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Nonsense. Hmmm. Then I guess this is nonsense, too. And a bit later in the same post,

It's not relative. You can measure your acceleration as a purely local experiment. Spacecraft, machinery that are subject to vibrations, hand controllers for some video games, and even cellphones are equipped with such a device. Accelerometers measure acceleration relative to a local inertial frame. (Or relative to "the fabric of space-time", a term I don't quite like.) Einstein used the equivalence principle as a guiding principle in his formation of general relativity. Note that it is not an axiom of relativity. (It can't be; it doesn't work for describing how gravitation causes light to curve.) Nonetheless, it was an important guiding principle in the development of general relativity and remains an important concept in the understanding and testing of general relativity. The equivalence principle says quite a bit about why acceleration is something special, and also why the Newtonian concept of an inertial frame is incorrect. Acceleration is not relative in the sense that velocity is relative.

Another case of first class evidence.Is this a joke? Of course it was a joke. Do people really need to use an emoticon every time they are being sarcastic? What dude in South America? After poking around the internet for dudes from South America who do faith healing by sticking things up peoples' noses, I found the aforementioned dude. njaohnt was referring to João Teixeira, who calls himself João de Deus or "John of God". He claims to channel 30 some dead people, including King Solomon. Shirley McClain loves him. So does Oprah. So did ABC "News", which did a Primetime Live on him back in 2005. ABC even gave James Randi 19 seconds to say something about Teixeira on that show (not even as much time as the History Channel gives to skeptics to argue against the nuttery of Giorgio Tsoukalos). James Randi does a nice job of ripping into this charlatan. See http://www.randi.org/jr/021805a.html.

Don't get hung up on the label "irrational". Mathematicians, until recently, were a rather stodgy lot. Look at the names they gave things. Irrational numbers: Numbers that don't make a bit of sense. The symbol preceding 2 in [math]\surd 2[/math], is the surd symbol: Short for absurd. Negative numbers: Things that aren't numbers. Imaginary numbers: Numbers that aren't quite "real". The designation of the real numbers as the real numbers was a back-formation to contrast the numbers that truly are "real" from those that are not. Nowadays mathematicians have embraced the rainbow and they have all kinds of numbers. These more recent inventions tend not have some derogatory label such as irrational, absurd, negative, imaginary, or complex. Instead they are merely given descriptive names such as transfinite numbers, p-adic numbers, quaternions, octonions, etc.

It's miracle, not miricle. If you have a modern browser it will highlight your misspelled words such as by underlining them in red. Pay attention! As far as how do we know that miracles aren't real? We don't. We also don't know if this is all a figment of my imagination (or if I am a figment of yours!). Religious miracles and solipsism are antithetical to science. This is a science discussion board.

Not at all! What you said is essentially the definition of pi, only holds in Euclidean geometry, and it is not a mystery. That the ratio of circumference to diameter is the same for all circles on a Euclidean plane is a mathematically provable concept. Science and mathematics are in a sense very different things. The proof that ratio of circumference to diameter is the same for all circles on a Euclidean plane is valid forever. Finding some non-Euclidean surface for which the ratio of circumference to diameter is not constant does not dislodge that proof. Scientific theories, unlike mathematical theorems, cannot be proven to be true. Scientific theories can however be demonstrated to be false. All it takes is one experiment that shows that the universe operates otherwise.

No! Imagine some object, call it object A, orbiting some primary. Now imagine some other object, call it object B, at the same distance from the primary as object A with a velocity that differs from that of object A. Does this mean that this object B is an "unstable, or unsustainable, orbit"? Of course not. It just means that it is in a different orbit. Pick up almost any text on orbital mechanics and it will go through excruciating detail on how to go from Keplerian elements to position and velocity, or from position and velocity to Keplerian elements. Any position, any velocity. There is no such thing as an unstable orbit in the Newtonian two body problem.

Interestingly enough, that is essentially how the IEEE floating point format works. You are most likely using IEEE floating point numbers whenever you use some product like Excel or whenever you do floating point computations in some common programming language. Positive 0.0 and negative 0.0 are different things in that format. They are not "equal". It would nonetheless be nice (very, very nice) if comparing +0 against -0 for mathematical equality yielded true. In fact, it's not just very nice; this is required behavior of a compliant implementation. The underlying machinery has to go out of its way to ensure that the result of such a comparison is true even though -0 and +0 have different representations.

FFS, read the original post! I'll embiggen the key phrases.

What do you mean by "stable orbit"? There's nothing magical about circular orbits. So long as an orbit lies well within the primary body's sphere of influence the object will remain bound to the primary. For example, the Earth's sphere of influence reaches out to the Sun-Earth L1 point, about 1.5 million kilometers from the Earth. Ignoring perturbations by the Moon, objects orbiting at 500,000 to 750,000 km or less will remain bound to the Earth. At periapsis (closest approach), so long as the orbit doesn't intersect the primary body or its atmosphere the orbit will remain stable. Objects don't spiral into planets (or even into our Sun) due to gravitation. You have to look at a couple of very big objects orbiting very close to one another before you start seeing that happen.

The only strike here is by you, and here it is: The context is in clearly terms of a complex valued functions. (Otherwise, why would the OP be taking the real part?) The context is also in terms of what Wolfram Alpha computes. Read the original post. Again. And then again. You need to lose your attitude.

Spiral in, how? Ignoring relativistic effects, gravitation alone does not make things "spiral". The Roche limit for the Moon is between 2.5 and 3.0 Earth radii. Using the 2.5 Earth radiui value, those fragments closest to the Earth would be in a 1450×7800 km orbit (perigee altitude by apogee altitude). Those furthest from the Earth, a 11300×31200 km orbit. The outer ones wouldn't even get to geosynchronous altitude. Those inner ones would eventually collide with the Earth due to atmospheric drag, but it would take a long time. At 3 Earth radii, that 1450 km becomes 4150 km, meaning it would take a long, long time for atmospheric drag to have any kind of effect at all.

That video was alluding to something called the Roche limit (wikipedia: http://en.wikipedia.org/wiki/Roche_limit) and in the extreme case of a black hole, something called spaghettification (wikipedia: http://en.wikipedia.org/wiki/Spaghettification). These tidal forces are exactly what broke Comet Shoemaker–Levy 9 into pieces a couple of years before it collided with Jupiter. Orbits aren't spirals. They are better looked at as perturbed ellipses. This Star Trek notion of unstable orbits is bad sci-fi nonsense. If you want to study orbits, you should start with understanding Keplerian orbits and then work your way up the ladder of complexity.

Well, sure. What can be done is to give a slightly deeper meaning. I was using "why" in the lay sense of "what makes that happen?" Physics has deep answers to these "why" kinds of questions with regard to electromagnetism -- but not yet with gravitation. That is what I meant when I wrote "General relativity describes what happens." That "other story" is (I think) what Santalum was asking about. Physicists don't know that "other story" yet. Right now we're stuck with descriptive models such as general relativity. Deadly accurate, but still just descriptive. @Santalum: That next step that people such as ajb are researching will most likely still be a descriptive model. Quantum mechanics is in a sense a deeper model than general relativity, but it still is ultimately just a descriptive model. Quantum mechanics, for example, can't answer why the fine structure constant or the other fundamental unitless constants have the particular values that they do have.

Space-time does not create gravity. A much better to look at it is "Mass tells space-time how to curve, and space-time tells mass how to move." (John Wheeler) As to why "mass tells space-time how to curve," general relativity doesn't say. General relativity describes what happens. Why this happens is a mystery left to future generations. Some physicists and mathematicians are working on this problem (I believe ajb is one such person), but the problem is a rather vexing one.

The rubber sheet analogy is just that -- an analogy aimed at the law audience. Don't try to read anything more into any one of those analogies than the limited thing the analogy is trying to depict. All of these analogies (the rubber sheet model, the expanding balloon model, the raisin bread model) are ultimately flawed in one way or another. In this case, what this rubber sheet analogy is trying to depict is that spacetime is not the simple model used Newtonian mechanics. Take it to say anything more than that and you are reading too much into it. If you want to go past the analogies you need to learn the mathematics and physics.

Maybe you ought to read up on modern astronomy. Despite the fact that the inertial nature of space in the vicinity of the Earth is highly localized (a little 10 meter ball 10 meters with reasonably accurate equipment), general relativity did not suddenly say that the star catalogs astronomers have built over the century are worthless. Astronomers have continued to build these catalogs for the almost 100 years since Einstein developed general relativity. The only change is that they now account for relativistic effects in these catalogs. You apparently read way too much into what I wrote. You appear to have read that I meant some globally true coordinates. Of course the exact validity of those coordinates are localized. That does not mean things beyond the tiny little ball over which space appears to be locally inertial cannot be given some meaningful coordinates. Those modern catalogs include pulsars, which are orders and orders of magnitude outside that little tiny local ball where space is locally inertial. It does not even mean that these coordinates can only be used inside that tiny little ball. That modern catalogs are specified in ICRF coordinates means that they are not exactly correct for an observer on or near the Earth. Aim a modern telescope with milliarcsecond pointing accuracy at the exact spot specified for some object and you won't see the desired object. You'll have to make some adjustments to those coordinates. Those solar system barycentric coordinates are still valid. They just aren't globally true. Astronomers have dealt with relativistic effects for almost 200 years before Einstein; they just didn't know that that is what they were doing. The classical explanation of stellar aberration is very close to the relativistic explanation given the small velocity at which the Earth orbits the Sun. And unlike what you said in an earlier post, it's not happenstance. This is nonsense. Of of Einstein's key goals in formulating general relativity was to ensure that it yielded Newtonian mechanics in the Newtonian limit. This was essential. He had no choice but to do this. Some future theory that unifies general relativity and quantum mechanics will similarly have to show that this new theory yields general relativity and quantum mechanics when looked at under the right conditions. Any new scientific theory must agree with existing science in the regimes where that existing science agrees well with observation. Failing to do that would immediately falsify this new theory. Of course. Why do you think I introduced the concept of an IMU? They are at the heart of the difference between the concept of a locally inertial frame in general relativity and the concept inertial frames in Newtonian mechanics and of special relativity. Once again, of course. What is it that makes you think I think otherwise, particularly when I said the exact same thing you did? My, my. You have quite an atrocious and quite repugnant attitude.

The first one is incorrect. For some unknown reason, calculator developers back in the late 1990s came up with their own precedence rules regarding implied multiplication. After having been met with criticism from scientists, engineers, mathematicians, and educators, most have since switched to using the standard precedence rules.

What fence are you talking about? Special relativity can handle non-inertial frames, in much the same manner as does Newtonian mechanics. The equations of motions just get a bit messier. Note very well: There's still an inertial frame with infinite extent lurking in the background somewhere. It is this global inertial frame lurking in the background to which Dr Rocket was referring he said "Special relativity works only in the context of an inertial reference frame." There are two ways to find an inertial frame in Newtonian mechanics. One is to compare to Newton's concept of Absolute Space (aka God's frame; Newton was a deeply religious man). An inertial frame is a frame of reference whose origin is not accelerating and whose axes are not rotating with respect to this absolute inertial frame. There's a problem with this notion: That absolute frame is unknowable. It is metaphysical concept. rather Newton goes into this in detail in his first scholium. Another approach is to use Newton's first law to test for the inertial nature of a frame. This removes the need for Newton's notion of absolute space and absolute time. One can do this with local experiments. Imagine a spacecraft far removed from any gravitational bodies outfitted with an inertial measurement unit (IMU) comprising an accelerometer and an inertial gyroscope. A frame based on that spacecraft is an inertial frame if those instruments register zero acceleration and zero rotational velocity. Constructing an inertial frame is no problem should those instruments yield non-zero readings. This IMU-based concept of an inertial frame carries over into general relativity. Nonetheless, there's a huge difference between the concept inertial frames in general relativity versus those in Newtonian mechanics and special relativity. Inertial frames of reference are local in general relativity but global in Newtonian mechanics. The concepts of frames of reference and coordinate systems are more or less synonymous in Newtonian mechanics. They are very distinct concepts in general relativity. Frames of reference are a physical concept. Coordinate systems are a mathematical concept. Just because inertial frames of reference are local in general relativity does not mean that one cannot attach space-time coordinates to some remote event in general relativity. You can.

The Moon is approaching apogee, so that might be what you are seeing. However, that you also see it as "bright" makes me think that something else is going on. The Moon is near apogee and is in the waxing crescent phase, only 15% of full. So it's definitely not "bright". I suspect that what you are seeing is the optical illusion known as the Moon illusion. The Moon appears to many to be much bigger shortly after moonrise / before moonset than it appears to be when it is higher in the sky. That apparently tiny Moon up in the sky looks bright for two reasons, one real and one that's in your head. It is brighter, for the same reason the Sun is brighter when it is high in the sky, due to the reduced amount of atmosphere the light has to pass through before it reaches your eyes. It also appears to be brighter because of two objects emitting the same amount of light, your mind perceives the smaller of the two as the brighter one.

It evaporated for me quite some time ago. Popularizes of science should do their best to explain, insofar as a lay person can understand. Cox, along with several others (e.g., Brian Greene, Michio Kaku), instead seem to hellbent to mysticizing physics. This episode iced the cake for me. He himself apparently misundersands the exclusion principle. No. He was trying to (mis)use the Pauli exclusion principle, which says that two fermions cannot occupy the same quantum state. There's more to state than just energy. He also apparently (mis)used quantum entanglement; how else did he get the instantaneous bit? Finally, he apparently used one particular sub-interpretation of one particular interpretation of quantum mechanics in getting to that "everything is connected" bit. That's the "wave function of the universe" nonsense that comes out of the multiple worlds interpretation. Quantum Woo.

That is assuming a uniform density Earth. That's a nice simplifying assumption for a freshman physics kind of problem, but not so good assumption of the real Earth. This assumption results in gravitational acceleration dropping linearly with increasing depth. This is not what happens inside the Earth. The D" layer (the core/mantle boundary) is 2890 km below the surface. Per this simplifying assumption, gravitational acceleration at that depth would be a bit over half of surface gravity. That is off by a factor of two! Gravitational acceleration is 10.6823 m/s^2 at that depth. A better model is that gravitational acceleration is constant at 10 m/s^2 from the surface down to halfway to the center of the Earth and then drops linearly from that point inward. This results in a 76.41 minute round trip as opposed to your 84 minutes. An even better model is to use the Preliminary Earth Reference Model (the source of that 10.6823 m/s^2 figure), but now you have to resort to numerical integration. I did this a while ago and got a 76.38 minute round trip. See this post for references on the Preliminary Earth Reference Model.

That is the last place you want to look to understand calculus. For one thing, there's a dirty little secret of science and math education. Unlike the humanities, we hardly ever look back to the original formulation of some concept. It took a couple of hundred years after Newton to get classical physics nailed down tight -- right about at the time physics was turned topsy-turvy by quantum mechanics and relativity. It took almost as long a time to get calculus nailed down tight. There's an even bigger problem with looking to Newton's Principia to get a handle on calculus. You will find that is almost completely devoid of calculus! There isn't even very much algebra. What you will see instead is wall-o'-text page after wall-o'-text page of geometric reasoning.

I apologize Mr.T. Had no idea I was doing something wrong and simply quoted verbatum from an article on wikipedia. Should have made a referral to it, and didn't. That post by swansont wasn't aimed at you, rigney. It was aimed at another member who posted his/her personal theory in this thread.

First off, you were quoting from the introductory paragraph of the current version of the wikipedia article on "Spacetime". You should have given a reference for those words that were not yours. That gibberish at the end is some very bad badness on the part of a wikipedia author way back in January, 2006. Just as you didn't site a reference for your quote, neither did that wikipedian. Those words are copied almost word-for-word straight out of the book "Pseudo-Riemannian Geometry, δ-Invariants and Applications" by Bang-Yen Chen. A reference would have been nice, particularly since those words were not MagnaMopus's words (the wikipedian in question). So what do those words mean? In the context of this wikipedia article, perhaps not much. In the context of that book, they mean a whole lot. That "whole lot" is written in mathematics, which is the language of science (particularly physics). It is anything but gibberish -- but only if you can grok the math. If you can't, that math just like someone took a shovelful of mathematical symbols and scattered them across the page.