate time that is the subject when discussion turns to time dilation. Coordinate time is only a local concept, somewhat artificial, but an effective surrogate for proper time over small distances in weak gravitational fields. Using these local charts and approximations one can make approximate sense of comparing clocks at separate locations, and hence one can talk sensibly about “time dilation” always remembering that we are only talking about approximations to proper time – and that clocks measure proper time and nothing but proper time. For people who are engaged in the development of very sophisticated clocks for the standardization of time these issues are relatively minor. Remember that the discussion has thus far involved only ideal clocks, which measure proper time with arbitrary precision. The scientists who develop atomic clocks do not have the luxury of ideal clocks. They are forced to use real clocks, made of real materials and subject to real quantum effects, which we have thus far ignored. Those clocks must be compared to one another (for a clock in isolation is useless) and hence the real distinction between coordinate time and proper time is blurred. The focus is on development of exquisitely accurate clocks that accurately measure their own proper time and verifying that accuracy by calculation and comparison with other clocks of comparable accuracy, which requires the use of coordinate time, which itself is based on somewhat arbitrary choices and agreements as to what coordinates are to be used. It is important to recognize that the “time” of everyday experience, in extreme conditions or when measured to extreme levels of precision is not as clearly and unambiguously defined as one might naively suppose. The time of general relativity is a local concept. Comparison of time at spatially separated locations requires approximation, and very sophisticated approximations when extreme precision is required. Naïve questions often run aground when the nature of the approximations is not recognized and people start talking about different notions of “time” without recognizing that they are doing so.

Instead of debating irrelevant statistics on how many people downloaded the article, why don't you try actually reading it ? The link is in my previous post. As I stated earlier, although I am personally skeptical of much of what has been published on global warming and might be predisposed to accept critical publications, that paper is junk. The editor should resign for publishing it. Anyone with any knowledge of dynamic feedback systems should have summarily rejected it -- no need to even send to referees. And those referees ought to be taken off of the list. GIGO

A pparently the paper itself is rather controversial http://www.physorg.com/news/2011-09-editor-remote-journal-resigns-citing.html In any case, here is the paper so one can judge for oneself http://www.mdpi.com/2072-4292/3/8/1603/pdf Added in edit: Having now had time to read through the paper quickly here is my take. While I am skeptical of a good deal of what has been published regarding global warming, my skepticism extends to both sides. I am not skeptical about the quality of this particular paper -- it is worthless. The single variable, first-order feedback model proposed is ludicrously simple and completely unrelated to basic physics. The satellite data deserves to be compared to climate code predictions, but it must be compared using real physics and mathematical models of sufficient sophistication and to be meaningful. As usual, the popular reporting is laughably sensationalized and inaccurate.

In order of priority 1. Choose the subject that most interests you. 2. You might look at the requirements for potential careers. But recognize that often people have careers that differ from their formal education. The person is more important than the specific degree in many cases -- government jobs can be exceptions. 3. My general impression is that many employers will better understand and appreciate a Biology degree than one in Environmental Science.

All that you have said is that if you take a list of primes, multiply them and add 1 then the resulting number is not divisible by any of the primes in the list. That is true, but trivial. In contradiction to your assertion, you need even confine the list to prime numbers, just integers other than 1. This is simply because if n divides both p and p+1 it must also divide 1. This is similar to the proof in my copy of The Elements, modulo recasting in less archaic language. This proof does not produce any identifiable sequence, as suggested by your OP, but merely shows that no finite list of primes can be complete. That is the important number theory result. But it has nothing to do with the production of your "series". Short version: This discussion is now just plain silly.

Absolutely correct.

Actually, no. And what I presented is the proof usually attributed to Euclid -- see for instance An Introduction to the Theory of Numbers by Hardy and Wright. The statement doesn't even make sense in the context of "any randomly-selected set of prime numbers". Moreover a product of primes, plus 1 is not necessarily prime. 3x5 + 1 = 16 which is most certainly not prime even though 3 and 5 are both prime.

no Euclid's proof goes as follows: Let [math]x_1,...,x_n[/math] denote the first n primes. I claim that there is a larger prime. So let [math]p=x_1x_2...x_n=1 [/math] and [math]q = p+1[/math] Now [math]q[/math] is not divisible by any of the [math]x_i[/math] so either q is prime or it is divisible by a prime between p and q. In either case there is a prime larger than [math]x_n[/math] The critical point is that one considers the product of ALL primes less than or equal to some given prime. The next number after 47 would be the product of all primes less than or equal to 43, plus one. That number is a lot larger than 1807.

Both are in free fall. Both have world lines that are spacetime geodesics. Both will experience the maximum propertime between any two points on their world lines. The idea of a "reference frame" in general relativity is purely a local concept. As an approximation, and only as an approximation, if Stan and Mary are close one could consider them as being in the same reference frame -- essentially floating near one another like two astronauts in the space station. In a pure sense one can only compare the readings of two clocks at points of intersection of their spacetime world lines and cannot compare two clocks at different locations. Time is also a local concept.

Any source, even a peer reviewed journal, can contain mistakes. There is no way to be sure except to acquire sufficient expertise to be able to determine for yourself whether what you read is correct, or even plausible. Some sources are better than others. Wikipedia is full of errors and opinions, but is a good place to start, if you read very skeptically. It is a good source for references for further investigation. Standard, time-tested text books are usually reliable, but you must still read critically and with understanding. Scholarpedia tends to be very reliable, though somewhat limited. Articles are written by invitation, and reviewed before publication. Authors and reviewers tend to be world-class experts. http://www.scholarpedia.org/article/Main_Page

There is a better chance that no one who understands the issue will bother to read it. Every math department regularly encounters someone who insists that he has solved the general trisection problem. It is one of the things that one encounters in an abstract algebra class on Galois theory. The trisection problem is impossible. This does not mean that no one has found the answer. It means that it has been proved rigorously that no classical straightedge and compass construction can exist that would trisect an arbitrary angle.

As with many universities, one of the degree requirements at Cornell is the ability to decipher the catalog and determine what is actually required to obtain the desired degree. If the catalog makes sense on first encounter, you are ready for either law school or an asylum.

This sort of question comes up from time to time, and often whoever wrote the question has some clever definite answer in mind. The plain truth is that there is no pat answer and the next number could be literally anything. It is quite possible, and in principle it is simple, to generate a polynomial whose value at 1,2,...,n is any set of n numbers. That does not guarantee that the next element in a sequence is given by that polynomial at n+1.

No, it does not. Energy is conserved in special relativity. Wrong. Work expended is exactly matched by energy gained by the system on which the work is done. You cannot possibly keep the current definition of work but change the definition of energy and still have conservation of energy. There is no point in calling your concept "energy" since whatever it is (you have assiduously avoided producing a definition), it is not the energy of classical mechanics or special relativity. Why not call it "Oscar' and avoid confusion with a theory that is known to work ? If you believe this, then explicitly produce the required definitions and prove that your assertion is correct. What you think is irrelevant. All that is relevant is what you can demonstrate using rigorous logic and verifiable empirical data. Classical mechanics and special relativity have met this criteria within known domains of validity. You have not even come close. Your understanding of the existing concepts appears to have been and continues to be seriously flawed.

A time-varying electromagnetic field is non-conservative.

That would be the usual convention.